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Deep-water redfish / Sebastes mentella / Djúpkarfi
03 Oct Deep-water redfish / Sebastes mentella / Djúpkarfi
Posted at 18:57h in Books by Þóra Sigurðardóttir 0 Comments
Djúpkarfi / Deep-water redfish / Sebastes mentella
Deep-water redfish looks very much like its cousin, the golden redfish. However, its stomach is redder and its whole colouring more even. Also the bony protrusion at the front of the lower jaw is larger and the lowest spikes of the chin bones bend forward. An easy way to find out if the fish in question really is a deep-water redfish is to slide the thumb backwards over the chin bones. If the thumb gets scratched, the fish is a deep-water redfish.
Deep-water redfish is found in the Northern Atlantic along the coast of eastern Greenland, south-west and south of Iceland, along the deep-sea ridge leading to the Faeroe Islands, around the islands to the north of Great Britain, off Lofoten and in the Barents Sea towards Svalbard. It is thought that the deep-water redfish off Greenland, Iceland and the Faeroe Islands are a different stock from the deep-water redfish found in the Barents Sea.
The main fishing grounds for deep-water redfish around Iceland are deep on the Faeroe Islands' ridge off the south-west coast of the country. A fully grown deep-water redfish is not commonly found off Northern Iceland.
Deep-water redfish keeps to the sea floor and the mid-ocean at a depth of five hundred to eight hundred metres, but it has been caught at a depth of about one thousand metres. It can reach the length of about seventy centimetres, but is most often thirty-five to forty-five centimetres in length. The longevity of this fish is not easily determined, but using isotope ratios, scientists have found that the deep-water redfish can reach the age of sixty-five to seventy-five years.
The deep-water redfish bears living young. The eggs are hatched inside the fish and the spawning occurs in March to May at a depth of five hundred to seven hundred metres in waters of about six degrees Celsius. The number of eggs for each female is between forty and four hundred, and when hatched the fingerlings are seven to eight millimetres in length. The fingerlings drift on the ocean currents towards Greenland, where they seek the sea floor on the underwater shelf at a depth of two to four hundred metres in waters of three to four degrees Celsius. Growth is slow, and pubescence is not reached until the fish has grown to thirty-seven to forty-two centimetres in length.
As the main habitat of the deep-water redfish lies quite far down in the ocean, it was not a common catch until late in the twentieth century, when trawlers got larger and more efficient for deep-water fishing. In later years this fish has been heavily caught, so much so that ichthyologists have started worrying about the state of the stock. However, as the deep-water redfish is mainly caught in international waters, it has proved difficult to reach agreements on regulatory measures.
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Proudly Made For You. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,689 |
namespace capnp {
class StructSchema;
class ListSchema;
struct DynamicStruct;
struct DynamicList;
template <typename T>
class Orphan {
// Represents an object which is allocated within some message builder but has no pointers
// pointing at it. An Orphan can later be "adopted" by some other object as one of that object's
// fields, without having to copy the orphan. For a field `foo` of pointer type, the generated
// code will define builder methods `void adoptFoo(Orphan<T>)` and `Orphan<T> disownFoo()`.
// Orphans can also be created independently of any parent using an Orphanage.
//
// `Orphan<T>` can be moved but not copied, like `Own<T>`, so that it is impossible for one
// orphan to be adopted multiple times. If an orphan is destroyed without being adopted, its
// contents are zero'd out (and possibly reused, if we ever implement the ability to reuse space
// in a message arena).
public:
Orphan() = default;
KJ_DISALLOW_COPY(Orphan);
Orphan(Orphan&&) = default;
Orphan& operator=(Orphan&&) = default;
inline typename T::Builder get();
inline typename T::Reader getReader() const;
inline bool operator==(decltype(nullptr)) { return builder == nullptr; }
inline bool operator!=(decltype(nullptr)) { return builder == nullptr; }
private:
_::OrphanBuilder builder;
inline Orphan(_::OrphanBuilder&& builder): builder(kj::mv(builder)) {}
template <typename, Kind>
friend struct _::PointerHelpers;
template <typename, Kind>
friend struct List;
friend class Orphanage;
};
class Orphanage: private kj::DisallowConstCopy {
// Use to directly allocate Orphan objects, without having a parent object allocate and then
// disown the object.
public:
inline Orphanage(): arena(nullptr) {}
template <typename BuilderType>
static Orphanage getForMessageContaining(BuilderType builder);
// Construct an Orphanage that allocates within the message containing the given Builder. This
// allows the constructed Orphans to be adopted by objects within said message.
//
// This constructor takes the builder rather than having the builder have a getOrphanage() method
// because this is an advanced feature and we don't want to pollute the builder APIs with it.
//
// Note that if you have a direct pointer to the `MessageBuilder`, you can simply call its
// `getOrphanage()` method.
template <typename RootType>
Orphan<RootType> newOrphan() const;
// Allocate a new orphaned struct.
template <typename RootType>
Orphan<RootType> newOrphan(uint size) const;
// Allocate a new orphaned list or blob.
Orphan<DynamicStruct> newOrphan(StructSchema schema) const;
// Dynamically create an orphan struct with the given schema. You must
// #include <capnp/dynamic.h> to use this.
Orphan<DynamicList> newOrphan(ListSchema schema, uint size) const;
// Dynamically create an orphan list with the given schema. You must #include <capnp/dynamic.h>
// to use this.
template <typename Reader>
Orphan<FromReader<Reader>> newOrphanCopy(const Reader& copyFrom) const;
// Allocate a new orphaned object (struct, list, or blob) and initialize it as a copy of the
// given object.
private:
_::BuilderArena* arena;
inline explicit Orphanage(_::BuilderArena* arena): arena(arena) {}
template <typename T, Kind = kind<T>()>
struct GetInnerBuilder;
template <typename T, Kind = kind<T>()>
struct GetInnerReader;
template <typename T>
struct NewOrphanListImpl;
friend class MessageBuilder;
};
// =======================================================================================
// Inline implementation details.
namespace _ { // private
template <typename T, Kind = kind<T>()>
struct OrphanGetImpl;
template <typename T>
struct OrphanGetImpl<T, Kind::STRUCT> {
static inline typename T::Builder apply(_::OrphanBuilder& builder) {
return typename T::Builder(builder.asStruct(_::structSize<T>()));
}
static inline typename T::Reader applyReader(const _::OrphanBuilder& builder) {
return typename T::Reader(builder.asStructReader(_::structSize<T>()));
}
};
template <typename T, Kind k>
struct OrphanGetImpl<List<T, k>, Kind::LIST> {
static inline typename List<T>::Builder apply(_::OrphanBuilder& builder) {
return typename List<T>::Builder(builder.asList(_::ElementSizeForType<T>::value));
}
static inline typename List<T>::Reader applyReader(const _::OrphanBuilder& builder) {
return typename List<T>::Reader(builder.asListReader(_::ElementSizeForType<T>::value));
}
};
template <typename T>
struct OrphanGetImpl<List<T, Kind::STRUCT>, Kind::LIST> {
static inline typename List<T>::Builder apply(_::OrphanBuilder& builder) {
return typename List<T>::Builder(builder.asStructList(_::structSize<T>()));
}
static inline typename List<T>::Reader applyReader(const _::OrphanBuilder& builder) {
return typename List<T>::Reader(builder.asListReader(_::ElementSizeForType<T>::value));
}
};
template <>
struct OrphanGetImpl<Text, Kind::BLOB> {
static inline Text::Builder apply(_::OrphanBuilder& builder) {
return Text::Builder(builder.asText());
}
static inline Text::Reader applyReader(const _::OrphanBuilder& builder) {
return Text::Reader(builder.asTextReader());
}
};
template <>
struct OrphanGetImpl<Data, Kind::BLOB> {
static inline Data::Builder apply(_::OrphanBuilder& builder) {
return Data::Builder(builder.asData());
}
static inline Data::Reader applyReader(const _::OrphanBuilder& builder) {
return Data::Reader(builder.asDataReader());
}
};
} // namespace _ (private)
template <typename T>
inline typename T::Builder Orphan<T>::get() {
return _::OrphanGetImpl<T>::apply(builder);
}
template <typename T>
inline typename T::Reader Orphan<T>::getReader() const {
return _::OrphanGetImpl<T>::applyReader(builder);
}
template <typename T>
struct Orphanage::GetInnerBuilder<T, Kind::STRUCT> {
static inline _::StructBuilder apply(typename T::Builder& t) {
return t._builder;
}
};
template <typename T>
struct Orphanage::GetInnerBuilder<T, Kind::LIST> {
static inline _::ListBuilder apply(typename T::Builder& t) {
return t.builder;
}
};
template <typename BuilderType>
Orphanage Orphanage::getForMessageContaining(BuilderType builder) {
return Orphanage(GetInnerBuilder<FromBuilder<BuilderType>>::apply(builder).getArena());
}
template <typename RootType>
Orphan<RootType> Orphanage::newOrphan() const {
return Orphan<RootType>(_::OrphanBuilder::initStruct(arena, _::structSize<RootType>()));
}
template <typename T, Kind k>
struct Orphanage::NewOrphanListImpl<List<T, k>> {
static inline _::OrphanBuilder apply(_::BuilderArena* arena, uint size) {
return _::OrphanBuilder::initList(arena, size * ELEMENTS, _::ElementSizeForType<T>::value);
}
};
template <typename T>
struct Orphanage::NewOrphanListImpl<List<T, Kind::STRUCT>> {
static inline _::OrphanBuilder apply(_::BuilderArena* arena, uint size) {
return _::OrphanBuilder::initStructList(arena, size * ELEMENTS, _::structSize<T>());
}
};
template <>
struct Orphanage::NewOrphanListImpl<Text> {
static inline _::OrphanBuilder apply(_::BuilderArena* arena, uint size) {
return _::OrphanBuilder::initText(arena, size * BYTES);
}
};
template <>
struct Orphanage::NewOrphanListImpl<Data> {
static inline _::OrphanBuilder apply(_::BuilderArena* arena, uint size) {
return _::OrphanBuilder::initData(arena, size * BYTES);
}
};
template <typename RootType>
Orphan<RootType> Orphanage::newOrphan(uint size) const {
return Orphan<RootType>(NewOrphanListImpl<RootType>::apply(arena, size));
}
template <typename T>
struct Orphanage::GetInnerReader<T, Kind::STRUCT> {
static inline _::StructReader apply(const typename T::Reader& t) {
return t._reader;
}
};
template <typename T>
struct Orphanage::GetInnerReader<T, Kind::LIST> {
static inline _::ListReader apply(const typename T::Reader& t) {
return t.reader;
}
};
template <typename T>
struct Orphanage::GetInnerReader<T, Kind::BLOB> {
static inline const typename T::Reader& apply(const typename T::Reader& t) {
return t;
}
};
template <typename Reader>
Orphan<FromReader<Reader>> Orphanage::newOrphanCopy(const Reader& copyFrom) const {
return Orphan<FromReader<Reader>>(_::OrphanBuilder::copy(
arena, GetInnerReader<FromReader<Reader>>::apply(copyFrom)));
}
} // namespace capnp
#endif // CAPNP_ORPHAN_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,947 |
Home » How Dubai Expo 2020 Will Change Business Opportunities in Dubai
How Dubai Expo 2020 Will Change Business Opportunities in Dubai
Dubai Expo 2020 is no less than a dream for business aspirants in Dubai and from around the world.
It will be the center stage for the global economy, art, culture, technology, science, diplomacy. A lot more for the whole six months from Oct 20, 2020.
What can be better when the world will be coming together to a place. Which is already a top business destination, well better will be the best.
Dubai Expo 2020 will shape the future of Dubai in fields other than oil and industries.
Dubai will be growing into various folds, and you can also grow with it!
This expo has generated a lot of business opportunities, employment, and infrastructure, and well, it is still generating and will be doing the same after it ends.
Therefore, there is a rise in business setup in Dubai in the SME sector.
The expo will end in 2021, but it will cater to opportunities for the future and in abundance.
Business opportunities after the Dubai Expo 2020
For Expo 2020, Dubai authorities have built almost a new city, the District 2020.
The neighborhood of Dubai, this place specially design with top-class infrastructure. To host the world for the very first time in the MEASA region.
The site is a network on its own, dedicated to various fields, activities, and operational hub for global operations for the expo.
Not only the exhibition but, this place will carry a legacy of the expo and will generate possibilities in the future.
The Expo District 2020, a city in itself has opened gates to business opportunities in these sectors:
Travel, tourism, and logistics
And, after the expo ends, the positive impact on the local economy of Dubai will generate new opportunities and attainability in all the sectors.
It will be the catalyst for growth with increased tourism, businesses, and employment, which was created during the expo.
The best example is the 1889 Exposition Universelle in Paris. The expo's legacy, the gateway arc, the Eiffel tower still attract tourism even after a century and consider as a global cultural icon.
In the same manner and the grand expo 2020 site will continue to generate tourism, which will eventually result in business growth in the travel and tourism business.
Likewise, different business verticals of the industry like hospitality, entertainment, human resources, infrastructure, energy, and more will receive the benefits and will grow.
The size, the impact, the attention of the world will bring various businesses into Dubai, and the welcoming economy of Dubai will welcome them with open hands.
How you can seize the opportunity
The significant investment in the grand affair of the expo the driving force of the sustainability of the business to establish in Dubai.
How? Well, the new frontiers owing to the exhibition will require various other establishments to take their legacy forward.
A key concern will be making use of the site afterward. There will be a whole new town and will require new businesses to keep it going.
Dubai is welcoming investors, small scale industries, and others and expects to do so in the time to come
The best time for company formation in Dubai or taking your business to new heights is now, during the expo, and after it also.
So, seize the once-in-a-lifetime opportunity through a business setup in Dubai. Or by benefiting your existing business and grow with the growing economy.
Consult the best business setup consultants in Dubai and get going.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,170 |
La Universidad de Potsdam (abrev.: UP, Alemán: Universität Potsdam) es la más grande de las tres universidades de Brandenburg y se encuentra en Alemania.
La universidad de Potsdam está situada a través de cuatro campus adentro Potsdam, Brandenburg, incluyendo el palacio nuevo de Sanssouci y el parque Babelsberg.
Campus
Palacio nuevo, Sanssouci: Facultades de filosofía, institutos de las matemáticas y deporte
Golm: Facultades de humanidad, de matemáticas y de ciencia
Potsdam-Babelsberg- en-Griebnitzsee: Facultades de la ley, de la economía y de los estudios sociales, institutos de la informática, el instituto de Hasso Plattner para la ingeniería de sistemas de software
Parque de Babelsberger
Referencias
Véase también
Potsdam
Universidad Humboldt de Berlín
Universidad Técnica de Berlín
Enlaces externos
Página oficial de la universidad (alemán/inglés)
Universidades de Brandeburgo
Postdam
Potsdam
Instituciones educativas establecidas en 1991
Universidades de Alemania fundadas en el siglo XX
Alemania en 1991 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,434 |
Hi all, new to the site. Hopefully will be new to Oldsmobiles soon too!!!
Looking for a decent 1970 or '71 Oldsmobile Vista Cruiser. Possibly a '72 if it's the right car. 350 or 455- doesn't need to be numbers matching, or even an Olds engine. Looking for something that's a nice cruiser as-is or only needs a few things... not looking for a major project but can handle anything. I hate paint/body but can do it.
Prefer black interior, colors other than tan or brown, and 3rd row seat.
Welcome Quan. I hope you have better luck than the last guy looking for a VC. How many pages later, how many leads, and I don't think the guy still got one.
Good luck with your search. Cool cars.
72 350 2bbl, brown interior...front buckets, rear split bench, no 3rd row seating. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,953 |
\section{
Background and Motivation
\label{sec:background}
}
Superconducting \ch{Nb} cavities are the enabling technology behind modern high-power and high-energy \glspl{linac}\cite{Padamsee2019,Padamsee2017}.
Electromagnetic \gls{rf} fields are stored in \gls{rf} cavities and are used to efficiently accelerate charged particle bunches (or beams).
To reduce \gls{linac} length, hence cost, the electric accelerating field can be increased, but with a proportional increase in the \gls{b-surf} parallel to the \gls{rf} cavity wall resulting in heat dissipation.
Large oscillating $B$-fields can result in vortex generation and movement of those vortices generates heat as explained below.
\gls{srf} \gls{linac} cavities (commonly made out of elemental \ch{Nb} metal) need to be operated in the superconducting state below the superconducting transition temperature $T_{c}$ (\SI{\sim 9.2}{\kelvin} for Nb), requiring extensive cryogenic infrastructure.
\gls{srf} cavities can only be operated in the Meissner state when \gls{b-surf} is lower than the \gls{b-vp}, which for \ch{Nb} is on the order of \SI{200}{\milli\tesla}.
In the Meissner state, magnetic fields are screened from the interior of the superconductor and can only penetrate on a \si{\nano\meter}-scale thin layer ($\sim$\SI{100}{\nano\meter} in \ch{Nb}) from the surface, defined by the \gls{lpd}.
As magnetic fields are increased further beyond \gls{b-vp} in \ch{Nb} (a type-II superconductor), a transition from the Meissner state to the vortex state occurs where quantized magnetic fluxes (circulated by vortices of supercurrents) enter the interior of the superconductor.
The rapid oscillation of these vortices in the \gls{rf} fields causes severe heat dissipation resulting in a sharp increase in the surface resistance\cite{Gurevich2017}.
\gls{rf} dissipation is contained within this surface layer and its response to the applied \gls{b-surf} has a significant impact on the overall performance of the \gls{linac}.
Intensive research at \gls{srf} facilities worldwide has also demonstrated that the performance of \gls{srf} cavities is very sensitive to (and can be enhanced by) various surface modifications (e.g., vacuum heat treatment\cite{Ciovati2004, He2021, Ito2021, Lechner2021}, dilute impurity diffusion\cite{Grassellino2013,Grassellino2017}, and higher \gls{b-vp} superconductor thin film coatings\cite{Kubo2016,Gurevich2006,Gurevich2015,Posen2015}).
Theoretical studies postulate that a "dirty" layer on a "clean" substrate (or more complicated layered structures) can help shield the underlayer from the formation of dissipative vortices\cite{Checchin_GL_theory,Gurevich2017,Kubo2016}.
A deeper understanding of how the Meissner screening deteriorates at \gls{b-surf} close to \gls{b-vp}, as well as how different surface modifications result in a reduced \gls{rf} dissipation and an enhanced \gls{b-vp} is required to improve cavity performance.
\Gls{dc} magnetic fields applied both perpendicular and parallel to the sample surface have been used as analogues of cavity operating conditions to characterize \gls{srf} materials with respect to the field of first flux penetration\cite{Junginger2018}. Perpendicular fields result in highly non-uniform surface fields when the sample is in the Meissner state, with flux accentuated at sample edges. Parallel fields on coin shaped or ellipsoid samples provide near uniform surface fields up to the field of first flux penetration. Measurements typically record the maximum local surface field (based on the applied field and the sample geometry) that can be reached before flux is detected in the bulk. Such measurements do not provide local details of the role of the surface layer(s) in the field of first flux entry.
Given the interest in layered structures and their precise role in enhancing the field of first flux entry, a diagnostic that provides the depth-resolved information of field attenuation through the London layer would provide new insight.
Useful tools to study \gls{srf} materials are techniques utilizing spin-polarized radioactive probes such as the \gls{mu} or radioactive \acrshort{li-8-neut} ions.
In either case, the projectile is implanted into the bulk or near surface where the probe's spin reorients according to the static (i.e., time-average) and dynamic (i.e., stochastic) components of the local magnetic field.
The $\beta$-decay emmissions of the probe are correlated with its spin orientation at the time of decay and the time evolution of the asymmetry of the $\beta$-decay provides information about the local magnetic field.
Early studies of \gls{srf} \ch{Nb} samples utilized the \gls{musr} facility at TRIUMF where magnetic fields were applied perpendicular to the face of the coin shaped samples of SRF material to characterize the field of first flux entry~\cite{Grassellino_muSR_TRIUMForig}.
Later, a spectrometer allowing the application of parallel fields was added\cite{Gheidi_SRF2015} and a number of measurements were performed both with ellipsoid and coin samples to characterize the role of various treatments on the field of first flux entry\cite{Junginger2018}.
The \gls{le-musr} facility at \gls{psi} has a much lower energy muon beam that allows depth-resolved studies in parallel fields~\cite{Salman2012,2008-Prokscha-NIMA-595-317,Morenzoni2000}. The low momentum muon projectile, however, is easily deflected by the applied parallel field on the sample and is limited to $\leq$\SI{30}{\milli\tesla}.
TRIUMF's implementation of the $\beta$-NMR technique~\cite{MacFarlane_ZPC2021,2015-MacFarlane-SSNMR-68-1} utilizes low-energy ($E\sim$ 20-30 \si{\kilo\electronvolt}) radioactive ions like \acrshort{li-8} that can be spin-polarized in-flight before delivery to a dedicated spectrometer.
Like \gls{le-musr}, the \gls{bnmr} technique can be used to ``soft-land'' the ion at the near surface using a sample biased at \gls{hv}. The variable \gls{hv} bias is used to decelerate the ion to energies varying from hundreds of \si{\electronvolt} to the full ion energy which in turn allows depth-resolved studies of the local magnetic field.
The TRIUMF \gls{bnmr} facility\cite{2014-Morris-HI-225-173,2018-Kreitzman-JPSCP-21-011056} is capable of studying samples in high perpendicular field (up to \SI{9}{\tesla}) or (prior to this work) in parallel fields up to a maximum $\sim$\SI{24}{\milli\tesla}.
Compared to low energy muons, the transverse deflection of the \gls{rib} can be more easily compensated and therefore an upgrade of the TRIUMF parallel field capability was proposed and realized.
The new facility provides ion deceleration (for depth-resolved studies) at high parallel fields up to \SI{200}{\milli\tesla} --- a capability unique in the world.
This paper describes the new facility, including the design considerations, the installed equipment, and the commissioning with both stable and radioactive beams.
\section{
\glsentryshort{bnmr} Technique and the Existing Facility at TRIUMF
\label{sec:existing-facility}
}
\gls{bnmr} is a type of \gls{nmr} technique which utilizes implanted spin-polarized radioactive nuclei as probes.
The main difference compared with conventional \gls{nmr} is the method of detection.
The local field can be monitored using the anisotropic emission of the $\beta$-particles (typically \si{\mega\electronvolt} energy), which is highly correlated with the direction of the nuclear spin at the time of the decay.
High energy $\beta$s are detected in two opposing simple detector telescopes (with pairs of scintillators in coincidence), oriented at 0 and $\pi$ radians relative to the direction of the initial spin polarization.
Due to the parity violation of the $\beta$-decay, the $\beta$ counts per unit time $C_0$ and $C_\pi$ differ by a factor proportional to the angular emission probability $W(\theta)$, i.e.,
\begin{align}
C_\theta (t) &= \frac{1}{2}\frac{N(t)}{\tau}W(\theta) = \frac{1}{2} \frac{N(t)}{\tau}
\begin{cases}
\{1+A_0 P(t)\} &\text{for $\theta = 0$,}\\
\{1-A_0 P(t)\} &\text{for $\theta = \pi$,}
\end{cases}\label{eq:L_R_counter}
\end{align}
where the total number of nuclei (with mean lifetime $\tau$) present at time $t$ after the beam has turned on and implanted at a constant rate $R_0$, is defined as:
\begin{equation}
N(t) = \int_{0}^{t}N(t,t')dt' = R_0\int_{0}^{t} \exp[-(t-t')/\tau]dt'.\label{eq:N_t_count}
\end{equation}
For the remaining terms in eq.~\ref{eq:L_R_counter}, $A_0$ is the proportionality constant which depends on both the properties (solid angle) of the detectors, ($\beta$ energy-dependent) detection efficiency, and the intrinsic asymmetry $A$ ($A_0\sim 0.1$ for \ch{^{8}Li})\cite{MacFarlane_ZPC2021}. $P(t)$ in eq.~\ref{eq:L_R_counter} is the degree of spin-polarization.
The count rates are combined to generate the experimental asymmetry:
\begin{equation}
A(t) \equiv \frac{C_0(t) - C_\pi(t)}{C_0(t) + C_\pi(t)} = A_0 P(t)\label{eq:asymmetry_count_rate},
\end{equation}
which is directly proportional to $P(t)$.
The time evolution of the (longitudinal) spin-polarization can therefore be measured as the probe spin interacts with the local magnetic field inside the sample.
This method of detection allows about ten orders of magnitude higher sensitivity (per nuclear spin) than conventional \gls{nmr}, allowing for the study of situations that are either extremely difficult or impossible with conventional approaches (e.g., thin films, ultra-dilute impurities, etc.)~\cite{2014-Morris-HI-225-173,2015-MacFarlane-SSNMR-68-1,MacFarlane_ZPC2021}.
The implantation energy of \gls{bnmr} probes can be varied (e.g., from $\sim$\SI{0.5}{\kilo\electronvolt} to $\sim$\SI{30}{\kilo\electronvolt}) to study the surface of materials on the \si{\nano\meter}-scale and in a depth-resolved manner.
Similar to \gls{musr}, the \gls{bnmr} technique does not rely on the presence of a suitable \gls{nmr} nucleus in the material and can therefore be applied to any material at any applied magnetic field (including zero field).
In contrast to \gls{le-musr}, where the \gls{t-lifetime} is \SI{2.2}{\micro\second}, typical \gls{bnmr} probes have a much longer \gls{t-lifetime} (e.g., $\tau=$ \SI{1.21}{\second} for \ch{^{8}Li}~\cite{Flechard2010}). It is therefore more sensitive to dynamics on a much longer timescale, making the two techniques complementary (rather than competitive).
Furthermore, \gls{bnmr} ions at TRIUMF need to be spin-polarized in-flight via optical pumping, whereas surface \glspl{mu} produced from two-body pion decay are nearly 100\% spin-polarized.
The $\beta$-NMR technique as implemented at TRIUMF requires a dedicated facility to produce high intensity \glspl{rib} with a high degree of spin polarization.
In the case of TRIUMF, the high intensity \glspl{rib} are produced at the \gls{isac} facility, which uses a \SI{500}{\mega\electronvolt} cyclotron as a driver to bombard a solid (radio)isotope production target with proton beams~\cite{Dilling_Hyperfine_ISAC}.
The resulting short-lived nuclides are then ionized, extracted, and separated on-line (i.e., using the so-called ISOL method) to produce isotopically pure \glspl{rib} before being sent downstream for various experiments.
\ch{^{8}Li^{+}}\, beams with an intensity of \SI{\sim e7}{\per\second} are routinely available.
Prior to being delivered to the \gls{bnmr} end-station, the \glspl{rib} at \gls{isac} are spin polarized in-flight using dedicated facility infrastructure (allowing for routine operation) via collinear optical pumping with a circularly polarized resonant laser~\cite{2014-Levy-HI-225-165}.
This step produces \SI{\sim 70}{\percent} nuclear spin polarization~\cite{MacFarlane2014_InitSpinPol}.
The TRIUMF \gls{bnmr} facility has two beamlines: the \gls{bnmr} leg, which allows for measurements at high perpendicular fields up to \SI{9}{\tesla}, and the \gls{bnqr} leg, which allows samples to be studied in parallel fields up to $\sim$\SI{24}{\milli\tesla} (prior to this work).
The NQR technique is a zero field analogue of NMR that depends on the quadrupole interaction of nuclear spins $>$1/2 in solids.
The spin-polarized \gls{rib} can be sent alternately to either of the two beamlines thanks to an electrostatic ``kicker'' that is used to ``pulse'' the beam.
In this way two experiments can run simultaneously.
At the end of each leg is a spectrometer assembly for sample measurements, containing the detectors, cold finger cryostat (for low temperature measurements), magnets, RF coil, \gls{hv} deceleration (\gls{hv} cage and deceleration electrodes), and \gls{ccd} imaging system.
In addition, in order to achieve depth-profiling the sample ladder and cryostat are raised to the bias of the \gls{hv} platform, which can be varied in order to decelerate the ion beam as it approaches the sample.
The bias capability spans from 0-35 \si{\kilo\volt} via a high stability (FuG Elektronik GmbH model HCL) \gls{dc} power supply.
A high voltage interlocked cage surrounds the biased equipment on a platform above the beamline (see the inset of fig.~\ref{fig:twospect}).
The \gls{bnmr} beamline with high (perpendicular-to-sample-surface) field spectrometer uses a superconducting solenoid to generate fields up to \SI{9}{\tesla}.
The detectors are oriented forward and backward relative to the sample, and the emitted $\beta$-particles penetrate through the sample to reach the front detector located downstream of the sample.
The back detector, located outside the \gls{hv} platform, contains a small aperture to allow passage of the \gls{rib}.
On the \gls{bnqr} beamline, the (parallel-field) spectrometer uses a normal conducting Helmholtz coil.
The detectors are oriented to the left and right side of the sample, and up to four samples can be loaded simultaneously into the \acrfull{uhv} chamber via a sample ladder load-lock system.
This paper reports the upgrade of the \gls{bnqr} leg with the addition of a 1m long extension of the beamline to include a parallel field spectrometer capable of up to \SI{200}{\milli\tesla} for unique depth-resolved studies of SRF and other materials.
\begin{figure*}[hbt!]
\centering
\includegraphics[width=0.9\textwidth]{beamline_layout.pdf}
\caption{
\label{fig:twospect}
Beamline layout of the upgraded $\beta$-NMR facility. Typical beam energy extracted from the source for \gls{li-8} is $\sim$20-30 \si{\kilo\electronvolt}. The \gls{rib} is spin-polarized in-flight (first neutralized with an alkali vapour cell, and later re-ionized by a \ch{He} vapour cell) using dedicated optical pumping infrastructure, allowing for routine operation. The fast electrostatic kicker allows for semi-simultaneous operation of two spectrometers (i.e., two experiments running at the same time off of a single \gls{rib}). At the \gls{bnmr} leg, a spectrometer equipped with \gls{uhv} cold-finger cryostat and superconducting solenoid allows measurements at high magnetic fields (0.5-9 \si{\tesla}) perpendicular to the sample surface. The upgraded \gls{bnqr} leg provides a new spectrometer for measurements up to \SI{200}{\milli\tesla} parallel to the sample surface, just downstream of the existing low-parallel-field (0-24 \si{\milli\tesla}) one. The \gls{bnqr} spectrometer is also equipped with a \gls{uhv} cold-finger cryostat. The inset shows the side view of the upgraded \gls{bnqr} beamline and the \gls{hv} platform above the beamline.
}
\end{figure*}
\section{
High-Parallel-Field Upgrade
\label{sec:technical-details}
}
The high-parallel-field upgrade on the \gls{bnqr} leg was completed in two phases. The scope of phase I was to modify and improve the existing low-parallel-field section. In phase II, an additional $\sim$\SI{1.02}{\meter} of new beamline was installed. Phase I and phase II were completed in June 2019 and in July 2021, respectively.
\subsection{Ion optics}
There are three main elements of ion optics used along the \gls{bnqr} beamline: electrostatic steerers, electrostatic quadrupoles, as well as a magnetic Helmholtz coil.
The steerers compensate for the vertical deflection of the ions as they pass through the fringe of the applied magnetic field and deflect the beam onto the sample; the quadrupoles are used to control the beam shape during transport and to focus the ions onto the sample; and the Helmholtz coil provides the magnetic field on the sample parallel to the surface.
An additional four-sector electrode close to the sample (within $\sim$\SI{50}{\milli\meter}) acts as a deceleration electrode, and allows independent horizontal and vertical steering which is used to compensate for the beam deflection and to re-center the beam on the sample.
The layout of the ion optics and Helmholtz coil in the \gls{bnqr} leg are shown in fig.~\ref{fig:diagnostics}.
There are 3 horizontal steerers (XCB) and 4 vertical steerers (YCB) and two sets of electrostatic quadrupole triplets.
\begin{figure*}[hbt!]
\centering
\includegraphics[width=0.9\textwidth]{beamline-optics_annotate.pdf}
\caption{
\label{fig:diagnostics}
Details of the upgraded \gls{bnqr} leg (see also fig.~\ref{fig:twospect}). Top: The final straight section of the beamline (XCB and YCB refer to horizontal and vertical steerers, FC to Faraday cups, Q to quadrupoles, LPM to Linear Profile Monitor).
Bottom: Enlarged view of the new beam line extension model, showing the electrostatic quadrupoles, steerers, ``skimmer'' plates (grounded electrodes with a defined aperture to limit the effective length of optical element), Faraday cup, thermal radiation baffle and final quadrant decelerating electrode. A representative beam trajectory (blue envelope), obtained using a particle tracking code (General Particle Tracer\cite{GPT}) for a \SI{200}{\milli\tesla} applied field is illustrated in the model.
}
\end{figure*}
Fig.~\ref{fig:diagnostics} also shows a model of the new beam line extension complete with a model beam trajectory for the case where the Helmholtz coil is producing \SI{200}{\milli\tesla} at the sample.
The magnetic fringe field diverts the beam $\sim$\SI{2.4}{\centi\meter} off-axis through a set of ``skimmers'', a vertical steerer, and a thermal radiation baffle. The vertical steerers adjust the position of the beam on the sample, while the radiation shield is used to reduce the heat load on the cryogenically cooled sample.
Even though the design was focused on an applied field of 200 mT, the two vertical deflectors can be operated independently to achieve the desired beam path at lower fields (all the way down to zero-field).
Beam focusing is achieved using two sets of quadrupole triplets to obtain an axially symmetric beam on sample.
Previously, an Einzel lens was used to focus the beam on the low-parallel-field (sample 1) spectrometer.
With the phase I upgrade, the first triplet replaced this Einzel lens and is now used to either obtain an optimum focus on the low-parallel-field spectrometer or can be used in tandem with the second triplet to provide a variable symmetric spot size at the second sample position (sample 2).
The optimal setpoints for both triplets can be obtained using the TRANSOPTR code~\cite{Heighway1981,TRANSOPTR_Manual} via a web application called \emph{Envelope} (see fig.~\ref{fig:Envelope}), which is hosted on a local server at TRIUMF.
\begin{figure*}[htb!]
\centering
\includegraphics[width=0.9\textwidth]{Envelope_sample2_whiteBG.png}
\caption{
\label{fig:Envelope}
Plot of the beamline envelope (beam going from left to right) obtained from the \emph{Envelope}
calculator based on the TRANSOPTR code\cite{Heighway1981,TRANSOPTR_Manual}, used to tune the beam dimensionns under flexible operating conditions via two quadrupole triplets (i.e., for various beamspot sizes, beam energies, \gls{hv} platform biases, etc). The high-parallel-field and low-parallel-field spectrometer samples are located at \SI{181.5}{\centi\meter} and \SI{76}{\centi\meter} away from the \gls{lpm}0 (at \SI{0}{\centi\meter}), respectively. The amplitude of the shaded areas located around 39 \si{\centi\meter}, 115 \si{\centi\meter}, and 180 \si{\centi\meter} indicate the focal strength of the first quadrupole triplet, second quadrupole triplet, and \gls{hv} bias, respectively.
}
\end{figure*}
Several \gls{hv} power supply modules providing bias to the electrostatic ion optics are installed in two separate crates.
Both crates and all the plug-in modules were purchased from \acrlong{wiener}, with the first crate (8U or 8 standard rack unit, corresponding to 14" in height) installed in a 19" rack at the beamline level, and the second smaller crate (3U or 5.25" in height) installed in another 19" rack inside the \gls{hv} platform.
The larger crate on the beamline level houses five modules for the \gls{bnqr} leg ion optics requiring \gls{hv} bias relative to ground, and the smaller crate housing two modules is used exclusively for the four-sector electrode biased relative to the \gls{hv} platform.
For the larger crate, one bipolar module uses 10 \gls{hv} channels to provide up to $\pm$\SI{500}{\volt} for the three horizontal steerers and the first two vertical steerers. Three additional unipolar modules provide six positive and six negative channels with biases up to \SI{6}{\kilo\volt} for the two quadrupole triplets.
All quadrupole electrodes with the same polarity are wired in parallel with \gls{uhv} compatible stripped, bare \ch{Cu} wires, as shown in fig.~\ref{fig:ion-optics-assembly}.
Not shown are the \gls{hv} feedthroughs installed in the external vacuum chamber with center conductors that connect to the internal \gls{hv} wires and electrodes (see also details in fig.~\ref{fig:diagnostics}).
One unipolar module (with 8 positive and 8 negative channels) uses 2 pairs of \gls{hv} positive and negative channels to provide up to \SI{3}{\kilo\volt} for each electrode of the last two vertical steerers, which is used to deflect the beam through the thermal radiation baffles.
Polarity reversal of this module is achieved with a switch. All beamline devices are remotely operated via an \gls{epics}~\cite{Dalesio1994,EPICS} interface.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{optics-box-annotated.pdf}
\caption{Ion optics ``boxes'' used in the upgraded \gls{bnqr} beamline. Feedthrough holes allow \ch{BeCu} extension posts connecting the \gls{uhv} feedthroughs to the \gls{hv} electrodes (see also details in fig.~\ref{fig:diagnostics}). The ``skimmer'' plates are grounded electrodes with apertures used to limit the effective length of the ion optics. Also shown are the stripped bare \ch{Cu} wires used to connect electrodes in parallel to the common \gls{uhv} feedthroughs.
Left: Ion optics box upstream of the low-parallel-field sample position, containing the first quadrupole triplet and steerer asssemblies.
Right: Ion optics box for the high-parallel-field section, containing the second quadrupole triplet and steerer assemblies. Note that the beam enters from the bottom side of the image.
\label{fig:ion-optics-assembly}
}
\end{figure}
\subsection{Beam Diagnostics and Beam Tuning}
Beam diagnostics are available at discrete locations to aid with the beam tuning via measurements of both the beam current and the beam profile (see fig.~\ref{fig:diagnostics}).
The beam current can be measured using three \glspl{fc}. The \glspl{fc} are driven by stepper motor actuators to allow precise positioning with respect to the beam trajectory.
All the \glspl{fc} are typically biased between 60-350 \si{\volt} with either negative or positive polarity, to either trap secondary electrons (i.e., for an absolute current reading) or to amplify the beam intensity during operation (e.g., when using a low-intensity \gls{rib}).
The driven position of the first two \glspl{fc} are aligned with the beamline center while the third \gls{fc} is positioned off-axis (from the beamline center) to allow maximum collection when the beam path follows the reference trajectory.
The reference values for the optimum motor positions were obtained by measuring the alignment of the \glspl{fc} using a theodolite via direct line-of-sight for the case where the Helmholtz coil is producing 200 mT at the sample. during beamline installation.
The beam profile at the sample location is obtained using a \gls{ccd} camera (Starlight Xpress) mounted on a viewport, by imaging the light emitted by a scintillator (typically sapphire\cite{Salman2014_sapphire}) on the sample ladder.
Imaging of the beam at the sample location is virtually limited to \gls{rib} as it is primarily the decay products and not the kinetic energy of the beam that induces scintillation (the $\beta$s and $\alpha$s produced during radioactive decay are in the \si{\mega\electronvolt} range, \num{\sim e3} times more energetic than the beam itself).
The beam profile (and centroid position) can also be measured using horizontal and vertical slits driven at \ang{45} in combination with a downstream \gls{fc}, the so-called \acrfull{lpm}.
The \gls{lpm} plate includes two slits (horizontal and vertical with respect to the beam axis) and three circular collimators of different diameters (8.0 \si{\milli\meter}, 5.6 \si{\milli\meter}, and 4.0 \si{\milli\meter}) on a copper block that allows beam size and position definition for reproducible tuning.
The \gls{lpm} is driven by a stepper motor, with the center of each slit or collimator (and corresponding stepper motor position) determined during the installation using a theodilite.
The tuning strategy and use of the diagnostics is as follows. The quadrupole triplets are set to theoretical values (from the \emph{Envelope} web application) for a given beam energy and desired spot size. The vertical steerers are set to the theoretical value for the chosen Helmholtz coil setting (i.e., applied field). The beam is first centered on axis and aligned using the LPM0 aperture and downstream \glspl{fc}.
A pilot beam using a stable isotope (e.g., \ch{^{7}Li^{+}}) is often used to establish the trajectory with the aid of the \SI{25}{\milli\meter} diameter \glspl{fc}. Fine tuning of beam spot size and position, particularly at reduced energies, is achieved with direct beam spot imaging using a scintillator (sapphire\cite{Salman2014_sapphire}) and a \gls{ccd} camera which views the sample ladder through a glass viewport from the downstream side.
The sapphire produces a bright image under irradiation by a \gls{rib}, even at the typical low beam intensity for \glspl{rib} of $10^6$ ions per second.
Another scintillator material (YAP:Ce, Proteus, Inc.) has also been used to image stable beam (e.g., during beam development using a non-radioactive ion source) but with a much duller image even at \SI{5}{\micro\ampere}.
\subsection{Ultra-high Vacuum System}
Ultra-high vacuum ($\sim 10^{-10}$ Torr) is essential to extend the lifetime of the clean surface of any sample under investigation, particularly during measurements at cryogenic temperatures, to prevent condensation of residual gases.
The UHV throughout the beamline is established by differential pumping using a single backing-pump and two turbo-pumps, which can provide pressures down to \SI{\sim e-9}{Torr}.
Somewhat lower pressures at the two sample locations are achieved using two separate cryopumps, mounted directly below the sample cryostat, providing base pressures down to \SI{\sim e-11}{Torr}.
The new high-parallel-field spectrometer is built on a stainless steel six-way cross with \gls{cf} flanges on each arm.
The diameter of the tube along the beam axis is 6" \gls{od}, which is a compromise between reasonable pumping speed and a more compact Helmholtz magnet.
The top port consists of an 8" \gls{cf} flange that provides access for the cryostat to be lowered into the beamline.
The side ports are 10" \gls{cf} flanges on 8" \gls{od} tube stubs, onto which re-entrant ports with welded thin (\SI{0.05}{\milli\meter}) stainless windows are bolted.
This window allows a large diameter space for mounting scintillation detectors within \SI{6.35}{\centi\meter} (2.5") of the sample.
The three existing UHV chambers on the low-parallel-field section have been modified to accommodate the new quadrupole triplet, the \gls{lpm}, as well as the new \gls{fc} and viewport for the \gls{ccd} camera.
Two additional vacuum chambers are also fabricated in-house for the beamline extension, which includes one six-way cross for the spectrometer chamber and one beamtube for the upstream steerers and quadrupoles.
All UHV chambers are electropolished after \gls{tig} welding and machining.
Subsequently, the chambers are leak-checked, and measured with a \gls{rga} after beamline assembly and pump-down.
\subsection{
Sample Environment
\label{sec:beam-dynamics}
}
\subsubsection*{Magnet Coils}
A copper conductor Helmholtz coil is used to provide a magnetic field of up to \SI{200}{\milli\tesla} at the sample position.
A field uniformity better than \SI{0.01}{\percent} is required for \gls{bnmr} experiment across a typical beamspot size, \gls{fwhm} of $\sim$3-4 \si{\milli\meter}.
The number of turns for each coil along the width and height is $8\times 12$, respectively, producing 56,080 \si{\ampere}-turns at a current of \SI{584}{\ampere}. The coil conductor is water-cooled with a 10 \si{\milli\meter} $\times$ 10 \si{\milli\meter} cross section from \gls{iacs}, consisting of \SI{100}{\percent} \ch{Cu}. Turn-to-turn insulation is provided by \SI{0.5}{\milli\meter} thickness double Dacron\textregistered\,glass, and \SI{1}{\milli\meter} thickness of additional fiberglass insulation. Each coil is composed of stacks of four pancakes encapsulated with vacuum impregnated clear epoxy resin. The resulting dimensions for a single coil are shown in fig.~\ref{fig:Coil-dim}, with a separation distance (between inner face of each coil) of \SI{154.9}{\milli\meter}.
The specified coils are sourced from Stangenes Industries Inc. (Palo Alto, CA).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Coil_dim.pdf}
\caption{Dimensions of the \gls{bnqr} modified Helmholtz coils. Left: Inner radius (IR) and outer radius (OR) of a single coil in the high-parallel-field magnet installed at the sample 2 position. Also shown are the cooling channel tails at the top of the model. Right: The dimensions of the coil pair, modified from a Helmholtz configuration to fit the \gls{uhv} spectrometer chamber (with $d_\text{coil}$ as the central separation and $d_\text{sep}$ as the inner face separation between the coil pair). The width (W) and height (H) are the overall dimensions of the cross-section of the conductor turns.}
\label{fig:Coil-dim}
\end{figure}
The magnetic fields along the coil pair axis (i.e., the parallel fields on the sample) and along the beamline axis were calculated using the \gls{fem} software OPERA-3d (TOSCA Solver for Static Current Flow problems)\cite{OPERA}. The simulated magnetic field homogeneity parallel to the sample surface (along the coil pair axis) and fringe fields along the beam axis are shown in fig.~\ref{fig:Coil-fields}.
The power supply provides up to \SI{600}{\ampere} at \SI{50}{\volt} and is specified to have a stability of 0.01\% ripple (\SI{60}{\milli\ampere} rms) with a digital resolution of 16-bit at \SI{200}{\milli\tesla}.
The power supply is sourced from Alpha Scientific Electronic (Stanton, CA) and installed with a water cooling system.
Remote control of the power supply is available through an \gls{epics} interface.~\cite{Dalesio1994,EPICS}
A photograph of the magnet installed around the cryostat is shown in fig.~\ref{fig:spectrometer-open}. The cryostat is biased at the platform voltage and the copper electrode to the right of center is the grounded anode defining the deceleration gap.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{B_profile.pdf}
\caption{Top: Relative field variation along the Helmholtz coil axis (i.e., across the sample) obtained by summing the measured field profile of a single coil. The sample is centered at $z = \SI{0}{\milli\meter}$, where the uniformity of the applied field (\SI{206}{\milli\tesla} here) is maximal.
The red dashed line indicates the required field uniformity across the typical \gls{fwhm} of the beamspot (grey shaded area).
Bottom: Calculated stray fields (transverse to the beam momentum and parallel to sample surface) along the beamline axis (i.e., the $x$-axis) from \gls{fem} simulation. The sample is located at the $x = \SI{0}{\milli\meter}$.
\label{fig:Coil-fields}
}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{HV_all.pdf}
\caption{Top: View of the \gls{bnqr} spectrometer through one of the beamline viewports with the stainless steel detector window removed. Note that the beam enters from the right side of the image. Shown here are the cryostat, the four-sector electrode (mounted on the cryostat heat-shield) and the \SI{200}{\milli\tesla} Helmholtz coil. Bottom: Calculated electric fields (using CST Studio)\cite{CST} with \SI{10}{\kilo\volt} bias applied to the sample (and the cryostat). The beam is decelerated to the desired implantation energy within a small (\SI{\sim 21}{\milli\meter}) gap between the ground anode and the four-sector electrode.}
\label{fig:spectrometer-open}.
\end{figure}
\subsubsection*{Cryostat Modification}
The existing cold-finger cryostat was modified to allow application of \gls{hv} bias for the four-sector electrode which is mounted on the cryostat radiation shields via SHAPAL\textsuperscript{TM} Hi-M soft spacers, the latter providing electrical isolation while maintaining high thermal conductivity.
The cryostat and the HV power supply for the four-sector steerers are housed inside the existing electrically isolated \gls{hv} platform (via a ceramic vacuum break) to allow \gls{hv} bias for beam deceleration.
\Gls{uhv} compatible wires, such as Kapton insulated coaxial cables and \gls{ofhc} bare copper wires, and lead-free solder joints are used for the assembly.
Non-magnetic components such as connectors and fasteners are also used in the vicinity of the sample.
To accommodate for thermal contraction during low temperature measurements, the cryostat is supported on motorized bellows, which can be used to adjust the vertical alignment of the sample. In practice, the alignment is done by using the CCD camera at lower temperature to determine the current position and compare to a reference image with optimal alignment (usually determined at room temperature).
The existing \gls{hv} platform is also modified to accommodate a new support structure for the cryostat.
A continuous flow of \gls{LHe} can be provided via a portable \gls{LHe} dewar, which is installed onto the HV platform prior to the experiment.
Operating temperatures between \numrange[range-phrase=--]{4.5}{300}\si{\kelvin} are routinely achievable.
Two temperature sensors (\ch{GaAlAs} diodes from Lake Shore Cryotronics, Inc.) are mounted on the side of an \ch{Al} ring that holds the sample ladder.
An additional platinum sensor (\ch{Pt}-100 from Lake Shore Cryotronics, Inc.) is also installed on the heat shield.
The temperature is controlled by adjusting the \gls{LHe} transfer line needle valve, the \ch{He} vapor mass flow to a roughing pump at the exhaust port of the cryostat, and the heater current for the cryostat (the latter is shown in fig.~\ref{fig:sample_config}).
Temperature stability is achieved with a Lake Shore model 336 controller driving a resistive heater on the cryostat cold block.
Operating at colder sample temperatures ($<$\SI{3}{\kelvin}) is foreseen in the near future with the addition of a cryo-head to be connected to the radiation shield.
Measurements at even lower temperatures are planned in the future with the addition of a \ch{He}-3 cryostat.
\begin{figure*}[hbt!]
\centering
\includegraphics[width=0.85\textwidth]{cryostat_all_test.pdf}
\caption{
Left: Radiation shield of the $\beta$-NQR cryostat with the four-sector electrode shown facing towards the incoming beam. Middle: Sample environment inside of the cryostat (with the radiation shield removed). Right: The sample ladder which is used to mount up to four samples via a \gls{uhv} load-lock. The first position of the sample ladder in the photo is reserved for a sapphire scintillator, used for beam imaging using a CCD camera. The cyan-colored border indicates a 8 \si{\milli\meter} $\times$ 8 \si{\milli\meter} viewing window of the CCD camera through the back of the sample ladder.\label{fig:sample_config}.}
\end{figure*}
\subsubsection*{Detector System}
Two scintillator telescope pairs (two on the left and two on the right of the cryostat position) are used to ``view'' the sample through thin (\SI{0.05}{\milli\meter}) stainless steel windows.
The scintillator pairs are spaced sufficiently far apart to define a solid angle accepting $\beta$s originating at the sample, but rejecting most cosmic events and any $\beta$s from ions stopped in nearby parts.
For the high-parallel-field spectrometer, Hamamatsu H6153-01 fine mesh dynode type \glspl{pmt} are used, as they are more resistant to the relatively high magnetic fields used during the experiments.
The detector assembly, consisting of the scintillator (Saint-Gobain BC412 plastic scintillator with dimensions of 10 \si{\centi\meter} $\times$ 10 \si{\centi\meter} $\times$ 0.3 \si{\centi\meter}), Lucite light-guide, and the PMTs mounted in single steel tubes that provide limited magnetic shielding, are shown in fig.~\ref{fig:spectrometer-closed}.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{PMT-annotated.pdf}
\caption{The detector assembly, consisting of the BC412 plastic scintillator, Lucite light-guide, and the PMTs (mounted in single steel tubes) on their mechanical support.
\label{fig:spectrometer-closed}}
\end{figure}
\section{
Commissioning and first results
\label{sec:comissioning}
}
As part of the instrument's commissioning, all devices, including their control interfaces and interlock logic, are tested without beam to establish that functional requirements are met.
\textit{In operando} tests using ion beams are organized into stable beam tests using light ions (e.g., \ch{^{7}Li^{+}}\, and \ch{^{12}C^{+}}) at \SI{20}{\kilo\electronvolt} and \gls{rib} tests using \ch{^{8}Li^{+}}\, at \SI{20}{\kilo\electronvolt}.
The stable beam tests are done to check that the optics (i.e., quadrupoles, steerers, etc.) are manipulating the beam as expected. The \gls{rib} is used to further check functionality, such as the final beam spot at the sample in order to confirm the optics and to check the detectors and \gls{daq} system. The Helmholtz coil field strength and direction are checked with a hand held probe (Gauss meter). A final confirmation is done by using polarized \ch{^{8}Li}\, beam and using the \gls{bnqr} signal to directly measure the magnetic field on the sample (as described below).
There are two basic types of \gls{bnmr} measurements: resonance and relaxation. The beam commissioning with \gls{rib} employed both methods to check the full functionality of the equipment.
A resonance measurement seeks to find the frequency (i.e., the resonance condition) that corresponds to an energy difference between a pair of the probe's magnetic sublevels.
At TRIUMF, resonances are performed (predominantly) using continuous beam delivery and a transverse \gls{cw} RF field $B_{1}$ to manipulate the probe spins.
In the \gls{cw} approach, the RF field is stepped slowly through a range of frequencies close to the probe's Larmor frequency:
\begin{equation}
\nu_{0} = \frac{\omega_{0}}{2\pi} = \frac{\gamma}{2\pi}B_{0} ,\label{eq:resonant_freq}
\end{equation}
where $\gamma / (2 \pi) = \SI{6.30221 \pm 0.00007}{\mega\hertz/\tesla}$~\cite{IAEA_TableNuclearMom_gyro_Stone2019} is the \ch{^{8}Li}\, gyromagnetic ratio and $B_{0}$ is the applied magnetic field.
On resonance, in the limit of sufficiently large $B_{1}$, the populations of the sublevels involved in the transition(s) become equalized, resulting in a reduction in spin-polarization (or asymmetry).
In a relaxation measurement, the temporal evolution of the probe's spin polarization (which is initially very far from thermal equilibrium) is monitored as it returns to thermal equilibrium.
The most commonly used type of such measurements is a so-called spin-lattice relaxation (SLR) measurement, wherein the applied field is parallel to the initial spin-polarization direction.
Spin-polarization of the probe is lost through an energy exchange with its surrounding environment (often called the ``lattice'' in NMR literature)\cite{SlichterBook}.
That is, spontaneous, stochastic fluctuations in the local field that are transverse to the probe spin direction serve to ``reorient'' it back to thermal equilibrium.
Real time data for either mode can be conveniently displayed during the running experiment and analyzed using specialized \gls{bnmr} analysis software such as bfit\cite{bfit_arXiV,bfit_paper}.
The following calibration measurements were used to check the functionality of the detector, new Helmholtz coil, and the \gls{daq} system.
The magnetic field of the new coil on the sample location was precisely measured using the \acrshort{li-8-neut} resonance (see above description).
A 99.99\% pure \ch{Au} foil (Alfa Aesar) was chosen as the calibration sample due to its relatively narrow resonance line and slow spin-lattice relaxation rate down to very low magnetic fields.~\cite{MacFarlane_Gold,Parolin_Gold}
The measured magnetic field on the \ch{Au} foil can be obtained from the frequency of the resonant peak, $\nu_{0}$ (eq.~\ref{eq:resonant_freq}), as a function of the current in the Helmholtz coil.
The results are shown in fig.~\ref{fig:field-calibration}, demonstrating a linear relation between the applied current and the measured magnetic fields, as expected.
For this calibration, stray fields originating from the superconducting solenoid at the \gls{bnmr} leg (operated at \SI{2.2}{\tesla} during calibration measurements) contribute to a small constant background field.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{HH6-calib-linear.pdf}
\caption{
\label{fig:field-calibration} Calibration of the high-parallel-field coil via the \ch{^{8}Li}\,\gls{nmr} frequency in \ch{Au} foil at different currents applied to the Helmholtz coils. The solid red line denotes a linear fit to the data, whose expression appears in the inset.
}
\end{figure}
During commissioning with \gls{rib}, various tunes (ion optics values along the beam path) at \SI{20}{\kilo\electronvolt} \gls{li-8} are established and their record is stored in a local database.
These tunes can be re-scaled for different beam energies, applied magnetic fields, and \glspl{rib}, as well as re-loaded (i.e., as a starting condition) to speedup future beam delivery.
An example of a typical beamspot image is shown in fig.~\ref{fig:beamspot}, obtained at \SI{200}{\milli\tesla} using a 12.5 \si{\milli\meter} $\times$ 12.5 \si{\milli\meter} sapphire plate at one of the sample positions on the sample ladder.
The average transverse beamspot dimensions (i.e., \gls{fwhm}) at the sample position for various applied magnetic fields are determined to be $\sim$3-4 \si{\milli\meter}.
Currently, \gls{slr} measurements on superconducting \ch{Nb} samples have been successfully performed at several magnetic fields up to \SI{200}{\milli\tesla}.
The \gls{slr} rate, commonly denoted as $1/T_1$, characterizes the time-constant in depolarization of the \ch{^{8}Li}\, nuclear spins after they are stopped inside the sample. The value of $1/T_1$ is extracted by fitting the measured time-dependent asymmetry $A(t)$ with a phenomenological depolarization function, e.g., $p(t,t';1/T_1) = \exp[-(t-t')/T_1]$ for a single exponential depolarization function, which is convolved with the square or rectangular beam pulse:
\begin{widetext}
\begin{align}
A(t) = A_0
\begin{cases}
\cfrac{R_0 \int_{0}^{t}\exp[-(t-t')/\tau] p(t,t';1/T_1) dt'}{N(t)} &\text{for $t\leq\Delta$,} \\
\cfrac{R_0 \int_{0}^{\Delta}\exp[-(t-t')/\tau] p(t,t';1/T_1) dt'}{N(\Delta)} & \text{for $t>\Delta$,}
\end{cases}
\label{eq:SLR_fitfunc}
\end{align}
\end{widetext}
where $\Delta$ is the beam pulse length (typically $\sim$1-4 \si{\second}), $\tau$ is the $^8$Li lifetime, and $N(t)$ is the total count rate as defined earlier in eq.~\ref{eq:N_t_count}.
Fig.~\ref{fig:nb-slr} shows a typical asymmetry spectrum and its fit to eq.~\ref{eq:SLR_fitfunc}, measured on a \gls{RRR}$\sim$300 \ch{Nb} sample (cut, etched, and annealed at \SI{1400}{\celsius}) at \SI{5}{\kelvin} using a \SI{4}{\kilo\electronvolt} \gls{li-8} beam.
Applications are available to display the raw counts, the calculated asymmetry, and a fit to various depolarization functions during the experiment\cite{bfit_arXiV,bfit_paper}.
\begin{figure}
\centering
\includegraphics[width=0.7\columnwidth]{Beamspot_200mT.pdf}
\caption{
\label{fig:beamspot}
Beamspot obtained at \SI{200}{\milli\tesla} on a sapphire scintillator. The image shown is looking upstream from the back side of the scintillator, viewed using the downstream CCD camera mounted on a viewport. The small (cyan) bounding box corresponds to the dimension of the visible area of the sapphire from the aperture at back of the sample ladder (see fig.~\ref{fig:sample_config}), while the larger (red) box indicates the real size of the scintillator.
}
\end{figure}
\begin{figure}[hbt!]
\centering
\includegraphics[width=\columnwidth]{SLR_fit_4keV_200mT.pdf}
\caption{
\label{fig:nb-slr}
\ch{^{8}Li}\,\gls{slr} normalized asymmetry spectrum, i.e., $A(t=0)=1$, measured at \SI{4.3}{\kelvin} and \SI{200}{\milli\tesla} applied field on a \ch{Nb} sample using \SI{4}{\second} beam pulses of \SI{4}{\kilo\electronvolt} \ch{^{8}Li^{+}}. The shaded area indicates the measured asymmetry during beam pulse on. The solid line is the normalized fit to eq.~\ref{eq:SLR_fitfunc}.
}
\end{figure}
At applied magnetic fields below $\sim$\SI{1}{\tesla}, the dominant relaxation mechanism in \ch{Nb} is due to cross relaxation between the \ch{^{8}Li}\, and the 100\% abundant host \ch{^{93}Nb}\, nuclear spins\cite{Parolin_Nb}, which gives a Lorentzian dependence of $1/T_1$ on the local magnetic field according to:
\begin{equation}
\frac{1}{T_1} \approx \frac{ ( \gamma B_d )^2 \cdot ( 1/\tau_c ) }{ (1 / \tau_c)^2 + [ \gamma B_{\text{loc}} ]^2 } ,\label{eq:SLR_Rate_Redfield}
\end{equation}
where $\gamma$ is the aforementioned \ch{^{8}Li}\, gyromagnetic ratio, $B_\text{loc}$ is the measured local magnetic field, $B_d$ is the magnitude of the fluctuating dipolar field (due to the host \ch{^{93}Nb}\, nuclear spins), and $\tau_c$ is the correlation time for the fluctuation in $B_d$.
The variation of magnetic fields in the superconducting \ch{Nb} due to Meissner screening can therefore be measured from the asymmetry spectra at different implantation energies (which corresponds to different implantation depths).
Experimental results on the same sample shown in fig.~\ref{fig:nb-slr} for five different implantation energies are shown in fig.~\ref{fig:nb-scds}.
Here, a \ch{Nb} sample is probed by \ch{^{8}Li}\, at five different depth distributions corresponding to ion energies of 4, 8, 12, 16 and 20 \si{\kilo\electronvolt} and average depths of 12, 22.5, 33, 44, 55 \si{\nano\meter}, respectively.
The average depths are computed using the \gls{srim} package~\cite{2010SRIM}.
The data set shows the expected signature of Meissner screening in superconducting \ch{Nb}\cite{Hossain2009} as the depolarization rate increases as the ion is implanted deeper into the surface.
Obtaining the accurate values of the local magnetic field requires a detailed analysis, taking into account the implantation distribution of the \gls{li-8} ions at various energies, which averages over fields at different depths.
The details of this analysis will be reported in a separate publication.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fit_SCDS_data.pdf}
\caption{
\label{fig:nb-scds}
Normalized fits to the \gls{slr} asymmetry spectra, i.e., $A(t=0)=1$, of a \ch{Nb} sample (\gls{RRR}$\sim$300) at various implantation energies measured at \SI{4.5}{\kelvin} and at \SI{98}{\milli\tesla}.
The higher relaxation rates deeper below the surface reflects the reduced magnetic field due to Meissner screening.
}
\end{figure}
\section{
Scientific Applications \& Future Capabilities
\label{sec:scientific-applications}
}
Just like conventional NMR using stable nuclei,
the breadth of applications of $\beta$-NMR is enormous.~\cite{2015-MacFarlane-SSNMR-68-1,MacFarlane_ZPC2021,2018-Kreitzman-JPSCP-21-011056,2014-Morris-HI-225-173}
The ability to operate at intermediate magnetic fields
(10s to 100s of \si{\milli\tesla})
opens new opportunities for scientific applications.
In addition to the \gls{srf} material investigations, some of the experiments enabled in this field regime are: studies of thermally activated dynamics (i.e., moving the Bloembergen-Purcell-Pound peak\cite{BPP_orig} via adjusting $B_{0}$)
~\cite{2014-McKenzie-JACS-136-7833, 2017-McKenzie-JCP-146-244903, 2017-McFadden-CM-29-10187, 2019-McFadden-PRB-99-125201, 2020-McFadden-PRB-102-235206}, and the study in soft condensed matter applications (see e.g.,~\cite{2014-McGee-JPCS-551-012039, 2015-McKenzie-SM-11-1755, 2018-McKenzie-SM-14-7324}) where the previously measured signal was completely wiped out below \SI{\sim 24}{\milli\tesla}, but access to fields an order of magnitude higher may ameliorate this difficulty.
Recent measurements on \ch{Nb} samples indicate vortex penetration at applied fields of \SI{98}{\milli\tesla} when \ch{Nb} samples are warmed close to \gls{tc}.
The mixed superconducting state is therefore also accessible and could be of interest for fundamental studies related to the superconducting vortex motion and dissipation~\cite{Vortex_Eley2021}.
\gls{bnmr} studies of the vortex lattice have been carried out in other superconductor materials such as \ch{YBa_2Cu_3O_{7-$\delta$}}\cite{vortex_YCBO_Saadaoui2009,vortex_YCBO_Saadaoui2009a} and \ch{NbSe2}\cite{vortex_NbSe2_Salman2007} using \gls{bnmr} spectrometer (with out-of-plane applied fields).
Further studies for \gls{srf} applications will benefit from lower sample temperatures in order to maintain the Meissner state at higher applied magnetic fields.
Installation of a procured cryocooler is ongoing and measurements of superconducting samples in this regime are expected in the near future.
\section{
Summary
\label{sec:summary}
}
The instrumentation at TRIUMF's \gls{bnmr} facility allows for the depth-dependent characterization of the local magnetic field near the surface of a sample. Both the infrastructure and instrumentation have been upgraded with a new beamline extension on the \gls{bnqr} leg, which has expanded previous capabilities from a maximum parallel magnetic field of \SI{24}{\milli\tesla} to \SI{200}{\milli\tesla}. This capability is targeted at testing \gls{srf} samples in a regime analogous to the magnetic field conditions in a \ch{Nb} cavity operating at the fundamental limit, but will be widely useful to other condensed matter research. This capability is unique in the world and we anticipate additional use for other material science investigations.
\begin{acknowledgments}
We thank:
L.~Merminga for involvement in the early stages of the project;
M.~H.~Dehn for assistance during the beamline assembly;
M.~Cervantes for sample preparation;
D.~Lang, J.~Keir, R.~Abasalti, B.~Hitti, B.~Smith, D.~Vyas, T.~Au, J.~Chow, T.~Hruskovec, N.~Muller, and M.~Marchetto for providing excellent technical support.
The hardware was funded through a Research Tools and Infrastructure grant from NSERC.
\end{acknowledgments}
\section*{Author Declarations}
\subsection*{Conflict of Interest}
The authors have no conflicts to disclose.
\subsection*{Author Contributions}
\textbf{Edward~Thoeng:} Data Curation (equal); Formal Analysis (equal); Investigation (equal); Validation (equal); Visualization (equal); Writing --- Original Draft Preparation (lead); Writing --- Review \& Editing (equal).
\textbf{Ryan~M.~L.~McFadden:} Data Curation (equal); Formal Analysis (equal); Investigation (supporting); Visualization (equal); Writing --- Review \& Editing (equal).
\textbf{Suresh~Saminathan:} Investigation (supporting); Methodology (equal); Validation (equal); Visualization (supporting).
\textbf{Gerald~D.~Morris:} Conceptualization (equal); Investigation (equal); Project Administration (supporting); Resources (equal); Supervision (supporting); Validation (equal).
\textbf{Philipp~Kolb:} Investigation (equal); Project Administration (supporting); Supervision (supporting) Validation (equal).
\textbf{Ben~Matheson:} Methodology (equal); Visualization (supporting).
\textbf{Md~Asaduzzaman:} Investigation (supporting).
\textbf{Richard~Baartman:} Conceptualization (equal); Supervision (supporting).
\textbf{Sarah~R.~Dunsiger:} Investigation (supporting).
\textbf{Derek~Fujimoto:} Formal Analysis (supporting); Investigation (supporting).
\textbf{Tobias~Junginger:} Formal Analysis (supporting); Investigation (supporting); Project Administration (supporting); Supervision (supporting).
\textbf{Victoria~L.~Karner:} Investigation (supporting).
\textbf{Spencer~Kiy:} Data Curation (equal); Investigation (supporting).
\textbf{Ruohong~Li:} Resources (equal).
\textbf{Monika~Stachura:} Investigation (supporting).
\textbf{John~O.~Ticknor:} Investigation (supporting).
\textbf{Robert~F.~Kiefl:} Conceptualization (equal); Funding Acquisition (supporting).
\textbf{W.~Andrew~MacFarlane:} Investigation (supporting); Supervision (supporting).
\textbf{Robert~E.~Laxdal:} Conceptualization (equal); Formal Analysis (supporting); Funding Acquisition (lead); Project Administration (lead); Supervision (lead); Writing --- Review \& Editing (equal).
\begin{comment}
Formal Analysis (lead)
Investigation(equal)
Software (lead)
Writing -- Original Draft Prepration (lead)
Writing -- Review \& Editing (equal)
Conceptualization (equal)
Funding Acquisition (equal)
Resources (equal) - Gerald
\end{comment}
\section*{Data Availability Statement}
The data that support the findings of this study are available from the corresponding author upon reasonable request. Raw data of the \gls{bnmr} experiments are available for download from: \url{https://cmms.triumf.ca}/
| {
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Il cross country maschile Elite è una delle prove inserite nel programma dei campionati del mondo di mountain bike. Si corre sin dalla prima edizione, 1990.
Albo d'oro
Aggiornato all'edizione 2022.
Medagliere
Note
Campionati del mondo di mountain bike | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 311 |
The Crow Reboot Now Being Produced by Davis Films
by Alex Welch
– on Nov 17, 2016
It is hard to think of a reboot that has had a harder time making its way to the big screen than the long-troubled remake of The Crow. After cycling through a number of directors throughout the years, and even more actors, like Bradley Cooper, Luke Evans, and Jack Huston, the likelihood of The Crow reboot ever seeing the light of day was slim, to say the least. More recently, the reboot finally caught a break after director Corin Hardy was brought on and Jason Momoa was rumored to be playing the role of Eric Draven, previously portrayed onscreen by the late Brandon Lee.
A report from several months ago announced The Crow reboot would go into production in January of 2017, with Momoa set to lead the reboot just after finishing principal photography on Zack Snyder's Justice League. After that, he was reportedly set to begin production on his 2018 standalone Aquaman film, with director James Wan.
However, THR is now reporting that The Crow reboot, titled The Crow Reborn, has officially moved from Relativity Media, and will be distributed, produced, and financed by Samuel Hadida's Davis Films, Highland Film Group, and Electric Shadow. In a worrying bit of news, the report goes on to state that it is unclear if either Corin Hardy or Jason Momoa will remain attached to the project following this transition.
For what it's worth, Edward R. Pressman, who produced the original 1994 film will now produce the reboot alongside Hadida, it has been revealed. To his credit as well, Hadida has a long history of dealing with beloved properties on the big screen, through his work on the Resident Evil and Silent Hill franchises.
Principal photography for the reboot is apparently still scheduled to take place next year, though the exact timeframe is no longer clear, which could lead to scheduling problems with cast and crew. After landing Momoa as its lead, and a promising directorial voice in Hardy, it seemed like The Crow Reborn had finally begun to gain traction behind the scenes for the first time in a long time. Now, depending on how this transition affects Hardy and Momoa's involvement in the project, it is entirely possible this could wind up another disappointing development for fans waiting to see The Crow return to the big screen in some form or another.
The Crow reboot does not currently have a release date. Keep checking back for more updates.
Source: THR
Tags: the crow | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 590 |
Yr.: Fr. Pos.: M Ht.: 5'9" Last School/Hometown: Northport / East Northport, N.Y.
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Yr.: Fr. Pos.: G Ht.: 5'5" Last School/Hometown: The Charter School of Wilmington / Wilmington, Del. | {
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package consulo.csharp.lang.psi.impl.light;
import com.intellij.openapi.util.text.StringUtil;
import com.intellij.psi.PsiElement;
import consulo.annotation.access.RequiredReadAction;
import consulo.csharp.lang.psi.impl.light.builder.CSharpLightTypeDeclarationBuilder;
import consulo.csharp.lang.psi.impl.source.resolve.type.CSharpTypeRefByQName;
import consulo.dotnet.psi.DotNetTypeDeclaration;
import consulo.dotnet.resolve.DotNetTypeRef;
import javax.annotation.Nonnull;
import javax.annotation.Nullable;
/**
* @author VISTALL
* @since 02.09.14
*/
public class CSharpLightAttributeWithSelfTypeBuilder extends CSharpAbstractLightAttributeBuilder
{
private final CSharpLightTypeDeclarationBuilder myType;
private final DotNetTypeRef myTypeRef;
@RequiredReadAction
public CSharpLightAttributeWithSelfTypeBuilder(PsiElement scope, String qualifiedName)
{
super(scope.getProject());
myType = new CSharpLightTypeDeclarationBuilder(scope.getProject(), scope.getResolveScope());
myType.withParentQName(StringUtil.getPackageName(qualifiedName)) ;
myType.withName(StringUtil.getShortName(qualifiedName));
myTypeRef = new CSharpTypeRefByQName(scope.getProject(), scope.getResolveScope(), qualifiedName);
}
@Nullable
@Override
public DotNetTypeDeclaration resolveToType()
{
return myType;
}
@Nonnull
@Override
public DotNetTypeRef toTypeRef()
{
return myTypeRef;
}
@Override
public String toString()
{
return "CSharpLightAttributeWithSelfTypeBuilder: " + myType.getVmQName();
}
}
| {
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{"url":"http:\/\/yozh.org\/2010\/11\/29\/mmm009\/","text":"The image is centered at $$0.1528886225+1.0395109505i$$. Once again, we see some of the repeated structures of the Mandelbrot set. At the center of the image, there is a miniature copy of the complete set. Moreover, the structure at the center of the image is repeated along filaments of the Mandelbrot set throughout the region under view. We see self-similarity upon self-similarity.","date":"2018-09-25 14:07:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8496206998825073, \"perplexity\": 276.7702530254386}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-39\/segments\/1537267161638.66\/warc\/CC-MAIN-20180925123211-20180925143611-00284.warc.gz\"}"} | null | null |
Beautiful Cul-de-Sac home features, 4 bedrooms, 2.5 bathrooms & 3 car garage. Step into the formal living room with two way fireplace accented with gray tiles, new shades & beautiful crystal chandelier that anchors the whole room. The Family room has a built in dark wood stained entertainment center, fireplace & ceiling fan. The open floor plan opens to the kitchen & eating area. The kitchen features brand new luxury vinyl plank flooring in washed oak dove, new built in microwave, granite counter tops, recessed lighting, kitchen island & walk in pantry. Downstairs also features a guest bathroom that has been remodeled with new tile tile flooring, baseboards & mirrored medicine cabinet. The laundry room with built in cabinets & storage room. Head upstairs to the open loft area currently used as another TV room & features custom paint design. The master bedroom is massive with beautiful new ceiling fan, crown molding, accent wall with master bathroom. The master bathroom features his & her sinks, separate shower & soaking tub. The second bedroom is also spacious with gorgeous accent lighting & new closet insert. The 2 remaining rooms are equally roomy & have new ceiling fans. The upstairs bathroom has dual sinks & shower & tub combo. The front & back yard have brand new sod & mulch with sprinkler system. The house features new lighting in nearly every room & fresh paint throughout. This home is just minutes from the lake, park, school & a short drive to the outlets. | {
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{"url":"http:\/\/accesspediatrics.mhmedical.com\/content.aspx?bookid=1462§ionid=85596774","text":"Chapter 95\n\nDIAGNOSIS\n\nOmphalocele and gastroschisis represent the 2 most frequently encountered abdominal wall defects requiring neonatal intensive care. As discussed previously in this book, these defects occur in roughly 1\u20133 per 10,000 live births. Although the incidence of omphalocele has remained constant in recent years, the incidence of gastroschisis has been increasing for unclear reasons.\n\nClinical Findings\n\nOmphalocele is associated with advanced maternal age and karyotype abnormalities; gastroschisis is associated with maternal age less than 20, smoking, and use of over-the-counter vasoactive drugs and salicylates during pregnancy.1, 2, and 3 In addition, illicit drug abuse and smoking may influence the severity of gastroschisis.4\n\nOmphalocele is characterized by the failure of the viscera to return to the abdominal cavity following physiologic midgut herniation during the 10th week of gestation. As a result, the omphalocele is contained within a protective membranous sac composed of amniotic epithelium lined by peritoneum, with the intervening space filled by Wharton\u2019s jelly. The stomach, small bowel, colon, and liver are frequently involved. Associated anomalies occur in 50%\u201370% of infants with omphalocele; cardiac defects are observed in 30%\u201350%. Karyotype abnormalities occur in 30% of cases, with trisomies 13, 18, and 21 most common.2\n\nGastroschisis, on the other hand, is characterized by prenatal evisceration through a defect in the anterior abdominal wall, almost always located just to the right of the umbilicus. This right-sided predilection is theorized to be caused by abnormal embryonic regression of the right umbilical vein. Importantly, the eviscerated abdominal contents do not have a protective membrane and are in direct contact with the amniotic fluid. The involved intestine is edematous, sometimes foreshortened, and almost always nonrotated. Of neonates with gastroschisis, 7%\u201310% will have an associated intestinal atresia. Unlike omphalocele, gastroschisis is not associated with karyotype abnormalities.2\n\nThe effective management of both omphalocele and gastroschisis hinges on early diagnosis and the involvement of appropriately trained staff (trained nurses, neonatologist, and pediatric surgeons). The embryologic and anatomic differences, however, lead to the differences in management depicted in the discussion that follows.\n\nConfirmatory (Diagnostic) Tests\n\nPrenatal Imaging\n\nThe sensitivity and specificity of prenatal ultrasound in identifying abdominal wall defects are 60%\u201375% and 95%, respectively.5,6 Once a fetus with an abdominal wall defect is identified, directed ultrasounds should be performed to look for associated anomalies and malformations. Fetal magnetic resonance imaging is gaining popularity and is now used as a reflex imaging study if initial ultrasound is suspicious for gastroschisis or omphalocele in many centers.\n\nLaboratory Tests\n\nThere is a marked elevation of maternal serum \u03b1-fetoprotein (AFP) in cases of gastroschisis; omphalocele tends to have a more modest AFP elevation (7\u20139 multiples of the mean compared to 4 multiples of the mean, respectively).7,8 As discussed previously in ...\n\nSign in to your MyAccess profile while you are actively authenticated on this site via your institution (you will be able to verify this by looking at the top right corner of the screen - if you see your institution's name, you are authenticated). Once logged in to your MyAccess profile, you will be able to access your institution's subscription for 90 days from any location. You must be logged in while authenticated at least once every 90 days to maintain this remote access.\n\nOk\n\nIf your institution subscribes to this resource, and you don't have a MyAccess profile, please contact your library's reference desk for information on how to gain access to this resource from off-campus.\n\nSubscription Options\n\nAccessPediatrics Full Site: One-Year Subscription\n\nConnect to the full suite of AccessPediatrics content and resources including 20+ textbooks such as Rudolph\u2019s Pediatrics and The Pediatric Practice series, high-quality procedural videos, images, and animations, interactive board review, an integrated pediatric drug database, and more.","date":"2017-03-25 03:54:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.22685863077640533, \"perplexity\": 6321.657787848256}, \"config\": {\"markdown_headings\": false, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-13\/segments\/1490218188785.81\/warc\/CC-MAIN-20170322212948-00608-ip-10-233-31-227.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
Microscopic mechanisms behind the formation of holographic grating are complicated. Respective nonlinear system of differential equations for the kinetics of the process, which includes many parameters that are used to characterize the photorefractive material, can be solved through an intricate numerical calculation only, along with making a series of simplifying assumptions. Therefore, it is interesting to construct simpler phenomenological models to cope with the problem.
A simpler approach can be developed towards the non-equilibrium steady states (NESS) that are reached when the diffraction efficiency of the grating changes no more under stable irradiation \cite{Collier}. This is a NESS that will be characterized here by a single mode of the electromagnetic field (the writing mode), and the respective modulation of the electric permittivity (the photorefractive response). The two latter quantities are interdependent, and should be obtained in a self-consistent manner in the theory. (One may think of some functional, which has extrema at NESS "points" of the system driven by irradiation.)
From the viewpoint of nonequilibrium thermodynamics, the driven open system that is represented here by the electromagnetic field coupled to the photorefractive medium, slowly evolves towards a NESS under fast external driving. The two time-scales differ in a dozen of orders in magnitude, and it is a natural phenomenological assumption that the modulation of the permittivity of the initially homogeneous medium is determined by the NESS mode itself. Based on this assumption is the discussion below.
An interesting feature of locally isotropic photorefractive medium - phase-sensitive birefringence effect, that is a manifest of the photonic bandgap and Bragg reflection, shows up.
\section{Exactly solvable model}\label{solve}
Vector nature of light is at the heart of the photorefractive effect. Fortunately, the two polarization modes can be treated separately, in the properly symmetric arrangements. In view of the holographic grating recording, of major interest are the modes of the interference pattern. Note that their observation can be perfectly "tuned in" due to Bragg reflection. By assuming that the active NESS mode is the electric component of the so-called E-wave,
\begin{equation}\label{ewave}
{E_y}(z,x,t)={F}\left(z\right){sin}({k_{x}} x-\omega t),
\end{equation}
Maxwell's equations are reduced to the second-order differential equation for the mode profile, $F(z)$:
\begin{equation}\label{maxwell}
{F''}+(k_0^2{\varepsilon} (z)-k_x^2){F}=0;
\end{equation}
for details see \cite{LLVIII}. $F(z)$ is referred to below as the "NESS mode".
It is assumed that the in-medium mode inherits i) the antiperiodicity of the driving field, $F(z+\lambda_z/2)=-F(z)$, and, similarly, ii) the reflection symmetry for the mode profile, $F(-z)=F(z)$, or $F(-z)=-F(z)$, and, consequently, the eq. (\ref{maxwell}) has to be solved under antiperiodic boundary condition for the even or the odd mode.
Perhaps the simplest model assumption for the NESS permittivity is that that its modulation follows the intensity of the NESS mode directly; then the transparent and nonmagnetic medium of the model, with periodic inhomogeneity in the $z$-direction, is described by its permittivity at a given frequency $k_0$ of the mode by
\begin{equation}\label{epsilon1}
{\varepsilon}(z)={\epsilon}-2{\eta}F(z)^2+{\eta}F_0^2;
\end{equation}
$F_0$ is the amplitude of the NESS mode, and it is assumed that the ${\eta}F_0^2$ is relatively small number: the modulation $-{\eta}F_0^2$$\leq$${\varepsilon}(z)-\epsilon$$\leq$${\eta}F_0^2$ , typically, is weak in holographic gratings. An expression similar to (\ref{epsilon1}) is used in nonlinear electrodynamics to study self-defocusing of the electromagnetic waves (see \cite{LLVIII}). Here its meaning is different: we consider the linear electrodynamics problem, and the "nonlinear" term describes the mode profile in the NESS. In view of the localization properties of electrons in photorefractive materials, which are known to be good insulators, the local approximation in (\ref{epsilon1}) seems to be reasonable: it is a natural phenomenological assumption that the modulation of the permittivity of the initially homogeneous medium, in the first approximation, is a quadratic function of the NESS mode amplitude.
After substituting $\varepsilon(z)$ as a function of the "brightness" of the mode, ${\varepsilon}(z)\rightarrow{\epsilon}-2{\eta}F^2+{\eta}F_0^2$, the linear equation (\ref{maxwell}) takes its nonlinear form:
\begin{eqnarray}
\label{maxwellness}
{F}''+(k_0^2{\epsilon}-k_x^2+{\eta}k_0^2 F_0^2-2{\eta}k_0^2 F^2){F}=0.\nonumber
\end{eqnarray}
A first integral can be written down explicitly:
\begin{eqnarray}
\label{first}
(F')^2+(k_0^2{\epsilon}-k_x^2+{\eta}k_0^2 F_0^2){F^2}-{\eta}k_0^2 F^4=(k_0^2{\epsilon}-k_x^2){F_0^2}.\nonumber
\end{eqnarray}
The wavelength of the mode is readily found:
\begin{eqnarray}
\label{firstg}
\frac{{\lambda}_z }{2}= \int_{-F_0}^{F_0} \frac{1}{\sqrt{(F_0^2-F^2)(k_0^2{\epsilon}-k_x^2-{\eta}k_0^2 F^2)}} \, {dF};\nonumber
\end{eqnarray}
see textbooks on classical mechanics. The mode of interest is, naturally, that of winding number 1. Resulting dispersion relation has the form:
\begin{eqnarray}
\label{drelation}
{\lambda}_z^2 (k_0^2{\epsilon}-k_x^2)=(4K(m))^2=\nonumber\\
(2\pi)^2(1+\frac{m}{2}+\frac{11 m^2}{32}+O\left(m^3\right)),
\end{eqnarray}
in contrast with the relation ${\lambda}_z^2 (k_0^2{\epsilon}-k_x^2)=(2\pi)^2$ for an isotropic and uniform medium. (We have avoided using the notation "$k_z$" along with the "$k_x$" to stress the fact that the former is actually a quasimomentum.) $K(m)$ is the complete elliptic integral of the first kind, and its modulus, $m$, is related to the parameters of the problem by the following equation:
\begin{eqnarray}
\label{modulus}
m=\eta F_0^2\frac{ k_0^2}{k_0^2 \epsilon-k_x^2}.\nonumber
\end{eqnarray}
The NESS mode has the form
\begin{eqnarray}
\label{sn}
F(z)=F_0{sn}\left(\left.\frac{4 K(m) z}{\lambda_z }\right|m\right)\nonumber
\end{eqnarray}
with the Jacobi elliptic function $sn$. The respective profiles of the permittivity modulation normalized to the effective NESS mode,
\begin{eqnarray}\label{prof}
\frac {{\lambda}_z^2 (k_0^2{\varepsilon} (z)-{k_x}^2)}{(2\pi)^2}=\frac{(4K(m))^2}{(2\pi)^2}\times\nonumber\\
\left(1-2m\left({sn}\left(\left.\frac{4 K(m) z}{\lambda_z }\right|m\right)\right)^2+m\right),
\end{eqnarray}
are shown in the \fref{lame}.
\begin{figure}
\includegraphics[width=3 in]{figure1.eps}\\
\caption{Permittivity modulations for $0\leq m \leq 0.02$. Plotted is the right-hand side of the eq. (\ref{prof}).}\label{lame}
\end{figure}
The eq. (\ref{maxwell}) with $\varepsilon(z)$ defined using (\ref{sn}) reduces to the well known equation of Lam\'{e}. With the brightest point of the interference pattern at $z=0$ the Jacobi $sn$ in (\ref{maxwellness}) and (\ref{sn}) will be replaced by the Jacobi $cd$. For references on Jacobi elliptic functions and Lam\'{e} equation see \cite{WW}.
Remarkably, the inverted brightness $(F^*(z))^2=F_0^2-(F(z))^2$ mode also exists as an eigenmode for the eq. (\ref{maxwell}),
\begin{eqnarray}
\label{cn}
F^*(z)=F_0{cn}\left(\left.\frac{4 K(m) z}{\lambda_z }\right|m\right);\nonumber
\end{eqnarray}
the ${cn}$ will be replaced by the $\sqrt{1-m}{sd}$ when the brightest point of the interference pattern is at $z=0$. The dispersion relation for these latter modes has the form
\begin{eqnarray}\label{drelationinv}
{\lambda}_z^2 (k_0^2{\epsilon}-{k_x^*}^2)=(1-m)(4K(m))^2=\nonumber\\
(2\pi)^2(1-\frac{m}{2}-\frac{5 m^2}{32}+O\left(m^3\right)),
\end{eqnarray}
and, therefore, the difference
\begin{eqnarray}
\label{birefrin}
{\lambda}_z^2 ({k_x^*}^2-k_x^2)=m(4K(m))^2=\nonumber\\
(2\pi)^2(m+\frac{m^2}{2}+O\left(m^3\right))
\end{eqnarray}
is small as expected. The picture is sensitive to the relative phase, and the dispersion relations (\ref{drelation}) and (\ref{drelationinv}) can be contrasted with those for the conventional birefringence case,
\begin{eqnarray}
\label{birefrino}
{\lambda}_z^2 ({k_0^2\varepsilon}_o-{k_{xo}}^2)=(2\pi)^2,{\lambda}_z^2 ({k_0^2\varepsilon}_e- {k_{xe}}^2)=(2\pi)^2\frac{{\varepsilon}_e}{{\varepsilon}_o},\nonumber
\end{eqnarray}
with the indices $o$ and $e$ denoting the ordinary and extraordinary waves, respectively.
The two solutions found belong to the band edges of the spectrum of the Hamiltonian associated with the periodic modulation of the $\varepsilon(z)$; the bandgap here manifests itself as a phenomenon resembling birefringence: the above expression (\ref{birefrin}) describes \emph{phase-dependent} birefringence; joint dispersion relation in parametric form is given by (\ref{drelation}) and (\ref{drelationinv}), and the small parameter $m$ is accounted for by (\ref{birefrin}). With the latter equation, experimental observation is straightforward, by measuring the difference between $k_x^*$ and $k_x$. Regarding such an experiment, as well as the symmetries of the problem, the remarks in the following section seem relevant.
\section{NESS, two basic geometries, and translation invariance}\label{ness}
\begin{figure}
\includegraphics[width=3 in]{figure2.eps}\\
\caption{Two complementary setups for holographic grating recording. The two incident plane waves dictate either $\lambda_z$ (on the left) or $k_x$ (on the right). }\label{setups}
\end{figure}
As regards the (relative phase-sensitive) "Bragg birefringence" effect mentioned at the end of the previous section, two setups shown in the \fref{setups} have to be considered. Full translation invariance in the bulk of the photorefractive medium is conserved along the horizontal axis only (the $x$-axis). With the boundaries of the plate parallel to the vertical axis (the $z$-axis), externally controlled is the quantity $\lambda_z$, and the quantity $k_x$ - in the next setup, with the boundaries parallel to the $x$-axis. Also notice that the parameter $\epsilon$ that enters the joint dispersion relation (\ref{drelation}) and (\ref{drelationinv}) is "renormalized" by the NESS grating, and actually also depends on the $m$, as a self-consistent quantity. Therefore, to use the dispersion relations, the small parameter $m$ has to be determined first, by using the relation (\ref{birefrin}).
This can be done, in this example, due to the reflection symmetry: by externally exciting the two modes, $F$ and $F^*$, separately.
Interference pattern created by two crossing plane waves in a homogeneous transparent medium - the intensity of the superposition mode of the electromagnetic field, is considered to be the "driving" field. For instance, the superposition mode (the two incident waves have opposite phases at the origin of the coordinate system)
\begin{eqnarray}
\label{trigs}
{E_y}\sin ({k_z} z) \sin ({k_x} x-\omega t)\nonumber
\end{eqnarray}
creates a one-dimensional interference pattern with intensity profile $\frac{1}{2}({E_y}\sin({k_z} z))^2$, and, similarly, the intensity profile is $\frac{1}{2}({E_y}\cos ({k_z} z))^2$ for the mode
\begin{eqnarray}
\label{trigc}
{E_y}\cos ({k_z} z) \cos ({k_x} x-\omega t)\nonumber
\end{eqnarray}
(for coinciding at the origin phases). As regards the interpretation of the (\ref{epsilon1}) in view of nonequilibrium processes for the E-wave, the entropy production in transparent materials is actually the Joule heat, and the modified $E_y^2$ is at place in the equation.
The period of the interference pattern is ${\lambda_z}/{2}$, and the bright and the dark strips alternate at half that length.
Translation by ${\lambda_z}/{4}$,
\begin{equation}\label{shifts}
F(z)\rightarrow \tilde{F}(z)\equiv F\left(z+{\lambda_z}/{4}\right),
\end{equation}
results in swapping them (recall that ${\lambda_z}/{2}$-antiperiodic modes are considered, and, therefore, the direction of the shift could be reversed as well). For the brightness of the interference pattern in a homogeneous medium, the following symmetry property holds:
\begin{equation}\label{symmetry}
{\tilde{F}(z)}^2={{{F}}_0}^2-{{F}}(z)^2.
\end{equation}
For the NESS, however, full translation invariance of the medium is broken due to the electric permittivity modulation: the "polaritonic" nature of the NESS mode should not be symmetric neither with respect to the bright and the dark, nor with respect to the ${\lambda_z}/{4}$-shifts. This is clear with the quartet of the modes obtained in the previous section, which solutions can be denoted as $F,\tilde{F},F^*,$ and $\tilde{F}^*$ in the view of the (\ref{shifts}). Both pairs of the modes, $F,F^*$, and $\tilde{F},\tilde{F}^*$, are described by a pair of the wavenumbers $k_x,k_x^*$, and the symmetry (\ref{symmetry}) is broken.
The symmetry with respect to shifts by ${\lambda_z/4}$ (see the definition (\ref{shifts})), obviously, is not for the homogeneous medium uniquely. With full translation invariance in the $z$-direction broken, there can exist modulations that preserve that symmetry. An entire family of such is presented by a symmetric modification of the eq. (\ref{epsilon1}):
\begin{eqnarray}
\label{goff}
\frac {{\lambda}_z^2 (k_0^2{\varepsilon} (z)-{k_x}^2)}{(2\pi)^2}=\frac{(4K(m))^2}{(2\pi)^2}\times\nonumber\\
\left(1-\frac{m}{2}+
\frac{3}{4}\left(\sqrt{1-m}+1\right)^2\left(\left(\frac{{F}^2+{\tilde{F}}^2}{{F_0}^2}\right)^2-1\right)\right)\nonumber
\end{eqnarray}
\begin{figure}
\includegraphics[width=3 in]{figure3.eps}\\
\caption{Symmetric modulations: $0\leq m \leq 0.1$.}\label{symm}
\end{figure}
The respective profiles of the permittivity modulation are clear from the \fref{symm}: plotted is the right-hand side of the above equation. The problem is equivalent to the so-called associated Lam\'{e} potentials \cite{Khare}.
In this symmetric case, the two modes have the same $k_x$: no Bragg birefringence is present. The modes, explicitly, are:
\begin{eqnarray}
\label{symmsANDc}
{F(z)=\frac{{F_0}(1-m)^{1/4}{{sn}\left(\left.u\right|m\right)}}{\sqrt{{dn}\left(\left.u\right|m\right)}}, \tilde{F}(z)=\frac{{F_0}{{cn}\left(\left.u\right|m\right)}}{\sqrt{{dn}\left(\left.u\right|m\right)}}}\nonumber
\end{eqnarray}
with $u\equiv\frac{4 K(m) z}{\lambda_z }$. The Wronskian $W(F,\tilde{F})$ (defined as ${\tilde{F}}{F'}-{F}{\tilde{F}'}$) is constant, because of the "degeneracy" mentioned above. This time the symmetry with respect to the inversion of the brightness does not hold, in contrast with the model presented by the eq. (\ref{epsilon1}): the respective mode $F^*(z)$ such that $(F^*(z))^2=F_0^2-(F(z))^2$ does not exist as an eigenmode for the given modulation of the permittivity; this is quite natural in the view of the degeneracy present in the second-order equation.
Gratings of this type can be of interest for the optically nonlinear photorefractive materials in the setups appropriate for the higher harmonic generation.
\section{Concluding remarks}\label{conclude}
In the view of the variety of possible mechanisms of the holographic grating formation in photorefractive media, it should be noted that even in a single material there may exist different mechanisms, due to the variations in the irradiation techniques (e.g., with pulsed irradiation, as compared to the cw irradiation, by varying the pulse duration and the intervals in between vs electronic relaxation times). It means that engineering the models is also possible.
In the case the holographic grating is recorded in a plate with $k_x$-controlled setup shown in the \fref{setups}, its thickness $l$ comes into play: because of reflections, it must match the phase of the interference pattern such that $2l=n\lambda_z$.
Similarly, in the case of using a mirror for recording, such considerations could help for better tuning the process, by slight variations of the angle of incidence and, if possible, the frequency of the incident beam.
In conclusion, it was discussed how the permittivity locally-isotropic modulation-induced anisotropy appears, leading to the Bragg birefringence effect, in the process of the recording of the holographic grating. The effect can be of interest for quantum devices, due to its phase-sensitivity.
\section*{Acknowledgments}
The work was supported by ISTC grant, Project A-1517.
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,090 |
Absalón Gechman (n. Mandato británico de Palestina, 4 de octubre de 1923 - f. Buenos Aires, Argentina; 11 de noviembre de 2006), mejor conocido como Ariel Absalón, fue un actor y director teatral con una amplia trayectoria en la escena argentina.
Carrera
Ariel Absalón se inició en el teatro independiente a los 11 años de edad. A los 22 egresó del Conservatorio Nacional de Arte Escénico, poseyendo además, amplios conocimientos de dibujo, pintura y artes plásticas. Tuvo por maestros a Antonio Cunill Cabanellas, Alfredo de la Guardia y Vicente Fatone, habiendo asimilado preceptivas de Stanislavsky, Antoine y Copeau, entre otros.
Pasó por distintas compañías profesionales y teatros independientes, ingresó al Cervantes en 1956 y actuó en radioemisoras comerciales y en canales de T.V., permitiéndole adquirir una sólida experiencia, la cuál despertó en él una irresistible atracción por manifestarse en las tareas directivas. Su capacidad directiva se manifestó en la puesta en escena de Las de Barranco, Ra-ta-ta-tá, entre otras.
En calidad de actor se lució notablemente durante la época dorada cinematográfica argentina. Habiendo debutado en 1949 con el film argentino-chileno Esperanza, protagonizado por Jacob Ben-Ami y Aída Alberti, filmó en más de 15 películas junto a distinguidas figuras de la escena nacional como Ignacio Quirós, Alfredo Alcón, Graciela Borges, Atahualpa Yupanqui, Olga Zubarry, Roberto Escalada, Alberto Olmedo, Ubaldo Martínez, Nelly Láinez, Enzo Viena, Alita Román, Narciso Ibáñez Menta, entre muchas otras.
En televisión dirigió varios espectáculos, conduciendo frente a las cámaras a Tito Alonso, Ricardo Trigo, Fernando Vegal, Beatriz Taibo, Luis Tasca, Fernando Labat y otras primeras figuras. Además, dirigió y protagonizó varias fotonovelas de la revista Cine primicias junto a Sandra Mahler, Héctor Armendáriz, Iris Alonso, entre otros.
Filmografía
Teatro
En 1949, forma parte de la compañía de Alberto Rodríguez Muñoz presentando en el teatro SODRE de Uruguay un breve ciclo de producciones contemporáneas: "Yo voy más lejos" del uruguayo Enrique Amorim, "La hermosa gente" del escritor armenio-americano William Saroyan y "Un día de Octubre", de Georg Káiser. Así lo describía el periódico uruguayo "El bien público":"El elenco de Rodríguez Muñoz agrupa a un conjunto de jóvenes elementos del teatro rioplatense, entre los que sobresalen la primera actriz argentina Lía Gravel, el actor Ariel Absalón, que acaba de efectuar un papel de importancia en la película "Esperanza", junto a Jacobo Ben Ami, y la actriz uruguaya Ofelia Gil San Martín."El mismo diario, anunciando la cercanía del estreno, escribía en septiembre de 1949:"El próximo jueves, en función nocturna, se presentará ante nuestro público el elenco del actor y director argentino Alberto Rodríguez Muñoz, con la obra del prestigioso escritor uruguayo Enrique Amorim intitulada "Yo voy más lejos". Además de la actriz Lía Gravel, primera figura femenina del conjunto, actuarán en esa obra en papeles protagónicos la actriz Ofelia Gil San Martín y el actor Ariel Absalón, cuya participación en realizaciones escénicas y cinematográficas en la vecina orilla fueron comentadas elogiosamente por la prensa porteña."En 1950 se incorpora al grupo "Don Muelsa" y trabaja con estrellas como Norma Giménez, Alfredo Alcón, Francisco de Paula, Adolfo Linvel, Carlos Bellucci, entre otros.
En 1954 debuta como director con "La loca del Cielo", de Lenormand, en el Teatro Patagonia. A partir de allí dirigirá varias obras como "Lluvia", de Somérset Maugham, en el Teatro Lassalle del empresario Arturo Puig, "Nuestra piel", de Lacour, en el Teatro Rossini (1956) y "Una rienda para el mar", de Solly.
Luego formó parte del conjunto del Teatro Nacional Cervantes dónde actuó en Facundo de la Ciudadela (1956), Los Mellizos (1957), Asesinato en la catedral (1957), Así es (si le parece) (1957) y en la comedia en tres actos titulada Los expedientes (1957) junto a reconocidos actores como Julio de Grazia, Juan José Edelman, Guillermo Bredeston, Miguel Bebán, María Elina Rúas, entre otros, bajo la dirección de Orestes Caviglia, Camilo Da Passano, Armando Discépolo y Osvaldo Bonet.
En 1959 trabaja como director de la obra Los dioses aburren de Malena Sandor, encabezada por Fernando Vegal, Leda Zanda y María Elina Rúas en el Teatro Lezama.
En la temporada 1961-1962 actuó en Il Corvo de Carlo Gozzi en el Teatro Caminito de La Boca. En este oportunidad interpretó a "Millo, el rey de Fratombosa" y compartió elenco con Jorge Luz, Aída Luz, Guillermo Helbling, entre otros.
En 1962 actuó en la obra Los hermanos Karamazov, junto a Miguel Narciso Bruse y Ricardo Trigo (h), presentado por "La Máscara", en el Colonial de Buenos Aires.
Entre otras de las comedias que dirigió y/o actuó se encuentran:
La isla de la gente hermosa (1950), de Román Gómez Masía - Teatro Odeon (Actor). En esta obra fue el debut profesional de Alfredo Alcón.
La loca del cielo (1955), de Miguel Luis Coronatto - Teatro Patagonia (Director y actor).
Facundo en la ciudadela (1956), de Vicente Barbieri - Teatro Cervantes (Actor).
Los mellizos (1957), de Plauto - Teatro Cervantes (Actor).
Asesinato en la catedral (1957), de T.S. Elliot - Teatro Nacional Cervantes (Actor).
Las de Barranco (1961), de Gregorio de Laferrere - Teatro La Plata (Director).
¡Despierta...Isabel! (1965), de Graciela Teisaire - Teatro Buenos Aires (Director).
La patada (1966), estrenada en el Teatro del Altillo.
Ra-ta-ta-tá (1966), de Miguel Luis Coronatto - Teatro Del Laberinto (Director).
Hay que respetar la corrupción (1966), de Miguel Luis Coronatto - Teatro Del Laberinto (Director).
Estas últimas dos obras cortas formaban parte del programa teatral Uno y dos presentado en el Teatro El Laberinto. Este espectáculo tuvo una crítica muy positiva en la revista Análisis:"Pero, fundamentalmente, Uno y dos destaca la lúcida labor del director Ariel Absalón: su montaje es esclarecedor, de ritmo ágil, ingenioso y registra auténticos hallazgos escénicos. El elenco de Absalón, formado por actores profesionales, rinde en un buen nivel general sobre todo en el caso de Eduardo Muñoz y Anadela Arzón, porque los demás sin desentonar, son apenas discretos."
Referencias
Actores de cine de Argentina
Actores de televisión de Argentina
Actores de teatro de Argentina
Actores de radio de Argentina
Directores de teatro de Argentina
Judíos de Argentina
Actores judíos | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,258 |
\section{Introduction}
Robots or stationary cameras when used for surveying and monitoring tasks collect large amounts of image data, which is often analyzed manually by human experts. At Woods Hole Oceanographic Institution (WHOI) and NOAA Fisheries for example, every year 1000s of hours of video is collected using AUVs and static cameras, and for every hour of video it current takes approximately 3-4 hours of manual processing time. Hence, there is a need for automated techniques that can speed up the analysis of such datasets by identify perplexing or anomalous observations. Through the use of such techniques we can focus the attention of the human expert on a small subset of the collected data that is most likely to contain relevant information. In this paper we explore the use of Bayesian non-parametric (BNP) topic modeling to detect and characterize such anomalies.
Compared to other kinds of sensor data, image data typically exists in millions of dimensions, corresponding to the number of pixels in the image, which makes it challenging to build an automatic anomaly detection technique.
Moreover, detecting anomalous events in a non-stationary unstructured environment, such as coral reefs is even more challenging due to its higher visual complexity, compared to urban scenes.
Our proposed approach to dealing with the anomaly detection problem is to first use a Bayesian non-parametric scene understanding technique to build a model of the scene, and then using this model identify observations that are perplexing for the model. BNP topic modeling techniques have been successful in building semantic models of the data that automatically grow in complexity with data complexity. We take advantage of these results, and extended our previous work on Realtime Online Spatiotemporal Topic Modeling (ROST) \cite{Girdhar2013IJRR} to incorporate Bayesian nonparametric priors. In this paper we refer to our resulting proposed scene modeling technique as BNP-ROST.
\section{Related Work}
\subsubsection{Topic Modeling}
Topic modeling techniques like Probabilistic Latent Semantic Analysis(PLSA) \cite{Hofmann:2001}, and Latent Dirichlet Allocation (LDA) \cite{Blei:2003,Griffiths:2004}, although originally developed for semantic analysis of text documents, they have been widely applied to other types of data such as images\cite{Bosch:2006, FeiFei:2005:CVPR, Wang2007}. The general idea behind topic modeling, as applied to image data is to describe each image in a dataset as a distribution over high level concepts, without having prior knowledge about what these concepts are.
Probabilistic Latent Semantic Analysis (PLSA)\cite{Hofmann:2001} models the probability of observing a word $w$ in a given document $m$ as:
\begin{eqnarray}
\Prob(w| d) = \sum_{k=1}^{K} \Prob(w | z=k) \Prob(z=k | d),
\end{eqnarray}
where $w$ takes a value between $1\dotsc V$, where $V$ is the vocabulary size; $z$ is the hidden variable or topic label for $w$ that takes a value between $1\dotsc K$, where $K$ is the number of topics, and is much smaller than $V$; and $d$ is the document number, which can take a value between $1\dots M$, where $M$ is the total number of documents. The central idea is the introduction of a latent variable $z$, which models the underlying topic, or the context responsible for generating the word. Each document $m$ in the given corpora is modeled using a distribution $\theta_m(k) = \Prob(z=k | d=m)$ over topics, and each topic is modeled using a distribution $\phi_k(v)= \Prob(w=v | z=k) $ over the set of vocabulary words. During the training phase, these distributions are learned directly using an EM algorithm.
The distribution of topics in a document gives us a low dimensional semantic description of the document, which can be used to compare it semantically with other documents. The problem with this approach is that since the dimensionality of the model is very large, a lot of training data is required. Moreover, it is easy to overtrain for a given data set.
Latent Dirichlet Allocation (LDA), proposed by Blei et al. \cite{Blei:2003} improves upon PLSA by placing a Dirichlet prior on $\theta$ and $\phi$, encouraging the distributions to be sparse, which has been shown to give semantically more relevant topics. Subsequently Griffiths et al. \cite{Griffiths:2004} proposed a Gibbs sampler to learn these distributions.
\subsubsection{Semantic Modeling of Image Data}
Topic modeling of images requires that the general idea of a textual word be replaced by visual words. One approach to generate these visual words from visual features is that described by Sivic et al.~\cite{Sivic:2006:videogoogle}.
Given visual word representation of scenes with multiple objects, topic modeling has been used to discover objects in these images in an unsupervised manner. Bosch et al. \cite{Bosch:2006} used PLSA and a SIFT based~\cite{Lowe:IJCV:2004} visual vocabulary to model the content of images, and used a nearest neighbor classifier to classify the images. Fei-Fei et al.\cite{FeiFei:2005:CVPR} have demonstrated the use of LDA to provide an intermediate representation of images, which was then used to learn an image classifier over multiple categories. Instead of modeling the entire image as a document, Spatial LDA (SLDA) \cite{Wang2007} models a subset of words, close to each other in an image as a document, resulting in a better encoding of the spatial structure. The assignment of words to documents is not done \emph{a priori}, but is instead modeled as an additional hidden variable in the generative process.
Summarizing benthic (sea floor) images is an especially difficult problem due to the general lack of order, symmetry, and orientation of the visual data. Steinberg et al. \cite{Steinberg2011} used a Gaussian mixture model to cluster benthic stereo images, while using a Variation Dirichlet Process~\cite{Kurihara2007} to automatically infer the number of clusters. Although this work did not use location information in the clustering process, the resulting cluster labels were shown to be spatially contiguous, indicating correctness. The computed labels were shown to outperform those obtained with spectral clustering and EM Gaussian mixture models, when compared with hand labeled ground truth data.
\subsubsection{Topic Modeling of Streaming Video Data}
BNP techniques have been previously used to characterize anomalous activities~\cite{Wang2007a, Hospedales2012}. In our own recent work \cite{Girdhar2013IJRR} we have described a realtime online topic modeling (ROST) technique that computes topic labels for observed visual words in a video while taking into account its spatial context in pixel space, and temporal context (frame count). ROST does this by generalizing the idea of a document to a spatiotemporal cell, and computing the topic label for the words in a cell in the context of its neighboring cells. In \cite{Girdhar2015a} we used ROST to identify interesting observations in a robot's view, and then used it to plan an adaptive path, which were shown to have higher information content than simple space filling paths.
\section{Bayesian Nonparametric (BNP) Scene Modeling}
Given a sequence of images or other observations, we extract discrete features $w$ from these observations, each of which has corresponding spatial and temporal coordinates $(x,t)$. In case of a simple 2D video the spatial coordinates would just correspond to the pixel coordinates, however in presence of 3D data, the spatial coordinates can be 3D.
Similar to ROST, we model the likelihood of the observed data in terms of the latent topic label variables $z$:
\begin{eqnarray}
\Prob(w| x,t ) = \sum_{k \in K_\text{active}} \Prob(w | z=k) \Prob(z=k | x,t).
\end{eqnarray}
Here the distribution $\Prob(w | z=k)$ models the appearance of the topic label $k$, as is shared across all spatiotemporal locations. The second part of the equation $\Prob(z=k | x,t)$ models the distribution of labels in the spatiotemporal neighborhood of location $(x,t)$. We say that a label is active if there is at least one observation which has been assigned this label. The set of all active labels is $K_\text{active}$.
Let $w_i = v$, be the $i$th observation word with spatial coordinates $x_i$, and time $t_i$, where $i\in[1,N)$, and the observation $v$ is discrete and takes an integer value between $[0,V)$. Each observation $w_i$ is described by latent label variable $z_i=k$, where $k$ again is an integer.
\begin{eqnarray}
\Prob(w_i=v | z_i=k) = \frac{n_{v,k} + \beta}{N+V\beta-1}.
\end{eqnarray}
Here $n_{v,k}$ is the number of times an observation of type $v$ has been assigned label $k$ thus far (excluding the $i$th observation), $N$ is the total number of observations, $V$ is the vocabulary size of the observations, and $\beta$ is the Dirichlet parameter for controlling the sparsity of the $\Prob(w|z)$ distribution. A lower value of $\beta$ encourages sparser $\Prob(w|z)$ with peaks on a smaller number of vocabulary words. This encourages topics to describe more specific phenomena, and hence requiring more topics in general to describe the data. A larger value of $\beta$ on the other hand would encourage denser distributions, encouraging a topic to describe more general phenomena in the scene.
In this work we assume that the set of all distinct observation words is known, and the set has size $V$, however, the number of labels used to describe the data $K$ is inferred automatically from the data. Through the use of Bayesian nonparametric techniques such as Chinese Restaurant Process (CRP), it is possible to model, in a principled way, how new categories are formed\cite{Teh:2006:HDP, Teh2010}. Using CRP, we model whether a word is best explained via an existing label, or by a new, previously unseen label; allowing us to build models that can grow automatically with the growth in the size and complexity of the data.
\begin{eqnarray}
\Prob(z_i=k | z_1,\dots, z_N ) =
\begin{cases}
\frac{n_{k,g_i} + \alpha}{C(i,k)} & k \in K_{\textit{active}} \\
\frac{\gamma}{C(i,k)} & k = k_{ \text{new}}\\
0 & \text{otherwise.}
\end{cases}
\end{eqnarray}
Here $n_{k,g_i} $ is the total number of observations in the spatiotemporal neighborhood of the $i$th observation, excluding itself; dirichlet prior $\alpha$ controls the sparsity of the scene's topic distribution; CRP parameter $\gamma$ controls the growth of the number of topic labels; and $C(i,k)= \sum_i^N (\Indi_{[n_k>0]}n_{k,g_i}+\alpha)+ \gamma -1$
is the normalizing constant.
We use the realtime Uniform+Now Gibbs sampler proposed in \cite{Girdhar2015Gibbs} to compute the posterior topic labels for the datasets. We update the sampler to use the Chinese Restaurant Process for automatic discovery of new labels.
\section{Anomaly Detection}
Given $\Prob(z|t)$, the topic label distribution of a each time step, we can compute the marginal distribution
\begin{eqnarray}
\Prob(z=k) = \sum_{t=1}^T\frac{ \Prob(z=k|t)}{T},
\end{eqnarray}
which defines the distribution of topic labels for the entire dataset. We can then define the perplexity score $S(t)$ of observations made at a given time-step $t$ as:
\begin{eqnarray}
S(t) &=& \exp (\Hx(\Prob(z=k|t), \Prob(z=k))\\
&=& \exp\left(- \sum_k(\Prob(z=k|t) \log \Prob(z=k)\right). \label{eq:ppx}
\end{eqnarray}
Here the function $\Hx(p,q) = -\sum_x p(x) \log q(x)$ computes the cross entropy of the two distributions $p$ and $q$. If we assume a normal scene has the topic distribution $\Prob(z)$, then $\Hx\left(\Prob(z=k|t), \Prob(z=k)\right)$ computes the average number of bits needed to encode time-step $t$, using codes optimized for distribution is $\Prob(z)$. Taking the exponential of the cross entropy then gives us the uncertainty in assigning topic labels to the given time step.
A time-step where most of the observations were labeled with a commonly occurring topic label will be given a low score, whereas if a time-step with rare topic labels will be given a high score.
\section{Experiments}
To demonstrate the effectiveness of the proposed BNP-ROST scene modeling for anomaly detection, we conducted two experiments on completely different kinds of datasets. The first dataset consists of images collected by a robot as it explores the seafloor, and the second dataset consists of a static camera set in a coral reef, observing a complex and dynamic scene.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\columnwidth]{seabed_deployment}
\caption{Jaguar AUV was used to collect the seafloor image data.}
\end{center}
\end{figure}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=1.8\columnwidth]{figs/d19_anomaly}
\caption{Galatheid crab dataset. (a) A stacked plot showing distribution of topic labels at each time step in the dataset, computed using the proposed technique. We see that Topic 0 and 1 are characteristic of the underlying terrain, whereas other topic labels are representative of other phenomena such as galatheid crab aggregations. (b) Shows the normalized perplexity scores for each time step. (c) Shows examples of images with high perplexity scores, corresponding to anomalous observations. Image with $t=116$ shows various animal species feeding off a fish carcass, and is the most anomalous scene in the dataset. Other anomalous observations are of galatheid crabs. (d) Shows examples of some typical images in the dataset, represented by their low perplexity scores.}\label{fig:d19anomaly}
\end{center}
\end{figure*}
\subsection{Fauna Detection in AUV Missions}\label{sec:expcrabs}
In this experiment we analyzed image data collected by the Jaguar AUV \cite{Kunz2009} as it explored the Hannibal seamount in Panama. The mission was conducted primarily at depths beyond 300 meters. At these depths, there is very limited visible fauna. This specific dataset was chosen because it contained observations of galatheid crab gatherings, which is an obviously anomalous phenomena.
Goal of this experiment was to see if the proposed BNP-ROST algorithm would be able to detect these observations of galatheid crabs, either by characterizing these observations with its own topic label, or by giving them high perplexity scores. Every third image in the dataset was hand-labeled by a team of expert biologists to mark the fauna in the images, which we used as the ground truth for the galatheid crab observations.
We ran the proposed BNP-ROST algorithm to compute topic distributions for each time step, and the perplexity score $S(t)$ described in Eq. \ref{eq:ppx}. The distribution $\Prob(z|x,t)$ was modeled using cellular approximation described in \cite{Girdhar2015a}, with cell size of 128x128 pixels. We used $\alpha=0.1, \beta=10, \gamma=1e-5$ for all our experiments. These parameters were chosen after a very sparse grid search in log space of the parameters.
The dataset presented here consists of 1737 images, taken once every three seconds, at an altitude of 4 meters above the seafloor by the Jaguar AUV. The seafloor depth varied between 300-400 meters.
We extracted the following different kinds of visual words from the data: Textons\cite{Varma2005} with four different orientations, and three different scales, quantized into 1000 different categories using the k-means algorithm; Oriented FAST and rotated BRIEF (ORB) \cite{RubleeE2011} features at FAST detected corners, quantized into 5000 categories; and hue and intensity pixel values distributed on a grid. For each image we extracted 16K texton words, 10K ORB words, and 4K pixel words.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=1.4\columnwidth]{figs/d19_GAL}
\caption{Comparison of the learned topic labels with with expert labeled ground-truth for galatheid crab observation. (a) Shows distribution of Topic 4 for each time step, computed automatically by the proposed scene modeling technique, without any supervision. (b) Shows ground normalized distribution of galatheid crabs across the same timeline, labeled by an expert biologist. We see that there are three regions in the AUV mission where the crabs were observed: around time steps 600, 1000, and 1600. All three regions are correctly characterized by the Topic 4 of the scene model, without any supervision. The peaks around timestep 100 correspond to detection of other kinds of crab (not galatheid), which are labeled with the same topic label as the galatheid crabs by the topic model, but were labeled with a different label by the biologist.}\label{fig:topiccrab}
\end{center}
\end{figure*}
\subsubsection*{Results}
Figure~\ref{fig:d19anomaly}(a) shows the distribution of topic labels over time for the galatheid crab dataset. We see that topic 0 and 1 are representative of the underlying terrain observed by the robot during the mission, and stay consistently represented throughout the timeline, whereas the other topics correspond to more episodic phenomena.
The plot in Fig.~\ref{fig:d19anomaly}(b) shows the perplexity score of each observation, which was computed given the topic distribution at that time step using Eq.~\ref{eq:ppx}.
A time-step with low perplexity score implies that it contains images that are characteristic of the entire dataset. Some examples of such images are shown in Fig.~\ref{fig:d19anomaly}(d). These images show the underlying terrain that is represented throughout the mission by topics 0 and 1. Some examples of images with high perplexity scores are shown in Fig.~\ref{fig:d19anomaly}(c). The highest scoring image corresponds to time step $t=116$, which show a feeding aggregation scene with several different species of crabs, squids and an eel eating a fish carcass. These kinds of feeding aggregations are rare in the deep ocean, where due to the lack of sunlight there is a limited supply of available nutrients to support life. The other two highest peaks corresponds to observations of the galatheid crab's mating aggregations.
We found out that distribution of topic 4 over the mission timeline (shown in Fig.~\ref{fig:topiccrab}(a)) matches closely with the distribution of galatheid crabs, annotated manually by a team of expert biologists (shown in Fig.~\ref{fig:topiccrab}(b)).
The Kolmogorov–Smirnov statistic for the topic 4 distribution, given the ground truth crab distribution, found to be $D=0.185$.
The substrate in this experiment is characterized by topics 0 and 1. Our hypothesis on on why two topics represent the underlying terrain and not one is as following. We use a constant Dirichlet parameter $\beta$ to represent the distribution of words for a given topic. This constant parameter implies that all topics are modeled to have word distributions with similar sparseness. Hence, if an entity is ideally represented by a more dense distribution of words, it is likely going to be represented using multiple topics under the current model. This is a limitation of the proposed approach.
\begin{figure*}[]
\begin{center}
\includegraphics[width=1.62\columnwidth]{figs/uhsi_auvdetect}
\caption{Coral reef dataset captured with a static camera. (a) Shows topic label distribution of the dataset across each time step. We see that the distribution is dominated with Topic 0, which characterizes the scene background. (b) Shows the normalized distribution of the next highest weighted topic. (c) Shows the perplexity score for each time step. We see that the perplexity scores correlate well with Topic 1 distribution, which is expected because Topic 1 is relatively rare in the dataset compared to the topic label representing the background. (d) Marks the hand labeled locations of events where the AUV was sighted, which corresponds to peaks 1,2 and 8. (e) Shows images corresponding to the eight highest peaks in the perplexity scores. We see that the perplexity model is able to detect all three events of an underwater vehicle passing in front of the camera. The other perplexity peaks correspond to sightings of barracuda, and schools of fish, both of which are anomalous.}\label{fig:auvdetect}
\end{center}
\end{figure*}
\subsection{Anomaly Detection with a Static Camera in a Complex and Dynamic Scene}
Coral reefs are busy and dynamic environments. As part of another experiment \cite{Campbell2015}, we set up several stationary cameras in the Gulf of Mexico, to characterize the fish behavior in presence of robots. In this experiment we analyzed the image data collected by the stationary seafloor cameras to see if the proposed technique is able to identify an underwater vehicle as anomalous, amongst other observations of fishes, constant flow of debris, lighting change due to wave action on the water surface, and sea plants moving continuously due to current. The dataset consists of an hour long video segment consisting of 17966 image frames, take at the rate of 5 frames per second.
To focus the topic modeling on the scene foreground, we used a mixture of Gaussian based background detection \cite{Z.Zivkovic2006} technique to compute a background mask for each time step. To characterize the constantly moving seafloor flora and ocean debris, due to the ocean currents, we use both the intensity values and optical flow values in the background model. We extracted Texton, ORB and intensity words for both the foreground and the background, however the background words were extracted with 1/4$th$ the density of foreground words, to give them less focus. Now to model the scene, we used the same BNP-ROST parameters as described in Sec.\ref{sec:expcrabs}, and computed topic labels for every time step.
\subsubsection*{Results}
Result of our unsupervised AUV detection experiments are shown in Fig.~\ref{fig:auvdetect}. We see that the topic distribution computed by the proposed technique is dominated by the first topic, which essentially characterizes the scene background. However if we plot the distribution of the second most weighted topic (shown in Fig.\ref{fig:auvdetect}(b)), we see a much more relevant temporal structure of the environment. We find that the peaks of the distribution of Topic 1 match the peak of the perplexity plot of the data (shown in Fig.~\ref{fig:auvdetect}(c)), which can be explained by the fact that Topic 1 is relatively rare (compared to the background Topic 0). In this experiment topic 1 models both the AUVs and the fishes. This is the result of the same problem of constant Dirichlet parameter, which resulted in the two different topic labels being used to characterize the substrate in experiment described in Sec. \ref{sec:expcrabs}.
The hand labeled AUV sighting events are shown in Fig.~\ref{fig:auvdetect}(d). We see that these events match perfectly with peaks 1,2 and 8. The other perplexity peaks correspond to sightings of barracudas and large schools of fishes.
\section{Conclusion}
In this paper we described a Bayesian non-parametric topic modeling technique for modeling semantic content of spatiotemporal data such as video streams, and then used it to identify anomalous observations. We applied the proposed technique to two different kinds of datasets containing observations from unstructured benthic environments. The first dataset containing image data collected by an AUV. We showed that the proposed technique
was able to automatically identify and characterize observations of galatheid crabs, and the computed distribution matched the hand-labeled distribution with Kolmogorov–Smirnov statistic $D=0.185$. The second data consisted of image data from a stationary camera set in a busy coral reef, where an underwater robot made three passes in front of the camera. The proposed algorithm was able to identify all three vehicle crossings as anomolous.The fact that the proposed unsupervised algorithm works well in two completely different scenarios, gives us confidence that this approach is well suited for a variety of applications. The Bayesian non-parametric nature of the approach insures that the anomaly models adapts automatically to the data, without requiring careful tuning of the hyper-parameters. Our ongoing efforts are to adapt the proposed technique to be useful with other kinds of data such as audio and sonar imagery. We are also working on using the proposed technique onboard a underwater robot, for context aware adpaptive data collection tasks.
\noindent \textbf{Acknowledgments:} \small{This work was supported by the Devonshire Foundation, the J. Seward Johnson fund, FQRTN postdoctoral fellowship, and NOAA CINAR.}
\bibliographystyle{IEEEtran_nourl}
| {
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{"url":"https:\/\/testbook.com\/question-answer\/the-absorptivity-of-blackbody-equals-to--6270c74fee0f31401a9fd37a","text":"# The absorptivity of blackbody equals to\n\nThis question was previously asked in\nBPSC AE Paper 4 (General Engineering Science) 25 March 2022 Official Paper\nView all BPSC AE Papers >\n1. 2\n2. 1\n3. 3\n4. 4\n\nOption 2 : 1\nFree\nEnglish Language Free Mock Test\n7.4 K Users\n15 Questions 15 Marks 12 Mins\n\n## Detailed Solution\n\nExplanation:\n\nKirchhoff\u2019s law\n\nThe emissive power of a body E is defined as the energy emitted by the body per unit area and per unit time. Assume that a perfectly black enclosure is available, i.e., one that absorbs all the incident radiation falling upon it. Now suppose that a body is placed inside the enclosure and allowed to come into temperature equilibrium with it. At equilibrium the energy absorbed by the body must be equal to the energy emitted; otherwise, there would be an energy flow into or out of the body that would raise or lower its temperature. At equilibrium we may write:\n\nEA = qi\n\nFor black body (absorptivity \u03b1 = 1):\n\nEbA = qiA(1)\n\n$$\\frac{E}{{{E_b}}} = \\alpha$$\n\nThis law states that the ratio of the emissive power of a body to the emissive power of a blackbody at the same temperature is equal to the absorptivity of the body. This ratio is defined as the emissivity \u03f5 of the body.\n\n$$\\frac{E}{{{E_b}}} = \\epsilon$$\n\n\u2234 \u03b1 = \u03f5\n\nThus, Kirchhoff\u2019s law states that the\u00a0emissivity of a body is equal to its absorptivity\u00a0when the body remains in\u00a0thermal equilibrium\u00a0with its surrounding.","date":"2023-02-07 02:47:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6633423566818237, \"perplexity\": 971.4910506053305}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500368.7\/warc\/CC-MAIN-20230207004322-20230207034322-00726.warc.gz\"}"} | null | null |
Alumni Global Volunteers, Global Volunteers Poland
Repeat Poland Volunteer Featured in Community Magazine!
Poland Volunteer Featured in Community Magazine: Judy of Woodbury, Minnesota was recently featured in her community's magazine, highlighting her several Global Volunteers service program including two in Poland!
Woodbury Magazine Article
Retired science teacher Judy T. of Woodbury decided to try a volunteer trip in 1997 for the opportunity to travel, use her teaching skills and do something worthwhile. Trepka has taken six two-week trips, serving in Greece, Romania, Spain, Hungary and Poland (twice). "In Crete, Greece I helped restaurant and hotel workers with conversational English skills, and in Iasi, Romania and Hodmezovasarhely, Hungary I taught English to students, " Judy says. "On my two trips to Poland, I helped business professionals improve their English skills as well as a retired University of Warsaw professor recovering from a stroke who had lost his English speaking skills. I worked with him for two weeks, 5-6 hours a day, and by the end he could speak English again. The Polish students told me 'It's a miracle, and his Polish is better too!'"
Judy traveled with Global Volunteers, a St. Paul-based organization supporting more than 100 host communities through year-round volunteer service, including caring for at-risk children, teaching conversational English skills, assisting local health education efforts and completing a wide array of local projects including painting, landscaping, building playgrounds and improving public works. "As is often the case with these programs, you get more than you give, " says Judy. "And you'll learn more about a country and its people than you will as a casual tourist. You're expected to put in at least a 40-hour volunteer week, but the weekends are free for exploring lovely out-of-the-way places that organized tours never get to."
Some highlights for Judy: seeing local World War II statues throughout Crete's small villages, and enjoying paella at a restaurant in Rota, Spain in late November, "a magical time when every business and institution brought out unique, exquisitely beautiful nativity scenes, " she says. But her true highlights were the cultural, human experiences gained through volunteering.
"In Hungary, the students would immediately rise and remain standing when I until I told them they could sit down; I had never experienced that as a teacher before, " says Judy. "The food in Reymontowka, Poland was cooked with fresh produce plus milk and meat from the three-generation farm next door. I've never had a drink I enjoyed more than the fresh black current nectar served at breakfast."
February 4, 2011 /0 Comments/by Michele Gran
https://205eev2oa0jm1t4yb914s1nw-wpengine.netdna-ssl.com/wp-content/uploads/2011/02/Volunteer-Judy-in-Poland..jpg 1125 1500 Michele Gran https://205eev2oa0jm1t4yb914s1nw-wpengine.netdna-ssl.com/wp-content/uploads/2016/02/2014-GlobalVolunteersLogo-Web.png Michele Gran2011-02-04 22:44:002016-11-23 12:02:00Repeat Poland Volunteer Featured in Community Magazine!
Learn about Queretaro! Gravel, Cloud Forests & Traditional Dance! | {
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Q: Как выводить текст, чтобы перенос строк сохранялся? Как выводить текст, чтобы перенос строк сохранялся?
Суть проблемы:
есть текст в Базе Данных, он бывает двух форматов - HTML (внутри содержит теги) и "просто текст"(внутри, как я понимаю, не теги, а \r, вероятно).
С сообщениями в формате HTML проблем нет.
Но если мы выводим сообщения формата "просто текст", к примеру:
1. Первая сторока
2. Второя сторока
3. Третья строка.
То все эти строки получаются слитно:
1. Первая сторока
2. Второя сторока
3. Третья строка.
Проблему можно решить, выводя этот текст командой alert(mess) (javascript).
В сообщение alert выводится правильно, т.е. с переносом строк.
Подскажите, как сделать чтобы перенос строк сохранялся и при выводе в виде HTML?
A: Есть цсс-свойство white-space, у него можно указать значение 'pre'. Это как закинуть под тег 'pre', но семантичнее
.myclass {
white-space: pre;
}
A: Можно закинуть все под тег <pre></pre>
А так используй символы переноса строк \n\r
| {
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How to Support the Diverse Books Movement
September 30, 2016 Dana Nuenighoff General, Publishing, Writing - Critiquing & Community 9 comments
We Need Diverse Books. Four simple words. This movement is not a trend or a fad, but a way of life. Diversity is all around us and as writers we have a responsibility to share that. However, it is not as simple as saying that your main character belongs to a marginalized group, especially if you, the writer, do not share that background. Writers from non-marginalized groups, like me, do not have the same experience as writers from marginalized groups.
There has been a lot of talk over social media platforms on whether non-marginalized writers should write marginalized characters. Many believe that writers from non-marginalized groups should not write characters who are from marginalized groups. Some call this "censorship," but it's not. Marginalized writers need their voices heard. When stories by non-marginalized writers are published with main characters who are from marginalized groups, then the writers from those group don't get that chance.
According to Publishers Weekly, 78% of the Publishing Industry is non-marginalized. Unfortunately, while the publishing industry as a whole says it wants to promote diversity, this demographic leads to a disconnect. Editors feel the need to connect with a story in order to publish and promote it. Many editors feel they do not connect with diverse stories that describe such a different experience from their own.
So, how do non-marginalized writers—who have the best intentions—join in without causing more harm than help?
Support Diverse Books.
This is so important. To change the industry, we have to support the voices of writers from diverse groups. Publishing is a business. Create demand for diverse books. Read them. Ask for them. Buy them. Review and recommend them. If you are looking for diverse books, Dahlia Adler and Malinda Lo both have lists. The site We Need Diverse Books has a whole long list of lists. You can also go onto Twitter and search the hashtag #buydiversebooks.
People don't have the obligation to educate you, so it's up to you to do some digging. Many writers are speaking up about diverse books. L.L McKinney and Justina Ireland are both great advocates to follow.
Ask Yourself if You are the Right Person to Tell This Story
No, seriously. If you are a non-marginalized writer, you probably wouldn't be the best person to write an issue book based on Black Lives Matter. It's not that you don't care about the issue. But, you aren't experiencing it firsthand. Yours isn't the voice to tell that story as if you understand it.
Sensitivity Readers
If you do decide to go on with a diverse book, one of the biggest things you will need is a sensitivity reader. This is probably the most important thing. You can have dozens of critique partners and beta readers, but if they are not from the culture, then they probably won't be able to tell you if you've misinterpreted or misrepresented something.
Sensitivity readers from that culture or background you are portraying can give you real, inside information. This can be a painful and long process. You have to be open to their criticism and realize it's not that they don't want you to write that particular book, they want it to be the best it can be so that others aren't offended. Think about your readers. Think about all the readers you're going to have and who could possibly be hurt by your story, if you get it wrong. You have a responsibility to tell this story right. Prioritize their feelings.
I hired a sensitivity reader for my most recent manuscript. Though I'm working in a fantasy world, my main character's culture has some basis in Polynesian and Hawaiian cultures. I'm nervous that it may not work. But I can't make my character any other culture. It wouldn't work with the story.
Be Prepared to Kill Your Darlings
Editing is all about killing your darlings, but in this case, you have to be open to harsh criticism from your sensitivity readers. Revision for sensitivity could require you hacking out chunks and reworking your characters, or it could include you shelving this book.
This is tough. But if feedback comes and your sensitivity reader says that the book isn't working, then you must have a serious talk with yourself. Are you willing to let this book go? I'm not going to lie. This is what I've been worrying about lately. My critique partners tell me I'm overthinking it, and they're both writers from marginalized groups, so they have experience with this. Still, I don't want to repeat J.K. Rowling's mistake with #MagicinNorthAmerica.
So, I research, I follow people, and I send my works-in-progress to appropriate readers. I hope that this will be enough. I hope that I'm making a valuable contribution to the diverse books movement. I hope that others will love my manuscript like I do, because I didn't write it just to be diverse. I didn't even think about the issue of diversity when I first imagined my characters. I just knew that there would be two vastly different cultures in it because that's the way the world is.
The diversity movement is a way of life. I will say it again and again. We all have an obligation to better the publishing industry by promoting diverse books and allow own voices writers to be heard.
What are your thoughts and experiences in writing diverse characters? Do you have advice to share? A great diverse book to recommend? Share your thoughts in the comments.
Related Posts on this site:
The Complexities of Diversity by Almitra Clay
Vicarious Contact by Dianna SanchezNaturalized Diversity by Joyce Audy Zarins
Diversity Part 2–to Make a Difference by Joyce Audy Zarins
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Great post with some good links to follow up with. Thanks for sharing. I'll be back again to explore some more.
Joyce Audy Zarins says:
Some elements of the publishing industry have used sensitivity readers for vetting some stories for many years. For example, Cobblestone and Faces children's magazines had a Native American advisory board to vet articles and artwork and truly made an effort to screen stories about other cultures for sensitivity. I once refused a juicy book project because the editor refused to vet for sensitivity and I believe she went ahead and hired another non-marginalized illustrator. Not good. And publishers of educational materials back as far as the 80s and 90s required illustrators to follow guidelines for what proportion of children of each marginalized group should be included in the art. They also included handicapped children in the artwork. But that is quite a different scale from a novel or other long work in which the protagonist and author are of different cultural backgrounds. We are supposed to write what we know. sometimes, though, it is also important to write about what there isn't enough of. Your point about finding sensitivity readers, in addition to regular beta readers, is an excellent one and so important to convey.
johnrberkowitz says:
"Otherwise we get books with all-white casts, and that, too, is a trend we'd like to stop." I should not have said "trend." Books with all-white casts are the norm, and we are only now beginning to move away from this.
I think it is perfectly fine for non-marginalized people to include characters that are marginalized. This is also part of the diversity conversation. Tessa Gratton said something to me: "All marginalized people ask is that people with privilege do no harm, and do everything in our power not to erase their experiences." Nobody is asking white people not to write about POC. They only ask that we do so respectfully and to be prepared for critical feedback from the people we represent. Otherwise we get books with all-white casts, and that, too, is a trend we'd like to stop. However I agree, those of us with privilege need to be sensitive to how marginalized people are represented in our own books. I think the idea of a sensitivity reader is tremendous. The key, as you point out, is to be open to criticism. And, indeed, WE NEED DIVERSE BOOKS.
hmmmmm says:
I started to comment, but then it turned into a blog post. Thanks for being the catalyst to get me to process this whole thing in writing… You can check it out at:
https://annaforrester.wordpress.com/2016/09/30/a-picture-book-diversity-conundrum/
Dana Nuenighoff says:
Love this post. I'm glad mine could inspire it!
Pingback: A PICTURE BOOK DIVERSITY CONUNDRUM | Hmmmmm
dnuenighoff says:
Reblogged this on Dana Nuenighoff and commented:
With my MS in the hands of a sensitivity reader, I had some time to jot down my thoughts on how to support diverse books!
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{"url":"https:\/\/indico.cern.ch\/event\/854124\/contributions\/4134893\/","text":"Indico has been upgraded to version 3.1. Details in the SSB\n\n# Initial Stages 2021\n\nJan 10 \u2013 15, 2021\nWeizmann Institute of Science\nAsia\/Jerusalem timezone\nSee you at IS2023 in Copenhagen in June 2023\n\n## Recent ATLAS measurements of correlations from small to large collision systems\n\nJan 12, 2021, 5:40 PM\n20m\nAndrea\u2019s room 1 (vDLCC)\n\n### Andrea\u2019s room 1\n\n#### vDLCC\n\noral Collective dynamics from small to large systems\n\n### Speaker\n\nSoumya Mohapatra (Columbia University (US))\n\n### Description\n\nMeasurements of two-particle correlations in $pp$ collisions show features that are strikingly similar to those seen in heavy-ion collisions, suggesting that a tiny droplet of the QGP is produced even in such collisions. In the $pp$ collisions models that attribute the correlations to semi-hard processes can also qualitatively reproduce the measurements. In this talk, we report on a series of ATLAS measurements exploring detailed properties of flow in small, medium, and large collision systems.\nNew ATLAS measurements of two-particle correlations with active selection on particles associated with jets from the event are performed to elucidate the origin of the long-range correlations. If the correlations are indeed generated by semi-hard processes, then the long-range correlations between particles associated with jets would be stronger than the inclusive hadron correlations, while removing jet-associated particles would weaken the correlations.\nAdditionally, measurements of the azimuthal anisotropy in $p$+Pb collisions reaching the transverse momentum of charged particles up to 50 GeV, in minimum-bias and jet-triggered events are presented. In the jet-triggered events, $v_2$ is non-zero over the entire kinematic range of the measurement, and is $\\approx 2$-$3$% at $p_\\mathrm{T} \\approx 50$ GeV.\nIn large collision systems, we focus on understanding the longitudinal structure of the initial-state in heavy-ion collisions as a key for modeling the early-time dynamics. This talk presents results of flow decorrelations in Xe+Xe and Pb+Pb collisions.\nIn $AA$ collisions, the decorrelations for $v_2$ show a strong centrality and $p_{\\mathrm{T}}$ dependence, while no such dependencies are observed for $v_3$ and $v_4$ decorrelations. Decorrelations in Xe+Xe collisions, when compared to Pb+Pb collisions, are found to be larger for $v_2$, but smaller for $v_3$. These system-dependent trends are not reproduced in current initial-state models when coupled with hydrodynamic evolution.\n\n### Primary author\n\nSoumya Mohapatra (Columbia University (US))\n\n### Presentation materials\n\n 2021-01-IS.pdf Recording","date":"2021-11-27 21:22:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3460933566093445, \"perplexity\": 4132.153146547116}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964358233.7\/warc\/CC-MAIN-20211127193525-20211127223525-00259.warc.gz\"}"} | null | null |
Q: A famous Inequality I know there is a famous inequality between $\Sigma_{i<j}^{n} x_{i}x_{j}$ and $x_{1}+x_{2}+...+x_{n}$ but I don't remember what exactly it is . For example in the case $n=2$ we have:
$x_{1}+x_{2}\ge 2(x_{1}x_{2})^{1/2}$.
I was wondering if anybody could help with that.
A: Are you talking about The inequality of arithmetic and geometric means?
A: $$f(x_1,\ldots,x_n)=\sum_{i\neq j}2x_i x_j= (x_1,\ldots,x_n)\,A_n\,(x_1,\ldots,x_n)^T \tag{1}$$
where the entries of $A_n$ equal one unless they are diagonal entries: in such a case they equal zero. $A_n+I_n$ is a rank-$1$ matrix with eigenvalues $n,0,\ldots,0$. It follows that the eigenvalues of $A_n$ are $n-1,-1,\ldots,-1$ and:
$$ \left|f(x_1,\ldots,x_n)\right|\leq (n-1)(x_1^2+\ldots+x_n^2).\tag{2} $$
By $(2)$, we have:
$$ \left|\sum_{1\leq i<j\leq n}x_i x_j\right|\leq \frac{n-1}{2}\cdot\sum_{i=1}^{n}x_i^2\tag{3} $$
and equality holds iff $x_1=x_2=\ldots=x_n$.
On the other hand, by the Cauchy-Schwarz inequality:
$$ \left(\sum_{i=1}^{n}x_i\right)^2\leq n\cdot\sum_{i=1}^{n}x_i^2\tag{4}$$
hence:
$$ 2\cdot\!\!\!\!\sum_{1\leq i<j\leq n}\!\!x_i x_j = (x_1+\ldots+x_n)^2 - (x_1^2+\ldots+x_n^2)\leq\left(1-\frac{1}{n}\right)\left(\sum_{i=1}^{n}x_i\right)^2.\tag{5} $$
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Armenian National Institute Website Now Includes 327 Armenian Genocide Memorials
WASHINGTON, DC – After extensive research and the gift of a major cache of photographs of Armenian Genocide memorials from around the world, the Armenian National Institute (ANI) website presently displays 327 memorials in 45 countries.
The 2015 centennial commemorations of the Armenian Genocide presented a somber occasion for many communities to install new memorials. The database previously accounted for 200 memorials in 32 countries. The new figures represent a 60% growth in the number of memorials in an additional 13 countries around the world. Also, 74 existing postings have been updated with new information. All these can be viewed in close detail with the 650 images that have been added to the existing database.
The database provides information about each memorial, a description of the type, its location, the year of installation, a transcription of important inscriptions, and additional details if available, such as the artist, sculptor, or architect, sponsors, official visitors, and amenities. The database also has a search function according to the type of memorial or the city in which a memorial is located.
Memorials are documented in as diverse a set of countries as Brazil and Bulgaria, Chile and Cyprus, Estonia and Ethiopia, Ireland and Italy, Singapore and Slovakia, and Ukraine and Uruguay. To the same extent that Armenian churches indicate the existence of diaspora communities, the Armenian Genocide memorials now also mark their location and attest to the depth of the Armenian people's commitment to honoring the memory of the victims of the 1915 atrocities.
Montebello, California, Armenian Genocide Memorial
Bikfaya, Lebanon, Armenian Genocide Memorial
The full list of countries on the ANI site includes: Argentina (7); Armenia (44); Australia (7); Austria (4); Belgium (2); Brazil (3); Bulgaria (6); Canada (9); Chile (2); Cyprus (4); Czechia (1); Denmark (1); Egypt (3); Estonia (1); Ethiopia (1); France (45); Georgia (3); Germany (14); Greece (2); Hungary (3); India (1); Iran (8); Ireland (1); Israel (3); Italy (8); Latvia (1); Lebanon (14); Mexico (1); Nagorno-Karabakh (Artsakh) (3); Netherlands (2); Poland (2); Romania (2); Russia (10); Singapore (1); Slovakia (2); Spain (4); Sweden (2); Switzerland (3); Syria (7); Ukraine (5); United Arab Emirates (1); United Kingdom (4); United States (74); Uruguay (4); Venezuela (1).
Vancouver, Canada, Armenian Genocide Memorial
Besides Armenia, where 44 memorials have been documented, 29 of which were recently added to the database, countries with a large number of memorials include France, with 45 documented sites, of which 29 were added, Germany with 14, most of which are recent installations, Lebanon with 14, some consisting of significant complexes, and the United States with 74 identified memorials, 25 of which were added.
Armavir, Armenia, Memorial to the Defense of Musa Dagh
Many of the memorials are monuments and the khachkar, the cross-stone in traditional Armenian design, and in varying sizes, has been adopted as a commonly-shared feature. This appears to be a more recent development in the installation trend, perhaps reflecting the origin of the community sponsoring the memorial, and its preference to rely upon an art form that references traditional Armenian memorial art.
Other sites have installed sculptures ranging from conventional figural representations all the way to completely abstract forms striving to capture the depth of the pain associated with the genocide experience. On the one hand, the figure of Komitas has become a symbol of the victim, with a noteworthy example in the middle of Paris, France, and another in Detroit, Michigan. On the other hand, there is a detectable change in styles where monuments raised in Armenia in the Soviet period usually resorted to abstraction with minimal direct reference to the Armenian Genocide, and where now memorials take more diverse forms since independence.
Paris, France, Memorial to Komitas
Whereas the well-known central memorial in Yerevan at the Tsitsernakaberd complex is the largest of its type, several other memorials in Armenia also involve substantial complexes, especially the Musa Dagh Memorial, located in Armavir, and the memorial in Nor Hadjin in Kotayk. Smaller-scale models resembling the Tsitsernakaberd memorial have been constructed in a number of locations, including a respectable replica in Fresno, California.
Yerevan, Armenia, Tsitsernakaberd Memorial Complex
Notably impressive memorial complexes have also been created in Lebanon and in the United States. The very large monument in Montebello, California, in the shape of an extended Armenian church cupola, dates from the semi-centennial commemorative activities of 1965. Also located in a public area is the Armenian Heritage Park in central Boston, Massachusetts, with its innovative sculptural installation that is adjustable and sits in a round pool across from a circular labyrinth. On the Mission Hills campus of the Ararat Home, the Los Angeles Armenian community's retirement facility for the aged, a number of memorials dedicated to specific purposes have emerged as another complex of notable monuments, including a memorial garden dedicated to Aurora Mardiganian, whose story has been recovered as the emblematic experience of the survivor. As for the campus of the seminary of the Armenian Catholicosate of Cilicia located in the town of Bikfaya in the Lebanon mountains, an entire series of monuments recall important aspects of Armenian history at the foot of its colossal bronze statue.
Boston, Massachusetts, Armenian Heritage Park
The memorial chapel on the grounds of the Catholicosate in Antelias, Lebanon, which serves as an ossuary, has also become another model, and other similar memorial chapels have been constructed elsewhere. The most elaborate of these constitute the chapel in the town of Der Zor in Syria, the site of the largest death camp during the Armenian Genocide.
The Der Zor complex was subjected to extensive damage and rendered to ruin at the hands of Islamic State terrorists who overran eastern Syria in 2014. The ISIS militants, who committed genocide against the Yazidi people of northern Iraq and persecuted the local Kurdish populations, were very likely to have inflicted damage to the memorial in service to their sponsors in Turkey.
In 1995, the Tsitsernakaberd memorial site was expanded with the construction of the first museum dedicated to the Armenian Genocide. The museum and associated research institute (AGMI) were once again sizably expanded for the 2015 centennial events. AGMI is now joined by other institutions providing physical exhibits such as the Armenian Museum of America in Watertown, Massachusetts, and the recently opened Armenian Museum in Moscow. Online exhibits and museums include Houshamadyan, which is dedicated to recovering and reconstructing the memory of Armenian life in the pre-genocide era, Land and Culture, which hosts a database of monuments destroyed or confiscated during the Armenian Genocide, and the Armenian Genocide Museum of America's interactive website. ANI also notes Research on Armenian Architecture, which is dedicated to the photographic recovery of Armenia's heritage subjected to destruction.
With the exception of many memorial sites in Syria, which have become difficult to access under conditions of civil strife in the country, very few memorials are actually placed at locations directly associated with the Armenian Genocide. There are none to speak of in Turkey, of course, where so many atrocities were committed all across Armenia and Anatolia, even though the first memorial service on April 24, 1919, was held in Istanbul. The only memorials in that part of the world are the unmarked and crumbling remains of ancient churches and abandoned homes.
A unique museum exists at an actual orphanage site in the town of Jbeil (also known as Byblos) in Lebanon. The orphanage structure is still intact and has been converted into the Armenian Genocide Orphans Museum that contains moving exhibits on the experience of child survivors who were housed in the facility. The plaza facing the building has also been converted into a memorial complex dedicated to the orphans, with a monument to the guiding light of the institution, Maria Jacobsen. Another orphanage building still intact is located in the small Swiss town of Begnins. It is marked by two modest plaques dedicated by the offspring of Armenian orphans who were housed and educated in that facility, and in recognition of the Swiss pastor Antony Krafft-Bonnard, who established the orphanage.
Begnins, Switzerland, Memorials to Armenian Genocide Orphans
Jbeil, Lebanon, Armenian Genocide Orphans Museum
Other humanitarians have been extended recognition through plaques and monuments dedicated in their memory. The Wall of Honor at the Tsiternakaberd Memorial recalls the most prominent, including Alma Johansson, Anatole France, Armin T. Wegner, Karen Jeppe, Henry Morgenthau Sr., Jakob Künzler, Johannes Lepsius, Pierre Quillard, James Bryce, Raphael Lemkin, Franz Werfel, and Fridtjof Nansen. Others are recognized with busts or memorials placed in their respective countries, including Franz Werfel in Austria, Karen Jeppe in Denmark, Anna Hedvig Büll in Estonia, and Johannes Lepsius in Germany.
The Armenian Genocide Memorials Database was first created as part of a memorials database project funded by the Rockefeller Foundation, with assistance from the United States Holocaust Memorial Museum. Robert Arzoumanian, Assistant to the ANI Director, conducted further research under the guidance of ANI Director Dr. Rouben Adalian to bring the database up-to-date to reflect and record the considerable efforts made by Armenian communities worldwide to honor the centennial of the Armenian Genocide by installing new memorials. These sites serve as gathering places for the annual commemorations that are held every year on the 24th of April, and have become tributes to the survivors who founded the communities of the Armenian Diaspora.
ANI extends its appreciation to its friends who have supported this ongoing effort to document Armenian Genocide memorials. ANI also acknowledges the public's input for the background information on many of the identified monuments. ANI continues to welcome public assistance with this undertaking, especially with any help it can receive concerning undocumented memorials, or the current condition of existing memorials.
For more information on ANI, please refer to the preceding announcements of "Armenian National Institute Website Now Includes 795 Official Records Affirming Armenian Genocide," and "Armenian National Institute Posts Database on Media Coverage of President Biden's Recognition of the Armenian Genocide and its Implications."
Founded in 1997, the Armenian National Institute (ANI) is a 501(c)(3) educational charity based in Washington, DC, and is dedicated to the study, research, and affirmation of the Armenian Genocide. The ANI website can be consulted in English, Turkish, Spanish, and Arabic. ANI also maintains the online Armenian Genocide Museum of America (AGMA).
Posted in Armenia, Armenian Church, Armenian Genocide, News, Turkey
Tags: ANI, Argentina, Armavir, Armenia, Armenian Genocide, Armenian National Institute, Artsakh, Australia, Austria, Begnins, Belgium, Bikfaya, Boston, Brazil, Bulgaria, California, Canada, Chile, Cyprus, Czech Republic, Denmark, Dr. Rouben Adalian, Egypt, Estonia, Ethiopia, France, Genocide Memorial, Georgia, Germany, Greece, Hungary, India, Iran, Ireland, Israel, Italy, Jbeil, Latvia, Lebanon, Mexico, Montebello, Nagorno Karabakh, Netherlands, Paris, Poland, Robert Arzoumanian, Romania, Russia, Singapore, Slovakia, Spain, Sweden, Switzerland, Syria, Tsitsernakaberd, Turkey, U.S., Ukraine, United Arab Emirates, United Kingdom, Uruguay, USA, Vancouver, Venezuela, Yerevan
Diaspora Must Do Better in Keeping Institutions
By Taleen Babayan
Among the maze-like streets of Bourdj Hammoud, where crossed wires hang on for dear life and the Armenian language swirls in the air exist buildings that evoke an integral part of Diasporan history. Once crown jewels, these landmarks now teeter on the edge of obsolescence — an unheard of development for the thousands who once called these corridors home.
Growing up, we all heard stories from our grandparents, who came of age during a significant make-or-break era. Genocide survivors themselves, or children of the brave who escaped the atrocities, they arguably helped save a race from succumbing to destitution. But the stories I heard from my grandfather were seldom set in his birthplace of Aintab. Instead, they took place in the port city of Beirut, in the hallways named after one of the Armenian people's most revered poets.
While it had been three decades since his black leather laced wingtips tapped the concrete floors of the Vahan Tekeyan School, I still felt my grandfather's presence in the classrooms and stairwells during a recent visit; the air tinged with his spirit and service. These were the very same pathways where he served as principal and taught his students, through both word and action, the pride in being Armenian and the importance of contributing to the nation and culture in a positive way. A lesson he made sure to instill in all his grandchildren.
At the height of the school's success — in the gilded Golden Age of Lebanon in the 1960s and early 1970s — my grandfather oversaw the education of thousands of students, each a mere generation or two removed from the crippling Armenian Genocide, which could have very well expunged the remaining survivors. But it was the Vahan Tekeyan School, akin to a handful of its counterparts around the Diaspora, that rose to the occasion — and lifted those students to their rejuvenated feet, citing countless successful alumni in its wake.
But now these venerable institutions are at a crossroads. And it is not just limited to Beirut and the political upheaval of the Middle East. Each carefully constructed Diasporan community, which built up communities and neighborhoods in their adopted countries, is vulnerable.
Armenians are capable of building — there is no doubt about that — but are they capable of preserving?
Far away from the turquoise ripples of the Mediterranean, the polluted East River envelopes the St. Vartan Cathedral, home to the Eastern Diocese, which is on its way to purportedly selling its air rights for $50 million. The Diocesan Council, ostensibly a democratically-elected body, has not been forthright in its public pronouncements of this rumored deal, which according to New York real estate savvy individuals, is well below market value.
What would the founders of the cathedral — those genocide survivors who barely made it to the U.S. with only the tattered clothes on their back — have to say about this charade? And how would they feel about the fact that the new wave of donors who picked up where their forefathers left off were shut out of this ill-conceived undertaking? The cathedral elevated the Armenian Diaspora to new heights in the New World — a golden dome smack dab in the middle of midtown Manhattan. To sell its coveted air rights, for a short-term fix, is shortsighted, to say the least.
Where is the vision that was once tirelessly pursued by leaders of our beloved institutions? It was only 50 years ago, that after decades of tireless fundraising, the doors swung open to thousands of faithful as the 12 Godfathers of St. Vartan Cathedral, their eyes brimming with tears, stood alongside His Holiness Vasken I, of blessed memory, as he consecrated the first Armenian cathedral in the Western Hemisphere.
Why is it that so few today are questioning the status quo and demanding better governance? Why are rational voices stifled as calls for a transparent conversation are neglected? Why are donors quickly called upon in times of financial need but not during times when their opinion should matter most? From top to bottom there is no accountability, discipline or leadership. Could this be a sign of national amnesia — a nation that is tired of itself?
My grandfather, though rooted in tradition, was a keen proponent of vision. In fact it was this vision early on in his tenure that expanded the Vahan Tekeyan School from an aluminum hovel to a robust, concrete building, all within the course of 10 years. What he was averse to, however, was mismanagement. With thousands of students and hundreds of faculty under his wing, he traveled around the world to fundraise — before, during and after the devastating Lebanese Civil War — inspiring donors to give to the educational institution during a time when the Diaspora wasn't even half as prosperous as it is now. So for those in decision-making positions to claim that these institutions can't be supported anymore, financially, is therefore, a complacent response. It should be the priority of leadership to be fiscally responsible. In this case, the Diocesan Council needs to shore up continued support and interest to sustain the Diocesan Center and St. Vartan Cathedral instead of throwing in the towel — and that can only be achieved by regaining trust in donors and ruling, not with an iron fist, but through transparency and consensus. Prudence and foresight — not egos and backroom deals — should lead commercial deals of this magnitude.
These historic institutions don't just support local communities; they support the Diaspora as a whole. And a strong Diaspora is a prerequisite for a strong Armenia. I vividly recall all of my grandfather's students who would visit him, years after he retired from a 60-year career in education and public service. He remembered each one by face, by name, by voice. Tekeyan wasn't just a school or a job to him. It was his life. His family. Each of those students were his children. So much so that 30, 40, even 50 years later, they still referred to him as their "Harkeli Baron Dnoren Yervant Babayan." My grandfather, who studied in Paris, spoke five languages and could have easily remained in Europe and entered a fruitful life in the private sector, gave his heart to the Vahan Tekeyan School and that is an integral reason for its bone-deep impact. In the case of the Eastern Diocese, there seems to be no business sense, but more dangerously, there seems to be no heart. And with no heart, there can be no pulse.
Lack of funds is not the real problem. Apathy is the evil. St. Vartan Cathedral and the Diocesan Complex were built by thousands of donors, large and small, and there are thousands of donors who still give, large or small, myself included. Yet it's being torn down by a few in front of our own eyes. I'm sure I'm not the only one who remembers the heyday of the dynamic activities of the cathedral and the Diocese: the headline-making all-inclusive One World Festival, the vibrant Armenian school programs, or the talented theater ensemble.
Based on contemporary history, the culturally rich Diasporan communities in the Middle East have faced political turmoil and conflict in addition to financial burdens. It is, to an extent, understandable that the most influential Diasporan communities — Beirut, Aleppo, Tehran, Baghdad, Cairo — don't have the sizable presence they once did, though they continue to possess innate vigor to keep their communities alive. But what about the so-called stable communities in, say, Paris, London or New York? Aren't they also on the verge of gradually dying out? Could the prospect of apathy and assimilation be even more menacing than that of political conflict?
Selling the air rights may spur a fatal domino effect, setting the precedent for New York City's St. Vartan Cathedral and Eastern Diocese to become an "in memoriam" like many other Diasporan institutions, that too, weren't "sustainable" anymore, vanishing into the annals of unrecorded history.
As Armenians, it is not that we can do better, it is that we must do better. Isn't it time to override the corroding dysfunction that plagues many Armenian organizations? We must expect more from those who sought to assume leadership positions. We must read their intentions, their motives and their abilities before entrusting them to make these critical decisions. And we must examine if they are indeed upholding their duties. Above all, we must honor and respect donors — those who have built, stone by stone, the very bedrock of our most revered Diasporan institutions.
I thought fondly of those donors and their sacrifices as I made my way out of the Vahan Tekeyan School, pleased to witness the dedicated staff in action. While stepping back onto the maze-like streets of Bourdj Hammoud, a teacher stopped me. He proudly revealed a book given to him by my grandfather, who had authored the tome. Our family ties went back — his father was a student of my grandfather's in Aleppo when he had served as principal of the Giligian School. And the teacher, himself a graduate of the Vahan Tekeyan School and a holder of a master's degree, chose to educate the next generation at his alma mater. As he clutched the book, he told me he still read through the pages for guidance and inspiration, my grandfather's lessons, too, ringing in his ears.
That, dear reader, is my legacy. What is yours? And more importantly – what are you doing to protect it?
Posted in Armenian Church, Arts & Culture
Tags: Armenian Church, Armenian Diaspora, Eastern Diocese, Lebanon, Taleen Babayan
The Forgotten Apogee of Lebanese Rocketry
Fifty-two years ago, as Soviet cosmonauts and U.S. astronauts were first venturing into space, another space program was also taking off—in Lebanon.
By Sheldon Chad
Yes, in the early 1960's, the country of 1.8 million people, one-quarter the size of Switzerland, was launching research rockets that reached altitudes high enough to get the attention of the Cold War superpowers.
But the Lebanese program was more about attitude than altitude: Neither a government nor a military effort, this was a science club project founded by a first-year college instructor and his undergraduate students. And while post-Sputnik amateur rocketry was on the rise, mostly in the U.S., no amateurs anywhere won more public acclaim than the ones in Lebanon.
Manoug Manougian, right, with members of the Haigazian College Rocket Society, which he founded in 1960. It later became the Lebanese Rocket Society.
But that is forgotten history now, says Manoug Manougian, now 77 and a mathematics professor at the University of South Florida in Tampa. He leads me into a conference room where he has set out on a table file boxes filled with half-century-old newspapers, photographs and 16mm film reels.
"When I decided to leave, no one was interested to take care of all this," says Manougian. "But I felt, even at that point, that it was a part of Lebanese history."
The Society launched its first "tiny baby rockets" at the mountain farm of one of its members.
Born in the Old City of Jerusalem, Manougian won a scholarship to the University of Texas, and he graduated in 1960 with a major in math. Right away, Haigazian College in Beirut was glad to offer him a job teaching both math and physics. The college also made him the faculty advisor for the science club, which Manougian reoriented by putting up a recruitment sign that asked, "Do you want to be part of the Haigazian College Rocket Society?"
He did this, he explains, because even as a boy, he loved the idea of rockets. He recalls taking penknife in hand and carving into his desk images of rocket ships flying to the moon. "It's the kind of thing that stays with you," Manougian says.
John Markarian, former head of the college, now 95, recalls thinking it was "a rather harmless student activity. What a wonderful thing it was." The first rocket, he says, "was the size of a pencil."
Manougian now teaches at the University of Southern Florida, where he keeps newspaper front pages on his office wall. "It was a part of Lebanese history," he says.
Six students signed up, and in November 1960, the Haigazian College Rocket Society (HCRS) was born. "It is not a matter of just putting propellant in the tube and lighting it," says Manougian. Former HCRS member Garo Basmadjian explains that at the time, "we didn't have much knowledge, so we looked at ways to increase the thrust of the rocket by using certain chemicals." After dismissing gunpowder, they settled on sulfur and zinc powders. Then they would pile into Manougian's aging Oldsmobile and head to the family farm of fellow student Hrair Kelechian, in the mountains, where they would try to get their aluminum tubes to do, well, anything.
"We had a lot of failures, really," says Basmadjian.
But soon enough "it did fly some distance," Manougian adds.
The Cedar launches were commemorated on this postage stamp issued on Lebanon's independence day.
The HCRS began using a pine-forested mountain northeast of Beirut to shoot off the "tiny baby rockets," as Manougian calls them, each no longer than half a meter (19″).
As they experimented, the rockets grew larger. By April 1961, two months after the first manned Soviet orbital mission, the college's entire student body of 200 drove up for the launch of a rocket that was more than a meter long (40″).
The launch tube aimed the rocket across an unpopulated valley, but at ignition, Manougian recalls, the thrust pushed the "very primitive" launcher backward, in the opposite direction, and instead of arcing up across the valley, the rocket blazed up the mountain behind the students.
1963 saw the launch of Cedar 3, a three-stage rocket that allegedly broadcast "Long Live Lebanon" from its nose cone as it rose.
Launches at the military site of Dbayea, overlooking the Mediterranean north of Beirut, drew crowds of spectators, journalists and photographers.
"We had no idea what lay in that direction," says Manougian. To investigate, the students started climbing, and on arrival at the Greek Orthodox Church on the peak, they came on puzzled priests staring at the remains of the rocket, which had impacted the earth just short of the church's great oaken doors. Manougian calculated that, even with the unplanned launch angle, considering thrust and landing point, the rocket had reached an altitude of about a kilometer (3300′), and he adds the bold claim that this was the first modern rocket launched in the Middle East.
Ballistics expert Lt. Youssef Wehbe (in uniform) began supporting the rocket society in 1961, initially by allowing it access to an artillery range on Mount Sannine.
The next day, Manougian got a call from Lieutenant Youssef Wehbe of the Lebanese military. He cautioned that the hcrs couldn't just go up any old mountainside and shoot off rockets. They could, however, do it as much as they wished under controlled conditions at the military's artillery range on Mt. Sannine. Wehbe, also in his 20's, was a ballistics expert, and he was more than intrigued. "Our first success," says Manougian, came there at Mt. Sannine, where the rocket they demonstrated for Wehbe soared 2.3 kilometers (7400′) into the air.
Newspapers got wind of the launches, and they reported that the "Cedar 2C" (named for the symbol of Lebanon) had reached 14.5 kilometers (47,500′). "Obviously, that's not yet the moon distance of 365,000 kilometers. But the Lebanese aren't after that, they're after technique," stated the report.
Under Wehbe's supervision, HCRS developed two-stage and then three-stage rockets, each bigger than the last and soaring higher and farther.
In the papers, the rocket men were portrayed as both brawny and brainy, and they were the talk of Lebanon. A fan club of prominent Lebanese—mostly women—formed the Comité d'encouragement du Groupe Haigazian. In the photos and films of the launches, one can see generals deferring to college kids in HCRS hardhats and eagerly posing in the press photos with them. Even Lebanese president Fuad Chehab invited Manougian and his students to the palace for a photo op.
"We were just having fun and doing something we all wanted to do," says Basmadjian. "When the president came into the picture and gave us some money, it took off."
"We were members of a scientific society. We felt good about it," says Simon Aprahamian, another former student. "But it didn't feel like what the us or ussr were doing. It's a small country, Lebanon. People felt, 'This is something happening in our country. Let's get involved.'"
Launches now drew hundreds of spectators to the site overlooking the Mediterranean Sea at Dbayea, north of Beirut, and Haigazian itself became known as "Rocket College." As the HCRS was now the country's pride, its name changed to the Lebanese Rocket Society (LRS).
Cedar 4 was the society's most powerful rocket. newspapers claimed that it reached a maximum height between 145 and 200 kilometers (90–125 mi), though the reality was surely much less. For Manougian, however, the rockets and their launches were not about setting records, but about teaching future scientists.
Lebanese weren't the only ones watching. Both superpowers, according to Manougian, had "cultural attachés" observing the launches, and he believes they did more than that. "My papers were always out of place on my desk," he says, and he recalls once leaving a note: "My filing cabinet I am leaving open. I have nothing to hide. But please don't mess up my desk!"
One night in 1962, Manougian was taken in the back of a limousine to a factory in the heart of downtown Beirut. There, he was introduced to Shaykh Sabah bin Salim Al-Sabah of Kuwait, who offered to fund Manougian's experiments generously if he moved them to Kuwait. Manougian hesitated, recalling the commitment he made to himself when he accepted the post at Haigazian: "Don't stay too long. You only have a bachelor's degree." More than a private lab, Manougian wanted to get back to Texas to get his master's.
Before Manougian left for Texas, however, he sat down with Wehbe to plan two launches for Lebanese Independence Day, November 21, in both 1963 and 1964. The rockets would be called Cedar 3 and Cedar 4, and each would have three stages.
They would dwarf what went before in both size and strength: seven meters (22′) long, weighing in at 1270 kilograms (2800 lbs) and capable of rising an estimated 325 kilometers (200 mi) and covering a range of nearly 1000 kilometers (about 620 mi), the rockets would generate some 23,000 kilograms (50,000 lbs) of thrust to hit a top speed of 9000 kilometers per hour (5500 mph). From the nose cone, a recording of the Lebanese national anthem would be broadcast.
On November 21, 1963, a model of Cedar 3 was paraded through Beirut's streets to great applause. The cover of the souvenir booklet shows a rocket overflying the city. For Cedar 4, Lebanon issued commemorative postage stamps showing the rocket leaving Earth's atmosphere. On launch day, 15,000 people showed up, along with generals and even the president.
In those years, Manougian recalls, the "rocket boys" were celebrities and Haigazian College was "rocket college." Above, Manougian answers a journalist's questions after a launch. The last rocket, Cedar 10, flew in 1967, after Manougian had returned to the us to earn his doctorate. Then, Cold War politics shut down the program.
The newspapers reported with national pride that the rockets flew into "space" and landed on the far side of Cyprus. The altitudes that were published varied from 145 to 200 kilometers (90–125 mi). The actual figures, however, are likely more modest. "That was totally wishful," says Ed Hart, the Haigazian physics professor who took over as faculty advisor to the LRS. "It never came close. We kept our mouths shut [because] it was not a student matter anymore. It had become a social, society kind of matter."
For Manougian, Wehbe told him that according to calculations, the rockets achieved their aims. Hart, whose specialty is science education, brings it back to empirical achievement: "We were teaching students a great deal, and that is what we came for: the mystery and structure of forces."
In 1964, master's degree in hand, Manougian returned to Lebanon, and again collaborated with Wehbe on a few more launches. By then, world powers were interested: France supplied the rocket fuel; the U.S. invited Wehbe to Cape Canaveral.
Cedar 8 was the last LRS rocket. Launched in 1966, it was a two-stage, 5.7-meter (18′) rocket with a range of 110 kilometers (68 mi)—a long way from the pencil-sized rockets of five years earlier. "We were launching in the evening, and we put lights on top of the second stage to be able to witness the separation. There were no hitches. It took off beautifully, the separation was fairly obvious, nothing exploded and it landed at the time it was supposed to land. To me that was a perfect launch," says Manougian, still in awe 50 years on.
Under Manougian's guidance, a new rocket society at USF is exploring rockets that use plasma engines.
By 1966 Manougian grew concerned about the extent of military involvement. "I'd accomplished what I'd come there to accomplish. It was time for me to get my doctoral degree and do what I love most, which was teaching," he says. He left in August, and the Lebanese Rocket Society was no more.
But under military auspices, a last Lebanese rocket, Cedar 10, flew in 1967. According to Manougian, Wehbe told him that French president Charles de Gaulle soon pressured President Chehab to shut down the rocket project for geopolitical reasons.
Decades of political turbulence followed, and the story of the LRS lay hidden away in Manougian's boxes.
Two years ago, science and engineering students at the University of South Florida approached Manougian to set up a rocket society. "My students did this 50 years ago," he replied, adding, "What can you do now that's innovative?" That's how he became faculty advisor of the Society of Aeronautics and Rocketry (SOAR), which is exploring rockets powered by electromagnetism and nano-materials. As in Beirut, he says, "the important thing is not the rocket. It is the scientific venture."
"Soar" is an apt metaphor for all involved. With the HCRS/LRS rocket projects, Lebanon punched well above its weight. Wehbe retired as a brigadier general. Manougian went on to win teaching awards, and he is loved by his students now as then. Many of the LRS students, and others inspired by them, went on to excel in scientific pursuits.
"Most of us come from very humble beginnings. But we had some brains and we studied hard," says Basmadjian.
"Did that experience help with regard to making new inventions?" asks another former student, Hampar Karageozian, who later studied at the Massachusetts Institute of Technology and founded several ophthalmological drug companies. "Yes, it did. Because it completely changed my attitude. The attitude that we could say that nothing is impossible, we really have to think about things, we really have to try things. And it might work!"
Sheldon Chad (shelchad@gmail.com) is an award-winning screenwriter and journalist for print and radio. From his home in Montreal, he travels widely in the Middle East, West Africa, Russia and East Asia. He will be reporting from Chad for his next story for Saudi Aramco World.
This story originally appeared on page 18 of the May/June 2013 Print Edition of Saudi Aramco World.
Posted in Arts & Culture, General Update, News
Tags: Beirut, Cape Canaveral, Charles de Gaulle, Cold War, Cyprus, Dbayea, France, Fuad Chehab, Garo Basmadjian, Haigazian, Haigazian College, Hampar Karageozian, John Markarian, Kuwait, Lebanon, Manoug Manougian, Massachusetts Institute of Technology, Mediterranean Sea, Middle East, Rocket, Simon Aprahamian, Society of Aeronautics and Rocketry, Soviet, Texas, University of South Florida, University of Texas, Youssef Wehbe | {
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Back to business today – fitness biz. Finally managed to go running for the first time since I got ill weeks ago and it was bloody knackering.. but very satisfying to be back out there.
After work Nic and I went swimming in Brockwell Lido. There was a small rain shower at one point but we swam for nearly an hour then had dinner in the lido cafe. It was a beautiful evening all round. | {
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Palmarès
Competizioni nazionali
Ajax: 1993-1994, 1994-1995, 1995-1996
Ajax: 1993, 1994, 1995
Competizioni internazionali
Ajax: 1994-1995
Ajax: 1995
Ajax: 1995
Collegamenti esterni
Calciatori della Nazionale olandese | {
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Q: Gaps between numbers of the form $pq$ Mathematicians keep improving Zhang's bound on gaps between primes. According to Wikipedia, there are infinitely many pairs of primes such that their difference is no more than 246. This is all very exciting.
What does it tell us about gaps between products of two distinct primes? Can we say that there are infinitely many pairs of numbers of the form $pq$, for distinct primes $p$ and $q$, such that their difference is less than some $K$? To be clear, I mean pairs of numbers $(a,b)$, where $a=p_1q_1$ and $b=p_2q_2$, where $p_i\neq q_i$, but it's okay if $p_i=q_j$, or $p_i=p_j$ when $i\neq j$.
Thanks in advance for any insights on this.
A: Goldston, Graham, Pintz, and Yildirim showed that one can take $K=6$ in your statement! (another link) Indeed they could show this even before Zhang's breakthrough.
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\section*{I. Introduction}
The study of the primary relaxation of supercooled liquids by means of dielectric techniques in the linear response regime is standard and allows investigations over an extremely broad frequency range\cite{Kremer02,Lunki:2000,Ranko:2014r}.
Apart from the detailed form of the spectra the nature of the dynamical heterogeneities has been studied using various frequency-selective
techniques\cite{G23,Ediger00,Israeloff00,Richert02}, including higher-dimensional nuclear magnetic resonance experiments\cite{SRS91,HWZS95,G13} and nonresonant dielectric hole-burning\cite{SBLC96,G16,Chamberlin:2018}.
The latter techniques allow the frequency-selective modification of the spectrum via the application of strong electric fields.
In the recent past, the nonlinear dielectric response of a number of glass-forming systems has been investigated and the results have been used to extract the length scale of the dynamical heterogeneities or the number of correlated particles $N_{\rm corr}$,
cf.\cite{Richert:2017, Gadige:2017}.
According to theoretical predictions, the modulus of the nonlinear susceptibilities (cubic and higher-order) exhibits a so-called hump, the height of which is directly related to $N_{\rm corr}$\cite{Bouchaud05,Albert:2016}.
It is thus assumed that the origin of the hump is intimately related to the existence of a growing amorphous order or some kind of domain structure.
On the other hand, it has been argued that the reorientational motion of individual molecules gives rise to a monotonous decay of the modulus from a finite low-frequency value to zero at high frequencies and this behavior is found for the model of isotropic rotational diffusion\cite{Albert:2016,Dejardin00,DD95,Kalmykov01}.
It is, however, to be mentioned that also some models lacking spatial correlations like the Box model and variations of this and other phenomenological models have been shown to exhibit a hump in the nonlinear susceptibilities\cite{Richert:2017,Richert06,Ladieu:2012,Buchenau:2017}.
We have computed the third-order and the fifth-order response functions for the well known asymmetric double well potential model of dielectric relaxation and for the simple Gaussian trap model for glassy relaxation and found a hump for certain values of the model
parameters\cite{G75, G87}.
Additionally, first results for the cubic response for the model of rotational random jumps have been presented in ref.\cite{G92}.
In the analyses of most nonlinear dielectric data it has been assumed that for very low frequencies the heterogeneous nature of the response becomes irrelevant and therefore at these long times the individual reorientational motion of the molecules determines the response.
As mentioned, usually the model of isotropic rotational diffusion has been used to calculate the corresponding resonse functions.
However, it is a well known fact that this model is not able to reproduce a number of aspects related to the noninertial molecular reorientations in supercooled liquids, see e.g.\cite{G42}.
Various models introducing finite angular jumps instead of diffusive motions have been proposed.
One of the first attempts to formulate a general stochastic model of rotational jumps has been provided by Ivanov\cite{Ivanov:1964} and since then a number of different treatments of the rotational motions in liquids have been presented\cite{anders:1972, Bagchi:1991, G31, ALessi:2001}.
Also models explicitly taking into account the dynamical heterogeneities can be used to reproduce a number of the findings related to the reorientational motion in the primary relaxation as monitored with different techniques\cite{G17,G21}.
In the present paper, we present results for the third-order and fifth-order nonlinear response for two models of molecular reorientations.
One model treats the isotropic rotations as random jumps on a sphere and the other model is the one of anisotropic rotational diffusion.
These models can both be viewed as limiting cases of more general rotational jump models\cite{G31}.
The remainder of the paper is organized as follows.
In the following chapter, we briefly review the models for molecular reorientations that will be used in the calculations of the nonlinear response afterwards.
After a discussion of the results for the third-order and the fifth-order dynamic susceptibilities we close with some concluding remarks.
\section*{II. Markovian reorientational jump models}
In general, the time-dependent orientation of a molecule is described in terms of the Eulerian angles
$\Omega(t)=(\phi(t),\theta(t),\psi(t))$ relating the axes of a molecular fixed frame and some laboratory fixed axes system.
If $P(\Omega,t|\Omega_0)$ denotes the probability to find an orientation $\Omega$ at time $t$ given that it was $\Omega_0$ at an earlier time $t=0$, the
master equation\cite{vkamp81} can be written in the form
\be\label{P.ME}
{\dot P}(\Omega,t|\Omega_0) = \int\!\!d\Omega'W(\Omega|\Omega')P(\Omega',t|\Omega_0)-\int\!\!d\Omega'W(\Omega'|\Omega) P(\Omega,t|\Omega_0)
\ee
with the probability $W(\Omega|\Omega')$ for a $\Omega'\to\Omega$-transition. The inital condition is given by $P(\Omega,t=0|\Omega_0) = \delta(\Omega-\Omega_0)$
and the probability of finding a given orientation is related to $P(\Omega,t|\Omega_0)$ via
$p(\Omega,t)=\int\!\!d\Omega_0P(\Omega,t|\Omega_0)p(\Omega_0)$.
The model of rotational diffusion is recovered in the limit of small rotation angles in which case the master equation corresponds to a Fokker-Planck equation.
In general, the conditional probability can be expressed in terms of Wigner rotation matrix elements
\be\label{P.Dlmn.allg}
P(\Omega,t|\Omega_0)=\sum_{l,m_1,m_2,n}\left(\frac{2l+1}{8\pi^2}\right)D_{m_1n}^{(l)*}(\Omega_0)D_{m_2n}^{(l)}(\Omega)F^{(l)}_{m_1m_2}(t)
\ee
with time dependent coefficients $F^{(l)}_{m_1m_2}(t)$ that are solutions of the respective equations.
In ref.\cite{G31}, we have introduced a reorientational jump model where rotational jumps with fixed angular width $\Delta\Omega$ have been considered.
The transition probabilities for this model can be written in the form:
\Be\label{W.kl}
&&W(\Omega_1|\Omega_2)=
\Gamma_R\sum_{l,m,n}\left(\frac{2l+1}{8\pi^2}\right)D_{mn}^{(l)*}(\Omega_1)D_{mn}^{(l)}(\Omega_2)\Lambda_{l,m}(\bar\theta,\bar\phi)
\nonumber\\
&&\hspace{-1.2cm}
\mbox{with}\quad
\Lambda_{l,m}(\bar\theta,\bar\phi)=\cos{(2m\bar\phi)}d_{mm}^{(l)}(\bar\theta)
\Ee
Here, $\bar\theta$ and $\bar\phi$ denote the jump angles and $\Gamma_R$ is the jump rate.
In a diagonal approximation, discussed in detail in ref.\cite{G31}, the $F^{(l)}_{mn}(t)$ in eq.(\ref{P.Dlmn.allg}) are given by $F^{(l)}_{mn}(t)=\delta_{m,n}F^{(l)}_m(t)$ with
\be\label{Fmt.gen}
F^{(l)}_m(t)=e^{-\Gamma_R\left(1-\Lambda_{l,m}(\bar\theta,\bar\phi)\right)t}
\ee
The exact result for the model of rotational diffusion is recovered for small jump angles and one has
\be\label{F.ARD}
F^{(l)}_{m}(t) = e^{-\{l(l+1)D_X + m^2(D_Z-D_X)\}t}
\ee
where it is assumed that the rotational diffusion coefficients $D_Y$ and $D_X$ are equal, but not necessarily equal to $D_Z$.
Eq.(\ref{F.ARD}) follows from $\Lambda_{l,m}(\bar\theta,\bar\phi)\simeq 1-\{l(l+1)-m^2\}(\bar\theta/2)^2+m^2(2\bar\phi^2)$ using
$D_X=\Gamma_R(\bar\theta/2)^2$ and $D_Z=\Gamma_R(2\bar\phi^2)$.
Only for $D_X\!\!=\!\!D_Z$ the second term in the exponential vanishes and the result for isotropic rotational diffusion,
\[
P_{IRD}(\Omega,t|\Omega_0)=\sum_{l}\left(\frac{2l+1}{8\pi^2}\right)D_{00}^{(l)*}(\Omega_0)D_{00}^{(l)}(\Omega)e^{-l(l+1)D_Xt}
\]
is recovered.
The model of rotational random jumps (with rate $\Gamma_{RJ}$) is obtained by averaging over all jump angles and one finds:
\be\label{F.RJ}
F^{(l)}_{m}(t) = e^{-\Gamma_{RJ}t}
\ee
Insertion of this result into eq.(\ref{P.Dlmn.allg}) yields the well known expression
$P_{RJ}(\Omega,t|\Omega_0)=\frac{1}{8\pi^2}+e^{-\Gamma_{RJ}t}[\delta(\Omega-\Omega_0)-\frac{1}{8\pi^2}]$\cite{Sillescu:1996}.
In this case, the coefficients are not only independent of $m$ but also do not depend on the rank $l$.
It has to be mentioned that the orientation $\Omega(t)$ is the orientation of the tagged molecule at a given time in a laboratory fixed frame.
For instance in case of anisotropic rotational diffusion, $\Omega(t)$ represents the orientation of the coordinate system of the diffusion tensor (D) in the laboratory fixed frame (L).
Experimentally, however, the orientation of the principal axes system (P) of the relevant interaction in the L-system is observed.
In case of dielectric relaxation, for instance, the P-system is defined by the orientation of the molecular dipole relative to the D-system.
This means that the expectation value of the dipole moment (the response) is written as:
\be\label{Mu.t}
\langle M(t)\rangle=M\langle D_{00}^{(1)}(\Omega_{PL}(t))\rangle
=M\sum_n D_{0n}^{(1)}(\Omega_{PD})\langle D_{n0}^{(1)}(\Omega_{DL}(t))\rangle
\ee
where we used the fact that $D_{mn}^{(l)}(\Omega_{PL})=\sum_{n'} D_{mn'}^{(l)}(\Omega_{PD})D_{n'n}^{(l)}(\Omega_{DL})$ and that
$\Omega_{PD}$ is a static quantity defined by the geometry of the molecule considered.
Here $M$ denotes the static value of the dipole moment and $M(\Omega_{PL})=M\cos{(\Omega_{PL})}$.
The expectation value of an orientation-dependent quantity $A(\Omega)$ can be expressed in terms of the solution of the master equation
\[
\langle A(\Omega(t))\rangle=\int\!d\Omega\int\!d\Omega_0 A(\Omega)P(\Omega,t|\Omega_0)p^{eq}(\Omega_0)
\]
where $p^{eq}(\Omega_0)$ denotes the equilibrium probability, in our case $p^{eq}(\Omega_0)=1/(8\pi^2)$ because all orientations are equally probable.
\section*{III. Nonlinear response theory for Markov processes}
The response of the system to an external $E$ field applied at time $t=0$ and measured by the moment $M(t)$ is given by eq.(\ref{Mu.t}), which in terms of the solution of the master equation in the presence of a field $E$ is written as
\be\label{Mu.t.G}
\chi(t)=\langle M(t)\rangle=M\sum_n D_{0n}^{(1)}(\Omega_{PD})\int\!d\Omega\int\!d\Omega_0 D_{n0}^{(1)}(\Omega) P^{(E)}(\Omega,t|\Omega_0)p^{eq}(\Omega_0)
\ee
Here, $\Omega$ is a shorthand notation for $\Omega_{DL}$.
The time-dependent perturbation theory for the conditional probability $P^{(E)}(\Omega,t|\Omega_0)$ has been discussed in detail in refs.\cite{G75,G87,G92}.
For the models considered in the present paper, it is sufficient to note that the field is coupled to the transition probabilities $W(\Omega_1|\Omega_2)$ in eq.(\ref{W.kl}) via the orientation dependent dipole moment $M(\Omega)$ in the following way:
\be\label{W.kl.E}
W^{(E)}(\Omega_e|\Omega_i)=W(\Omega_e|\Omega_i)e^{-\beta E(t)(\mu M(\Omega_i)-(1-\mu)M(\Omega_e))}
\ee
where $\mu$ is a model parameter that determines how the system couples to the field.
For $\mu=1$, the coupling takes place via the initial orientation of the transition ($\Omega_i$) and for $\mu=0$ only the destination orientation ($\Omega_e$) is relevant. The particular choice $\mu=1/2$ is important for small step reorientations because in this case $(M(\Omega_i)-M(\Omega_e))\sim dM/d\Omega$ and therefore represents the force acting on the system via the application of the field.
As detailed in ref.\cite{G75,G87}, eq.(\ref{W.kl.E}) is used in the master equation for $P^{(E)}(\Omega,t|\Omega_0)$ and the time-dependent perturbation theory is obtained from a series expansion of $W^{(E)}(\Omega_e|\Omega_i)$.
From this, the conditional probabilities are obtained in any desired order in the field amplitude.
We will compute the experimentally relevant response to a sinusoidal field of the form $E(t)=E_0\cos{(\omega t)}$.
In the following, we will write for the corresponding susceptibilities monitored after the decay of initial transients:
\Be\label{Chi.om.def}
\chi^{(1)}(t)
&&\hspace{-0.6cm}=
{E_0\over2}\left[e^{-i\omega t}\chi_1(\omega)+c.c.\right]
\nonumber\\
\chi^{(3)}(t)
&&\hspace{-0.6cm}=
{E_0^3\over2}\left[e^{-i\omega t}\chi_3^{(1)}(\omega)+e^{-i3\omega t}\chi_3^{(3)}(\omega)+c.c.\right]
\\
\chi^{(5)}(t)
&&\hspace{-0.6cm}=
{E_0^5\over2}\left[e^{-i\omega t}\chi_5^{(1)}(\omega)+e^{-i3\omega t}\chi_5^{(3)}(\omega)+e^{-i5\omega t}\chi_5^{(5)}(\omega)+c.c.\right]
\nonumber
\Ee
where $c.c.$ denotes the complex conjugate.
In the present paper, we will concentrate on the discussion of the response functions $\chi_k^{(k)}(\omega)$, i.e. we focus on the highest frequency component in a given order (third order or fifth order).
We note that we only consider systems that are in thermal equilibrium and therefore the well-known fluctuation-dissipation theorem (FDT) relating the linear response to a short field pulse, $R^{(1)}(t)$, and the autocorrelation function of the dipole moment holds, see e.g.\cite{Chandler87}.
\section*{IV. Results for simple models of molecular reorientations}
Here, we consider the Brownian rotational motion of molecules in terms of the simple models of rotational random jumps (RJ) and anisotropic rotational diffusion (ARD) in addition to the model of isotropic rotational diffusion (IRD).
In the terminology of ref.\cite{CrausteThibierge10,Brun11} these models describe the 'trivial' dynamics of individual molecules in a glass-forming liquid without any so-called glassy correlations and therefore cannot account for the nontrivial features like the observed hump in the nonlinear response.
We do not go into technical details of the calculations, which are lengthy but straightforward.
The results for $\chi_k^{(k)}(\omega)$, $k$=1,3,5, for the three models considered in the present paper are explicitly given in the Appendix.
The linear dielectric susceptibility for the IRD model and the RJ model can be written in the form:
\be\label{Chi1.iso}
\chi_{1,Z}(\omega)=\Delta\chi_1{1\over1-i\omega\tau_{10}}\quad\mbox{with}\quad \Delta\chi_1=\beta{M^2\over3}
\ee
where the relaxation time is $\tau_{10}=1/(2D_X)$ if $Z$=IRD and $\tau_{10}=1/\Gamma_{RJ}$ for $Z$=RJ.
Furthermore, $\beta=1/(k_BT)$ denotes the inverse temperature and we will set the Boltzmann constant $k_B$ to unity in all following expressions.
The static linear response (corresponding to $\Delta\epsilon$ in the dielectric terminology) is denoted by $\Delta\chi_1$.
For the ARD model, one finds:
\be\label{Chi1.ard}
\chi_{1,ARD}(\omega)=\Delta\chi_1\left({\cos^2{(\Theta)}\over1-i\omega\tau_{10}}+{\sin^2{(\Theta)}\over1-i\omega\tau_{11}}\right)
\ee
with $1/\tau_{1m}=2D_X+m^2(D_Z-D_X)$ and $\Theta\equiv\Theta_{PD}$ denoting the angle between the z-axes of the molecular axis system (P-system) and of the principal axis system of the diffusion tensor (D-system).
Thus, in this case one has a superposition of two Lorentzians with weights depending on the value of $\Theta$.
Only for $\Theta=0,\pi/2$ one is left with one Lorentzian.
However, the fact that the spectrum is given by a superposition of two Lorentzians is not relevant in the present context because
in supercooled liquids usually distributions of relaxation times exist that give rise to very broad spectra.
In the general case, the relaxation time is most meaningful defined via the decay time of the normalized dipole autocorrelation function,
\be\label{tau1.def}
\tau_1=\int_0^\infty\!\! dt \langle M(t)M(0)\rangle_n=\cos^2{(\Theta)}\tau_{10}+\sin^2{(\Theta)}\tau_{11}
\ee
which reduces to $\tau_1\equiv\tau_{10}$ for the IRD model and the RJ model.
In the following, we will present all spectra as a function of $\omega\tau_1$ with the consequence that $\chi_1(\omega)$ for the RJ model and IRD model coincide. Furthermore, for the ARD model the spectra for $\Theta=0$ and $\Theta=90^o$ are identical to the one for the IRD model.
In the past, experimental results of nonlinear dielectric spectra have either been presented in terms of real and imaginary part of the susceptibility or, alternatively, the modulus and the phase have been considered.
In particular, it has proven meaningful to scale the modulus by the squared static linear response in the following way:
\be\label{X35.def}
X_3(\omega)={T\over(\Delta\chi_1)^2}\left|\chi_3^{(3)}(\omega)\right|
\quad\mbox{and}\quad
X_5(\omega)={T^2\over(\Delta\chi_1)^3}\left|\chi_5^{(5)}(\omega)\right|
\ee
These definitions eliminate the trivial temperature dependences, $\chi_3^{(3)}\propto\beta^3$ and $\chi_5^{(5)}\propto\beta^5$.
Using $\Delta\chi_1=\beta M^2/3$ and the expressions given in the Appendix, one can write the moduli in terms of the spectral functions for each of the models considered:
\be\label{X35.rots}
X_{3,Z}(\omega)={3\over20}\left|S_{3,Z}(\omega\tau_1)\right|
\quad;\quad
X_{5,Z}(\omega)={9\over560}\left|S_{5,Z}(\omega\tau_1)\right|
\ee
where $Z$ is an abbreviation for IRD, ARD or RJ.
The low-frequency limits for all models considered coincide and are given by:
\be\label{X35.limits}
X_{3,Z}(\omega\to0)={1\over20}
\quad\mbox{and}\quad
X_{5,Z}(\omega\to0)={1\over280}
\ee
However, for high frequencies the limiting behavior for the RJ model differs from that for the diffusion models.
One finds $X_{3,Y}(\omega\to\infty)\sim\omega^{-3}$ and $X_{5,Y}(\omega\to\infty)\sim\omega^{-5}$ for $Y$=IRD, ARD while for the RJ model one has $X_{k,RJ}(\omega\to\infty)\sim\omega^{-1}$ for both $X_3$ and $X_5$.
However, we will not further discuss this high frequency behavior as it does not appear to be observable in supercooled liquids due to the existence of other relaxation phenomena such as the so-called wing or secondary processes at higher frequencies\cite{Lunki:2000}.
In Fig.\ref{Fig1} we show the real and the imaginary part of the third-order and the fifth-order response for the model of isotropic rotational diffusion and for the RJ model.
\begin{figure}[h!]
\centering
\includegraphics[width=15cm]{Rot1}
\vspace{-0.25cm}
\caption{Real and imaginary part of the cubic susceptibility $\chi_3^{(3)}(\omega)$ (left) and the fifth-order susceptibility
$\chi_5^{(5)}(\omega)$ for the models of rotational diffusion (red dashed lines) and rotational random jumps for $\mu=1$ (black full lines) and $\mu=0$ (blue dot-dashed lines).
The green dotted lines represent the linear response, cf. eq.(\ref{Chi1.iso}), scaled by a factor of ten.
}
\label{Fig1}
\end{figure}
In the latter case, we used two values for the parameter $\mu$ that describes the coupling to the external field, cf.
eq.(\ref{W.kl.E}).
For $\mu=1$ (black full lines) the coupling takes place via the initial state of a rotational transition and for $\mu=0$ (blue dot-dashed lines) only the destination state is relevant.
For the RJ model, it appears that $\mu=1$ is the more natural choice because the idea underlying the model is that every single transition completely decorrelates the orientation in the sense that starting from a given orientation any other can be reached in a single step.
It is then meaningful to assume that starting with a coupling to an initial orientation according to $(-E\cdot M(\Omega_i))$,
an average over all possible destination orientations is to be performed, $\langle(-E\cdot M(\Omega_e))\rangle$, which vanishes.
Therefore, according to eq.(\ref{W.kl.E}), this corresponds to choosing $\mu=1$.
The overall behavior of the results for the IRD model and the RJ model is quite similar for both response functions, in particular if $\mu=0$ is chosen in case of the RJ model (blue dot-dashed lines).
For $\mu=1$ (black full lines), the deviations from a monotonous behavior of the real parts are stronger.
In Fig.\ref{Fig2} the modulus and the phase are shown for the same parameters as in Fig.\ref{Fig1}.
\begin{figure}[h!]
\centering
\includegraphics[width=16cm]{Rot2}
\vspace{-0.5cm}
\caption{upper panels: $X_3(\omega)$ (left) and $X_5(\omega)$ (right) for the models of rotational diffusion (red dashed lines) and rotational random jumps (black full lines ($\mu=1$) and blue dot-dashed lines ($\mu=0$)).
The insets show the relative maximum value of the hump, $\hat X_k=X_k^{\rm max}/X_k(0)$, as a function of $\mu$.
Lower panels: Phase $\vartheta_k(\omega)=\rm acos{(\chi_k^{(k),,}(\omega)/\chi_k^{(k),}(\omega))}$ (in deg.) as a function of frequency.
}
\label{Fig2}
\end{figure}
It can be seen that that in both cases, the third-order and the fifth-order response, the RJ model with $\mu=1$ gives rise to a hump in $X_k(\omega)$.
The insets in the upper panels of Fig.\ref{Fig2} show the relative magnitude $\hat X_k=X_k^{\rm max}/X_k(0)$ of the hump as a function of the parameter $\mu$.
For values $\mu\lesssim1/2$, no hump appears.
For larger values of $\mu$ one observes a clear hump with a maximum at a frequency somewhat smaller than $\omega\tau_1\simeq1$.
This means that the simple model of isotropic rotational random jumps in which each transition completely destroys the orientational correlations yields results similar to the nonlinear response observed for supercooled liquids.
However, the behavior observed in Fig.\ref{Fig2} does not depend on temperature, since all spectral functions are only dependent on the scaled frequency $\omega\tau_1$, cf. the expressions given in the Appendix.
Thus, the experimentally observed decrease of the height of the hump with increasing temperature can only be modelled by changing the parameters of the model, in particular the value of $\mu$.
In Fig.\ref{Fig3} we show $X_k(\omega)$ for the model of anisotropic rotational diffusion for various values of the angle $\Theta$.
\begin{figure}[h!]
\centering
\includegraphics[width=16cm]{Rot3}
\vspace{-0.25cm}
\caption{$X_3(\omega)$ (left) and $X_5(\omega)$ (right) for the models of isotropic rotational diffusion (red dashed lines) and anisotropic rotational diffusion for parameters as indicated in the figure.
Here, $\Delta=D_Z-D_X$.
}
\label{Fig3}
\end{figure}
For $\Theta=0$, the results are identical to those for the IRD model.
The same holds for vanishing 'diffusional anisotropy', $\Delta=0$.
For small values of $\Theta$, there are only minor differences between the results for the two models.
With increasing $\Theta$, a shoulder or a peak at higher frequencies develops depending on the value of $\Delta$.
This is clearly observable for $\Theta=40^o$, where a shoulder is found for $\Delta=10$ and a peak for $\Delta=50$.
This behavior with a varying height of the high-frequency peak is observed up to angles of approximately $\Theta=70^o$ (for $\Delta=10$).
In this regime also the linear susceptibility is composed of two Lorentzian with comparable intensities.
For higher values of $\Theta$ the peak shifts to lower frequencies and a single hump is observed in the moduli $X_3$ and $X_5$ as is most prominently seen for $\Theta=90^o$ in Fig.\ref{Fig3}.
Note that for $\Theta=90^o$, the scaled linear response of the ARD model coincides with the corresponding one for the IRD model.
This does not hold for the nonlinear response functions.
The overall behavior of $X_5(\omega)$ is very similar to the one of $X_3(\omega)$.
In both cases the position of the hump shifts to slightly smaller frequencies with increasing $\Delta$ and at the same time it broadens somewhat.
As an example for the behavior at intermediate angles $\Theta$, we plot $X_3(\omega)$ for $\Theta=60^o$ and various values of the diffusional anisotropy $\Delta$ in Fig.\ref{Fig4}.
\begin{figure}[h!]
\centering
\includegraphics[width=8cm]{Rot4}
\vspace{-0.5cm}
\caption{$X_3(\omega)$ for the model of anisotropic rotational diffusion for $\Theta=60^o$.
The values of $\Delta$ are $\Delta=1,\,5,\,10,\,20,\,50$ in the direction of the arrow.
The inset shows the value of the maximum of the high-frequency peak.
}
\label{Fig4}
\end{figure}
It is obvious how for smaller values of $\Delta$ a shoulder evolves that turns into a secondary peak for larger anisotropy.
The inset in Fig.\ref{Fig4} shows the increase of the height of the secondary peak with increasing $\Delta$.
This behavior is similar to the corresponding one for those values of $\Delta$ for which a hump is observed.
This fact is detailed in Fig.\ref{Fig5}, where we present the maximum height of the hump for $\Theta=90^o$.
\begin{figure}[h!]
\centering
\includegraphics[width=8cm]{Rot5}
\vspace{-0.5cm}
\caption{Maximum height scaled to the value at zero frequency, $\hat X_k=X_k^{\rm max}/X_k(0)$ for $\Theta\!=\!90^o$.
Lower black line: $\hat X_3$, upper blue line: $\hat X_5$ and dashed green line: $(\hat X_3)^2$.
}
\label{Fig5}
\end{figure}
It is apparent that the relative magnitude of $\hat X_5=X_5^{\rm max}/X_5(0)$ is larger than the corresponding third-order quantity.
In green (dashed line), we show the square of the relative amplitude of $X_3$ as one should approximately have $X_5\sim X_3^2$ under some additional assumptions\cite{Albert:2016}.
As for the model of rotational random jumps, also in the present case the results are independent of temperature.
\section*{V. Conclusions}
While linear susceptibilities are related to equilibrium time correlation functions via the FDT and well known relations among the real and imaginary parts of $\chi_1(\omega)$ exist, the situation is different for nonlinear response functions.
In order to learn about the information content of the nonlinear susceptibilities one has to consider the results of explicit model calculations.
As mentioned in the Introduction, a number of such calculations have been performed for different models exhibiting glassy relaxation in the past and in some cases a behavior similar to what is observed in supercooled liquids was obtained.
The interpretations of the hump observed in the nonlinear susceptibilities of glass-forming liquids in terms of growing amorphous order are mainly concerned with the phenomena in the frequency range on the order of the inverse primary relaxation time of the system.
At longer time scales or smaller frequencies it is assumed that the response does not reflect the dynamic heterogeneities but is due to the rotational motion of individual molecules because exchange processes average out the heterogenous nature of the relaxation.
This is the origin of the mentioned 'trivial' contribution to the nonlinear response functions.
Usually, the model of isotropic rotational diffusion (IRD) is used for the computation of the nonlinear response due to this trivial relaxation.
In the present paper, we have also considered the rotational motion of individual molecules in supercooled liquids.
In addition to the IRD model, we considered two models for Brownian rotational motion, namely the model of isotropic rotational random jumps (RJ) and the model of anisotropic rotational diffusion (ARD).
These models are considered because it is known for a very long time, that the IRD model does not give a reasonable description of the rotational motion in supercooled liquids and that models using finite jump angles yield more reliable results.
In addition, most molecules showing a significant glass-forming ability are not adequately described as spherical and the rotational motion might show deviations from isotropy.
For both, the RJ model and the ARD model, we find a hump in the moduli of the nonlinear susceptibilities, $X_3$ and $X_5$, for some values of the model parameters.
This means that also models for the 'trivial' contribution to the response can show features similar to what is observed in experiments on supercooled liquids at some temperature.
A temperature dependent height of the hump can only be modelled via changing relevant model parameters.
We only discussed the intrinsic features of the response functions and did not attempt to model the response typically observed in supercooled liquids.
In order to do so, one would use one of the models for the rotational motion and fit the linear susceptibility using a distribution of relaxation times. The nonlinear response could then be fitted by adjusting the other model parameters like $\mu$ in case of the RJ model and $\Theta$, $\Delta$ for the ARD model.
On the other hand, if one assumes a different model for instance including cooperativity for the primary relaxation and uses the model for the reorientational motion solely for the trivial contribution, one has to be careful when extracting quantities related to the height of the experimentally observed hump in $X_3$ or $X_5$.
The calculations presented here clearly indicate that it is not straightforward to extract informations from nonlinear response functions.
As shown earlier, in some cases model calculations can help to discriminate among different models\cite{G87}.
However, general arguments regarding the detailed behavior of nonlinear susceptibilites are rare and more theoretical effort will be required to obtain conclusive results.
\section*{Acknowledgement}
Useful discussions with Roland B\"ohmer, Gerald Hinze, Francois Ladieu and Jeppe Dyre are gratefully acknowledged.
\begin{appendix}
\section*{Appendix A: The model of isotropic rotational diffusion}
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}
The linear and nonlinear dielectric spectra for the model of isotropic rotational diffusion have been calculated\cite{Dejardin00, Albert:2016} and the corresponding expressions are repeated here for convenience.
The method used in ref.\cite{Dejardin00} is slightly different from the time-dependent perturbation theory that is used in the present approach.
The results, however, agree up to a constant, which is defined indirectly by the definition of the susceptibilities,
cf. eq.(\ref{Chi.om.def}).
The linear response is determined by the expression given in eq.(\ref{Chi1.iso}).
For the cubic response, we consider the $3\omega$-component, which for this model is given by
\be\label{chi33.IRD}
\chi_{3,IRD}^{(3)}(\omega)=-{1\over60}\beta^3M^4S_{3,IRD}(\omega\tau_1)
\;;\;
S_{3,IRD}(x)=\frac{1}{(1-ix)(1-i3x)(3-i2x)}
\ee
Here, $x=\omega\tau_1$ with $\tau_1=1/(2D_R)$.
The $5\omega$-component of the fifth-order response is given by:
\Be\label{chi55.IRD}
\chi_{5,IRD}^{(5)}(\omega)=&&\hspace{-0.6cm}
{1\over1680}\beta^5M^6S_{5,IRD}(\omega\tau_1)
\nonumber
\\
S_{5,IRD}(x)=&&\hspace{-0.6cm}
\frac{4-i5x}{(1-i5x)(3-i4x)(1-i3x)(3-i2x)(1-ix)(2-ix)}
\Ee
\section*{Appendix B: The model of ansotropic rotational diffusion}
\setcounter{equation}{0}
\renewcommand{\theequation}{B.\arabic{equation}}
For anisotropic rotational diffusion, we proceed in the following way.
We start the calculation from the rotational jump model discussed in Section II and perform the limit of small jump angles in the end of the calculations. Furthermore, we use $\mu=1/2$ as this value is the relevant one in the diffusive limit of the master equation.
Technically, this means that in the time dependent perturbation expansion of the propagator ${\bf G}^{(E)}$, the matrix of
$P^{(E)}(\Omega,t|\Omega_0)$, only the terms containing the linear perturbation ${\cal V}^{(1)}$ have to be considered in eq.(8) of
ref.\cite{G87} because all other terms vanish in the diffusive limit.
In the notation used there, this can be written as ${\bf G}^{(n)}={\bf G}\otimes\left[{\cal V}^{(1)}\otimes{\bf G}\right]^n$ with
$n=3$ or $n=5$. The symbol $\otimes$ indicates the convolution
${\bf G}\otimes{\cal V}^{(1)}\otimes{\bf G}\equiv\int_{t_0}^t\!dt'{\bf G}(t,t'){\cal V}^{(1)}(t'){\bf G}(t',t_0)$.
As mentioned in the text, in the case of anisotropic reorientational motions, the results do not only depend on the overall value of the molecular dipole moment, $M$, but also on the orientation of the diffusion tensor relative to the applied electric field, cf. eq.(\ref{Mu.t}), which we write as ($\Omega(t)\equiv\Omega_{DL}(t)$):
\be\label{xi.def}
\langle M(t)\rangle=M\sum_n \xi_n\langle D_{n0}^{(1)}(\Omega(t))\rangle
\quad\mbox{with}\quad \xi_n=D_{0n}^{(1)}(\Omega_{PD}).
\ee
Without going into the details of the lengthy calculations, we simply will present the results in a compact form.
We define the following function:
\Be\label{Y.def}
Y(L_1,M_1;L_2,M_2)&=&\sum_N\xi_N(\Gamma_{L_2,M_2}+\Gamma_{1,N}-\Gamma_{L_1,M_1})(-1)^{M_2}\times
\nonumber\\
&&\hspace{0.5cm}\times
\left(2L_2+1\right)
\left(\begin{matrix}
1 & L_1 & L_2\\
N & M_1 & -M_2
\end{matrix}\right)
\left(\begin{matrix}
1 & L_1 & L_2\\
0 & 0 & 0
\end{matrix}\right)
\Ee
Here,
$\left(\begin{smallmatrix}
1 & L_1 & L_2\\
N & M_1 & -M_2
\end{smallmatrix}\right)$ denotes a Wigner 3-j symbol and the rates are defined by:
\be\label{Gam.lm}
\Gamma_{L,M}=L(L+1)D_X+M^2(D_Z-D_X)
\ee
cf. the discussion in the context of eq.(\ref{F.ARD}).
With this, we find for the cubic response:
\be\label{chi33.ARD}
\chi_{3,ARD}^{(3)}(\omega)={\beta^3M^4\over60}\!\!\!\sum_{m_1,m_2,m_3}\!\!\!(-)^{m_1}\xi_{-m_1}\xi_{m_3}\Gamma_{1,m_3}
Y(1,m_3;2,m_2)Y(2,m_2;1,m_1)G_{m_1,m_2,m_3}(\omega)
\ee
with
\be\label{G3.def}
G_{m_1,m_2,m_3}(\omega)={5\over4}\left[(\Gamma_{1,m_1}-i3\omega)(\Gamma_{2,m_2}-i2\omega)(\Gamma_{1,m_3}-i\omega)\right]^{-1}
\ee
For the fifth-order response, one has:
\Be\label{chi55.ARD}
\chi_{5,ARD}^{(5)}(\omega)={\beta^5M^6\over1680}\!\!\!\sum_{L=1,3}\sum_{m_1\cdots m_5}\!\!\!(-)^{m_1}\xi_{-m_1}\xi_{m_5}\Gamma_{1,m_5}
Y(1,m_5;2,m_4)Y(2,m_4;L,m_3)
\!\!\times
\nonumber\\
&&\hspace{-9.5cm}\times
Y(L,m_3;2,m_2)Y(2,m_2;1,m_1)G_{m_1,m_2,m_3,m_4,m_5}(\omega)
\Ee
\Be\label{G5.def}
&&G_{m_1,m_2,m_3,m_4,m_5}(\omega)=
\nonumber\\
&&\hspace{0.5cm}{35\over16}
\left[(\Gamma_{1,m_1}-i5\omega)(\Gamma_{2,m_2}-i4\omega)(\Gamma_{L,m_3}-i3\omega)(\Gamma_{2,m_4}-i2\omega)(\Gamma_{1,m_5}-i\omega)\right]^{-1}
\Ee
\section*{Appendix C: The model of isotropic rotational random jumps}
\setcounter{equation}{0}
\renewcommand{\theequation}{C.\arabic{equation}}
In this case, one has to consider a ME and one has to fix the value of $\mu$ in eq.(\ref{W.kl.E}) as discussed in the text.
Also in this case, we do not present details of the lengthy calculations and only present the results.
The linear response is given by the same expression as for the model of rotational diffusion, eq.(\ref{Chi1.iso}), with the replacement $\tau_1=1/\Gamma_{RJ}$.
Also the third-order response can be written in a form that is very similar to eq.(\ref{chi33.IRD}). However, the spectral function is quite different and this gives rise to a different behavior.
\be\label{chi33.RJ}
\chi_{3,RJ}^{(3)}(\omega)={1\over60}\beta^3M^4S_{3,RJ}(\omega/\Gamma_{RJ})
\;;\;
S_{3,RJ}(x)={1\over3}\left[I_{3,0}(x)+4\mu I_{3,1}(x)+4\mu^2 I_{3,2}(x)\right]
\ee
with $x=\omega/\Gamma_{RJ}$.
Here, the individual terms are given by:
\Be\label{I3.mu}
I_{3,0}(x)=&&\hspace{-0.6cm}
-\frac{2+i3x}{2(1-ix)(1-i3x)}\nonumber
\\
I_{3,1}(x)=&&\hspace{-0.6cm}
\frac{ix}{(1-i2x)(1-i3x)}
\\
I_{3,2}(x)=&&\hspace{-0.6cm}
\frac{-2x^2+ix}{2(1-ix)(1-i2x)(1-i3x)}\nonumber
\Ee
For the fifth-order susceptibility, we find:
\Be\label{chi55.RJ}
\chi_{5,RJ}^{(5)}(\omega)=&&\hspace{-0.6cm}
{1\over1680}\beta^5M^6S_{5,RJ}(\omega/\Gamma_{RJ})
\nonumber
\\
S_{5,RJ}(x)=&&\hspace{-0.6cm}
{1\over9}\left[{1\over8}I_{5,0}(x)-2\mu I_{5,1}(x)+\mu^2 I_{5,2}(x)-4\mu^3 I_{5,3}(x)+\mu^4 I_{5,4}(x)\right]
\Ee
where the spectral functions are given by:
\Be\label{I5.mu}
I_{5,0}(x)=&&\hspace{-0.6cm}
\frac{16-27x^2+i69x}{(1-ix)(1-i3x)(1-i5x)}
\nonumber\\
I_{5,1}(x)=&&\hspace{-0.6cm}
\frac{40x^2+ix(15+36x^2)}{(1-i2x)(1-i3x)(1-i4x)(1-i5x)}
\nonumber\\
I_{5,2}(x)=&&\hspace{-0.6cm}
\frac{59x^2+144x^4-ix(3+128x^2)}{(1-ix)(1-i2x)(1-i3x)(1-i4x)(1-i5x)}
\\
I_{5,3}(x)=&&\hspace{-0.6cm}
\frac{32x^2+36x^4-ix(3+34x^2)}{(1-ix)(1-i2x)(1-i3x)(1-i4x)(1-i5x)}
\nonumber\\
I_{5,4}(x)=&&\hspace{-0.6cm}
\frac{-57x^2+72x^4-ix(4+226x^2)}{(1-ix)(1-i2x)(1-i3x)(1-i4x)(1-i5x)}
\nonumber
\Ee
For convenience, we give the expression for the particular choice $\mu=1$:
\be\label{S3R.mu1}
S_{3,RJ}^{(\mu=1)}(x)=\frac{-2-6x^2+i13x}{6(1-ix)(1-i2x)(1-i3x)}
\ee
and
\be\label{S5R.mu1}
S_{5,RJ}^{(\mu=1)}(x)=\frac{16-1629x^2+216x^4-ix(227+2070x^2)}{72(1-ix)(1-i2x)(1-i3x)(1-i4x)(1-i5x)}
\ee
\end{appendix}
| {
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**EARLY BIRD BOOKS**
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The Crystal Gryphon
A Witch World Book
Andre Norton
To S.A.G.A.
(Swordsmen and Sorcerers Guild of America)
In recognition of their encouragement
in our chosen field of
Ensorcelling.
Contents
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
1
Here Begins the Adventure of Kerovan, Sometime Lord-Heir in Ulmsdale of High Hallack.
I was one born accursed in two ways. Firstly, my father was Ulric, Lord of Ulmsdale in the north. And of his stock there were told dire tales. My grandfather, Ulm the Horn-Handed, he who led his people into this northside dale and chartered the sea-rovers who founded Ulmsport, had looted one of the places of the Old Ones, taking the treasure sealed within. All men knew that this was no ordinary treasure, for it glowed in the dark. And after that looting not only Ulm, but all those who had been with him on that fearsome venture, were visited by a painful sickness of body from which most of them died.
When I was born, my father was already in middle years. He had taken two ladies before my mother and had of them children. But the children had been either born dead or had quitted this world in their early years, sickly creatures one and all. He had sworn, however, to get him a true heir, and so he set aside his second lady in favor of my mother when it seemed as if he would get no son of her.
My mother's lineage also laid me under a curse. She was the Lady Tephana, daughter to Fortal of Paltendale, which lies farther to the northwest. There are those who even now make off-warding signs at dalesmen from those parts, saying that, when our folk moved thither to settle, there were still Old Ones, seeming like ourselves; and that our people—the Borderers—entered into a blood-mixing with these, the offspring therefrom being not altogether human.
Be that as it may, my father was desperate for an heir. And Tephana, lately widowed, had borne already a goodly child who was now in his second year—Hlymer. My father was willing to forego dowry, to close his ears to any rumor of mixed blood, and to welcome the lady with full honor. By all accounts I have heard, she was willing, even unto risking the curse laid on my father's family by their treasure theft.
My birthing came too early and under strange circumstances, for my mother was on her way to Gunnora's shrine to give offerings for a son and a safe delivery. When she was yet a day's journey away, her pains came on her very swiftly. There was no hall, not even a landsman's dwelling near enough, and a mighty storm was brewing. Thus her women and guards took her for shelter into a place they would have normally shunned, one of those strange and awesome remains of the Old Ones, the people of uncanny power who held the dales in the dim past before the first of our blood wandered up from the south.
This building was in good repair, as is often true of the constructions left by that unknown race. For the Old Ones seem to have used spells to bind stones together in such a way that even time cannot devour them, and thus some buildings look as if they were abandoned only yesterday. What purpose this one might have served none could guess. But there were carvings of men and women, or those who had such seeming, on the inner walls.
My mother's travail was hard, and her ladies feared that they might not save her. After I was born they half-wished that they had failed to, for asking to look upon the babe, she saw me full and gave a great cry, losing her senses and near her wits. She wandered in some mind maze for several weeks thereafter.
I was not as other children. My feet were not with toes, like unto human kind; rather they were small hoofs, split, covered with horn such as make up the nails upon fingers. In my face my eyebrows slanted above eyes that were the color of butter amber, the like of which are not seen in a human countenance. Thus, all gazing upon me knew that, though I seemed far stronger of wind and limb than my unfortunate half-brothers and sisters before me, in me the curse had taken another turning. I did not sicken and die, but thrived and grew.
But my mother would not look upon me, saying I was a demon changeling, implanted in her womb by some evil spell. When those about her brought me nigh, she became so disordered in her wits that they feared her state would be permanent. Soon she declared she had no true child but Hlymer—and later my sister Lisana, born a year after me, a fair little maid with no flaw. In her my mother took much pleasure.
As for me, I was not housed at Ulmsdale Keep, but sent out at nurse to one of the foresters. However, though my mother had so disowned me, my father was moved, not by any affection—for that I was never shown by those closest to me in blood—but rather by his pride of family, to see that my upbringing was equal to my birth. He gave me the name of Kerovan, which was that of a noted warrior of our House, and he saw that I was tutored in arms as became a youngling of station and shield, sending to me one Jago, a keepless man of good birth who had served my Lord as Master of Menie until he was disabled by a bad fall in the mountains.
Jago was a master of the arts of war, not only with the lesser skills that can be battered into any youngling with a strong body and keen eyes, but also those more subtle matters that deal with the ordering of bodies of men great and small. Crippled and tied to a way of living that was only a half-life for a once-active man, he set his brain to labor as he had once ordered his body. Always he searched for new lore of battle, and sometimes at night I would watch him with a strip of smoothed bark before him, patiently setting out in labored and crooked script facts concerning the breaking of sieges, the ordering of assaults, and the like, droning on to me the while, emphasizing this point or that by a fierce dig into the bark with the knife he used for a pen.
Jago was far more widely traveled than most dalesmen, who perhaps in a whole lifetime know little beyond four or five dales outside their own birthplace. He had been overseas in his first youth, traveling with the Sulcar Traders, those dangerous sea-rovers, to such half-fabled lands as Karsten, Alizon, and Estcarp—though of the latter nation he said little, appearing uneasy when I besought him to tell of his travels in detail. All he would say was that it was a land where witch spells and ensorcellment were as common as corn in a field, and that all the women were witches and held themselves better and apart from men, so that it was a place where one kept one's eyes to oneself and walked very quietly and mum-tongued.
There was this which makes me remember Jago well and with gratitude. In his eyes I was apparently like any other youngling, and not a young monster. So when I was with him I could forget my differences from my fellows and rest content.
Thus Jago taught me the arts of war—or rather such as a dale heir should know. For in those days we did not know the meaning of real war, giving that name to our petty skirmishing between rival lords or against the Waste outlaws. And of those we saw many in the long winters when starvation and ill weather drove them against us to plunder our granaries and try to take our warm halls and garths. War wore a far grimmer face in later years, and men got full bellies of it. It was no longer a kind of game which was played by rules, as one moves pieces back and forth across a board on a winter's eve.
But if Jago was my sword tutor, the Wiseman Riwal showed me there were other paths of life in the world. It had always been held that only a woman could learn the ways of healing and perform the spells my people draw upon in their time of need in body and spirit. Thus Riwal was as strange to his fellows as I. He had a great thirst for knowledge, which was in him as a longing for bread might be in a starving man. At times he would go roving, not only in the forest country, but beyond, into the Waste itself. When he returned he would be burdened by a pack like any peddler who carried his own stock in trade.
Being kin to the Head Forester, he had taken without formal leave one of the cots nearby. This he made snug and tight by the work of his own hands, setting above its door a mask carved of stone, not in the likeness of our people. Men looked askance at Riwal, yes—but let any animal ail, or even a man keep his bed in sickness that could not be named—then he was summoned.
About his cot grew all manner of herbs, some of those long-known to every housewife in the dales. But others were brought from afar with masses of soil bundled up about their roots, and he set them out with care. Everything grew for Riwal, and the farmer who had a wish for the best of crops would go cap in hand at sowing time and ask the Wiseman to overlook his land and give advice.
Not only did he bring green life, but he also drew that which wings over our heads or pads on four feet. Birds and animals that were hurt or ailing came to him of their own wills. Or else he would carry them to his place gently and tend them until they were able once more to fend for themselves.
This was enough to set any man apart from his fellows. But it was also well known that Riwal went to the places of the Old Ones, that he tried to search out those secrets our blood had never known. And for that, men did fear him. Yet it was that which drew me to him first.
I was as keen-eared as any child who knows that others talk about him behind their hands. And I had heard the garbled stories of my birth, of that curse which lay upon the blood of Ulm, together with the hint that neither was my mother's House free of the taint of strange mixture. The proof of both was perhaps in my flesh and bone. I had only to look in the mirror of Jago's polished shield to see it for myself.
I went to Riwal, boldly perhaps in outward seeming, but with an inward chill that, young as I was, I fought to master. He was on his knees setting out some plants which had long, thin leaves sharply cut, like the heads of boar spears. He did not look up as I came to him, but rather spoke as if I had already spent the morning in his company.
"Dragon's Tongue, the Wisewomen call this." He had a soft voice with a small tremor, not quite a stammer. "It is said to seek out the putrid matter in unhealing wounds, even as a tongue might lick such hurts clean. We shall see, we shall see. But it is not to speak of plants that you stand here, Kerovan, is it, now?"
"It is not. Men say you know of the Old Ones."
He sat back on his heels to look me eye to eye.
"But not much. We can look and finger, search and study, but of their powers—those we cannot net or trap. One can only hope to brush up a crumb here and there, to speculate, to go on everseeking. They had vast knowledge—of building, of creating, of living—beyond our ken. We do not even know why they were near-gone from High Hallack when the first of our ancestors arrived. We did not push them out—no, already their keeps and temples, their Places of Power were emptied. Here and there, yes, a few lingered. And they may still be found in the Waste and beyond the Waste in that land we have not entered. But the most—they were gone, perhaps long before men, as we know them, arrived. Still—to seek what may still lie here—it is enough to fill a lifetime and yet not find a tenth of a tenth of it!"
In his sunbrowned face his eyes were alight with that same spark I had seen in Jago's when he spoke of a trick of swordplay or a clever ambush. Now Riwal studied me in turn.
"What seek you of the Old Ones?" he asked.
"Knowledge," I answered. "Knowledge of why I am as I am—not man—yet neither—" I hesitated, for my pride would not let me voice what I had heard in whispers.
Riwal nodded. "Knowledge is what every man should seek, and knowledge of himself most of all. But such knowledge I cannot give you. Come."
He arose and started toward his dwelling with his swinging woodsman's stride. Without further question I followed after. So I came into Riwal's treasure house.
I could only stand just within the door and stare at what lay about me, for never before had I seen such a crowding of things, each enough to catch the eye and demand closer attention. For in baskets and nests were wild animals, watching me with bright and wary eyes, yet seeming, in this place, to feel such safety that they did not hide in fear. There were shelves in plenty on the walls. And each length of roughly hewn, hardly smoothed board was crammed with a burden of clay pots, bundles of herbs and roots, and bits and fragments that could only have come from the places of the Old Ones.
There was a bed, and two stools were so crowded upon the hearth that they sat nearly in the fire. The rest of the dwelling was more suited for storage than for living. In the middle of the room Riwal stood with his fists planted upon his hips, his head turning from side to side as if he tried to sight some special thing among the wealth of objects.
I sniffed the air. There was a mingling of many odors. The aromatic scent of herbs warred with the musky smell of animals and the suggestion of cooking from a pot still hanging on the boil-chain in the fireplace. Yet it was not in any way an unclean or disgusting smell.
"You seek the Old Ones—look you here, then!" Riwal gestured to one shelf among the many.
I skirted two baskets with furry inhabitants and came closer to see what he would show me. There I found set-out fragments, one or two being whole, of small figures or masks—bits which in some instances Riwal had fitted together to form broken but recognizable figures.
Whether these indeed represented various beings among the Old Ones, or whether they had had life only in the imagination of their creators, no one might know. But that they had beauty, even when they tended toward the grotesque, I could see for myself.
There was a winged figure of a woman, alas lacking a head; and a man of humanoid proportions, save that from the forehead curled two curved horns. Yet the face below was noble, serene, as if he were a great lord by right of his spirit. There was a figure with webbed feet and hands, plainly meant to suggest a water dweller; and a small one of another woman, or at least a female, with long hair covering most of her body like a cloak. These Rival had managed to restore in part. The rest were fragments: a head, crowned but noseless, the eyes empty pits; a delicate hand that bore an intricate ring of metal on both thumb and forefinger, those rings seemingly a part now of the hand, whose substance was not stone but a material I did not know.
I did not touch; I merely stood and looked. And in me was born a longing to know more of these people. I could understand the never-ending hunger that kept Riwal searching, his patient attempts to restore the broken bits he found that he might see, guess, but perhaps never know—
So Riwal also became my teacher. I went with him to those places shunned by others, to search, to speculate; always hoping that some find might be a key that would open to us the doors of the past, or at least give us a small glimpse into it.
My father made visits to me month by month, and when I was in my tenth year, he spoke to me with authority. It was plain he was in some uneasiness of spirit when he did so. But I was not amazed that he was so open with me, for always he had treated me, not as a child, but as one who had good understanding. Now he was very sober, impressing me that this was of import.
"You are the only living son of my body," he began, almost as if he found it difficult to choose the words he must use. "By all the right of custom you shall sit in the High Seat at Ulmskeep after me." He paused then, so long I ventured to break into his musing, which I knew covered a troubled mind.
"There are those who see it differently." I did not make that a question, for I knew it to be a statement of fact.
He frowned. "Who has been saying so to you?"
"None. This I have guessed for myself."
His frown grew. "You have guessed the truth. I took Hlymer under my protection, as was fitting when his mother became Lady in Ulm. He has no right to be shield-raised to the High Seat at my death. That is for you. But they praise me now to hand-fast Lisana with Rogear, who is cousin-kin to you."
I was quick enough to understand what he would tell me and yet loath to hear it. But I did not hesitate to bring it into the open myself.
"Thus Rogear might claim Ulmsdale by wife-right."
My father's hand went to his sword hilt and clenched there. He rose to his feet and strode back and forth, setting his feet heavily on the earth as if he needed some firm stance against attack.
"It is against custom, but they assault my ears with it day upon day, until I am well-nigh deafened beneath my own roof!"
I knew, with bitterness, that his "they" must be mainly that mother who would not call me son. But of that I did not speak.
He continued. "Therefore I make a marriage for you, Kerovan, an heir's marriage so that all men can see that I do not intend any such offense against you, but give you all right of blood and clan. This tenth day Nolon rides to Ithkrypt, carrying the proxy axe for your wedding. They tell me that the maid Joisan is a likely lass, lacking two years of your age, which is fitting. Safe-married, you cannot be set aside—though your bride will not come to you until perhaps the Year of the Fire Troll."
I counted in my mind—eight years then. I was well content. For marriage had no meaning for me then, save that my father deemed it of such importance. I wondered, but somehow I did not dare at that moment to ask, whether he would tell this Joisan, or her kinsmen who were arranging our match, what manner of lord she would meet on her true bride-day—that I was what I was. Inside I shrank, even in thought, from that meeting. But to a boy of my years that fatal day seemed very far away, and perhaps something might happen to make sure it would never occur.
I did not see Nolon set forth to play my role in axe marriage, for he rode out of Ulmskeep where I did not go. It was only two months later that my father came to me looking less unhappy, to tell me that Nolon had returned, and that I was indeed safely wed to a maid I had never seen, and probably would not see for at least eight more years.
I did not, thereafter, think much of the fact that I had a lady, being well-occupied, with my studies and even more with the quests on which I went with Riwal. Though I was under the guardianship of Jago, he made no protest when I spent time with Riwal. Between those two came to be an odd companionship, in spite of their being so dissimilar in thought and deed.
As the years passed, that stiffness which had come from my tutor's old hurt grew worse, and he found it difficult to face me in open contest with sword or axe. But with the crossbow he was still a skilled marksman. And his reading of maps, his discussion of this or that battle plan, continued. Though I saw little use then for such matters in my own life, I paid him dutiful attention, and that was to be my salvation later.
But Riwal did not appear to age at all, and his long stride still carried him far distances without tiring. I learned early to match his energy. And, while my knowledge of plants was never as great as his, yet I found a kinship with birds and animals. I ceased to hunt for sport. And I took pleasure in the fact that his wild ones did not fear me.
Best of all, however, were our visits to the places of the Old Ones. Riwal prospected further and further over the borders of the Waste, seeking ever to find something intact from the ancient days. His greatest hope, as he confided in me, was to discover some book roll or rune record.
When I suggested that the reading of such could well be beyond his skill, for surely the Old Ones had not our tongue, he nodded in agreement. Still I felt he opposed that thought, sure that if he did find such, the Power itself would aid him to understand it.
It was in the Year of the Spitting Toad that I had been wed. As I came closer to manhood, the thought of that distant lady began now and then to trouble me oddly. There were two lads near my years in the foresters' hold, but from the first they had not been playmates, or later companions. Not only did rank separate us, but they had made me aware, from the beginning of my consciousness of the world about me, that my non-human appearance cut me off from easy friendships. I had given my friendship to only two men—Jago, old enough to be my father, and Riwal, who could have been an older brother (and how I sometimes wished that was the truth!).
But those forester lads went now to the autumn fair with lass-ribbons tied to the upper latches of their jerkins, whispering and laughing about the adventures those led them to. This brought to me the first strong foreboding that when it did at last come time to claim the Lady Joisan in person, she might find me as ill a sight as had my mother. What would happen when my wife came to Ulmsdale and I must go to bide with her? If she turned from me in open loathing—
Nightmares began to haunt my sleep, and Riwal at last spoke to me with the bluntness he could use upon occasion. When he demanded what ill thought rode me, I told him the truth, hoping against hope that he would speedily assure me that I saw monsters where there were only shadows, and that I had nothing to fear—though my good sense and experience argued on the side of disaster.
But he did not give me that reassurance. Instead he was silent for a space, looking down at his hands, which had been busied fitting together some of his image fragments, but now rested quiet on the table.
"There has ever been truth between us, Kerovan," he said at last. "To me who knows you well—above all others would I choose to walk in your company. But how can I promise you that this will turn to happiness? I can only wish you peace and—" he hesitated. "Once I walked a path that I thought might end in hand-fasting and I was happy for a little. But while you bear your differences to others openly, I bear mine within. Still, there they be. And the one with whom I would have shared Cup and Flame—she saw those differences, and they made her uneasy."
"But you were not already wed," I ventured, when he fell silent.
"No, I was not. And I had something else."
"That being?" I was quick to ask.
"This!" He spread out his hands in a gesture to encompass all that was about him under that roof.
"Then I shall have this also," I said. Marry I had, for the sake of custom and my father's peace of mind. What I had seen and heard of marriages among the dale lords did not set happiness high. Heirs and lords married to increase their holdings by a maid's dowry, to get a new heir for the line. If inclination and liking came afterward, that was happiness, but it certainly did not always follow so.
"Perhaps you can." Riwal nodded. "There is something I have long thought on. Perhaps this is the time to do it."
"Follow the road!" I was on my feet, as eager as if he meant to set out upon that beckoning mystery this very moment. For a mystery it was, and beckon it did.
We had come across it on our last venture into the Waste, a road of such building as put any dale's effort to shame, making our roads seem like rough tracks fit only for beasts. The end of the road we had chanced upon was just that, a sharp chopping-off of that carefully laid pavement, with nothing about the end to explain the why-for. The mystery began nearly on our doorstep, for that end point was less than a half day's journey from Riwal's cot. The road ran on back into the Waste, wide, straight, only a little cloaked here and there by the drift of windborne soil. To find its other end was a project we had indeed long held in mind. The suggestion that we set out on this journey quite pushed from my mind the thought of Joisan. She was just a name anyway, and any meeting between us was still years ahead, while the following of the road was here and now!
I was answerable to none but Jago for my actions. And this was the time of year when he made his annual trip to Ulmskeep, where he kept festival with old comrades-in-arms and reported to my father. Thus I was free to follow my own wishes, which in this case meant the road.
2
Here Begins the Adventure of Joisan, Maid of Ithkrypt in Ithdale of High Hallack.
I, Joisan of Ithkrypt, was wed at harvest time in the Year of the Spitting Toad. By rights that was not considered a year for new beginnings; but my uncle, Lord Cyart, had the stars read three times by Dame Lorlias of Norstead Abbey (she who was so learned in such matters that men and women traveled weary leagues to consult her), and her report was that my wedding was written as a thing needful to my own fortune. Not that I was aware of much more than the stir the question caused, for I was thereupon the center of long and tiring ceremonies that brought me close to tears for the very tiredness they laid upon me.
When one has no more than eight years, it is hard to judge what occupies most the thoughts and plans of those in the adult world. I can remember my wedding now mostly as a bright picture in which I had a part I could not understand.
I remember wearing a tabard stiff with gold-thread stitchery that caught up a pattern of fresh-water pearls (for which the streams of Ithdale are rightly famous). But I was more occupied at the time with keeping to Dame Math's stem warning that I must not spot or wrinkle my finery; that I must be prudent at the feast table lest I spill and so mar the handiwork of long and patient hours. The robe beneath was blue, which did not please me over-much as it is a color I do not fancy, liking better the dark, rich shades such as hue the autumn leaves. But blue is for a maiden bride, so it was mine to wear.
My new lord was not present to drink the Life Cup and light the House Candle with me hand to hand. In his place stood a man (seeming ancient to me, for his close-cropped beard was frost-rimmed with silver), as stern as my uncle in his look. His hand, I remember, bore a scar across the knuckles that had left a raised banding of flesh of which I was acutely aware as he clasped my fingers in the ceremony. And in the other hand he held a massive war axe that signified my true lord who was about to twine my destiny with his—though that lord was at least a half-dozen years or more away from being able to raise that axe.
"Lord Kerovan and Lady Joisan!" The guests shouted our names together, the men unsheathing their knives of ceremony so that the torchlight flashed upon the blades, vowing to uphold the truth of this marriage in the future, by virtue of those same blades, if need be. My head had begun to ache with the noise, and my excitement at being allowed to attend a real feast was fast ebbing.
The elderly Lord Nolon, who stood proxy at the wedding, shared a plate with me politely throughout the feast. But, though he asked me with ceremony before making a choice from all offered platters, I was in too much awe of him to say "no" to what I liked not, and his choices were mainly of that nature. So I nibbled at what my taste rebelled against and longed for it to come to an end.
It did, much later, when the women with great merriment laid me, wearing only my fine night shift, in the great, curtained bed. And the men, headed by my uncle, brought in that awesome axe and bedded it beside me as if it were indeed my lord. That was my wedding, though afterward it did not seem too strange, just one of those things difficult for a child to understand, something to be dismissed to the back of one's mind.
Only that axe, which was my partner in place of a flesh-and-blood bridegroom, was a stark prophecy of what was to come—not only to me but to all the country that was my home: High Hallack of the many dales.
After the departure of Lord Nolon, life soon returned to what I had always known, for by custom I would continue to dwell under my birthroof until I was of a suitable age for my lord to claim me.
There were some small changes. On high feast days I sat at the left hand of my uncle and was addressed ceremoniously by my new title of Lady of Ulmsdale. My feast-day tabard also no longer bore only one House symbol, but two, being divided in the center vertically with a ribbon of gold. To the left, the leaping Gryphon of Ulmsdale was worked in beads that glittered like gems. On the right was the familiar Broken Sword of Harb, that mighty warrior who had founded our line in High Hallack and given all his kin fame thereafter when he had defeated the dread Demon of Irr Waste with a broken blade.
On my name-day, or as near to that as travel conditions permitted, would come some gift sent by my Lord Kerovan, together with proper greetings. But Kerovan himself was never real to me.
Also, since my uncle's lady was dead, he looked to his sister Dame Math for the chatelaine's duties in Ithkrypt. She took over the ordering of my days, to secret sighs and stifled rebellion on my part. This and this and this must be learned, that I be a credit to my upbringing when I indeed went to order my lord's household. And those tasks, which grew with my years, induced in me sometimes a desire never to hear of Ulmsdale or its heir; a longing in all my being to be unwed and free. But from Dame Math and her sense of duty I had no escape.
I could not remember my uncle's lady at all. For some reason, though he lacked an heir, he made no move through the years to wed again. Perhaps, I sometimes thought, even he dared not think of lessening in any part Dame Math's authority. That she was an able chatelaine, bringing peace and comfort to all she had dominion over, could not be denied. She kept those about her in quiet, sobriety and good order.
In her long-ago youth (it was almost impossible to think of Dame Math as ever being a maid!) she had been axe-wed in the same fashion as I to a lord of the south. But before he could claim her, the news came that he had died of a wasting fever. Whether she thereafter regretted her loss, no one ever knew. After the interval of mourning she retired to the House of Dames at Norstead, an establishment much-revered for the learning and piety of its ladies. But the death of her brother's lady had occurred before she took vows of perpetual residence, and she had returned to the mistress's role at Ithkrypt. She wore ever the sober robe of Dame, and twice a year journeyed to Norsdale for a period of retreat. As I grew older, she took me with her.
My uncle's heir was still undecided, since he had made no binding declaration. He had a younger sister also—one Islaugha, who had married and had both son and daughter. But since that son was heir to his father's holding, he was provided for.
I was the daughter of his younger half-brother, but not being male, I could not inherit save by direct decree—the which my uncle had not uttered. My dowry was such to attract a husband, and my uncle, should he wish, had also the right—no, even duty, to name that husband heir, but only when he declared it so would it be binding.
I think Dame Math would have liked to see me in the House of Dames, had the marriage with Kerovan not been made. And it is the truth that I did find my visits there pleasant. I was born with an inquiring mind and somehow attracted the notice of Past-Abbess Malwinna. She was very old, but very, very wise. Having talked with me several times, she directed that I be given the right to study in the library of the House. The stories of the past which had always enchanted me were as nothing to the rolls of chronicles and travels, dale histories, and the like, that were on the shelves and in the storage boxes in that room.
But what held me most were the references to the Old Ones, those who had ruled this land before the first of the dalesmen came north. I knew well that such accounts as I found were not only fragmentary, but perhaps also distorted, for the larger numbers of the Old Ones had already withdrawn before our forefathers arrived. Those our ancestors had contact with were lesser beings, or perhaps only shadows, left as one would discard a threadbare cloak.
Some were evil as we judged evil, in that they were enemies to humankind—like the demon Harb had slain. There were still places that were filled with dark enchantment, so that any venturing unwisely into such could be enwebbed. Other such beings could grant prayers and gifts. Such was Gunnora—the Harvest Mother—to whom all women were loyal, and whose mysteries were as great in their way as the Worship of the Cleansing Flame to which the House of Dames was dedicated. I myself wore an amulet of Gunnora—her sheath of wheat entwined with ripened fruit.
Yet others seemed neither good nor ill, being removed from the standards of humankind. At times they manifested themselves capriciously, delivering good to one, evil to another, as if they weighed men on some scales of their own and thereafter dealt with them as they saw fit.
It was chancy to deal with any of the Old Ones save Gunnora. The accounts I found at Norstead were full of instances where humans had awakened from long slumber powers that never should have been disturbed. At times I would seek out Abbess Malwinna in her small garden and ask questions, to which she gave answers if she could. If she could not, she admitted her ignorance frankly. It was on my last such meeting with her that I found her sitting with a bowl upon her knee.
The bowl was of green stone, wrought so finely that the shadow of her fingers about it showed through the substance. It had no ornamentation but its beauty of line, and it was very beautiful indeed. Within was enough wine to cover the bottom and come a bit up the sides.
I knew it was wine, for the heady smell reached me. The warmth of her fingers about it was releasing the scent of the grape. She turned it slowly around and around, so the liquid washed back and forth, but she did not watch it. Instead she looked at me so searchingly that I felt discomfort, as if I had been found wanting in some necessary quality. I searched my conscience hurriedly for any fault I might recently have shown.
"It is long," she said, "since I have tried this, Joisan. But this morning I awoke with the need for doing so, and for you. In my youth I had the gift of farseeing—for gift it is, though some shrink from it. They are afraid of that which they cannot touch, see, taste, hear, or otherwise clearly perceive. It is a gift that cannot be controlled. Few who have it can summon it at will; they must wait until the time it draws them to action. But if you are willing, this day I can use it for you—for how much or how well, that I cannot tell."
I was excited, for of farseeing I had heard. TheWisewomen could use it—or some of them could. But, as the Past-Abbess said, it was not a talent that could be sharpened for use and then put ready to hand like a man's sword or a woman's needle—it must be seized upon when it came, and there was no use in trying to control it. However, with my excitement there was also a tiny chill of fear. It was one thing to read, to listen to, stories of the Power. It was, I understood now, another to see it in action, and for one's own self. Yet at that moment I do not think even panic would have kept me from saying "yes" to her offer.
"Kneel before me, Joisan. Take this bowl within your two hands and hold it level and steady."
I did as she bade, cupping my palms, one on either side of the bowl, holding it as one might hold a firebranch that might be ignited at any moment. Then she leaned forward and touched the fingers of her right hand to my forehead.
"Look upon the wine; think of it as a picture—a picture—" Oddly enough her voice sounded farther and farther away. As I looked down into the bowl, I was no longer seeing only dark liquid. It was rather as if I hung suspended in the air above a wide, borderless expanse of darkness, a giant mirror with none of the brilliance a true mirror possesses.
There came a misting, a change on that surface. Tendrils of the mist became shadow forms. I saw a round ball that glinted and, entombed in that, a form familiar to me—that of a gryphon gleaming white.
At first the ball was very large, near filling the whole of the mirror. Then it shrank swiftly, and I saw it was fastened to a chain. The chain swung from a hand, so that the ball revolved. The gryphon in it sometimes faced me, sometimes faced away. But there grew in me the knowledge that this ball was of great importance.
It was very small now, for the hand that dangled it was also shrinking. The arm to which it was attached, and then the body belonging to the arm, appeared. Now a man stood there. His face was turned from me, hidden. He wore war mail, the hood drawn up about his throat. There was a battle sword girded to him, and over his shoulder I saw the arch of a crossbow. But he wore no House tabard, nothing to identify him, only that swinging ball. Then he left, tramping away as if he had been summoned elsewhere. The mirror was dark and empty; nor did any more shadows gather there.
Malwinna's hand fell from my forehead. As I raised my eyes to blink and blink again, I saw a woeful pallor on her face. So I quickly set aside the bowl and dared to take her hands within mine, striving to help her.
She smiled weakly. "It draws the strength—the more when one has little strength left. But it was laid on me to do this thing. Tell me, my daughter, what did you learn?"
"You did not see it, then?" I was surprised.
"No. It was not a farseeing for me, that I knew. It was yours only."
I told her what I had seen: the gryphon englobed and a man in battle dress holding it. And I ended, "The gryphon is the badge of the House of Ulm. Did I then see the Lord Kerovan to whom I am wed?"
"That may be so," she agreed. "But it is in my mind that the gryphon is that which is of the greatest importance to your future. If such ever comes to your hand, my daughter, do you guard it well. For it is also to be believed that this is a thing of the Old Ones and a focus of some power they once knew. Now, call Dame Alousan, for I have need of one of her strengthening cordials. But speak not of what we have done here this morning, for farseeing is a private thing and not to be talked of lightly."
I said naught to any of the Dames, nor to Math. And the Past-Abbess allowed them to believe that she was merely a little wearied, so they fussed about her, for she was greatly loved. No one paid any attention to me. I had taken the bowl with me into the guesting room and put it on the table there.
Though I continued to look into it now and again I saw nothing but the wine; no dark mirror, no shadows moving. Yet in my mind was so vivid a picture of the crystal gryphon that I could have painted it, had I any skill in limning, in every small detail. And I speculated as to what it might mean. The gryphon so enclosed had differences from the one that appeared as Ulm's badge. A gryphon by rights had the wings and forepart of an eagle: its front legs end in a bird of prey's strong talons. But the rear, the tail, the hind paws are those of a lion, one of the beasts known to the south alone. On its bird's head a lion's ears stand upright.
In the ancient learning the gryphon symbolizes gold: the warmth and majesty of the sun. Ofttimes in legends it is the guardian of hidden treasure.
Thus the gryphon is mainly pictured in red and gold, which are sun colors. Yet the one enclosed in the globe was the white of ice—a white gryphon.
Shortly after that farseeing, Dame Math and I returned home to Ithkrypt. But we did not remain there long. For in this Year of the Crowned Swan I had reached the age of fourteen, and Dame Math was already preparing my bride clothes and the furnishings I would take with me when Kerovan would send for me, as was the custom, in the next year or two.
So we went on to Trevamper, that town set at the meeting of highway and river where all merchants in the north show their wares upon occasion. Even the Sulcarmen, who are searovers and seldom come far from wind and wave, travel to Trevamper. For there is the interior trade. And by chance we met there also my Aunt Islaugha, her son Toross, and her daughter Yngilda.
She came to pay a call on Dame Math, but I felt it was one of duty only and there was little liking between these sisters. However the Lady Islaugha presented a smiling face and spoke us fair, congratulating me on the fine marriage that had united me to the House of Ulm.
Yngilda pushed closer to me when our elders had turned their attention back to their own concerns, and I thought she stared rudely. She was a stout girl, bundled in rich clothing down which her braids rippled, their ends bound in ribbons hung with little silver bells meant to chime sweetly as she moved. Such a conceit did not suit her broad, flatfish face, with its too-small mouth always pursed a little as if she chewed upon a spicy secret she debated over sharing.
"You have seen the likeness of your lord?" she asked almost abruptly.
I stirred uneasily under the probing of her eyes. I knew her then for unfriend, though why she should be so when we hardly knew each other, I could not guess.
"No." As always when such uneasiness with others was in me, I was wary. But the truth is better than any evasion which may later trip one up. And for the first time I wondered.a little at a matter I had never considered before. Why had Kerovan not caused to be sent a likeness of himself? That such was done in axe marriages I knew.
"A pity." Her gaze seemed to have some manner of triumph in it now. "Look you here—this is my promised lord, Elvan of Rishdale." She brought out of her belt pocket an oblong of wood with a face painted on it. "He sent it with his bride gift two years ago."
The painted face was that of a man of middle years, no boy. And it was not a pleasant countenance to my thinking, but perhaps the limner had either not been skillful or had some reason not to flatter this Elvan. That Yngilda was proud of it was plain.
"He would seem a man of authority." I did the best I could in way of praise. My disliking for the pictured face grew stronger the longer I regarded it.
She took that, as I had hoped, as a compliment to her promised lord.
"Rishdale is an upper dale. They are wool people, and the trade is rich. Already my lord has sent me this and this—" She patted an amber necklace which lay above her tabard and thrust her hand out to me that I might look upon a massive thumb ring of a serpent with eyes that were flecks of red gem-fire.
"The serpent is his House badge. This is his own ring, sent for a welcome gift. I go to him next harvest time."
"I wish you happy," I answered.
Her pale tongue swept out over her lower lip. Again she was in two minds over some speech to make. At last she brought herself to it, bending her head even closer, while I had all I could do not to withdraw at her approach, for her close company did not please me.
"I would I could say the same to you, kinswoman."
I knew I should not encourage her now, yet something made me ask, "And why not, kinswoman?"
"We are not so far from Ulmsdale as you. We have heard—much." And she strove to give such a dire accent to that last word that she did indeed make an impression on me. For all my prudence and distrust, I could not now deny her this confidence.
"Much of what, kinswoman?" My tone made a challenge of that, one she was quick to note and that pleased her, I am sure.
"Of the curse, kinswoman. Did they not tell you that the Heir of Ulmsdale lies under a double cursing? Why, his own mother has refused to look upon his face since his birth hour. Have they not told you that?" she repeated with open relish. "Alack, that I should spoil your dreaming about a brave young lord. He is a monster thing, they say, sent to live apart because all men shrink from—"
"Yngilda!" That saying of her name was as sharp as a whip crack, and under it she flinched as if indeed some lash had bitten into her body. Dame Math stood over us, and it was plain in her face she had heard those words.
So open was her wrath that at that moment I knew Yngilda had indeed spoken the truth, or at least come so close to it as to shake my guardian. Only the truth could have aroused her ire so greatly.
She said no more, only eyed Yngilda menacingly until the girl edged back, her full cheeks blanching a little in her fright. She gave a kind of squeak and scrambled away. But I sat where I was and met Dame Math eye to eye. Within me the cold grew, setting me to shivering.
Cursed—a monster whom even his mother could not bear to look upon! By the Heart of Gunnora, what had they done to me, to give me in marriage to that? I could have screamed my terror aloud, but I did not. For in that much I kept my control. I only said slowly, forcing my voice to be level, determined to know the full of it here and now, "By the oath of the Flame you serve, Lady, tell me now the truth. Are her words that truth? Am I wed to one who is not like other men?" For I could not bring myself to say "monster."
I think up until that moment Dame Math might have covered with fair words. But now she sat beside me, her face grave, as the flush of anger faded.
"You are no longer a child, Joisan. Yes, I will give you what truth I know. It is true that Kerovan dwells apart from his kin, but he is not a monster. There is a curse laid on those of the House of Ulm, and his mother comes from the updates, from a family rumored to have inter-wed with Old Ones. Thus he has such blood within him. But he is not monstrous—of this Lord Cyart made sure before he would consent to the marriage.
"Yet he dwells apart from his kin. Is it true that his mother will not look upon him?" The cold within me was such now I could hardly control myself.
Still she was frank with me. "That is true because of the manner of his birthing, and she is a fool!" Then she told me an unusual tale of how the Lord of Ulm had taken wives and had no living heir because of the curse. How he wed a third time with a widow, and how she had been taken on the road before her time with birth pains and had borne her son within the walls of one of the Old Ones' buildings. And of how thereafter she had turned her face from him because she was so filled with fear that the babe was of the Old Ones' sending. But he was sound and no monster. His father swore to that by the Great Oath for which there can be no breaking.
Because she told it all so plainly, I believed her and was less shaken.
Then Dame Math added, "Joisan, be glad that you take a young lord. Yngilda, for all her prating, goes to one already wed once, a man old enough to be her father, and one who will have little patience with any youthful follies. She will find him far less indulgent to her whims and laziness than her mother, and she will perhaps rue the day she left her own keep for his.
"Kerovan by all accounts is one you will well company with—for he is learned in rune scrolls as well as in sword-play, which so occupies the minds and bodies of most men. He has a liking for searching out old things, such as you have also. Yes, you have much to think right in your wedding, and little to see of shadows. You are a maid of good mind and not easily shaken. Do not let the envious words of this foolish wench overset your reason. I swear, if you wish it, by the Flame—and you well know the meaning of such an oath for me—that I would not stand by without protest and see you wed to any monster!"
Knowing Dame Math, that reassurance was indeed all I needed. Yet during the days that followed I did think again and again of the strange upbringing Kerovan must have had. That a mother had turned her face from her child was hard to believe. Still, giving birth in a place of the Old Ones might have poisoned her mind against the cause of her pain and fear as she lay therein. And I knew well from my reading at the Abbey that many such places had malignant atmospheres that worked subtly upon mankind. She could well have fallen prey to such influences during her hours of labor.
For the rest of our stay in town my aunt and her daughter did not come near us. Perhaps Dame Math had made plain her views on what Yngilda had told me. I was well content not to see her full face, her pursed mouth, and her probing eyes again.
3
Kerovan
To most dalesmen the Waste is a fearsome place. Outlawed men were driven to refuge there, perhaps coming to regard it in time as they had their native dales. And there are hunters, wild as any outlaws in their own fashion, ranging it to bring back packloads of strange furs as well as lumps of pure metal congealed into odd shapes: not native ores, but substances that had been worked and then reduced to broken pieces.
Such lumps of metal were greatly prized, though smiths had to rework them with care. Swords and mail made from this metal were stronger, more resistant to weathering. On the other hand, sometimes it had fearsome properties, exploding in vast configurations to consume all nearby—as if some power had struck it. A metal-smith both yearned to use it for the promise of fine craftsmanship and feared that each piece he brought to the forge might be one of the cursed bits.
Those who found such metal and traded in it were notoriously close-mouthed about the source. Riwal believed that they mined, not the earth, but places of the Old Ones wherein some ancient and unbelievably horrible conflict had fused metal into these lumps. He had attempted to win the confidence of one Hagon, a trader, who had twice passed through our forest territory. But Hagon refused to talk.
So it was not only the broken-off road that beckoned us. There were other secrets to be uncovered. And I found this venture well to my liking.
We reached the broken-off end of the road by mid-morning and stood studying it before we set foot on its earth-drifted surface. It was indeed a puzzle, for that break was as clean-cut as if some giant swordsman had brought down his blade to sever the masonry. Yet, if some such action had occurred, where was the rest? For beyond the break there was not even a trace of old rubble to suggest it had ever run beyond this point. And why would any road come to such a purposeless ending? It may be true that the purposes of the Old Ones were not the same as those of men, and we cannot judge their actions by ours.
"How long ago since men walked here, Riwal?" I asked.
He shrugged. "Who knows? If it were men who did so. But if the road ends thus, the beginning may be of more interest."
We were riding the small, desert-bred horses used by Waste rovers, tough beasts with an inherited ability to go far on a minimum of drink and forage. And we led a third horse with our supplies in a pack. We went clothed as metal traders, so that any spying upon us could believe we were of the Waste ourselves. We traveled alert to sign and sound, for only he who is ever-watchful can hope to best the traps and dangers of such a land.
The Waste is not pure desert, though much is arid land with a scant covering of small, wind-beaten shrubs and sun-dried grass in ragged clumps. At times, dark copses of trees grow so thick they huddle trunk to trunk. And outcrops of stone stand like pillars.
Some of these had been worked, if not by man, then by creatures who used stone for monuments. But the pillars had been so scoured by years of winds that only traces of the working remained. Here a wall could be seen for a bit; there a pair of columns suggested a past building of some pride.
We passed such a place soon after we took to the road, but there was not enough left to explore. In the open there was silence, for this was a windless day. The clop-clop of our mounts' feet on the pavement seemed to echo, making far too loud a sound, so that I found myself looking from side to side, and now and then over my shoulder. The feeling grew stronger that we were being watched—by outlaws?
In spite of myself I found my hand straying ever in the direction of the sword hilt, ready to defend against attack. Yet when I glanced at Riwal, I saw him riding easy, though he also watched right and left.
"I feel"—I urged my mount closer to his—"that we are watched." Perhaps I humbled my pride to admit that, yet this was more his land than mine, and I relied on him.
"It is ever so—in the Waste," he returned.
"Outlaws?" My fingers closed about the hilt now.
"Perhaps. But more likely other things." His eyes did not quite meet mine, and I sensed he was at a loss to explain. Perhaps he, too, feared to display some weakness before me, a younger and less-tried venturer.
"Is it the truth then that the Old Ones left guardians?"
"What man among us knows?" He countered my question with another. "This much is so: when one ventures into their ways, there is often this feeling of being watched. Yet it has never been with me more than just watching. If they left guardians, as you say, those are now too old and tired to do more than watch."
I found that hardly reassuring. And still I continued to watch—though nothing stirred out in that flat land across which the road hammered a straight and level path.
At nooning we drew to the side of the pavement, ate and drank, and gave our horses to drink also from the water skins we carried. There was no sun, and the sky over us was gray; still I could see no clouds gathering to threaten storm. But Riwal sniffed the air, his head up to the sky.
"We must seek shelter," he said, and there was urgency in his voice.
"I see no storm clouds."
"Storms come unheralded and swiftly in the Waste. There—" He had been surveying the countryside around, and now he pointed ahead to where there was a pile beside the road, perhaps another cluster of time-eroded ruin.
We pushed on, to discover that sight-distance was deceptive in this place. There was a haze that seemed to rise from the ground so that things appeared closer than they were. But at length we reached the spot he had appointed. And none too soon, for the sky was no longer the gray of a gloomy day, but had darkened now into twilight come hours too soon.
Chance had brought us to shelter. Though the ruins at the outset of the road had been so formless as to only suggest they had once had purpose, this ancient building was in better preservation. There was actually part of a room or hall among the jumble of stone blocks with a portion of roof over it. And into that we crowded both ourselves and our animals.
Now the wind blew, whirling up the grit, hurling it in marching columns to fill eyes, mouths, nostrils. We had to fight to gain the last few strides to cover. Once inside, when we turned to look out, it was to see a curtain of dust.
That did not last long. Overhead sounded the rumble of thunder as if an army with a siege train marched. And the lash of lightning followed with force enough to suggest it had struck not too far away. Then came rain—quickly beating down the dust, yet not clearing any path for our vision; rather providing a second curtain, this time of moisture, not grit.
Water ran in a stream across the pitted floor, so we crowded back into the farthest comer of the ruin. The horses whinnied and snuffled, rolling their eyes, as if they found this fury of nature frightening. But to me our corner gave an illusion of shelter, though I flinched when the lightning struck again.
Such fury deafened us. We were reduced to the point of simple endurance and we kept hold of the reins, lest our mounts break out into the storm. As they began to quiet, no longer tossing their heads or stamping, I relaxed a little.
The dark was close to that of true night, and we had no torch. So crowded were we that Riwal's shoulder rubbed mine whenever he moved even slightly, yet the rain was so tumultuous we could not have heard each other without shouting, which we did not do.
What had been the original purpose of the ruin? Built so beside the road, could it have been an inn? Or was it a guard post for some patrol? Or even a temple? As Riwal had said, who knew the purposes of the Old Ones.
With one hand I explored the wall. The surface of the stone was smooth, not pitted as the more exposed portions were. My fingers could detect no seam or joining, yet those blocks had been set together somehow. Suddenly—
Men sleep and dream. But I will swear any oath I did not sleep. And if I dreamed, then it was unlike any dream I had ever known.
I looked out upon the road, and there were those moving along it. Yet when I tried to see them through what appeared to be a mist, I could not. They remained but shapes, approximating men. Could they be men?
Though I could not see them clearly, their emotion flowed to me. They were all moving in one direction, and this was a retreat. There was a vast and overwhelming feeling of—no, it was not defeat, not as if some enemy had pressed them into this withdrawal, but rather that circumstances were against them. They seemed to long for what they left behind, with the longing of those torn from deep rooting.
Now I knew that they were not all alike or of one kind. Some as they passed gave to me their sense of regret, or loss, as clearly as if they had shouted it aloud in words I could understand. But others were less able to communicate in this fashion, though their emotions were none the less deep.
The main press of that strange and ghostly company was past. Now there was only a handful of stragglers, or of those who found it the hardest to leave. Did I or did I not hear the sound of weeping through the rain? If they did not weep in fact they wept in thought, and their sorrow tore at me so I could not look at them any longer, but covered my eyes with my hands and felt on my dusty cheeks tears of my own to match theirs.
"Kerovan!"
The shadow people were gone. And so was the force of the storm. Riwal's hand was heavy on my shoulder, as if he shook me awake from sleep.
"Kerovan!" There was a sharp demand in his voice, and I blinked at what I could see of him in the dusk.
"What is the matter?"
"You—you were crying out. What happened to you?"
I told him of the shadow people withdrawing in their sorrow.
"Perhaps you have the sight," he said gravely when I had done. "For that might well have happened when the Old Ones left this land. Have you ever tried farseeing or tested a talent for the Power?"
"Not I!" I was determined that I would not be cut off from my fellows by a second burden. Different I might be in body because of the curse laid on me before my birth, but I needed not add to that difference by striving to follow those paths trod by Wisewomen and a few men such as Riwal. And he did not urge me, after my quick denial. Such a way must be followed by one wholly willing; not by one led into it by another. It has its disciplines that are in some ways more severe than any warrior training, and its own laws.
After the storm the day lightened again, and we were able to set out at a brisk pace. The water still settled in pools and hollows, and we refilled our smaller water bag, letting the horses drink their fill before we moved on.
I wondered, when we rode that way, if I would have the sensation of the company of those I had seen in the vision or dream. But that was not so. And shortly I forgot the intensity of the emotion that I had shared with them. For that I was thankful.
The road, which had run so straight, made a wide curve heading toward the north and the greater unknown of the Waste. Now ahead we caught sight of heights making a dark blue line across the sky of evening, as if we headed for a mountain chain.
Here also the land was more hospitable. There were trees where before had been mostly shrubs and stretches of grassland. We came to where the road arched in a bridge over a stream of some size. And it was beside that running water that we camped for the night. In fact Riwal settled us, not on the bank of the stream, but on a bar which thrust out into it. The water was high from the storm, and there was flotsam carried with it, piled around the rocks edging that bar.
I eyed his choice with some disfavor. To my mind he had deliberately selected a site which would give us little room and which appeared dangerous from the sweep of water. He must have read my expression for he said, "This is chancy land, Kerovan. It is best to take the common precautions when within it—some uncommon ones too."
"Common precautions?"
He gestured at the stream. "Running water. That which is ill-disposed to us, if it be of the Power and not human, cannot cross running water. If we camp so, we have only one front to defend."
So reasoned, it was common sense. Thus I pushed rocks and pulled loose drift to clear a space between for us and the horses. Nor did Riwal deny us a fire made from the driest of the drift. The river was falling, but the current was still swift. It held life also, for I saw a dark shape of a length to suggest that the fish of this country were of a huge size, though I was teased by the disturbing suggestion that that shadow beneath the surface did not altogether resemble any fish I knew. I decided that in the Waste it was better not to probe too deeply into the unknown.
We set a watch, as we would in enemy country. At first, during my tour of duty, I was so uneasily alert that I found myself peopling each shadow with an intruder, until I took my fancies in hand and forced control over them.
Though the day had been sunless, we did not lack a moon. Its rays were particularly strong, making the landscape all black and silver—silver in the open, black in the shadows. There was life out there, for once I heard the drum of hoofs, and our horses nickered and tugged at their tethers, as if some of their wild kin had pounded by. Once I heard a distant, mournful cry, like the howl of a hunting wolf. And something very large with wings planed noiselessly over our camp as if to inspect us. Yet none of these were frightening in themselves, for all men know that there are wild horses in the Waste, and wolves run through the dales as well. And there are winged night-hunters everywhere.
No, it was not those sounds that disturbed me. It was what I did not hear. For I was as certain as if I could see it that out there in the black and silver land lurked something, or someone, who watched and listened with the same intensity that I did. And whether it was of good or ill I could not guess.
Sun and morning banished such fancies. The land was open, empty, in the daylight. We crossed the hump of the bridge and headed on, while before us the mountains grew sharper to the sight.
By nooning we were in the foothills, which were ridges sharper than our dales, more like knife slashes in soil and rock. No longer was the road straight. It narrowed to a way along which two of us might still ride abreast, but no wider, and it twisted and curved, ran up and down, as if its makers had followed always the easiest route through this maze of heights. Here, too, the Old Ones had left their mark. Carved on the walls of rock were faces, some grotesque, some human-seeming and benign, and often bands of runes that Riwal busied himself to copy.
Though no one could read the script of the Old Ones, Riwal had hopes that someday he would be able to do so. We had dawdled so while he copied the runes that noon found us in a narrow vale where we took our rest under the chin of a vast face that protruded strongly from the parent cliff of which it had been carved.
I had studied it as we came up, finding it in something vaguely familiar, though what that was I could not say.
Oddly enough, though we were here surrounded by the work of those who had vanished, I felt free of that watching, as if whatever had been here once was long gone and had left no trace. And my spirits rose as they had not since the storm caught us.
"Why all these carvings?" I wondered. "The farther we go, the more they are clustered on the walls."
Riwal swallowed a mouthful of travel bread in order to answer. "Perhaps we now approach some place of importance; a shrine, even a city. I have gathered and sifted the stories of traders for years, yet I know of none who have come this way, into the foothills of the mountains."
That he was excited I could see, and I knew that he anticipated some discovery that would be far greater than any he had made during his years of wandering in the Waste. He did not linger over his food, nor did I, for his enthusiasm grew to be mine also. We did not pause beneath that giant chin for long, but rode on.
The road continued to weave through the foothills, and the carvings grew more complex. There were no more heads or faces. Now runes ran in complex patterns of lines and circles. Riwal reined in before one.
"The Great Star!" His awe was plain to see.
Surveying the complexity of that design, I could at last make out a basic five-point star. But the star was overlaid with a wealth of other curves and bits, so it took careful examination to make it out at all.
"The Great Star?" I asked.
Riwal had dismounted and gone to the rock face in which that pattern was so deeply chiseled, running his fingers along the lines as far up as he could reach, as if he wished to assure himself by touch that what his eyes reported was true.
"It is a way, that much we know, of calling upon one of the highest of the Powers," he said, "though all save the design has been lost to us. Never before have I seen it in so complex a setting. I must make a drawing of this!"
Straightway he brought out his horn of ink, tight-capped for journeying; his pen; and a fresh piece of parchment on which he began to copy the design. So lost was he in the task that I grew restless. At last I felt I could no longer just sit and watch his slow stroke upon stroke as he studied each part of the design to set it down.
"I shall ride on a little," I told him. He grunted some answer, intent upon his labors.
Ride on I did, and the road took a last turning—to the end!
Before me a flat rock face bore no sign of any gateway or door. The pavement ended flush with that cliff. I stared in disbelief at such an abrupt and seemingly meaningless finish to our quest. A road that began nowhere and ended thus—? What had led to its making? What could its purpose have been?
I dismounted and went to run my fingertips along the surface of the cliff. It was real, solid rock—die road ran to it and ended. I swung first to one side and then to the other, beyond the boundaries of the pavement, seeking some continuation, some reason. There were two pillars standing, one on either hand, as if they guarded some portal. But the portal did not exist!
I advanced to lay hand upon the left pillar, and, as I did so, at its foot I caught a glimpse of something. It was a faint glimmer, near-buried in the gravel. Straightway I was on my knees, using first my fingers and then the point of my knife, to loosen my find from a crack in which it had been half-buried.
The gleaming object I had cupped in my hand was a strange find. It was a ball, a small globe of crystal, a substance one might have thought would have been shattered among these harsh rocks long since. Yet it did not even bear a scratch upon its smooth surface.
Within it was a tiny image, so well-wrought as to be the masterpiece of some gem-cutter's art—the image of a gryphon, the beast that was my own House symbol. The creature had been posed with one eagle-clawed foot raised, its beak open as if it were about to utter some word of wisdom to which it bade me listen. Set in the globe directly above its head was a twisted loop of gold, as if it had once been so linked to a chain for wearing.
As I stood with it cupped in my hand, the glimmer of light that had led me to its discovery grew stronger. And I will swear that the crystal itself became warm, but only with such warmth as was pleasing.
I held it on the palm of my hand, level with my eyes, that I might study the gryphon closely. Now I could see that there were small flecks of crimson in the head to mark the eyes. And those flecks sparkled, even though there was no outer light to reflect within them, almost as if they had life of their own.
Long had I been familiar with all the broken bits on Riwal's shelves, but never before had such a thing been found intact—save for the brokers loop at the top, and that, I saw, could be easily repaired. Perhaps I should offer it to Riwal. And yet as I felt its warmth against my flesh, saw the gryphon's stance of wisdom and warning within, I had the belief that this was meant for me alone and that its finding was not by mere chance but by the workings of some purpose beyond my knowledge. If it were true that my mother's House had intermated with Old Ones, then it could well be that some small portion of such blood in my own veins made me find the crystal globe familiar and pleasant.
I took it back to Riwal. When he saw it, there was vast amazement on his face.
"A treasure—and truly yours," he said slowly, as if he wished what he said were not so.
"I found it—but we share equally." I made myself be fair.
He shook his head. "Not this. Is it mere chance that brings a gryphon to one who wears that badge already?" Reaching forth, he touched the left breast of the jerkin above my mail, on which was discreetly set the small gryphon head I always wore.
He would not even take the globe into his hand, though he bent his head to study it closely.
"This is a thing of Power," he said at last. "Do you not feel the life in it?"
That I did. The warmth and well-being that spread from it was a fact I could not deny.
"It will have many uses." His voice was low, and I saw that his eyes were now closed, so he was not viewing it at all. "It shall bind when the need is for binding; it shall open a door where there is want of a key; it shall be your fate, to lead you into strange places."
Though he had never said he could farsee, in that moment I knew that he was gripped by a compelling force which enabled him to envision the future uses of the thing I had found. I wrapped it within a scrap of his parchment and stowed it against my flesh within my mail for the greatest safety that I could give it.
About the bare cliff Riwal was as puzzled as I. All the signs suggested a portal of some importance, yet there was no portal. And we had, in the end, to be content with what we had discovered and to begin the trek back from the Waste.
Never during that journey did Riwal ask to see the gryphon again, nor did I bring it forth. Yet there was no moment during the return that I was not aware of what I carried. And the two nights that we lay encamped on the return road, I had strange dreams, of which I could remember very little save that they left an urgency upon me to return to the only home I had ever known, because before me lay a task of importance.
4
Joisan
Though I had little liking for Yngilda, I found her brother Toross unlike her. In the autumn of that year, soon after we returned to Ithkrypt, he came riding over the hills with a small escort, their swords all scabbarded with peace-strings, ready to take part in the fall hunt that would fill our winter larder after the kills were salted down.
Differing from his sister in body as well as in mind, he was a slender, well-set youth, his hair more red than the usual bronze of a dalesman. He possessed a quick wit and a gift of song that he used to advantage in the hall at night.
I heard Dame Math say to one of her women that that one, meaning Toross, could well carry a water horn through life to collect the tears of maids sighing after him. Yet he did nothing to provoke such admiration; never courted their notice, being as ready in riding and practice of arms as any of the men, and well-accepted by them.
But to me he was a friend such as I had not found before. He taught me the words of many songs and how to finger his own knee-harp. Now and then he would bring me a branch of brilliant leaves clipped at their autumn splendor, or some like trifle to delight the eye.
Not that he had much time for such pleasures, for this was a bustling time when there was much to be done for the ordering of supplies against the coming of cold days. We stewed some fruits and set them in jars with parchment tied firmly over the mouths; dried other such; brought forth heavy clothing and inspected it for the need of repairs.
More and more of this Dame Math left to my ordering, as she said that now I was so nigh in years to becoming the lady of my lord's household I must have the experience of such ways. I made mistakes, but I also learned much, because I had no mind to be shamed before strangers in another keep. And I felt more than a little pride when my uncle would notice with approval some dish of my contriving. He had a sweet tooth, and rose and violet sugars spun artfully into flowers were to him an amusing conceit with which to end a meal, and one of my greater triumphs.
Though I busied myself so by day, and even a little by lamplight in the evening when we dealt with the clothing, yet I could not altogether thrust out of mind some of the thoughts Yngilda had left with me. Thus I did something in secret that otherwise only a much younger maid would have thought on.
There was a well to the west in the dale that had a story about it—that if one went there when the full moon was reflected on its water surface and cast in a pin, then luck would follow. Thus, not quite believing, yet still drawn by some small hope that perhaps there was luck to be gained by this device, I stole away at moonrise (which was no small task in itself) and cut across the newly harvested fields to the well.
The night was chill, and I pulled high the hood of my cloak. Then I stood looking down at the silvered reflections in the water and I held out my pin, ready to drop it into the disk mirrored there. However, before I released it, the reflection appeared to shiver and change into something else. For a long moment I was sure that what I had seen there had been far different from the moon, more like a crystal ball. I must have dropped the pin without being aware, for suddenly there was a troubling of the water, and the vision, if vision it had been, was gone.
I was so startled that I forgot the small spell-rhyme I should have spoken at that moment. So my luck-bringing was for naught, and I laughed at my own action as I turned and ran from the well.
That there is ensorcellment and spell-laying in the world we all know. There are the Wisewomen who are learned in such, as well as others, such as the Past-Abbess, who have control over powers most men do not understand. One can evoke some of these powers if one has the gift and the training, but I had neither. Perhaps it is better not to dabble in such matters, Only—why at that moment had I seen again (if I had in truth seen it) that englobed gryphon?
Gryphon—beneath the folds of my cloak my fingers sought and found the outlines of that beast as it was stitched upon my tabard. It was the symbol of the House of Ulm, to which I was now bound by solemn oath. What was he like, my thoughts spun on, this unseen, unknown lord of mine? Why had he never sent to me such a likeness of himself as Yngilda carried? Monster—Yngilda had no reason to speak spiteful lies, there must lie some core of truth in what she had said to me. There was one way—
Gifts came yearly from Ulmsdale on my name-day. Suppose that when they were brought this year, I sought out the leader of the party bringing them, asked of him a boon to be carried to his lord: that we exchange our likenesses. I had my own picture, limned by uncle's scribe, who had such a talent. Yes, that was what I would do!
It seemed to me in that moment that perhaps the well had answered me by putting that thought into my head. So I sped, content, back to the hall, pleased that none there had marked my absence.
Now I set to work upon a project of my own. That was making a suitable case for the picture drawn on parchment. As deftly as I could, I mounted it on a piece of polished wood.
For it I then worked a small bag, the fore-part embroidered with the gryphon, the back with the broken sword. I hoped my lord could understand my subtle meaning: that I was dutifully looking forward to Ulmsdale; that Ithkrypt was my past, not my future. This I did in secret and in stolen moments of time, for I had no mind to let others know my plan. But I had no time to hide it one late afternoon when Toross came upon me without warning.
The mounted picture lay before me in the open, as I had been using it to measure. When he saw it he said sharply, "There is one here, kinswoman, who sees you well as you are. Whose hand limned this?"
"Archan, my uncle's scribe."
"And for whom have you had it limned?"
Again there was that sharp note in his voice, as if he had a right to demand such an answer from me. I was more than a little surprised, and also displeased, that he would use such a tone, where before he had been all courtesy and soft speeches.
"It is to be a surprise for my Lord Kerovan. Soon he will send my name-day gifts. This I shall return to him." I disliked having to spread my plan before him, yet his question had been too direct to evade.
"Your lord!" He turned his face a little from me. "One forgets these ties exist, Joisan. Do you ever think what it will mean to go among strangers, to a lord you have never seen?" Again that roughness in his speech, which I could not understand. I did not think it kind of him to seize so upon a hidden fear this way and drag it out before my eyes.
I put aside my needle, took up the picture and the unfinished case, and wrapped them in the cloth wherein I kept them, without answering him. I had no intention of saying "yes" or "no" to that question which he had no right to ask.
"Joisan—there is the right of bride-refusal!" The words burst from him as he stood there with his head still averted. His hands were laced upon his sword belt, and I saw his fingers tighten and press.
"To so dishonor his House and mine?" I returned. "Do you deem me such a nothing? What a poor opinion you carry of me, kinsman! What have I done to make you believe I would openly shame any man?"
"Man!" He swung around to face me now. There was a tautness to his mouth, an expression about his eyes I had never seen before. "Do you not know what they say of the heir of Ulmsdale? Man—what was your uncle thinking of when he agreed to such a match? Joisan, no one can hold a maid to such a bargain when she has been betrayed within its bonds! Be wise for yourself and think of refusal—now!''
I arose. In me anger grew warm. But it is in my nature that when I am most in ire I am also the most placid seeming outwardly. For which, perhaps, I should thank fortune, for many times has it given me good manner.
"Kinsman, you forget yourself. Such speech is unseemly, and I know shame that you could think me so poor a thing as to listen to it. You had better learn to guard your tongue." So saying I left him, not heeding his quick attempt to keep me there.
Then I climbed to my own small chamber and there stood by the northward window, gazing out into the dusk. I was shivering, but not with the cold; rather with that fear I thought I had overcome in the weeks since Yngilda had planted it in my mind.
Yngilda's spite, and now this strange outburst from Toross, who, I had not believed, could have said such a thing to me! The right of bride-refusal, yes, that existed. But the few times it had been invoked in the past had led to death feuds between the Houses so involved. Monster—Yngilda had said that. And now Toross—repeating the word "man" as if it could not be applied to my lord! Yet my uncle would not wish to use me ill, and surely he had considered very well the marriage proposal when it had first been made to him. I had also Dame Math's solemn oath.
I longed all at once for the garden of the Past-Abbess Malwinna. To her alone could I speak of this matter. Dame Math's stand I already knew; that my lord was a victim of misfortune. This I could believe more readily than that he was in any way not a man. For after sworn oaths between my uncle and his father, such a thing could not be. And I heartened myself by such sensible council, pinning additional hope on my plan to send Kerovan the picture.
But thereafter I avoided Toross as much as I could, though he made special attempts several times to have private conversation with me. I could claim duties enough to keep me aloof, and claim them I speedily did. Then there came a day when he had private conversation with my uncle, and before the day's end he and his men rode out of Ithkrypt. Dame Math was summoned to my uncle, and thereafter Archan came to bring me also.
My uncle was scowling as I had seen him do at times when he was crossed in some matter. And that scowl was turned blackly on me as I entered.
"What is this boil of trouble you have started, wench?" His voice was only slightly below a roar, aimed at me when I was scarcely within the doorway. "Are you so light of word that you—"
Dame Math arose from her chair. Her face was as angercast as his, but she looked at him, not me.
"We shall have Joisan's word before you speak so!" Her lower tone cut across his. "Joisan, this day Toross came to your uncle and spoke of bride-refusal—"
It was my turn to interrupt; my anger also heated by such an accusation from my uncle, before he had asked my position in the matter.
"So did he speak to me also. I told him I would not listen; nor am I an oath-breaker! Or do you, who know me well, also believe that?"
Dame Math nodded. "It is as I thought. Has Joisan lived under your eyes for all these years without your knowing her for what she is? What said Toross to you, Joisan?"
"He seemed to think evil concerning my Lord Kerovan, and that I should use bride-refusal not to go to him. I told him what I thought of his shameful words and left him, nor would I have any private speech with him thereafter."
"Bride-refusal!" My uncle brought his fist down on the table with the thump of a war drum. "Is that youngling mad? To start a blood feud, not only with Ulmsdale, but half the north who would stand beside Ulric in such a matter! Why does he urge this?"
There was frost in Dame Math's eyes, a certain quirk to her lips which suggested that she was not altogether displeased at his asking that.
"I can think of two reasons, brother. One stemming from his own hot blood. The other placed in his mind by—"
"Enough! There is no need to list what may or may not have moved Toross to this folly. Now listen, girl," he swung on me again. "Ulric took oath that his heir was fit to be the lord of any woman. That his wife was disordered in her wits when the lad was born, that all men know. She so took such a dislike to the child she named him monster, which he is not. Also Ulric spoke with me privately upon a matter which has much to do with this, and which I tell you now, but you shall keep mum-mouthed about it hereafter—remember that, girl!"
"I shall do," I gave him my promise when he paused as if expecting that assurance from me.
"Well enough. Then listen—there is always something behind such wild tales when you hear them, so learn in the future to winnow the true from the false.
"The Lady Tephana, who is your lord's mother (and a fine mother she has been to him!), had an elder son Hlymer, by her first marriage. Since he got no lands from his father, she brought him with her to Ulmsdale. In addition she has had a daughter—Lisana—who is but one year younger than your lord.
"This daughter she has seen betrothed to one of her own House. And the daughter she dotes upon with all the affection to equal her distaste for Kerovan. Thus Ulric of Ulmsdale has reason to believe that within his own household the seeds of trouble for his heir—for Hlymer makes common cause with Lisana's betrothed, and they see a lord to come who is not Kerovan. Ulric can make no move against them, for he has no proof. But because he would not see his son despoiled when he could no longer protect him, he wished some strong tie for Kerovan, to unite him with a House that would support him when the time comes that he needs shields raised for him.
"Since no man can sit in the high seat of a keep who is not sound of body and mind, how better create doubt in possible supporters for a threatened heir than by bending rumor to one's use, spreading tales of 'monsters' and the like? You have seen what happens when such tales come into the hearing of those who do not guess what may lie behind their telling. Toross came to me with such a story—he is filled with it. Since I am sworn not to reveal, save to the parties most deeply concerned, any of Ulric's fears for the future, I bade Toross ride forth if he could not hold his tongue. But that you might have listened to him—"
I shook my head. "It was he who came to me with it. But I had already heard such a tale in greater detail from his sister in Trevamper."
"So Math told me." The flush had faded from my uncle's face. Now I knew he was slightly ashamed of the way he had greeted me, not that he would ever say so. But such things had always been understood between us.
"You see girl," he continued, "how far this story has spread. I do not think Ulric is altogether wise in not better ordering his household. But each man is lord in his own keep and needs must face his own shadows there. But know this—your wedded lord is such a man as you will be proud to hand-fast when the time comes—as it will soon now. Take no heed of these rumors, knowing their source and purpose."
"For which knowing I give thanks," I replied.
When Dame Math and I left his company together, she drew me apart into her own chamber and looked at me searchingly, as if by that steady gaze alone she could hunt out every unspoken thought within my mind.
"How chanced Toross to speak to you on this matter? He must have had some reason—one does not so easily break custom. You are a wedded lady, Joisan, not an unspoken-for maid who allows her eyes to stray this way or that."
So I told her of my plan. To my surprise she did not object nor seem to think what I was doing was beneath the dignity of my station. Instead she nodded briskly.
"What you do is fitting, Joisan. Perhaps we should have arranged such an exchange ourselves long ago. That would have broken such rumors. Had you had a picture of Kerovan in your belt-purse when Yngilda spoke to you, it would have answered well. So Toross was angered at what you did? It was past time when that youth should have returned to those who sent him to make trouble!" She was angry again, but not with me. Only what moved her now she did not explain.
So I finished the picture case, and Dame Math approved its making as an excellent example of my best needlecraft. Making all ready, I laid it away in my coffer against the arrival of the party from Ulmsdale.
They were several days late, and the party itself was different from the earlier ones, for the armsmen were older, and several of them bore old, healed wounds which would keep them from active field service. Their leader was crooked of back and walked with a lurch and a dip.
Besides a casket that he delivered with ceremony to me, he bore a message tube sealed with Ulric's symbol for my uncle and was straightway taken into private conversation with him, as if this were a matter of great import. I wondered if my summons to Ulmsdale had come at last. But the nature of the bearer was such that I could not accept that. My lord would have come himself as was right, and with a retinue to do me honor through those lands we must cross to his home.
Within the casket was a necklet of northern amber and gold beads, with a girdle to match. Truly a gift to show me prized. Yet I wished it had been just such a picture as I had ready to return to him. I knew that Dame Math would make opportunity to let me speak alone with this Jago who commanded the Ulmsdale force, that I might entrust him with my gift. But it appeared he had so much to say to my uncle there was little time for that, for he did not come out of the inner chamber until the hour for the evening meal.
I was glad he was seated beside me, for it gave me a chance to say that I would see him privately, that I had something to entrust to him. But he had a speech in return.
"Lady, you have had Ulmsdale's gift, but I have another for you from the hand of Lord Kerovan himself which he said to give to you privately—"
Within me I knew then a rise of excitement, for I could conceive of nothing save that we had been of one mind, and what Jago had for me was also a picture.
But it was not so. When Dame Math saw that we came together in a nook between the high seat and the wall, what he laid in my eager hand was not a flat packet, but a small, round one. Quickly I pulled away the covering of soft wool to find that I held a crystal globe and within it a gryphon—even as I had seen it at the House of Dames! I nearly dropped it. For to have something of the Power touch into one's life so was a thing to hold in awe and fear. Set in its surface just above the gryphon's head was a ring of gold, and there was strung a chain so one could wear it as a pendant.
"A wondrous thing!" Somehow I found my tongue and hoped that I had not betrayed my first fear. For to no one could I explain the momentary panic I had felt. The more I studied it now, the clearer became its beauty, and I thought that it was truly a treasure, finer than any that had ever been sent to me in any casket of ceremony.
"Yes. My lord begs you accept of this, and perhaps wear it sometimes, that you may know his concern for you." That sounded like some set speech which he had memorized. And I decided swiftly to ask no questions of this man. Perhaps he was not too close to my lord after all.
"Tell my lord I take great joy in his gift." I found the formal words easier than I would have done a moment earlier. "It shall abide with me night and day that I may look upon it, not only for my pleasure in its beauty, but also because of his concern for me. In return," I hurriedly brought forth my own gift, "do you place this in my lord's hand. Ask of him, if he wills, to send its like to me when he may."
"Be sure that your wish is my command, Lady." Jago slipped it into his belt-purse. Before he could say more, if there was more to be said between us, there came one of my uncle's men to summon him again to that inner room, and I did not see him further that night.
Nor did we have more than formal speech together during the two remaining days that he was at Ithkrypt. I gave him ceremonious farewell when he rode forth, but by then all within the keep knew of the news that had come with the men from Ulmsdale.
By birth and inclination dalesmen are not sea-rovers. We have ports for trading set up along the coast, and there are villages of fisherfolk to be found there. But deep-sea ships do not sail under the flag of any dales lord. And those who trade from overseas, such as the Sulcarmen, are not of our blood and kin.
News from overseas is long old before it reaches us. But we had heard many times that the eastern lands were locked in a struggle for power between nation and nation. Now and then there was mention of a country, a city, or even some warlord or leader whose deeds reached us in such garbled form they were already well on the way to becoming a tale more fancy than fact.
Of late, however, there had been new ships nosing along our shores. The Sulcarmen had suffered some grievous defeat of their own two years since in the eastern waters. And so we had not the usual number of their traders coming for our woolen cloth, our wonder-metal from the Waste, our freshwater pearls. But these others had put in to haggle, driving hard bargains, and they seemed over-interested in our land. Often when they had discharged a cargo, before taking on another, they would lie in harbor, and their crews would ride north and south as if exploring.
Our thoughts of war never encompassed more than the feuds between dale and dale, which could be dark and bloody at times, but which seldom involved more than a few score of men on either side. We had no king or overlord, which was our pride, but also in a manner our weakness, as was to be proven to us. Sometimes several lords would combine their forces to make a counter-raid on Waste outlaws or the like. But such alliances were always temporary. And, while there were several lords of greater following than others (mainly because they held richer and more populous dales), none could send out any rallying call all others would come to.
This must have been clear to those who spied and went—that we were feeble opponents, easy to overrun. However, they misread the temper of the dales, for a dalesman will fight fiercely for his freedom. And a dalesman's loyalty to his lord, who is like the head of his own family, is seldom shaken.
Since Ulmsport lay at the mouth of that dale, it had recently been visited by two ships of these newcomers. They called themselves men of Alizon and spoke arrogantly of the size and might of their overseas land. One of their men had been injured inland. His companion from the ship had been killed. The wounded man had been nursed by a Wisewoman. By her craft she could tell true from false. And, while he wandered in a fever, talking much, she listened. Later, after the coming of his comrades to bear him away, she had gone to Lord Ulric. He had listened carefully, knowing that she knew of what she spoke.
Lord Ulric was prudent and wise enough to see that there was that to it which might come to overshadow our whole land, as it did. Speedily he sent accounts of what had been learned to all the neighboring dales, as well as to Ithkrypt.
It seemed from the babbling of the wounded man that he was indeed a spy, the scout of an army soon to be landed on us. We realized that those of Alizon had decided that our rule was so feeble and weak in its nature they could overrun us at their pleasure, and this they moved to do.
Thus was the beginning of the great shadow on our world. But I nursed the crystal ball in my hand, uninterested in Alizon and its spies, sure that somehow the Lord Kerovan would do as I willed, and I would look upon the picture of a man who was no monster.
5
Kerovan
To my great surprise I discovered Jago had returned before Riwal and I came out of the Waste. His anger with me was such that, had I been younger, I think he would have cut a switch from the nearest willow and used it for my discipline. I saw that that anger was fed, not wholly from my supposedly ill-advised foray into the dubious territory, but also from something he had learned at Ulmskeep. Having spoken his mind hotly, he ordered me to listen, with such serious mien that I lost the defiance his berating had aroused.
On two occasions in the past I had been to Ulmskeep, both times when my mother was away visiting her kin. So it, and the lower part of the dale, was not unknown to me. Also on those times when my father had come to me, he had taken patience to make me aware of the spread of our lands, the needs of our people, those things that it was necessary for me to know when the day came for me to take his place.
But the news Jago brought I had not heard before. For the first time I learned of the invaders (though they were not termed so then, being outwardly visitors on leave from their ships at Ulmsport).
With what contempt they regarded us we quickly learned, for we are far from stupid—at least in that way. Dalesmen may insist too much upon their freedom, having a strong disliking for combining forces, save in time of immediate and pressing need. But we can sniff danger like wild things when it treads our land.
They had first come nosing into our ports, up river mouths, a year or so earlier. Then they had been very wary, circumspect, playing the roles of traders. Since the stuffs they had to offer in return for our native wool, metal, and pearls were new and caught the eye, they found a welcome. But they kept much to themselves, though they came ashore in twos and threes, never alone. Once ashore they did not linger in port, but journeyed inland on the pretext of seeking trade.
As strangers they were suspect, especially in those districts lying near the Waste, even though it was known that their origin was overseas. Men met them courteously and with guest right, but as they looked and listened, asked a question here, another there with discretion, so did others watch and listen. Soon my father, gathering reports, could see a pattern in their journeying which was not that of traders, but rather, to his mind, the action of scouts within new territory.
He sent privately to our near neighbors: Uppsdale, Fyndale (which they had visited under the pretext of the great fair held there), Flathingdale, and even to Vastdale, which also had its port of Jorby. With all these lords he was on good terms, for we had no feuds to separate us. And the lords were ready to listen and then set their own people to watching also.
What grew out of all this was the now-strong belief that my father had judged the situation rightly, and these overseas strangers were prowling our country for some purpose of their own, one meaning no good to the dales. It was soon to be decided whether or not the lords would make common cause and forbid hew landings to any ship from Alizon.
However, to get the lords to make common cause on any matter was a task to which only a man with infinite patience might set himself. No lord would openly admit that he accepted the will of another. We had no leader who could draw the lords under one banner or to one mind in action. And this was to be our bane.
Now there were to assemble at Ulmskeep five of the northern lords to exchange their views upon the idea. They needed some excuse for such a gathering, however, for it must be a festival of a kind to keep people talking in such a way that the strangers might hear a false excuse. My father had found a cause in the first arming of his heir, bringing me now into the company of my peers as was only natural at my age.
So far I followed Jago. But at his flat statement that I was to be the apparent center for this gathering, I was startled. For so long had I accepted my lot apart from the keep and from the company of my kin, that this way of life seemed the only proper one to me.
"But—" I began in protest.
Jago drummed with fingertips upon the table. "No, he is right, Lord Ulric is. Too long have you been put aside from what is rightly yours. He needs must do this, not only to give cover to his speech with the lords, but for your own sake. He has learned the folly of the course he has followed these past years."
"The folly of—?" I was astounded that Jago spoke so of my father, since he was so stoutly a liegeman of Ulric's as to think of him with the awe one approached an Old One.
"Yes, I say it—folly!" The word exploded a second time from his lips, as might a bolt from a crossbow. "There are those in his own household who would change matters." He hesitated, and I knew without words what he hinted at: that my mother favored my sister and her betrothed for the succession in Ulmsdale. I had never closed my ears to any rumor brought to the foresters' settlement, deeming I must know the worst.
"Look at you!" Jago was angry once more. "You are no monster! Yet the story spreads that Lord Ulric needs must keep you pent here in chains, so ill-looking a thing, so mind-damaged, that you are less than a man, even an animal!"
His heat struck a spark from me. So this was what was whispered of me in my own keep!
"You must show yourself as you are; be claimed before those whose borders march with Ulmsdale as the proper heir. Then none may rise to misname you later. This Lord Ulric now knows—for he has heard some whispering, even challenged those whisperers to their faces. And one or two were bold enough to tell him what they had heard."
I got up from the table which stood between us and went to Jago's great war-shield where it hung upon the wall. He spent long hours keeping it well-burnished so that it was like a mirror, even though the shape distorted my reflection.
"As long as I keep on my boots," I said then, "perhaps I will pass as humankind."
Those boots were cunningly made, being carefully shaped so my cloven hoofs appeared normal feet. When I went shod, no man might be aware of the truth. The boots had been devised by Jago himself and made from special leather my father sent.
Jago nodded. "Yes, you will go, and you will keep your boots on, youngling, so you can prove to every whisperer in the dales that your father sired a true heir, well able to take lord's oath. With weapons you are as good, perhaps even better, than those who are keep armsmen. And your wit is keen enough to make you careful."
Which was more praise than he had ever given me in our years together.
Thus mailed and armed (and most well-booted) I rode with Jago out of the exile which had been laid upon me and came at last to take up life in my father's keep. I did so with inner misgivings, having, as Jago pointed out, some store of wit, and well-surmising that I was far from welcome by some members of that household.
I had little chance to speak again with Riwal before I went—though I longed for him to offer to go with me, knowing at the same time that he never would. In our last meeting he looked at me in such a way that I felt he could somehow see into my mind and know all my uncertainties and fears.
"You have a long road to ride, Kerovan," he said.
"Only two days," I corrected him. "We but go to Ulmskeep."
Riwal shook his head. "You go farther, gryphon bearer, and into danger. Death stalks at your shoulder. You shall give, and, in giving, you shall get. The giving and the getting will be stained with blood and fire—"
I realized then that he was farseeing, and I longed to cover my ears, for it seemed to me that his very words would draw down upon me the grim future he saw.
"Death stalks at the heels of every man born," I summoned my courage to make answer. "If you can farsee, tell me what shield I can raise to defend myself."
"How can I?" he returned. "All future is fan-shaped, spreading out in many roads from this moment. If you make one choice, there is that road to follow; if you make another yet a second path, a third, a fourth—But no man can outstride or outfight the given pattern in the end. Yours lies before you. Walk with a forester's caution, Kerovan. And know this—you have that deep within you, if you learn to use it, that shall be greater than any shield or sword wrought by the most cunning of smiths."
"Tell me—" I began.
"No!" He half-turned from me. "So much may I say, but no more. I cannot farsee your choices, and no word of mine must influence you in their making. Go with the Peace." Then he raised his hand, and, between us in the air, he traced a sign. I half-started back, for his moving finger left a faint glimmering which was gone almost as quickly as I marked it. And I realized then that in some of his seeking Riwal must have been successful, for his sign was of the Power.
"Until we meet again then, comrade." I spoke friend-farewell.
He did not face me squarely, but stood, his hand still a little raised to me. I know now that he understood this was our last meeting and perhaps regretted it. But I was not cursed with that sight which can hurt far more than it shields. What man wishes to see into the future in truth when there are so many ills lying there in wait for us?
As we traveled to Ulmskeep, Jago talked steadily, and that this was of purpose, I shortly understood. He so made known to me the members of my father's household, giving each his character as he himself saw it, with unspoken shadings which allowed me to perceive that this one might be well-disposed, that one not. I think he saw me as a child fated to make some error that would bring about disaster and was doing what he could to give me some manner of protection against any outspoken folly.
My elder half-brother, who had come to Ulmskeep as a very young child, had, when he reached the age suitable for instruction in arms, been sent to our mother's kin. But within the past year he had returned, riding comrade-in-arms to our kinsman Rogear who was my sister's betrothed. I could well guess that he was to me an unfriend and with him I must be wary.
My mother had her followers among others at the keep, and Jago, with what delicacy he could summon, named those to me, giving short descriptions of each and of the positions they held. On the other hand my father's forces were in the greater number, and among those were the major officers—the Master of Armsmen, the marshal, and others.
It was a divided household, and such are full of pitfalls. Yet on the surface all seemed smooth. I listened carefully, asking some questions. Perhaps these explanations were Jago's idea; perhaps my father had suggested that I be so cautioned before I came to face friend and unfriend, that I might tell one from the other.
We rode into the keep at sunset, Jago having blown a signal with his approach horn so that we found the door guard drawn up to do us honor. I had marked the residence banners of Uppsdale and Flathingdale below our gryphon on the tower, and knew that two of those my father had summoned were already here. Thus, from the beginning, I would be under the eyes of the curious as well as those of the covertly hostile.
I must play my role well, seemingly unaware of any cross-current; bear myself modestly as became a candidate for arms, yet be far from a fool. Was I able to do this? I did not know.
The guards clanged swords together as we dismounted. My father, wearing a loose robe of ceremony over his jerkin and breeches, came forth from the deep shadow of the main portal to the hall. I fell to one knee, holding out my sword by the point that he could lay fingers lightly on its hilt in acknowledgment.
Then he drew me to my feet in a half-embrace, and, with his hand still on my arm, brought me out of the open into the main hall where a feasting board had already been set up, and serving men and maids were busied spreading it across with strips of fair linen and setting out the plates and drinking horns.
There were two other men of early middle age, robes of ceremony about their shoulders. My father made me known to Lord Savron of Uppsdale and Wintof of Flathingdale. That they regarded me keenly I was well aware. But I was strengthened in my role by the knowledge that I made, in my mail and leather over-jerkin, no different appearance than their own sons might display. At this moment no man might raise the cry of monster. They accepted my proper deference as if this were an ordinary meeting and I had been absent from Ulmskeep for perhaps only an hour or two, not for all my life.
Since I was still considered but a boy and, by custom, of little importance, 1 was able speedily to withdraw from the company of my elders and go to that section of the barracks where the unmarried men of the household were quartered. In deference to my heirship, I was given a small room to myself—a very bare room in which there was a bed, narrow and hard, two stools, and a small table, far less comfortable than my room in die forester's holding.
A serving lad brought in my saddle bags. I noted that he watched me with open curiosity when he thought I did not mark him, lingering to suggest that he bring me hot water for washing. When I agreed, I thought that he was determined to make the most of his chance to view the "monster" at close quarters and perhaps report his discoveries to his fellows.
I had laid aside my jerkin and mail by the time he returned, standing in my padded undercoat, sorting through my clothing for my best tabard with its gryphon symbol. He sidled in with a ewer from which steam arose, watching me as he placed it and a basin on the table, and twitched off a towel he had borne over one shoulder to lay beside it.
"If you need service, Lord Kerovan—"
I smoothed out the tabard. Now I deliberately unhooked my undercoat, letting it slide from my body, which was thus bared to the waist. Let him see I was not misshapen. As Jago said, if I clung to my boots, no man could miscall me.
"You may look me out a shirt—" I gestured toward my bags and then saw he was staring at me. What had he expected? Some horribly twisted body? I glanced down and saw what had become so much a part of me I had forgotten I still wore it—the crystal gryphon. Lifting its chain over my head, I laid it on the table as I washed and saw him eyeing it. Well enough, let him believe that I wore a talisman. Men did so. And one in the form of the symbol of my House was proper.
He found the shirt and held it for me as I redonned the gryphon. And he hooked my tabard, handed me my belt with its knife-of-pride before he left, doubtless with much to tell his fellows. But when he had gone, I drew the gryphon from hiding and held it in my hand.
As usual when I held it so, it had a gentle warmth. With that warmth came a sensation of peace and comfort, as if this thing, fashioned long before my first ancestor had ridden into the south dales, had waited all these centuries just to lie in my hand.
So far all had gone as it should. But before me still lay that ordeal from which my thoughts had shied since first I had known that I must come to Ulmsdale, the meeting with my mother. What could I say to her, or she to me? There lay between us that which no one could hope to bridge—not after all these years.
I stood there, cupping close the crystal ball and thinking of what I might do or say at a moment which could be put off no longer. Suddenly it was as if someone spoke aloud—yet it was only in my mind. I might have looked through a newly unshuttered window so that a landscape hitherto hidden lay before me.
The scene was shadowed, and I knew it was not mine to ever walk that way, just as there could be no true meeting between us which would leave us more than strangers. Yet I felt no sense of loss, only as if a burden had been lifted from me, to leave me free. We had no ties; therefore I owed her no more than she willed me to. I would meet her as I would any lady of rank, paying her the deference of courtesy, asking nothing in return. In my hand the globe was warm and glowing. But a sound at the door made me slip that into hiding before I turned to face who stood there.
I am only slightly over middle height, being slender-boned and spare-fleshed. This youth was tall enough that I must raise my eyes to meet his. He was thick of neck and shoulder, wide of jaw. His hair was a mat of sandy curls so tight he must battle to lay it smooth with any comb. He had a half-grin on his thick lips. I had seen such an expression before among the armsmen when a barracks bully set about heckling some simpleton to impress his standing upon his fellows.
His tabard of state was wrought with fine needlework and stretched tightly over the barrel of his chest. And now he was running his fingers up and down its stiff fronting as if striving to draw attention to it.
Big as he was, he did not altogether fill the doorway, for there was a smaller and slighter figure beside him. I was for an instant startled. The oval of that face was like enough to mine to stamp us kinsmen, just as his dark hair was also mine. His expression was bland, nearly characterless, but I guessed that behind it lurked a sharp wit. Of the two I would deem him the more dangerous.
From the first I knew these for the enemy. Jago's descriptions fitted well. The giant was closer-kin to me than the other, for he was my half-brother Hlymer, while his companion was my sister's betrothed, my cousin Rogear.
"Greetings, kinsmen." I spoke first.
Hlymer did not lose his grin; it grew a little wider. "He is not furred or clawed, at least not as can be seen. I wonder in what manner his monster marking lies, Rogear." He spoke as if I were a thing and not able to hear or sense his meaning. But if he meant to arouse some sign of anger he could play upon, he was a fool. I had taken his measure early.
Whether he would have carried on along that theme if left alone I was not to know, because Rogear then answered, not Hlymer's comment, but my greeting, and courteously in kind, as if he never meant to do otherwise.
"And to you greeting, kinsman."
Hlymer had a high voice that some big men own and a slight hesitancy of speech. But Rogear's tone was warm and winning. Had I not known him to be what he was, I might have been deceived into believing that he had indeed sought me out to make me welcome.
They played my escort to the great hall. I did not know whether to count it as a relief or not when I saw that there were no chairs placed for ladies, that this meal at least was clearly intended only for the men. Undoubtedly my mother had chosen to keep to her own apartments. Since all knew the situation, none would comment on it.
I saw my father glance sharply at me now and then from his High Seat. My own place was down-table, between Hlymer and Rogear (though whether they had purposefully devised that or not, I did not know). If my father was not satisfied, there was little he could do without attracting unwelcome attention.
My companions' tricks began early. Hlymer urged me to empty my wine horn, implying that any moderation on my part marked me in that company. Rogear's smooth flow of talk was clearly designed to point up the fact that I was raw from some farmyard, without manners or wit. That neither accomplished their purposes must have galled. Hlymer grew sullen, scowling, muttering under his breath words I did not choose to hear. But Rogear showed no ill nature at the spoiling of whatever purpose he had in mind when we sat down.
In the end Hlymer was caught in his own trap—if he considered it a trap—and grew muddle-headed with drink, loud in his comments, until some of those around turned on him. They were young kinsmen of the visiting lords and, I think, frank in their desire for no trouble.
So began my life under my father's roof. Luckily I did not have to spend much time within the range of Hlymer and Rogear. My father used the fact of my introduction and confirmation as his heir to keep me much with him, making me known to his neighbors, having me tutored in those details of the ceremony that occurred on the third day of the gathering.
I swore kin-oath before a formidable assemblage of dale lords, accepted my father's gift-sword, and so passed in an hour from the status of untried and unconsidered youth to that of man and my father's second in command. As such I was then admitted to the council concerning the men from Alizon.
Though all were agreed that there seemed to be some menace behind the coming of these men from overseas, there was sharp division as to how the situation was to be handled. In the end the conference broke, as such so often did in the dales, with no plan of action drawing us together after all.
Because of my new status in the household, I rode part way down the dale with the lord of Uppsdale to give him road-speed. On my way back my career as Heir of Ulmsdale was nearly finished before it had rightfully begun.
In courtesy to our guest, my sword was in peace-strings, and I had gone unmailed. But in the moment of danger I was warned. For there flashed into my mind such a sudden sensation of danger that I did not wait to loose the cords from my sword. Instead I plucked free my knife, at the same time throwing myself forward, so that the harsh hair of my mount's mane rasped my cheek and chin. There was the sharp crack of a bolt's passing, a burn across my shoulder—by so little I escaped death.
I knew the tricks the foresters used in savage infighting when the outlaws came raiding. So I threw my knife, to be answered with a choked cry from that man who had arisen between the rocks to sight on me a second time. Now I brought my sword around, charging a second man who had emerged, steel in hand. One of the battle-trained mount's hoofs crushed down, and the man was gone, screaming as he rolled across the ground.
We had made an important capture, we discovered, for though these two wore the dress of drifting laborers who make their way from dale to dale at harvest time, they were indeed the very invaders we had been discussing at such weary length. One was dead; the other badly injured. My father called the Wisewoman of the dale and had her attend him, and light-headed with fever, he talked.
The purpose of their attack on me we did not learn. But there was much else of useful knowledge, and the threat lying over us grew darker. My father had me in with Jago and his trusted officers. Now he spoke his mind.
"I do not pretend to farseeing, but any man with wit in his head can understand that there is purpose and planning behind this. If we do not look ahead, we may—" He hesitated. "I do not know. New dangers mean new ways of dealing with them. We have always clung to the ways of our fathers—but will those serve us now? It may be that the day will soon come when we need friends to hold shields with us. I would draw to us now all assurances of those as we can.
"Therefore"—he smoothed out on the table around which we sat the map of the dalesside—"here is Uppsdale and the rest. To them we have already made clear what may come. Now let us warn the south—there we may first call upon Ithkrypt."
Ithkrypt and the Lady Joisan. For long periods of time I had put her out of mind. Had the day now arrived when my father would order ours to be a marriage in fact? We had both reached an age when such were common.
I thought of my mother and the sister who had kept themselves stubbornly immured in their own apartments since I had come to Ulmskeep. Suddenly I knew that I would not have my Lady Joisan join them there as she would do if she came now. How could she help but be swayed by their attitude toward me? No, she must come willingly to me—or not at all!
But how could I make sure that she did so?
As that sudden warning of danger had saved my life, so again came an answer as clear and sharp as if spoken aloud.
Thus after my father had cautioned Jago as to what to say to Lord Cyart when he delivered the name-day gift to my lady, I spoke privately with my old tutor. I could not say why I had to do this; I did not want to; yet it was laid upon me as heavily as—a geas—is put upon some hero who cannot thereafter turn aside from it. I gave him the crystal gryphon, that he might put it in Lady Joisan's own hand. Perhaps it was my true bride-price. I would not know until that moment when we did indeed at long last stand face to face.
6
Joisan
One difference did the news brought to my uncle from Ulmsdale make in my own plans for the future. It was decided that I would not be going there to join my lord this season as had been heretofore thought, but I must wait upon a more settled time. For if spies had been sent into Ulmsdale with such boldness, the enemies' forethrust might soon be delivered. My uncle sent such a message with Jago. There came no protest from Lord Ulric or Lord Kerovan in return, so he deemed they agreed. Thereafter he sat sober-faced, talking with his armsmen and with messengers he sent in turn to Trevamper and those dales where he had kinship or old friendship.
It was a time of spreading uneasiness. We harvested more closely that year than any time I could remember, plundering all the wild berries from the field bushes, taking nuts from the woods trees, laying up what manner of stores we could. It was as if some foreshadowing of the starving years to come already lay across the land.
And in the next summer my uncle ordered the planting increased, with more fields cleared and sown. The weather was as uncertain as the threatening future, for there were a number of storms of great severity. Twice the roads were washed out, and we were isolated until men could rebuild them.
We had only ragged scraps of news when some lord's messenger found his way to us. No more spies had come into Ulmsport. From the south there were rumors of strange ships that did not anchor openly in any dale port, but patrolled the coast. Then these were seen no more for a space, and we took a little comfort in that.
It appeared that my uncle feared the worst, for he sent his marshal to Trevamper, to return with two loads of the strange metal from the Waste and a smith who straightway set about fashioning arms and repairing the old. Much to my astonishment, my uncle had him take my measurements and prepare for me a coat of mail.
When Dame Math protested, he stared at her moodily, for his good humor was long since fled.
"Peace, sister. I would give the same to you, save that I know you would not wear it. But listen well, both of you. I believe that we face darker days than we have ever known. If word comes that these invaders come in force, we may find ourselves beaten from dale to dale. Thus—"
Dame Math had drawn a deep breath then, her indignation fading, another emotion on her face, an expression I could not read.
"Cyart—have—have you had then—?" She did not finish, but her apprehension was such that fear uncoiled within me.
"Have I dreamed? Yes, Math—once!"
"Spirit of Flame shelter us!" Her hands went to the set of silver hoops slung together at her girdle. She turned them swiftly in her fingers, her lips shaping those formal prayers that were the support of those in the House of Dames.
"As it was promised"—he looked at her—"so have I dreamed—once.''
"Twice more then to come." Her coifed head came up, her lips firm. "It is a pity that the Warning does not measure time."
"We are lucky, or perhaps cursed, to have it at all," he returned. "Is it better to know there is blackness ahead and so live in foreshadow? Or be ignorant and meet it unwarned? Of the two I choose the warning. We can hold Ithkrypt if they come by river or over the hill ways—perhaps." He shrugged. "You must be prepared at the worst to ride—not to the coast or southward—but to Norsdale, or even the Waste."
"Yet no one has yet come save spies."
"They will, Math. Have no doubt of that. They will come!"
When we were back in our own quarters I dared to ask a question. "What is this dream of which my lord speaks?"
She was standing by the window, gazing out in that blind way one does when one regards thoughts and not what lies beyond. At my words she turned her head.
"Dream—?" For a moment I thought she was not going to answer. Then she came away from the window, her fingers busy with her prayer hoops as if she drew comfort from them.
"It is our warning." Obviously she spoke reluctantly. "To those vowed to the Flame such things are—ah, well, I cannot gainsay that it happens, and it is not of our doing. A generation ago Lord Randor, our father, took under his protection a Wisewoman who had been accused of dealing with the Old Ones. She was a quiet woman who lived alone, seeking out none. But she had a gift of tending animals, and her sheep were the finest in the dale. There were those who envied her. And as my lord has said concerning ugly stories, malice can be spread by tongue and lip alone.
"After the way of her calling she went alone into the wild places seeking herbs and strange knowledge. But if she knew much others did not, she made no parade of it, or used it to the hurt of any. Only the poisoned talk turned the dale against her. They arose one night to take her flock and drive her forth.
"Lord Randor had been in Trevamper, and they thought him still there, or they would not have dared. As they set torch to her roof place, he and his men came riding. He used his own whip on those who meant her harm; set her under his shield for all men to see.
"She said then that she could not stay, for her peace, so broken, could not be reclaimed. But she asked to see our mother, who was heavy with child. And she laid her hands upon the Lady Alys' full belly, saying that she would have ease in the birthing—which was thankful hearing, for the lady had had one ill birth and a child dead of it, to her great sorrow.
"Then the Wisewoman said also that it would be a son, and he would have a gift. In times of great danger he would farsee by the means of dreaming. That with two such dreams he could take measures and escape whatever fate they foretold, but the third time would be ill.
"She went then from Ithkrypt and from the dale, and no man saw her go. But what she foretold came true, for our mother was safely delivered within a month of your uncle, the Lord Cyart. And your uncle did dream. The last time he dreamed, it was of the death of his lady, which happened when he was in the south and unable to come to her, though he killed a horse trying. So—if he dreams—we can believe."
Thus I learned to wear mail because my uncle dreamed. And he taught me also how to use a light sword that had been his as a boy. Though I was not too apt a pupil with that, I proved myself apt with the bow and won the title of marksman. In days to come I was to thank my uncle for such skills many times over—when it was too late for him to know that he had truly given me life by his forethought.
So passed the Year of the Moss Wife, which should have seen me with my Lord Kerovan in Ulmsdale. Sometimes I took into my hands the crystal gryphon and held it, thinking of my lord and wondering what manner of man he was. In spite of all my hopes, no messenger riding between Ulmsdale and Ithkrypt brought me the picture that I desired. At first that angered me a little; then I made excuses, thinking that perhaps they had no one in Ulmsdale with the art of limning out a face, for such talent is not widely given. And in the present chancy times he could not seek afar for something of such small importance.
Though we had stores in plenty, we used them sparingly that winter, scanting even on the Yule feast as had never been done in the past, for my uncle was ever on guard. He had his own scouts riding the frontiers of our dale and awaited all messengers impatiently.
The Month of the Ice Dragon passed, and that of the Snow Bird in the new year was well begun when the news we had awaited came from the lips of a man who had battled through drifts to come to us, so stiff with cold he had to be lifted from a horse that thereafter fell and did not rise again.
Southward was war. The invasion had begun, and it was of such a sort as to startle even those lords who had tried to foresee and prepare. These devils from overseas did not fight with sword and bow after the manner of the dales. They brought ashore from their ships great piles of metal in which men hid, as if in monsters' bellies. And these manmade monsters crawled ahead, shooting flames in great sweeps from their noses.
Men died in those flames or were crushed under the lumbering weight of the monsters. When the dalesmen retreated into their keeps, the monsters bore inward with their weight against the walls, bringing them down. Such a way of war was unknown, and flesh and blood could not stand against it.
Now that it was too late, the rallying call went forth. Those who lingered in their keeps to be eaten up one by one were the thick-headed and foolish ones. Others gathered into an army under the leadership of four of the southern lords. They had already cut off three of the monsters, which needed certain supplies to keep them running, and destroyed them. But our people had lost the coastline. So the invaders were pouring in more and more men, though their creeping monsters, happily, appeared not so numerous.
The summons came to the northern dales for men to build up a force to contain the invaders, to harass them and restrain them from picking off each dale in turn as one picks ripe plums from a tree.
Hard on the heels of that messenger came the first of the refugees, ones who had blood-claim on us. A party of armsmen escorted a litter and two women who rode, guided through the pass by one of my uncle's scouts—the Lady Islaugha, Yngilda, and, in the litter, delirious with the fever of a wound, Toross, all now landless and homeless, with naught but what they could carry on their persons.
Yngilda, wed and widowed within two years, stared at me almost witlessly, and had to be led by the hand to the fire, have a cup placed in her fingers, and ordered firmly to drink. She did not seem to know where she was or what had happened to her, save that she was engulfed by an unending nightmare. Nor could we thereafter ever get any coherent story from her as to how she had managed to escape her husband's keep, which was one of those the monsters had battered down.
—Somehow, led by one of her lord's archers, she had made her way to the dale's camp and there met the Lady Islaugha, come to tend her son, who had been hurt in one of the attacks against the monsters.
Cut off from any safety in the south, they had turned for shelter to us. Dame Math speedily took over the nursing of Toross, while his mother never left him day or night if she could help.
My uncle was plainly caught between two demands—that of the army battering the invaders and his inborn wish to protect his own, which was the dalesman's heritage. But, because he had from the first argued that combination against the foe was our only hope, he chose the army.
He marshaled what forces he could without totally stripping the dale, leaving a small but well-trained troop to the command of Marshal Dagale. There was a thaw at the beginning of the Month of the Hawk and, taking advantage of that, he left.
I watched him alone from the gatetower (for Dame Math had been summoned to Toross, who had taken a turn for the worse), just as I had stood in the courtyard minutes earlier pouring the spiced journey drink from the war ewer into the horns of the men to wish them fair fortune. That ewer I still held. It was cunningly wrought in the form of a mounted warrior, the liquid pouring through the mouth of his horse. There was a slight drip from it into the snow. Red it was, like blood. Seeing it, I shivered and quickly smeared over the spot so that I might not see what could be a dark omen.
We kept Ithkrypt ever-prepared, not knowing—for no messengers came now—how went the war, only believing that it might come to us without warning. I wondered if my uncle had dreamed again, and what dire feelings those dreams had brought him. That we might never know.
There was one more heavy snow, closing the pass, which gave us an illusion of safety while it lasted. But spring came early that year. And with the second thaw a messenger from my uncle arrived, mainly concerned with the necessity for keeping the dale as much a fortress as we could. He said little of what the army in the south had done, and his messenger had only gloom to spread. There were no real battles. Our men had to turn to the tactics of Waste outlaws, making quick raids on enemy supply lines and camps, to do all the damage they could.
He had one bit of news: That Lord Ulric of Ulmsdale had sent a party south under the command of Lord Kerovan, but he himself had been ill. It was now thought that Lord Kerovan might return to take control at Ulmsdale if the invading army forced a landing at Ulmsport as they had at Jorby and other points along the coast.
That night, when I was free of the many duties that now were mine, I took up the crystal gryphon, as I had not for a long time. I thought on him who had sent it to me. Where did he lie this night? Under the stars with his sword to hand, not knowing when the battle-horn might blow to arms? I wished him well with all my heart, though I knew so little of him.
There was warmth against my palm, and the globe glowed in the dusky room. It did not in that moment seem strange to me; rather it was comforting, easing for a moment my burdens.
The globe was no longer clear. I could not see the gryphon. Rather there was a swirling mist within it, forming shadows—shadows that were struggling men. One was ringed about, and they bore in upon him. I cried out at the sight, though none of it was clear. And I was afraid that this talisman from my lord had now shown me his death. I would have torn the chain from my neck and thrown the thing away; yet that I could not do. The globe cleared, and the gryphon watched me with its red eyes. Surely my imagination alone had worked that.
"There you are!" Yngilda's voice accused me. "Toross is very uneasy. They wish you to come."
She watched me, I thought, jealously. Since she had come from behind the curtain of shock that had hidden her on her arrival, she was once again Yngilda of Trevamper. Sometimes I needed all my control not to allow my tongue sharpness in return when she spoke so, as if she were mistress here and I a lazy serving maid.
Toross mended slowly. His fever had yielded to Dame Math's knowledge, but it left him very weak. We had discovered that sometimes I was able to coax him to eat or to keep him quiet when the need arose. For his sake I was willing to serve so. Yet lately I had come to dislike the way his hand clung to mine when I sat beside him and the strange way he looked at me and smiled, as if he had some claim on me that no one could deny.
This night I was impatient at such a summons, though I obeyed it, for I was still shaken by what I had seen, or imagined I had seen, in the globe. I willed myself not to believe that this had been a true farseeing. Though with men at desperate war, and my lord with them, I could well accept he was so struck down. At that moment I wished that I did have the farsight, or else that we had a Wisewoman with such skill. But Dame Math did not countenance the uses of that power.
That party of kinsmen had been only the first of the refugees to find their way hither. If my uncle had farseen that we might have to open our doors and supplies to such, he had not mentioned it. I wondered, marking each morn as I measured out the food for the day (which had become one of my more serious duties since Dame Math had the overseeing of the sick and wounded) whether we could eat even sparingly until harvest, if this continued.
For the most part the newcomers were lands-people, women and children, with a sprinkling of old or wounded men, few of them able to help us man any defense that might be needed. I had spoken with Dame Math and the marshal only the evening before, the three of us deciding that as soon as the weather lightened sufficiently, they would be sent on to the west, where there were dales untouched by war, even to the House of Dames at Norstead. We dared not keep the burden of useless hands here.
Now as I came into the chamber where Toross lay, I was trying to occupy my mind with plans for such an exodus, rather than allow myself to think of the shadows in the globe. About that I could do nothing. I must consider what I could do.
He was braced up by pillows, and it seemed to me that he looked better than he had since they brought him here. If this was so, why had they sent for me? Such a question was actually on my lips when the Lady Islaugha arose from a stool by the bed and moved away, he holding out his hand in welcome to me.
She did not glance in my direction, but took up a tray, and with a hasty murmur left the room.
"Joisan, come here where I may truly look upon you!" His voice was stronger also. "There are shadows beneath your eyes; you drive yourself too hard, dear heart!"
I had come to the stool, but I did not sit. Instead I studied his face closely. It was thin and white, with lines set by suffering. But there was reason in his eyes, not the cloudiness of a fevered mind. And that uneasiness I had felt with him filled me.
"We all have duties in plenty here, Toross. I do no more or less than my share."
I spoke shortly, not knowing whether to comment on his using an endearment he had no right to give to me who was a wedded wife.
"Soon it will all be over," he said. "In Norstead the war and its ugliness cannot touch you—"
"Norstead? What mean you, Toross? Those who go to Norstead are the refugees. We cannot keep them here, our supplies will not allow it. But our own people do not go. Perhaps you will ride with them—"
As soon as I said that, I felt again a slight lightening of burden. Life in Ithkrypt would be easier without these kinsfolk ever at my side.
"But you shall go also." He said that as calmly as if there could be no question. "A maid has no place in a keep as good as besieged—"
Dame Math—surely she had not planned so behind my back? No, I knew her better. Then I lost that touch of panic—Toross had not the least authority over me. At my uncle's orders, or at Kerovan's, would I leave, but for no other.
"You forget, I am not a maid. My lord knows I am at Ithkrypt. He will come for me here. So I stay until that hour."
Toross' face flushed. "Joisan, do you not see? Why do you cling loyally to that one? He has not claimed you within the marriage term; that is already two months gone, is that not so? You can now give bride-refusal without any breaking of oaths. If he wanted you, would he not have come before this?"
"Through the ranks of the enemy, no doubt?" I countered. "Lord Kerovan leads his father's armsmen in the south. This is no time to say that the days of agreement be strictly kept. Nor do I break bond unless my lord himself says he does not want me!" Perhaps that was not quite so, for I had as much pride as any woman. But I wished Toross to understand, not to put into blunt words what could not be unsaid. If he went farther, it would end all friendship between us, and I had liked him.
"You are free if you wish it," he repeated stubbornly. "And if you are truthful, Joisan, you know that this is your wish. Surely that I—what I feel for you and have since first I saw you is plain. And you feel the same, if you will allow yourself to—"
"Untrue, Toross. What I tell you now is as strong as if I took Flame Oath—as I shall if you need that to make you understand. I am Lord Kerovan's wife and so I will remain as long as we live and he does not deny it. As a wife it is honor-breaking and unfitting that I listen to such words as you have just said. I cannot come to you again!"
I turned and ran, though I heard him stir and give a gasp of pain, then cry out my name. I did not look back, but came into the great hall. There was the Lady Islaugha ladling broth from a pot into a bowl, and I went to her swiftly.
"Your son needs you," I told her. "Do not ask me to go to him again."
She looked up at me, and I could see in her face that she knew what had happened and that she hated me furiously because I had turned from him. To her he was her heart's core, to which all must be allowed and given.
"You fool!" she spat at me.
"Not such a fool as I would be if I listened." That much of a retort I allowed myself, and then stood aside as, with the slopping bowl still in her hands, she hurried towards Toross' room.
I remained by the fire, stretching out my chilled hands to it. Was I a fool? What had I of Lord Kerovan to keep with me? A bauble of crystal—after eight years of marriage which was no marriage. Yet for me there had been no choice, and I did not regret what I had just done.
7
Kerovan
It seemed that I could hardly remember a time when there had not been war, so quickly does a man become used to a state of constant alarm, peril, and hardship. When the news of the invasion came, my father made ready to march southward at the summons of those most beset. But before he could ride, he thought better of it for two reasons. He still believed that Ulmsport was now one of the goals of the enemy fleet, and he was not well. He had taken a rheum that did not leave him, and was subject to bouts of fever and chills that were not for an armsman in the field.
Thus it was that I led those who marched under the gryphon banner when we went to the aid of our kinsmen. Jago pled to ride with me, but his old hurt was not such as would allow it, and I went with Marshal Yrugo.
My half-brother and Rogear had returned to the dale of my mother's kin, as their true allegiance was to the lord there. I was not sorry to see them ride before I left. While there had never been an open break, nor had Hlymer, after the first few days of my coming to Ulmsdale, sought to provoke me, yet I remained uneasy in their company, knowing they were no friends.
In fact I had lain but little in Ulmskeep in those days; rather moving about the dale, staying in Ulmsport, collecting information for my father while he was confined to his bed. And in this I served two purposes: not only to act as his eyes and ears, but to learn more of this land and its people where, if fortune favored, I would some day govern.
At first I was met by covert hostility, even a degree of fear, and I knew Jago's warnings had deep roots; the rumors of my strangeness had been used to cause a stir. But those who saw me during my casting up and down the dale, who reported to me, or took my orders, soon were as relaxed in my presence as they would be with any marshal or master. Jago told me after a space that those who had been in such contact now cried down any mutterings, saying that anyone with two eyes in his head could see I was no different in any manner from the next lord's heir.
My half-brother had already done some disservice to any cause he might have wished to foster by his own defects of character. I had marked him as a bully on our first meeting, and that he was. To his mind no one of lesser rank had any wit or feelings and could be safely used as a man uses a tool. No, not quite, for the expert workman has a respect for a good tool and treats it carefully.
On the other hand Hlymer was able with weapons, and, for all his bulk, he was an expert swordsman with great endurance. He had a long reach, which put him to advantage over a slighter opponent such as myself. And I do not think in those days I would have cared to meet him in a duel.
He had a certain following within the household whom he relished parading before me now and then. I never went out of my way to attach any man to me, keeping to the circumspect role that I had taken when warned by Jago. Having been reared to find my company mainly in myself, I knew none of those small openings for friendship that could have led to companionship. I was not feared; nor was I loved. Always I was one apart.
I sometimes wondered during those days what my life might have been had not the invasion come. Jago, returning from his mission to Ithkrypt, had sought me out privately and put in my hand an embroidered case less than the length of my palm, made to contain a picture. He told me that my lady begged such of me in return.
Giving him my thanks, I waited until he was gone before I slipped the wooden-backed portrait out into the light and studied the face. I do not know what I expected—save I had hoped perhaps oddly, that Joisan was no great beauty. A fair face might make her the more unhappy to come to such a one as I was after being flattered and courted. There are certain types of beauty that attract men even against their wills.
What I looked upon now was the countenance of a girl, unmarked as yet, I thought, by any great sorrow or emotion. It was a thin face, with the eyes over-large in it. And those eyes were a shade that was neither green nor blue, but a mixture of both, unless he who had limned that picture had erred.
I believed that he did not, for I think he had not flattered her. She must be here as she was in life. No, she was no beauty, yet the face was one I remembered, even when I did not look upon it. Her hair, like my own, was darker than usual, for the dalesmen tend to be fair and ruddy. It was the brown of certain leaves in autumn, a brown with a red undernote.
Her face was wider at the brow than the chin, coming to a point there, and she had not been painted smiling, but looking outward with sober interest.
So this was Joisan. I think that holding her portrait so and looking upon it made me realize in truth and sharply, for the first time, that here was one to whom my life was bound and from whom I could not escape. Still that seemed an odd way to regard this thin, unsmiling girl—as if in some manner she threatened to curb my freedom. The thought made me a little ashamed, so I hurriedly slipped the picture back into its case and thrust it into my belt-pouch to get it both out of sight and out of mind.
Jago had told me she wished one in return. Her desire was natural. But even if I desired—which somehow I did not—to honor her request, there was no way of doing so. I knew no one in the dale who had the talent for limning. And somehow I did not want to ask any question to discover such a one. So to my lady's first asked boon I made no reply. And in the passing days, each with a new burden of learning or peril, I forgot it—because I wanted to, perhaps.
But the picture remained in my pouch. Now and then I would look upon its casing, even start to slide out the picture, yet I never did. It was as if such looking might lead me to action I would later regret.
By all custom Joisan herself should have come to me before the end of the year. But custom was set aside by the rumors of war. And the next season found me fighting in the south.
Fighting—no, I could not claim to so much! My forester training made me no hero of battles, but rather one of those who skulked and sniffed about the enemy's line of march, picking up scraps of information to be fed back to our own war camp.
The early disasters, when keep after keep along the coast has fallen to the metal monsters of Alizon, had at last battered us into the need of making a firm alliance among ourselves. That came very late. In the first place, the enemy, showing an ability to farsee and outguess us that was almost as superior to ours as their weapons were, had removed through murder several of the great southern lords who had personal popularity enough to serve as rallying points for our soldiers.
There were three remaining who were lucky or cautious enough to have escaped that weeding out. They formed a council of some authority. Thus we were able to present a more united front and we stopped suffering defeat after defeat, but used the country as another weapon, following the way of battle of Waste outlaws who believe in quick strikes and retreats without losing too many of their men.
The Year of the Fire Troll had seen the actual beginning of the invasion. We were well into the Year of the Leopard before we had our first small successes. Yet about those we dared not be proud. We had lost so much more than any gain, save slamming the invaders back into the sea again, would mean. The whole of the southern coast was theirs, and into three ports poured ever-fresh masses of men. It would seem, though, that their supply of such fearsome weapons as they had used in the first assaults—those metal monsters—was limited. Otherwise we could not have withstood them as long as we did, or made our retreat north and west any more than the disorganized scramble of a terrified rabble.
We took prisoners, and from some of those learned that the weapons we had come to fear the most were not truly of Alizon at all, but had been supplied by another people now engaged in war on the eastern continent where Alizon lay. And the reason for the invasion here was to prepare the way in time for these mightier strangers.
The men of Alizon, for all their arrogance, seemed fearful of these others whose weapons they had early used, and they threatened us with some terrible vengeance when the strangers had finished their own present struggle and turned their full attention on us.
But our lords decided that a fear in the future might be forgotten now. It was our duty to defend the dales with all we had, and hope we could indeed drive the invaders back into the waves. Privately I think none of us in those days was sure that we were not living in the last hours of our kind. Still no one spoke of surrender. For their usage of captives was such that death seemed more friendly.
I had returned from one of my scouts when I found a messenger from the battle leader of this portion of the country—Lord Imgry—awaiting me with an urgent summons. Bone-weary and hungry, I took a fresh mount and grabbed a round of dried-out journey bread, without even a lick of cheese to soften it, to gnaw on while I rode.
The messenger informed me that a warning of import had been flashed overland by the torch-and-shield method, and at its coming he had been sent to fetch me.
At least our system of placing men in the heights to use a torch against the bright reflection of a shield to signal had in part speeded up the alerts across country. But how I could be involved in such a message I could not guess. At that moment I was so achingly tired my wits were also sluggish.
Of the lords who comprised our war council, Lord Imgry was the least approachable. He was ever aloof. Still his planning was subtle and clever, and to him we owed most of our small successes. His appearance mirrored what seemed to be his inner nature. His face wore a cold expression. I do not think I ever saw him smile. He used men as tools, but did not waste them, and his care for his followers (as long as they served his purposes) was known. He saw there was food for their bellies and shelter if possible, and he shared any hardships in the field. Yet he had no close tie with anyone in his camp, nor, I believed, with any of the other lords either.
Imgry was respected, feared, and followed willingly by many. That he was ever loved I could not believe.
Now, as I came into his camp, a little dizzy from lack of sleep, long hours of riding, and too little food, I tried not to stagger as I dismounted. It was a point of honor to face Imgry with the same impassive front as he himself always presented under the most harrowing conditions.
He was not as old perhaps as my father, but he was a man one could never conceive of as having been truly young. From his cradle he must have been scheming and planning, if not for his own advancement (which I suspected), then for the advancement of some situation about him. There was a fire in the landsman's rude cottage where he had his headquarters, and he stood before it, gazing into the flames as if there lay some scroll for his absorbed reading.
The men of his menie were camped outside. Only his armsman sat on a low stool polishing a battle helm with a dirty rag. A pot hung on its chain over the flames, and from it came a scent to bring juices into my mouth, though in other days I might have thought such a stew poor enough fare.
He turned his head as I shut the door behind me, to regard me with that sharp, measuring look that was one of his principal weapons against his own kind. Tired and worn as I was, I stiffened my will and went to meet him firmly.
"Kerovan of Ulmsdale." He did not make a question of that, rather a statement.
I raised my gloved hand in half-salute as I would to any of the lords commanding.
"Herewith."
"You are late."
"I was on scout. I rode from camp at your message," I returned levelly.
"So. And how went your scout?"
As tersely as I could, I told him what my handful of men and I had seen.
"So they advance along the Calder, do they? Yes, the rivers make them roads. But it is of Ulmsdale that I would speak. So far they have only landed in the south. But now Jorby has fallen—"
I tried to remember where Jorby might lie. But I was so tired it was hard to form any map picture in my mind. Jorby was port of Vastdale.
"Vastdale?" I asked.
Lord Imgry shrugged. "If it has not yet fallen it cannot hold out. But with Jorby in their hands they can edge farther north. And Ulmsport is only beyond the Cape of Black Winds. If they can strike in there and land a large enough force, they will come down from the north and crack us like a marax shell in a cook's chopper!"
This was enough to push aside the heavy burden of my fatigue. The force I had brought south with me was a small one, but every man in it had been a grievous loss to Ulmsdale. And since then there had been five deaths among our number, and three so sorely wounded they could not raise weapons now, if ever. If the enemy invaded at Ulmsport, I knew that my father and his people would not retreat, but neither could they hope to hold for long against the odds those of Alizon would throw against them. It would mean the ruinous end of all I had known.
As he spoke, Lord Imgry took a bowl from the table, scooping into it with a long-handled ladle some of the simmering stew. He put the steaming bowl back on the table and made a gesture.
"Eat. You look as if you would be the better for it."
There was little grace about that invitation, but I did not need much urging. His armsman rose and pushed his stool over for me. On that I collapsed rather than sat, reaching for the bowl, too hot yet to dip into, but, having shed my riding gloves, I warmed my chilled hands by cupping them about its sides.
"I have had no news out of Ulmsdale for—" How long had it been? One day in my mind slid into another. It seemed that I had always been tired, hungry, cold, under the shadow of fear—and this had gone on forever.
"It would be wise for you to ride north." Imgry had gone back to the fire, not turning his head toward me as he spoke. "We cannot spare you any force of men, not more than one armsman—"
It rasped my pride that he would deem me fearful of traveling without an escort. I thought that my services as scout must have proved that I could manage such a ride without detaching any force save myself from his company.
"I can go alone," I said shortly. And began sipping at the stew, drinking it from the bowl since there was no spoon offered me. It was heartening and I relished it.
He made no protest. "Well enough. You should ride with the morn. I shall send a messenger to your men, and you can remain here."
I spent the rest of the night wrapped in my cloak on the floor of the house. And I did indeed ride with the first light, two journey cakes in a travel pouch, and a fresh mount that Lord Imgry's armsman brought to me. His lord did not bid me farewell, nor did he leave me good-speed wishes.
The way north could not be straight, and not always could I follow any road if I would make speed, taking mainly sheep tracks and old cattle paths. There were times when I dismounted and led my horse, working a way along steep dale walls.
I carried a fire torch with me and could have had a fire to warm and brighten the nights I sheltered in some shepherd's hut, but I did not. For this was wild country, and we had already heard rumors that the wolves of the Waste were raiding inland, finding rich pickings in the dales where the fighting men had gone. For my mail and weapons, my mount, I would be target enough to draw such.
Mainly I spent the nights in dales, at keeps where I was kept talking late by the leaders of pitiful garrisons to supply the latest news, or in inns where the villagers were not so openly demanding but none the less eager to hear.
On the fifth day, well after nooning, I saw the Giant's Fist, that beacon crag of my own homedale. There were clouds overhead, and the wind was chill. I thought it well to speed my pace. The rough traveling was wearing on my horse, and I had been trying to favor him. But if I dropped down to the trader's road, I would lose time now, so I kept to the pasture trails.
Not that that saved me. They must have had their watchers in the crags ready for me to walk into a trap. And walk into it I did, leading my plodding horse, just at the boundaries of Ulmsdale.
There was no warning given me as there had been that other time when death had lain in ambush. So I went to what might have been slaughter with the helplessness of a sheep at butchering time.
The land here was made for such a deed, as I had to come along a narrow path on the edge of a drop. My horse threw up its head and nickered. But the alert was too late. A crashing blow between my shoulders made me loose the reins and totter forward. Then, for a moment of pure horror, I was falling out and down.
Darkness about me—dark and pain that ebbed and flowed with every breath I drew. I could not think, only feel. Yet some instinct or need to survive set me scrabbling feebly with my hands. And that urge worked also in my darkened mind, so that even though I could not think coherently, I was dimly aware that I was lying face-down, my head and shoulders lower than the rest of me, jammed in among bushes.
I believe that my fall must have ended in a slide and that those bushes saved my life by halting my progress down to the rocks at the foot of the drop. If my attackers were watching me from above, they must have thought I had fallen to my death, or they certainly would have made a way down to finish me with a handy rock.
Of such facts I was not then aware, only of my pain of body and a dim need to better my position. I was crawling before I was conscious of what I must do. And my struggles led to another slide and more dark.
The second time I recovered my senses it was because of water, ice cold with the chill of a hill spring as it washed against my cheek. Sputtering, choking, I jerked up my head, trying to roll away from that flood. A moment later I was head-down once more, lapping at the water, its coldness adding to my shivering chill, but still clearing my head, ordering my thoughts.
How long I had lain in my first fall I did not know, but it was dark now, and that dusk was not a figment of my weakened brain I was sure. The moon was rising, unusually bright and clear. I pulled myself up to a sitting position.
It had not been Waste outlaws who had attacked me, or they would have come to plunder my mail and weapons and so finished me off. The thought awoke a horror in me. Had Lord Imgry's suggestion already come to a terrible conclusion here? Had the invaders moved in to occupy Ulmsdale, and had one of their scout parties met me?
Yet that attack had so much of an ambush about it, had been delivered in so stealthy a fashion, that I could not believe it had been launched by the enemies I had faced in the south. No, there was something too secret in it.
I began to explore my body for hurts and thought I was lucky that no bones seemed to be broken. That I was badly bruised and had a lump on my head was the worst. Perhaps my mail and the bushes in which I had landed had protected me from worse injury. But I was shaking from shock and chill, and found when I tried to drag myself to my feet I could not stand, but had to drop down again, clutching to a rocky spur to steady myself.
There was no sign of my horse. Had it been taken by those who had thrown me over? Where were they now? The thought that they might be searching for me made me fumble to draw sword and lay it, bare-bladed, across my knee. I was not too far from the keep. If I could get to my feet and get on I would reach the first of the pasture fields. But every movement racked me so with pain that my breath hissed between my teeth, and I had to bite down upon my lower lip until I tasted my own blood before I could steady myself.
I had been much-favored by fortune in escaping with my life. But I was in no manner able to defend myself now. Therefore, until I got back a measure of strength, I had to move slowly and with all caution.
What I heard were the usual night sounds—birds, animals, such as were nocturnal in their lives. There was no wind, and the night seemed to me abnormally still, as if waiting. Waiting for what—or whom?
Now and then I shifted position, each time testing my muscles and limbs. At last I was able to struggle to my feet and keep that position, in spite of the fact that the ground heaved under me. The quiet, except for the continued murmur of the water, continued. Surely no one could come near without revealing himself.
I essayed a step or two, planting my boots firmly on the rocking ground, looking ahead for hand-holds to keep me upright. Then I saw a wall, the moon making its stones brightly silver. Toward this I headed and then along it, pausing ever to listen.
Soon I reached a section without cover, and there I was to my hands and knees, creeping along the stones, still alert to all around me.
Some distance away sheep grazed, and that peaceful sight was reassuring. Had there been raiders in the dale, certainly this field would have been swept bare. Or were those real sheep? The wintertime tales of the landsmen came to mind, of phantom sheep and cattle coming to join the real. And of how on certain nights or misty mornings, no herder could get the same count twice of his flock. If that were the case, then he could not, above all, return them to the fold; for to pen the real and the phantom together was to give the phantom power over the real.
I pushed aside such fancies and concentrated on the labor at hand: to win the end of the field and wall. And then to head for the keep.
When I did reach the end of the wall I could look directly at the keep where it stood on its spur base jutting out over the road to Ulmsport. In the moonlight it was clear and bright, light enough to let me see the lord's standard on its tower pole.
That did not seem to be as it should. And then, as if to make all plain, there came a light wind from the east, lifting the edge of what hung on the pole, pulling out to display the standard widely if only for a long moment—but enough to let me see.
I do not know whether I uttered any sound or not. But within me there was a cry. For only one reason would a lord's banner ever hang at night, tattered, in such ragged strips. And that was to signify death!
Ulmsdale's banner slashed, which meant that my father was—
I caught at the wall against the weakness that strove to bring me to my knees.
Ulric of Ulmsdale was dead. Knowing that, I could guess, or thought I could, why there had been an ambush set up in the hills. They must have been expecting me. Though if my father's death had been sent as a message, it had missed me on the way. Those who wanted to prevent my arrival must have had men at every southern entrance to the dale to make sure of me.
To proceed now might well be to walk into dire danger which I was not yet prepared to face. I must make sure of my path before I ventured along it.
8
Joisan
Though I had willed Toross and his kinswomen to be out of Ithkrypt, their going was not so easily accomplished, for Toross still kept to his bed. Nor could I suggest that he be taken away by litter. But I did avoid his chamber. That I had garnered the ill will of Islaugha and Yngilda went without saying. Luckily there were duties enough to keep me out of their way.
In riding skirt, with a packet of cheese and bread for my nooning, I rode with an armsman in the morning, inspecting the fields; visiting our outposts in the hills. I wore mail now and that sword my uncle had given me, and none raised their voices to say such a guise did not become me, for these were times when each turned hand to what must be done.
A sickness had come upon us without warning, bringing fever and chills and deep, racking coughs. By some favor of the Flame I escaped the worst of this, and so into my hands came more and more authority. For Dame Math was one of the early stricken. And, while she also left her bed among the first, she was plainly weakened, though she attended to her duties with little care for herself.
Marshal Dagale was also among the sick, and during those days his men turned to me for orders. We manned the lookout posts as best we might and tried also to get in the crops. It was a hard season, for there was much to be done and few on their feet for the doing of it. Days and nights were lost to me in a general sea of weariness from which there was no rest.
All who could labor, did. Even the little children dropped seeds into the waiting furrows left by plows their mothers guided. But we could only do so much, planting less than the year before. By midsummer day, instead of celebrating by a feast, I rather selected those who must leave us for Norsdale and saw them off, mainly afoot, for we could not spare mounts.
Toross did not go with them. His hurt had mended now enough so he could get about, and I hoped he would have the courtesy to leave. But he did not. Rather he fell into companionship with Dagale, acting as his second in command when the Marshal once again took up his duties.
I was never happy during those weeks. Though Toross did not seek me out, yet I felt his eyes ever upon me, his will like an invisible cord striving to draw me as he wished. I could only hope that my will to resist was as strong. I liked Toross for himself, as I had from our first meeting. In those days he had had a gaiety of temper that was in contrast to the somber life I had always known. He was gentle and considerate and talked amusingly. His face was comely, and he knew well how to make himself agreeable in company. I had seen the eyes of the maids in the household follow him and had also felt his charm.
His wound had sobered him somewhat. Still he lightened our hearts in those days, and I did not deny that he had much to give. But his quiet confidence that I would go to him, yield—that I could not understand.
I know that dalesmen look upon women as possessions, perhaps to be wooed and indulged for a season, and then, once won, to be a part of the household like a hawk, hound, or horse. We are bartered by our kin for our dowry rights, for alliances between dales. And in such matters we have no voice to oppose what we may fear or hate.
For a woman to set herself up in opposition to any alliance made for her is to suggest she may have some commerce with a dark power. And if accused of that she can be in dire danger, even from those to whom she has the closest blood ties. But this does not make it an easy portion to swallow.
My way had been relatively easy in such matters—until now. First, Dame Math was a woman of presence and spirit, one of the Dames who had the respect of men and a place of her own in the dales. Her brother had made her the head of his household and deferred to her, taking her counsel in many matters.
Being discreet, she had worked within the frame of custom, not in open opposition to it. She had seen to it that I learned much that was forbidden or deemed unnecessary for most maids. I could read and write, having been tutored at the House of Dames. And I had not been set to small tasks elsewhere when she and Lord Cyart conferred about important matters, but had been encouraged to listen. Dame Math sometimes thereafter quizzed me as to the decision I might have made on this matter or that, always impressing on me that such knowledge was needful for a lady of a keep.
My uncle had taken obedience to his decrees as his right, but in addition he had often explained the reason for them, not given orders only, though he had a nasty temper and could be sharp. But as I grew older, he asked my will in small matters, and allowed me to have it.
I knew that there was speculation among the dalesmen concerning me. I had heritage from my father, but not in land, as he was second son and half-brother to Cyart. Cyart could, by custom, name me heir, even though I was a girl, but the choice could also fall on Toross because of his sex.
Until the southland had been overrun, Toross had been heir in the direct line to his father's dale. Now he was as lacking in lordship as any second or third son. And his continued assumption that I would come to him was, a small nagging doubt told me, perhaps not because he was moon-struck with my person (for I had no such vanity) but that he might so have a double claim upon Cyart as heir-to-be.
Perhaps I did him wrong in that, but that this thought moved in Islaugha I am sure. It made her try to veil her dislike of me and strive rather to throw us together and foster a closer relationship. I began to feel much as a hare coursed by two hounds during those summer days, and clung closer and closer to duties I could use as a screen.
The leaving of the first refugee party at midsummer was a relief, though not as great a one as I had hoped for. I had an additional worry concerning Dame Math. Though she kept to her tasks, I was well aware that she tired very easily; that beneath her coif her face grew thinner, her skin more transparent. She often kept her hands clasped tight about her prayer hoops now, and, in spite of that tight grip, her fingers shook in a way she could not control.
I spent all the time I could with her, between us a question that neither of us, I believe, wanted to ask or answer. But she talked more than she had in all the years before, as if she had a very short time left in which to impart to me so much. The lore of healing and herbs I already knew a little, since that had been a part of her lessons since my childhood. Besides that she spoke of other things, some of it strange hearing, and so I learned much of what perhaps was seldom shared between one generation and the next.
That we lived in a haunted land we all knew, for a person need only turn head from one side to another to sight some remnant of the Old Ones. They were old dangers that could be stirred into life by the unwary—that, too, all knew. Children were warned against straying—venturing into places where there was an odd stillness, more like waiting than abandonment.
I was drawn into the edge of a secret that was not of my seeking, nor of Dame Math's save that duty, by which she ordered her life, urged it on her. The Flame to which the Dames gave homage was not of the Power as the Old Ones knew it. And at most times those who professed the Flame shunned what lay in the hills, invoking their own source of Power against the alien one. But it seemed that even one of the House of Dames could be driven in times of stress to seek aid elsewhere.
She came to me early one morning, her poor face even more worn and haggard as she stood plucking nervously at her prayer hoops, gazing over my head at the wall as if she did not want to meet my eyes.
"Joisan, all is not well with Cyart—"
"You have had a message?" I wondered why I had not heard the way horn. In those days such were always used between friend and friend on the approach to any keep.
"None by word of mouth, or in runes," she answered slowly. "I have it here." She allowed the hoops to dangle from her belt-chain and lifted her thin fingers to smooth the band across her forehead.
"A dream?" Did Math share that strange heritage?
"Not as clear as a dream. But I know ill has come to him, somewhere, somehow. I would go to the moon-well—"
"It is not night, and neither is it the time of the full moon," I reminded her.
"But water from that well can be used—Joisan, this I must do. But—but I do not think I can go alone—that far—"
She swayed and put her hand to the wall to steady herself. I hurried to her, and her weight came against me so I had trouble guiding her to a stool.
"I must go—I must!" Her voice rose, and there was in it an undercurrent of alarm that frightened me. When one who has always been rock-firm becomes unstable, it is as if the very walls are about to topple.
"You shall. Can you ride?"
There were beads of moisture showing on her upper lip. Looking at her straightly in that moment I saw that Dame Math had become an old woman. As if overnight all the weight of years had crushed down upon her, which was as frightening as her unsteadiness.
Some of her former determination stiffened her shoulders, brought her head upright again.
"I must. Get me one of the ponies, Joisan."
Leaning on me heavily, she came out into the courtyard, and I sent a stable boy running for a pony: those placid, ambling beasts we kept mainly for the carrying of supplies. By the time he returned, Dame Math was as one who has sipped a reviving cordial. She mounted without too much difficulty, and I led the pony across the fields to that very well where I had once slipped in the nighttime to ask a question of my own.
If any marked our going, they left us alone. The hour was early enough so that I think most were still at their morning food. As I tramped beside the pony, I felt the pinch of hunger at my own middle.
"Cyart—" Dame Math's voice was hardly above a whisper, yet it was as if she called and hoped to have an answer to her calling. I had never thought much about the tie between those two, but the ring of that name, uttered in her voice, told me much at that hour. For all their outward matter-of-fact dealing with each other there was deep feeling too.
We came to the well. When I had been there at night before, I had not seen clearly those traces by which others had often times sought out a sign of the Power that was said to be there. There were well-worn stones rimming in the well, and beyond those, bushes. To the bushes things were tied. Some were merely scraps of ribbon, color lost through the action of wind and weather. Others were crudely fashioned of straw or twig—manikins or stick horses, sheep—all twirling and bobbling within sight of the water, perhaps set there to signify the desire of the petitioners.
I helped Dame Math from the pony's back, guiding her forward a step or two until she pulled free from my aid and walked as might one who needed no help. With her goal in sight, a semblance of strength flowed back into her.
From a deep skirt pocket she brought forth a bowl no larger than could be fitted into the hollow of her hand. It was of silver, well-burnished. And I remembered that silver was supposed to be the favorite metal of the Old Ones, just as opals, pearls, jade, and amber were their jewels.
She gestured me to stand beside her and pointed to a plant that grew at the lip of the well itself. It had wide leaves of dark green veined with white, and I did not remember ever seeing its like before.
"Take a leaf," she told me, "and with it dip to fill this bowl."
The leaf, pinched, gave forth a pleasant aroma, and it seemed to twist almost of its own will into a cup, so I might spoon water into the bowl. The water in the well was very high, its surface only a little below the level of its stone rim. Three times did I dip and pour before she said, "Enough!"
She held the bowl between her hands and raised it, blowing gently on the liquid within so it was riffled by her breath.
"It is not water of the Ninth Wave, which is the best of all for this purpose, but it will do."
She ceased to puff, and the water was smooth. Over it she gave me one of those compelling looks that had always brought my obedience.
"Think of Cyart! Hold him as a picture in your mind."
I tried to draw a mind picture of my uncle as last I had looked upon him, and when he had drunk the stirrup cup of my pouring before riding south. I was surprised that the months between had dulled my memory so quickly, because I found it hard to recall him with any clarity. Yet I had known him all my life long.
"There is that about you"—Dame Math looked at me narrowly—"which gainsays this. What do you have on you, Joisan, which obstructs the Power?"
What did I have about me? My hand went to my bosom where the crystal gryphon lay in hiding. Reluctantly, urged to this by the stern eyes of Dame Math, I brought forth the globe.
"Hang it over there!"
Such was her authority that I obeyed her, looping the chain near one of those straw people lashed to a branch. She watched and then turned her gaze again to the bowl.
"Think of Cyart!" she demanded once again.
Now it was as if a door opened and I could see him, clear in every detail.
"Brother!" I heard Dame Math cry out. Then there were no more words, only a desolate sound. She stared down into the small bowl, her face very bleak and old.
"So be it." She took one step and then another, turned over the bowl and let the water splash back into the well. "So be it!"
Harsh, startingly clear, a sound tore the morning air, the alarm gong from the keep tower! That which we had feared for so long had come upon us—the enemy was in sight!
The pony whinnied and jerked at its tether, so I reached for the reins. As I struggled to control the frightened animal, the gong continued to beat. Its heavy ring sounded in echoes from the hills like the thunder of a rising storm. I saw Dame Math hold out the bowl as if offering it to some unseen presence, allowing it to drop into the well. Then she came to me.
The need for action was like youth poured into her frail body. Yet her face was one knowing hope no longer, looking forward into a night without end.
"Cyart has dreamed his final dream," she said, as she mounted the sweating pony. Of him she did not speak again; perhaps because she could not. For a moment or two I wondered what she had seen in the bowl. Then the alarm shook everything from my mind save the fact that we must discover what was happening at the keep.
The news was ill indeed, and Dagale broke it to us as he marshaled his men for what all knew could be no defense, only a desperate attempt to buy time for the rest of us. The invaders were coming upriver, the easiest road to us from the coast. They had boats, our scouts reported, that were not sailed or oared, but still moved steadily against the current. And we of Ithkrypt had little time.
We had long ago decided that to remain in the keep and to be battered out of it was deadly folly. It was better for those who could not fight to take to the hills and struggle westward. So we had even rehearsed such retreats.
At the first boom of the gong, the herdsmen had been on the move, and the women and children also, riding ponies or tramping away with their bundles, heading west. I went swiftly to my chamber, pulled on with haste my mail coat and my sword, and took up my heavy cloak and the saddle bags in which I had packed what I could. Yngilda was gone, garments thrown on the floor, her portion of our chamber looking as if it had already been plundered.
I sped down the hall to the short stairs up to Dame Math's room. She sat in her high-backed chair, resting across her knees something I had never seen in her hands before, a staff, or wand. It was ivory white, and along its surface were carven runes.
"Dame—your cloak—your bag—" I looked about me for those that we were to have ever-ready. But her chamber was as it had always been; there was no sign she meant to quit it.
"We must be off!" I hoped she was not so weak she could not rise and go. I could aid her, to be sure, but I had not the strength to carry her forth.
She shook her head very slowly. Now I saw her breath came in gasps as she could not draw enough air into her laboring lungs.
"Go—" A whispering voice came with visible effort from her. "Go—at once—Joisan!"
"I cannot leave you here. Dagale will fight to cover our going. But he will not hold the keep. You know what has been decided."
"I know—and—" She raised the wand. "For long I have followed the Flame and put aside all that I once knew. But when hope is gone and the heart also, then may one fight as best one can. I do now what I must do, and in the doing perhaps I may avenge Cyart and those who rode with him." As she spoke, her voice grew stronger word by word, and she straightened in her chair, though she made no effort to rise from it.
"We must go!" I put my hand on her shoulder. Under my touch she was firm and hard, and I knew that, unwilling, I could not force her from her seat.
"You must go, Joisan. For you are young, and there may still be a future before you. Leave me. This is the last command I shall lay upon you. Leave me to my own reckoning with those who will come—at their peril!"
She closed her eyes, and her lips moved to shape words I could not hear, as if she prayed. But she did not turn her prayer hoops, only kept tight hold on the wand. That moved as if it had the power to do so of itself. Its point dropped to the floor and there scratched back and forth busily as if sketching runes, yet it left no marks one could see.
I knew that her will was such she could not now be stirred. Nor did she look up to bid me any farewell when I spoke one to her. It was as if she had withdrawn into some far place and she had forgotten my existence.
Loath to go, I lingered in the doorway, wondering if I could summon men and have her carried out by force, sure she was not now responsible in word or deed. Perhaps she read my thought in my hesitancy, for her eyes opened wide once again, and in her loose grip the wand turned, pointed to me as a spear might be aimed.
"Fool—in this hour I die—I have read it. Leave me pride of House, girl, and let me do what I can to make the enemy sorry he ever came to Ithkrypt. A blood-debt he already owes me, and that I shall claim! It will not be a bad ending for one of the House of the Broken Sword. See you do as well when your own time is upon you, Joisan."
The wand twirled as it pointed to me. And I went, nor could I do otherwise, for this was like a geas laid upon me. A will and power greater than my own controlled me utterly.
"Joisan!" The gong no longer beat from the watch-tower, so I heard that call of my name, "Joisan, where are you?"
I stumbled down the steps and saw Toross standing there, his war hood laced in place, only a portion of his face visible.
"What are you waiting for?" His voice was angry and he strode forward, seized me by the shoulder and dragged me toward the door. "You must mount and ride—as if the night friends themselves were upon us—as well they may be!"
"Dame Math—she will not come—"
He glanced at the stair and then at me, shaking his head.
"Then she must stay! We have no time. Already Dagale is at arms on the river bank. They—they are like a river in flood themselves! And they have weapons that can slay at a greater distance than any bolt or arrow can fly. Come—"
He pulled me over the doorsill of the great hall and into the open. There was a horse there, a second by the gate. He half-threw me into the saddle.
"Ride!"
"And you?"
"To the river, where else? We shall fall back when we get the shield signal that our people are in the upper pass. Even as we planned."
He slapped my mount upon the flank so that the nervous beast made a great bound forward and I gave all my attention to bringing it once more under control.
I could hear far-off shouting, together with other sounds that crackled, unlike any weapon I could imagine. By the time I had my horse again under control, I could see that Toross was riding in the opposite direction toward the river. I was tempted to head after him, only there I would have been far more of a hindrance than a help. To encourage those who fled, to keep them going, was my part of the battle. Once in the rougher ground of the heights, we would split apart into smaller bands, each under the guide of some herder or forester and so, hopefully, win our way westward to whatever manner of safety would be found in High Hallack now.
But before I came to the point where the trail I followed left the dale bottom a stab of memory caught me. The crystal gryphon—I had left it snared on the bush beside the well! And I had to have it. I swung my mount's head around, sending him across a field of ripe grain, not caring now that he trampled the crop. There was the darker ring of trees marking the well-site. I could pick up the gryphon, angle in a different direction, and lose very little time.
Taking heed of nothing but the trees around the well and what I had to find there, I rode for it and slid from the saddle almost before the horse came to a full halt. But I had enough good sense to throw my reins over a bush.
I pushed through the screen of growth, setting jogging and waving many of those tokens netted there. The gryphon—yes! A moment later it was in my hand, safe again. How could I ever have been so foolish as to let it go from me? I could not slip the chain over my mail coif and hood, but I loosened the fastening at my throat long enough to thrust my treasure well within.
Still tugging at the lacings to make them fast, I started for my horse. There was a loud nicker, but I was too full of relief at finding the gryphon to pay the heed I should have to that. So I walked straight into danger as heedlessly as the dim-witted.
They must have seen me ride up and set their trap in a short time, favored by the fact that I was so intent upon the bauble that had brought me here. As I reached for the reins of my horse, they rose about me with a skill suggesting this was not the first time they had played such a game. Out of nowhere spun a loop that fell neatly over my shoulders and was jerked expertly tight, pinning my arms fast. I was captive, through my own folly, to those of Alizon.
9
Kerovan
So my father was dead, and I had been left for dead. Who now ruled in Ulmskeep? Jago—my mind fastened on the only friend I might now find within those walls ahead. During the months I had spent here as my father's deputy, I had acceptance but no following to which I might look now for backing. But I must somehow learn what had happened.
I drew into a screen of brush at the fence corner. The night wind was chill, and I shivered, being unable to stop that trembling of my body any effort produced. The keep would be closed at this hour except for—
Now I could think more clearly. Perhaps the shock of seeing the tattered banner had cleared my head. There was the Escape Way—
I do not know what brought our forefathers up from the south. They left no records, only a curious silence concerning the reason for their migration. But the fortifications they built here, their way of life, hinted that they had lived in a state of peril. For the petty warrings they engaged in here after their coming could never have been so severe as to necessitate the precautions they used.
They did not have to fight against the Old Ones for the dales. Why then the keeps—one strong one built in each dale—with those secret exit points known only to each lord and his direct heir? As if each need look forward to some time of special danger when such a bolt-hole would be in need.
Therefore, Ulmskeep had an entrance open to me, my father having shown it to me secretly late one night. I had a way into the heart of what might now be enemy territory and, if I were to learn anything, that I must take. There was this also—I licked my lips tasting blood, a sorry drink for me—there was this: perhaps the last place they would search for me would be within that grim building with its tattered, drooping banner.
I took my bearings from the keep and began to move with more surety now that I had a goal in mind, though I did not relinquish any of my care not to be seen. It was some distance I must go, working my way carefully from wall to wall, from one bit of cover to the next. There were lights in the keep windows and in those of the village. One by one those winked out as I kept on at a snail's pace, for I had schooled myself to patience, knowing that haste might betray me.
A barking dog at a farmhouse, well up-slope, kept me frozen with a pounding heart until a man shouted angrily and the brute was still. So it took me some time to reach the place I hunted.
Ulmsdale was freer of those relics of the Old Ones than most of the northern dales. In fact it was only here, in the shadow of the Giant's Fist, that there were signs any had found their way into this valley before the coming of my own race. And the monument to the past was not an impressive one—merely a platform leveled among the stones of these heights, for what purpose no man might say.
The only remarkable thing about this smoothed stretch of stone was that deep-carven in it was that creature from which the first lord of Ulmsdale had taken his symbol—a gryphon. Even in this uncertain light the lines of the creature's body were clear enough to give me the bearings I needed.
So guided, I scrambled up the slope a little farther, my bruised, stiff body protesting every action, until I found that place in the wall of the valley where care had been taken generations ago to set stones about a cunningly concealed break.
I edged past those into a dark pocket. Until that moment I had not realized the difficulties of this path without a light. Drawing my sword, I used it to sound out walls and footing, trying to remember as clearly as I could what lay before me now.
All too soon the sword met empty space, and I had found my destination. I sheathed my blade and crouched to feel about with my bare hands. Yes, this was the lip of the vent down which I must go. I considered the descent. In the first place the boots fashioned to hide my feet were built only for ordinary service. I distrusted them when I had to use toe holds in the dark. In fact I was not even sure my hoofed feet would serve there, but at least they would be better free. So I wrested off my boots and fastened them to my belt.
The substance of my hoofs was not affected by the chill of the stone, seeming not to have the sensitivity of flesh, and somehow with my feet free I felt secure enough to swing over and test beneath me for footholds. I need not have worried; my hoofs settled well into each and, heartened, I began the descent. I could not recall how deep I must go. In fact when my father had brought me hither we had not climbed this; he had only shown it to me from below.
Thus I went down into the dark, and the space seemed to be endless. It was not. A reaching hoof touched solid surface, and very cautiously I placed the other hoof beside it. Now—a light—
Fumbling in my pouch, I brought out my strike-light, keeping it ready in one hand while I felt along the wall with the other. My fingers caught at a knob of wood. I snapped the light, and the torch flared, dazzling me with sudden illumination.
Not stopping to put on my boots, for I relished more and more the freedom of my hoofs, hitherto so cramped by concealment, I started along a downward-sloping way which would bring me under the dale-floor to the keep. It was a long way and, I think, more than half of it was a natural fault, perhaps the bed of some stream diverted by nature or man. The roof was low, and in several places I went to hands and knees to pass.
But here I did not have to fear discovery, and I made the fastest pace I could over the sand and gravel. The slope went sharply downward for a space; then it leveled out, and I knew that I was now in the valley. The keep could not be too far ahead.
My torch shone on a break in the wall of the passage, crude steps going up at a steep angle—though the passage kept on—into a sea-cave of which my father had told me. I thus had two ways of escape.
I began to climb, knowing this stair was a long one. It went up not only through the crag on which the keep was built, but within the wall of that to my father's own chamber. Halfway up I paused and rubbed out the torch on the wall. Now I needed both hands for the holds here, and there were peepholes along the way where light might betray me.
The first of these was in the barracks. A cresset burned low against the far wall, leaving the room much-shadowed. There were some men asleep here, but only a handful.
I climbed again and looked now into the great hall from a position somewhere behind my father's high seat. There was a fire on the hearth which was never allowed to go out. A serving-man nodded on a bench near it and two hounds were curled up there—nothing else. This was normal enough at this hour.
The end of the passage was before me, and I could no longer put off reaching it—though I dreaded what might lie ahead.
Men freely use the word "love" to cover both light emotions, such as affection and liking, and viler ones such as lust, or strong attachments that last through life. I had never been one to use it at all—for my life in youth had been devoid of much emotion—fear, awe, respect were more real to me than "love." I did not "love" my father. In the days that I had spent with him after he had given me public acknowledgment, I respected him and was loyally attached to his service.
Yet there always stood between us the manner of my upbringing; that I had been hidden away. Though he had come to see me during those years, had brought me small gifts such as boys delight in, had provided well for me, yet I had always sensed in him an uneasiness when we were together. I could not tell if that came from his reaction to my deformity, or whether he reproached himself for his treatment of me and yet could not bring himself to defy my mother's feelings to name me openly son. I knew only, from very early years, that our relationship was not akin to that of other fathers and sons. And for a long time I thought that the fault was mine, so I was ashamed and guilty in his presence.
Thus we built a wall stone by stone, each adding to it, and we could not break it down. Which was a great loss, I know, for Ulric of Ulmsdale was such a man as I could have "loved," had that emotion ever been allowed to grow in me. Now as I went to his chamber through the dark of this hidden way, I felt a sense of loss such as had never emptied me before. As if I had once stood at the door of a room filled with all the good things of this world and yet had been prevented from entering in.
My hand was on the latch of the panel that opened inward, concealed by the back of the huge, curtained bed. I inched this open, listening. Almost I swung it shut again, for I heard voices and saw the gleam of lamplight. But I remembered that so well-concealed was this way, that unless I crawled around the bed to boldly confront the speakers, they would not know of my presence. And certainly this was a chance to learn exactly what might be going on. Thus I squeezed through the door and edged around the head of the bed, the stiff, embroidered folds of the curtains providing excellent cover, until I found a slit to let me see as well as hear.
There were four in the room, two using for a seat the long chest against the wall; one on a stool; and the last in the high-backed chair in which my father had sat when I bid him farewell.
Hlymer and Rogear. On the stool a girl. I caught my breath, for her face—leaving out those points of difference that were due to her sex—could have been my own! And on the chair—I had no doubt that for the first time in my life I was looking upon the Lady Tephana.
She wore the ashen gray robes of a widow, but she had thrown back the concealing veil, though the folds of it still covered her hair. Her face was so youthful she could have been her daughter's elder sister by only a year or two. There was nothing in her features of Hlymer. By her cast of countenance I was indeed her son.
I felt no emotion, only curiosity, as I looked at her. Since I had reached the age of understanding, I had been aware that for all purposes of living, I was motherless, and I had accepted that fact. She had not even kin-tie to me as I watched her now.
She was speaking swiftly. Her hands, long-fingered, and with a beauty that drew the eyes, flashed in and out in quick gestures as she spoke. But what I saw and resented, was that on her thumb was the gleam of my father's signet, which only the ruler in Ulmsdale had the right to wear and which should have weighted my own forefinger at this moment.
"They are fools! And because they are fools, should we be also? When the news comes that Kerovan has been killed in the south, then Lisana will be heir, and her lord"—she nodded to Rogear—"will command here in her name. I tell you that these invaders offer good terms. They need Ulmsport, but they do not want to fight for it. A fight will gain nothing for us, for we cannot hold long against what they can land. Who gains by death and destruction? The terms are generous; we save this valley by such bargaining—"
"Willingly will I be Lisana's lord and Ulmsdale's," Rogear answered, as she paused for breath. "As to the rest—" He shook his head. "That is another matter. It is easy to make a bargain. To keep it does not always suit the one in power. We can open gates but not close them again thereafter. They know just how weak we are."
"Weak? Are we? Say you that, Rogear?" Lady Tephana gazed at him directly. "Foolish boy, do you then discount the inherited might of our kin? I do not believe that these invaders have met their like before."
He was still smiling, that small, secret smile which had always led me to think that he carried within him some belief in himself that far outreached what others saw in him. It was as if he could draw upon some secret weapon as devastating in its way as those the invaders had earlier sent against us.
"So, my dear lady, you think to invoke those? But take second thoughts or even third upon that subject. What may answer comes as its own whim and may not easily be controlled if it takes its own road. We are kinsmen, but we are not truly of the blood."
I saw her face flush, and she pointed her finger at him. "Do you dare to speak so to me, Rogear!" Her voice rose higher.
"I am not your late lord, my dear lady." If she threatened him with that gesture, he did not show it. "His line was already cursed, remember, thus making him easily malleable to anything pertaining to them. I have the same countermeasures bred in me that you have. I cannot be so easily shaped and ordered. Though even your lord escaped you in the end, did he not? He named his body-heir in spite of your spells and potions—"
Her face changed in a subtle way that made me suddenly queasy, as if something had sickened my inner spirit. There was evil in this room. I could smell it; see it sweep in to fill that vessel waiting to hold it—that form of woman I refused to believe had ever given me life.
"What did you deal with, my dear lady, in that shrine when you bore my so-detested cousin, I wonder?" Rogear continued, still smiling, though Hlymer drew away along the bench as if he expected his mother to loose some blast in which he did not want to be caught. "What bargain did you make—or was it made before my cousin's birth? Did you cast a spell to bring Ulmsdale's lord to your bed as your husband? For you have had long dealings with them, and not with those on the White Path either. No, do not try that on me. Do you think I ever come here unprotected?"
Her pointing finger had been drawing swift lines. Just as Riwal, when he bade me farewell, had gestured in the air, and as I had seen thereafter a faint gleam of light marking the symbol he had traced in blessing, so did her finger leave behind a marking, or pattern. The marking was smoky-dark. Still it could be seen in the subdued light of the chamber, as if its darkness had the evil, black quality.
Rogear's hand was up before his face. He held it palm-out, and all those hand lines that we bear from birth and that are said by the Wisewomen to foretell our futures stood out on his flesh as if they had been traced in red. Behind that hand he still smiled faintly.
I heard a short, bitten-off cry from the Lady Tephana, and her hand dropped back into her lap. On her thumb the ring looked dull, as if its honest fire had been eaten out by what she had done moments earlier. I longed to free it from her flesh.
"Yes," Rogear continued, "you are not the only one, my dear lady, to go seeking strong allies in hidden places. It is born into us to have a taste for such matters. Now, having made sure that we are equally matched, let us return to the matter at hand. Your amiable son—" He paused and nodded slightly at Hlymer, though Hlymer looked anything but amiable. He sat hunched-over, darting glances first at his mother and then at Rogear, as if he feared one and had begun to hate the other.
"Since your amiable son has rid us of the other barrier standing in the way of possession of Ulmsdale, we must indeed make our plans. But I do not altogether agree that we should deal with the invaders."
"And why not?" she demanded. "Do you fear them? You, who have that"—she nodded to his hand—"to stand to your defense?"
"No, I do not fear them personally. But neither do I intend to give tamely into their hands any advantage. I believe, my dear lady, that you can indeed summon thunder from the hills to counteract any treachery that they might plan. But that which can be so summoned will not take note of selective destruction, and I do not propose to lose Ulmsdale in defending it."
"You will lose it anyway then." For the first time Lisana broke silence. "Also, dear Rogear"—there was little liking in her voice as she named him so—"we are not yet handfasted. Are you not a little beforehand in naming yourself lord here?"
She spoke coolly, and regarded him straight-eyed, measuringly, as if they were not betrothed but, rather, opponents at a gamingboard.
"True spoken, my sweeting," he agreed amiably. Had I been Lisana I would not have found that amiability pleasing, though. "Do you intend to be lord as well as lady here?"
"I intend not to be any piece in your gaming, Rogear," she returned swiftly, and there was no sign of uneasiness in her.
He stared at her as one who studied some new and perhaps unaccountable thing. I thought I saw his eyes narrow a fraction. And then he looked not to her but to her mother.
"Congratulations, my dear lady. So you have made sure of your power this way also."
"Naturally. Did you expect any less?" She laughed. Then he echoed that laughter.
"Indeed not, my dear lady. Ah, what a happy household we shall be. I can see many amusing evenings before us, trying this spell and that, testing each other's defenses."
"There will be no evenings at all," growled Hlymer, "unless we unite upon what is to be done to hold Ulmsdale. And I see little chance of that where the great lords have failed. Ulmsport is open—they need only to bring up a goodly force and land. The keep can hold out for a day, mayhap two—but—" He shrugged. "You have heard all the tales; we shall end like the rest."
"I wonder." Rogear had lost that shadow-smile. He glanced from Lady Tephana to Lisana and back again. "What if they cannot land? Wind and wave, wind and wave—"
Lady Tephana was intent, regarding him in the same searching way he had earlier looked upon Lisana.
"That takes the Power."
"Which you have in part, and my sweeting"—he nodded to the girl—"has in part, and to which I can add. Wind and wave have this advantage also. It will seem a natural catastrophe and one they will not fault us for. We shall be blameless. Follow in part your suggestion, my dear lady, but do not treat, only seem to treat. Then wind and wave—"
She moistened her lips with tongue tip. "It is a mighty summoning."
"Perhaps one beyond your powers?"
"Not so!" She was quick to answer. "But it will take the three of us truly united to do such a thing, and we must have life force to draw upon."
He shrugged. "It is a pity that we cleared the path so well of your lord's devoted followers. Hate can feed such a force, and we could have used their hatred. That grumbler Jago, for example."
"He drew steel on me!" Hlymer shrilled. "As if a broken man could touch me!"
"A broken man, no," Rogear agreed. "Had he been the man he once was—well, I do not know, brother-to-be. At any rate there are others to lend us life force. If we decide—"
Lisana had lost her cool withdrawal. I saw her eyes shine with an avid hunger.
"We will!" she cried out. "Oh, we will!"
For the first time Rogear showed a faint shadow of uneasiness, and he spoke to the Lady Tephana rather than to her daughter.
"Best curb your witchling, my dear lady. Some rush where prudence walks with double care."
Lisana was on her feet so suddenly, her stool spun away as her skirts caught it.
"Do not lesson me, Rogear! Look to your own Power, if you have as much as you claim!"
"We shall all look to our Power," Lady Tephana replied. "But such a plan takes preparation, and that we must turn to now." She arose, and Hlymer went quickly to her side, offering his arm in clumsy courtesy. It was almost, I thought, as if he would rather be in her company than Rogear's. Lisana followed, and Rogear was left alone.
My hand went to my sword hilt. What I had heard here filled me with horror, though it explained much. That these were working with Dark Power was plain, and they were not fresh come to such dealings either. That they—or at least the Lady Tephana (for never again in my mind did I think of her as my mother)—were old in such work, they had admitted. If my father had been ensorcelled, as Rogear had hinted, that explained much, and I could now forgive him all. The wall was broken—too late for me to tell him so.
What they would enter upon now was some great summoning of the Power. Perhaps it could save Ulmsdale—for their own purposes. But dared I set my own love of country against them? If they aroused some of the ancient forces of this brooding land against the enemy successfully, then, even though I hated them, still I must count them allies at this moment. So I watched Rogear go out from that chamber, and I did not challenge him.
There was one thought only in my mind. I had no idea what might be the consequences of the act they spoke of. In spite of the signs set on me at birth, I had none of the talent those three appeared to possess. There was only one in this land now from whom I could seek enlightenment, to learn whether I must let them do this to keep the dale free, or else turn what I could set against them to ensure that they fail. Which was the greater danger—the summoning of the Power or allowing the invaders foothold here? I could not judge, but perhaps Riwal, with his learning in those matters, might.
There was nothing in my father's keep now but a trap, and the sooner I was out of it the better, not only for my own safety, but perhaps for the future safety of the dale itself. However, as I went, I cherished in my mind those words concerning Jago. That Hlymer had forced him to a fight I did not doubt. And someday he would account to me for that.
By the time I reached the gryphon-marked stone on the hillside, the night was far-advanced. Weariness added to the pain of bruises and of my aching head, yet I was driven by time. The sooner I reached Riwal the better. To do so on foot would be a lengthy journey, and hunger gnawed at me.
How I might have fared had I not met the trader, I do not know. But as I skulked along the dale rim, I came upon one of those traces, not really a road, but in the summer seasons were used by hunters and traders—especially those bound for the Waste.
I had taken to cover when I heard the clip-clop of hoofs on the stone outcrop that comprised part of the trail, since I was fairly sure that, even if he who came were not an enemy, he might report me where it would do the least good. Only when I saw the manner of man who rode one rough-coated pony and led another, was I a little easier.
He did not pass, but halted directly opposite the thicket wherein I had taken cover and raised a peeled branch he carried in his hand. Not quite a staff, nor yet a whip, for it was too thick for that.
"Lord Kerovan—" He did not raise his voice. In fact he spoke in a low tone, yet the words carried clearly to where I lay.
He wore the leather and rough wool of a poor traveler. Now he pushed back his hood as if in revealing himself fully he would make himself known to me. Yet I did not recall seeing his face before.
Unlike most of the traders he was clean-jawed, his skin showing little beard-marking. And his features had a slightly strange cast, not quite those of a dalesman. His hair was clipped short and stood out from his skull very thickly, more like the fur of an animal pelt than any hair I had ever seen. Also it was brindled in color, neither gray nor brown nor black, but a mixture of those colors.
"Lord Kerovan!" He repeated my name and this time he beckoned with his wand.
I could not resist that gesture. Whether I would or no, I had to go to him, rising to my feet and pushing through the bushes that I might show myself to one who might be a deadly enemy.
10
Joisan
Captive to Alizon! All those tales of dark horror that the refugees told made me expect to be surrounded by demons as I was jerked farther into the open by the rope that bound my arms to my body. Still, these were only men, save that there was that in their faces which made my mouth go dry with fear. A quick death can be faced, but there are other things—
They spoke among themselves, laughing, and their tongue was strange. He who seemed leader among them came to me and pulled at the still-loose neck-thongs of my mail, bringing off my hood so my hair, loosened by his roughness, spilled out across my shoulders. He stroked that, and I longed for a dagger in my hand. But they were careful to make fast their noose, and I had no chance.
Thus they brought me back to Ithkrypt, or nearly there. Before we entered the courtyard where the enemy clustered, there was a flash of light from the tower, followed by a thunderous sound. Then, my head ringing from the fearful noise, near-shaken to the ground, I saw such a thing as I would not have believed. Ithkrypt's walls began to sway; great gaps appeared between the blocks of which they were built. And the walls toppled outward, catching and crushing many of the enemy, while a choking dust arose.
I heard screams and cries through that cloud, and I tried to run. But fortune was against me, for the end of the rope that bound me kept me helplessly anchored.
Had this been the work of those monsters the invaders commanded? I was sure not, for they would not so have killed their own men.
Dame Math! But how had she done this? I was dumbstruck by the evidence of such Power as I believed only one of the Old Ones could conjure; the Old Ones or some Wisewoman who had had dealings with the forbidden. Wisewoman—Dame—such were opposed. But before she had surrendered her will to the Flame and the Sisterhood that paid it homage, what else had Math been? At any rate the blood-price she had taken here was worthy of our House, as she had promised. In spite of my own fear I rejoiced that this was so. Valiant had our men always been. My own father had fallen facing five Waste outlaws, taking four of them with him. And now these from overseas would know that our women could fight also!
But this was my only chance to get away. I tugged against the rope. The murk of the dust was clearing, enough to show what held me prisoner. He who had led me here lay with the rope about his own waist, and he was face-down, a broken hunk of stone between his shoulders.
I thought him dead, and that made me more frenzied in my attempt to free myself. For to be bound to a dead man was the last horror. Yet the cord held, and I was as well-tethered as a horse at an inn door.
So they found me when they came reeling out of the rubble of Ithkrypt and pounding up from the river. Of our men there were no signs, save for three bodies we had passed on our way here. I could only believe that Dagale's force had paid for our escape time with their lives.
Only for a short space were the enemy so disorganized. I regretted bitterly that we could not have taken advantage of that; wreaked more damage among them while they were shocked by what happened. Now I did not doubt that any prisoners they might take would pay for this unexpected slaughter. Such fear rode me as to freeze my body, half-stupify my mind.
Dame Math had had the best of it. She had gone to her death, yes, but in a fitting manner. It was plain I would not be allowed such an ending, though they did not strike me down when they came upon me still bound to my captor. Instead they slashed through the rope and dragged me with them out of the ruin of Ithkrypt down to the river where there was a group of officers.
One among them had the speech of the dales, though he mouthed words gutturally. I was still so deafened by that terrible blast that I could hardly hear him. When I did not reply to his questions, he slapped my face viciously, one side and then the other.
Tears of pain spilled from my eyes and I knew shame that these should see them. I summoned what small pride was left me and tried to face them squarely and with my head up, as became a daughter of my House.
"What—was—there?" He thrust his face close to mine, and his breath was foul. He had a brush of beard on his chin, and his cheeks above that were splotched with red, his nose veined and swollen. His eyes were keen and cruel, and he was not, I was sure, a stupid man.
There was no need to conceal what I thought I knew, perhaps more reason to say it, since even these invaders must know that High Hallack held many secrets, most of them not to be plumbed by humankind.
"The Power," I said.
I think he read in my face that I spoke a truth I believed. One of his companions asked a question in that other tongue, and he made answer, though he did not look from me to the questioner. A moment later came his second demand.
"Where is the witch?"
Again I told him the truth. Though we did not use that term, I understood his meaning.
"She was within."
"Well enough." Now he did turn away and make his report to the others, and they spoke for some time among themselves.
I felt very weak and tired and wished I could drop to the ground. My head hurt still, as if the assault of sound upon my ears had injured something far inside my skull, and the despair of my captivity was a leaden cloak about me. Yet I held as well as I could to my resolve to keep my pride.
He faced me again, this time looking me up and down searchingly. In his beard his thick lips grinned in some ways like those men who had first taken me.
"You are no farm wench, not with mail on your back. I am thinking we have caught us a prize. But more of that later."
So I was allowed a respite, for at his orders I was left on the river bank, where their boats clustered and men were still leaping ashore. To my eyes they seemed as many as the stalks of grain in the fields, and there was no end to them. How could our small force have hoped to halt them even for an instant, any more than a single pebble might halt the flow of a spring flood.
There I also saw what had happened to our men. Some had fallen in battle. Those were the lucky ones. For the rest—no, I shall try to hold the doors of memory against what happened to them. I believed now the invaders were not humans but demons.
I think they took pains for me to see all this in order to break me. But in that they judged me wrong, for it stiffened rather than bent me to any will of theirs. It is not how a man dies, but how he bears that final act that has meaning. And the same is true for a woman of the dales. In me grew a coldness like the steel from the Waste, twice-forged and stronger than any other thing in our world. I swore that Dame Math was right, and I must contrive to make my own passing count against the enemy.
But it seemed they now forgot me. I was still bound, and they made the rope fast to one of the boat-chains. Men came and looked at me from time to time as if I were a prisoned beast. Their hands were on my hair and my face, and they jabbered at me in their own tongue, doubtless warning of much I would certainly rue. But none did more than that. It was coming night, and they had set up a line of campfires and were driving in sheep and cattle, slaughtering some of them.
A mounted party had gone down-dale in pursuit of our people. I besought the Flame that those fleeing had won the rugged land where their guides could lead them by routes only dalesmen knew.
Once I saw a small party return, heard women screaming for a while, and knew that some of our people had been run down. I tried to close my ears, shut off thought. This was that place of Outer Darkness come to earth, that place to which evil crawls and from which it issues forth again—and to that evil there is no end.
I tried to plan my own ending before they could turn to me for amusement. The river—could I hurl myself into the waters there? If I could edge along the chain to which my rope had been tied, surely there was a chance.
Toross—dully I wondered what had become of him. I had not seen his body with those others. Perhaps he had escaped at the last. If so, I could still find a small, unfrozen part in me that wished him well. Oddly at that thought his face was vivid in my mind, as sharp as if he stood before my eyes. Within my mail and under my jerkin something was warm.
The gryphon. That unlucky bauble which had doomed me to this state. The gryphon was growing ever warmer, almost like a brand laid to my skin. From it flooded not only that heat, but something else, a strength, a belief in myself against all the evidence that reason provided for my outer eye and ear. It was like some calm voice assuring me that there was a way out, and that my delivery was at hand, though I knew this could not be.
Fear became a small, far-off thing, easy to put aside. My sight was more acute, my hearing keener. My hearing—!
Even through the clamor of the camp I detected that sound. There was something coming—down-river!
How I knew this, I could not have told. But I knew I must be ready. Perhaps I was only dazed by fatigue, by my fear and despair. Yet I was as certain of this rescue as I was that I still lived and breathed.
"Joisan!" A thread of whisper, but to my alerted ears so near a shout, I feared all the camp might hear it.
I was afraid to answer aloud, but I turned my head a fraction in the direction from which they had come, hoping the gesture would signal my recognition.
"Edge—this—way—" The words came from the river. "If—you—can—"
Torturous and slow were my efforts to obey. I kept watch on all about me, my back to where that voice whispered, lest I somehow betray this hope. Now I felt wet hands reach up to mine, the swing of a knife against the rope. That fell away, my stiffened hands were caught and held; my rescuer was half in the water.
"Slip over!" he ordered.
I wore mail and the dragging skirt. I could not hope to swim so burdened. Yet it would be better to meet death cleanly. Now I waited as a party of the invaders tramped along the path. They did not glance in my direction. Then I rolled over and down into the water, felt hands catch and drag me even as I gulped air and went under.
We were in the grip of the current where the stream ran the swiftest, and my struggles would not surface me for long. I choked and thought this was the end. Still that other fought for me, with what poor aid I could give. We came against a rock where he clung and held my head above the flood.
His face was close to mine, and it was no surprise to see that Toross held me so.
"Let me go. You have given me clean death, kinsman. For that I thank you—"
"I give you life!" he replied, and there was stark determination in his face. "Hold you here, Joisan!" He set my hands to the rock and I held. He crawled and pulled himself up into the air, and then by main force, for I had little strength left, dragged me up beside him.
By fortune we had come across the river into rough pastureland, downstream from the ruins of Ithkrypt, and between us and the western hills was now the full invader force. Toross was shivering, and I saw he had stripped to his linen undershirt, his mail gone. There was a raw slash across his cheek where blood welled.
"Up!" He caught my arm, pulling at me. My long skirt, so water-soaked, was like a trap about my legs, and the mail was a deadweight upon me. But I stumbled forward, unable to believe that we had actually achieved this much and that the alarm for my escape had not yet sounded.
Thus we reached some rocks and fell, rather than dropped, behind them. I fumbled with the lacings of my mail, wanting to be rid of that, but Toross caught my hands.
"No, you may need that yet. We are far from out of the dragon's jaws."
Of that I needed no reminding. I had no weapon, and, as far as I could see, Toross carried none except the knife. Perhaps he had found a sword too weighty for that swim. We were reduced to that single blade and perhaps stones snatched from the ground if we were cornered.
"We must take the hills." In the growing dark he gestured up-slope. "And try to work our way around those butchers to join our people. But we had better wait for full dark."
Something in me urged action, to get as far from those fires, the noise across the river, as we could. Yet what he said made sense. To draw upon one's patience was a new ordeal, I now discovered.
"How—how did you come alive from the river battle?" I asked.
He touched that slash on his face, which had now clotted again, giving him a bloody mask to wear. "I took this in the last charge. It stunned me enough to make them think me dead. When I realized that I must be dead to escape, I played that part. Then I got away—but I saw them bring you from Ithkrypt. What happened there, Joisan? Why did they turn their destruction against their own men?"
"That was not done by the invaders; it was Dame Math. She used the Power."
For a moment he was silent and then he demanded, "But how can such a thing be? She was a Dame—a Dame of Norstead Abbey."
"It seemed that before she swore to the Flame she had other knowledge. It was her choice to call upon that in the end. Do you not think we can move now, Toross?" I was shivering in my wet clothing, trying hard to control the shaking of my body. Though it was late summer, this eve brought with it the foretaste of autumn.
"They will be waiting." He was on his knees, peering back to the river.
"The enemy—have they already climbed so high in the dale?" It appeared our small gift of fortune was fast-dwindling.
"No—Angarl, Rudo." Toross named the armsmen who had come with him out of the south. "My mother sent them back, and they were to press me into going into the hills. Had I not seen you in the hands of our enemies, the Hounds of Alizon, I might have yielded to them."
That we were not alone, if we could reach his men, gave me a spark of comfort, though both men were old, sour and dour. Rudo was one-eyed, and Angarl had lost a hand many years before.
So we began a stealthy withdrawal. I could not honestly understand why we were not sighted, though we took to all the cover the rough ground afforded. That there had come no outcry on my escape was also a mystery. I had expected them early on my trail—unless they believed I had perished in the river.
We found a narrow track winding upward, and Toross pushed the pace here. I would not admit that I found the going increasingly difficult, but strained my energy to the utmost to keep up. That he had laid me under debt to him weighed on me also. I was grateful, as anyone would be grateful for their life given back to them. Yet that Toross was the giver could cause future difficulties. But there was no reason to look forward to those—what mattered now was that we continued to claw our way through the dark.
Though I had lived in the dale all my life, I had only a general idea of where we might be heading. We needed to get west, but to do so and pass any enemy patrol, we must first head south. My wet boots became a torture to my feet. Twice I stopped and wrung out my skirt as best I could. Now it was plastered to my legs, chafing my skin.
Toross headed on and up with confidence, as if he knew exactly where he was bound, and I could only follow and trust in him.
We struck another track, faint, but at least easier footing than the way we had climbed, and this angled west. If the enemy had not combed this high, then we were heading around them. Once we heard more screams out of the night from the opposite bank of the river. Again I tried to close my ears to what I could in no way help. Toross did not falter in stride as those sounds reached us, but padded on. If they moved him to anger I did not know it. We did not talk as we went, saving all our breath for our exertions.
In spite of every effort, we could not move silently. There were scrapes of footfall on rock, the crack of a branch underfoot, the swish of our passing through brush. And after each of those small betrayals we froze to wait and listen.
Still our fortune held. The moon was rising—a full moon like a great lantern in the sky. It could show us the pitfalls before us, but it could also display us to a hunter. Toross halted. Now he caught my arm and drew me close to whisper in my ear. "We must cross the river at the traders' ford. It is the only way to reach the hill-paths."
He was right, of course, but to me that was our death blow. There was no way we could cross that well-known ford without being sighted. Even if by some miracle we could get across—why, then we had a long distance through open fields to traverse.
"We cannot try the ford; they will see us."
"Have you a better plan then, Joisan?"
"None save that we keep west on this side of the river. It is all sheep pasturage and steep hillsides where they cannot ride us down without warning."
"Ride us down!" He made a bitter sound that was not laughter. "They need only point one of their weapons at us from afar and we die. I have seen what I have seen!"
"Better such a death than to fall into their hands. The ford is too great a risk."
"Yes," he agreed. "But I do not know this way. If you do, Joisan, it will be you who will lead us."
What I knew of this upper dale was little enough, and I tried hard to call it to mind. My hope was a wooded section that was like a cloak. This had none too good a reputation with the dalesmen and was seldom entered, mainly because it had been rumored to cover some ruin of the Old Ones. Such tales were enough to ward off intruders, and, had we been fleeing any dales pursuit, to gain the edge of that wood would have given us freedom. But the invaders had no such traditions to stay them. Now I said nothing of the legend, only that I thought it would give us shelter. And if we could make our way through it, we might then continue about the rim of the dale and straight northwest to join our kin.
As we went, the effort slowed our pace. I fought the great weariness of my body, made bone and muscle answer my will alone. How it was with Toross I did not know, but he was not hurrying as we stumbled on.
The fires of the enemy camp were well behind us. Twice we lay face-down, hardly breathing, on the hillside, while men moved below, hoping we could so blend in with the earth. And each time, while my heart beat wildly, I heard them move on.
So we came to the edge of the wood, and there our luck failed us just when hope was the strongest. For there was a shout behind and a harsh crack of noise. Toross cried out and swept me on before him, pushing me into the underbrush of the ill-omened place. I felt him sag and fall, and turned to catch him by the shoulders, half-dragging, half-leading him on with me.
He stumbled forward, almost his whole weight resting on me. In that moment I thought of Dame Math. Oh, that I had her wand in my hand, the Power strong in me so that I could blast those behind.
Fire burned on my breast, so hot and fierce a flame, I staggered and loosed my grip on Toross so that he fell heavily to the ground moaning. I tore at the lacings of my mail shirt to bring out what was causing that torment.
The gryphon globe in my hand was burning hot. I would have hurled it from me, but I could not. Before me as I stood came the sounds of men running, calling out. The glow of the globe; that would reveal us in an instant! Yet I could not throw or drop it. I could only stand and hold it so, a lamp to draw death to us.
Still the running feet did not come. Rather they bore away, downhill along the fringe of the wood. I heard an excited shout or two from the lower slope—almost as if they harried some quarry. But how could they when we were here with the globe as a beacon?
My ears reported that they were indeed drawing off. I could hardly believe that was the truth. With the globe as a battle torch this thin screen of brush could not conceal us. But we were free, with the chase going away.
Toross moaned faintly, and I bent over him. There was a growing stain on his shirt, a thread of blood trickling from his half-open mouth. What could I do? We must not stay here—I was sure that at any moment they would return.
I dropped the globe to my breast, where it lay blazing. There were no burns on my flesh where I had cupped that orb against my will, though in those moments when I had held it I might have been grasping a red-hot coal.
"Toross!" To move him might do his wound great harm; to leave him here certainly meant his death. I had no choice. I must get him on his feet and moving!
The blaze of the globe lit his crumpled body. As I bent over him, setting my hands in his armpits, he stirred, opened his eyes and stared straight up, not seeming to see me at all.
As my hands tightened on him, I had a curious sensation such as I had never experienced before. Spreading out from that blazing crystal on my breast came waves of energy. They coursed and rippled down my arms, through my fingers—
Toross moaned again and coughed, spewing forth blood and froth. But he wavered upward at my pull. When he was on his feet I set my shoulder under his, drew his arm about me, and staggered on. His feet moved clumsily, and most of the weight of his body rested on me, but I managed to keep him moving.
What saved us was that, though a screen of brush ringed the wood, there was comparatively little undergrowth beneath the trees, so we tottered along, heading away from the dangers in the open. I did not know how far I could manage to half-carry Toross, but I would keep going as long as I could.
I am not sure when I noticed that we were following a road—or at least a walk of stones that gave us almost level footing. In the light of the globe, for it continued to blaze, I could see the pavement, moss-grown, but quite straight. Toross coughed again with blood following. And we came out under the moon's rays, the woods a dark wall as we stood in a paved place into which that white-silver radiance poured with unnatural force, as if it were focused directly on us with all the strength the moon could ever have.
11
Kerovan
On the hill-slope I, Kerovan of Ulmsdale, faced the wayfarer in trader's clothing, who was no trader, as I knew when his staff beckoned me out of hiding against my will. I put my hand to sword hilt as I came, but he smiled, gently, tolerantly, as one might upon a frightened child.
"Lord Kerovan, no unfriend faces you." He dropped the point of the staff.
Instantly I was freed from that compulsion. But I had no desire to dodge back and away again, for there was that in his face which promised truth.
"Who are you?" Perhaps I demanded that more abruptly than courtesy allowed.
"What is a name?" he returned. The point of his wand now touched the ground and shifted here and there, though he did not watch it, as if he wrote runes in the dust. "A traveler may have many names. Let it suffice for now that in these dales I am called Neevor."
I thought he gave me a quick, searching look, as if to see if I knew that name. But my want of understanding must have been plain to read. I thought he sighed, as if regretting something lost.
"I have known Ulmsdale in the past," he continued. "And to the House of Ulric I have been no unfriend—nor do I stand aside when one of his blood needs aid. Where do you go, Lord Kerovan?"
I began to suspect who—or what—he might be. And I was awed. But because he stood in the guise he did I felt no fear.
"I go to the forest lodge, seeking Riwal."
"Riwal—he was a seeker of roads, worshiping knowledge above all things. Though he never entered the wide door, he stood on its threshold, and those I serve did not deny him."
"You say of him 'was.' Where is he now?"
Again that wand-tip, which had come to rest, scrabbled across the earth.
"There are roads amany. Understand only that the one he had taken hence is not yours to follow."
I snatched at what might lie behind that evasion, believing the worst, because of all I had seen and heard, not only this night but in the months in the south.
"You mean he is dead! And by whose hand?" Once more that cold anger possessed me. Had Hlymer also taken this friend from me?
"The hand that dealt the blow was but an instrument—a tool. Riwal sought certain forces, and there were those who stood in opposition. Thus he was removed."
Neevor apparently did not believe in open speech, but was fond of involvements that veiled the question rather than revealed it.
"He turned to the Light, not Dark!" I spoke for my friend.
"Would I be here otherwise, Lord Kerovan? I am a messenger of those forces he sought, to which he was guiding you before the war horns sounded. Listen well. You are one poised upon a mountain peak with before you two paths. Both are dark with danger; both may lead you to what those of your blood speak of as death. It is in your fate that you can turn to either from this night onward. You have it in you to become as your kin-blood, for you were born in the Shrine of—" Did he utter some name then? I believe that he did. Yet it was one not meant to be spoken by man. I cowered, putting my hands to my ears to shut out the awful echoes from the sky above.
He watched me closely, as if to make sure of my reaction. Now his wand-staff swung up, pointing to me, and down its length came a puff of radiance that floated from its tip through the air and broke against my face before I could dodge the touch, though I felt nothing.
"Kinsman," he said, in his gentle voice, losing that majesty of tone he had held a moment earlier.
"Kinsman?"
"It seems that when the Lady Tephana wrought her bargain, she did not understand what she achieved. However, she sensed it; yes, she sensed it. You were a changeling, Kerovan, but not for her purposes. In that she read aright. She had set to fashion an encasement of blood, bone, and flesh for her use. Only the spirit it enclosed was not of her calling. It does not advantage one to take liberties with Gunnora. I do not know who looks through your eyes. I think that he yet sleeps, or only half-wakes. But the time will come when you shall remember, at least in part, and then your heritage shall be yours. No, not Ulmsdale—for the dales will no longer hold you—you shall seek and you shall find. But before that you must play out what lies here, for you are half dalesman."
I was trying to understand. Did he mean that the Lady Tephana had worked with some Power before my birth, to make my body a vessel into which to pour some manifestation of the Dark? If so my hoofed feet might be the mark. But—what was I?
"Not what you fear in this moment, Kerovan," he answered my unspoken thought swiftly. "Halfling you are, and your father's son, though he was under ensorcellment when he begot you. But where that seeker of Dark Knowledge strove to make a weapon to be used for her own purposes, she gave entry to another instead. I cannot read the rune for you. The discovery of what you truly are, and can be, you must make for yourself. You can return now, ally yourself with them, and find she cannot stand against you. Or—"
His wand indicated the barren hillside. "Or you can walk into a world where the Dark and what you call death will sniff at your heels, ever seeking a way for which there is no guide. The choice is yours."
"They speak of calling wind and wave to defeat the invader," I said. "Is this good or ill for Ulmsdale?"
"To loose any Power carries great risk, and those who strive to follow the old ways but are not of the blood, risk double."
"Can I prevent them then?"
He drew back. I thought his voice colder as he answered, "If you so wish."
"Perhaps there is a third way." I had thought of it once or twice as I climbed these slopes. "I can claim kin-right from Ithkrypt and gain a force to retake Ulmsdale before the enemy comes." But even as I spoke, I knew how thin a chance I had. Lord Cyart was fighting in the south and must have stripped his dale of forces, save for a handful of defenders. There would be none there to be spared, even if I went as a beggar.
"The choice is yours," Neevor repeated. And I knew he would give no advice.
My duty in Ulmsdale was part of my training. If I turned my back now upon my father's land, made no attempt to save those dwelling in it from disaster, either by enemy hand or the spells of that witch and her brood, then I would be traitor to all bred in me.
"I am my father's heir. I cannot turn my back upon his people. Nor do I take part in their witchery. There may be those to follow me—"
He shook his head. "Try not to build a wall out of shifting sand, Kerovan. That Dark biding within the walls of Ulmskeep has spread. No armsman will rise to your summons."
I did not doubt that he knew exactly what he said. There was that about Neevor which carried full belief. So—it must be Ithkrypt after all. At least I would find shelter there from which to gather men and support. Also I must send a message to Lord Imgry.
Neevor thrust his wand-staff through his belt. Then he turned to his led pony, tumbling from its back the small pack that had been lashed there.
"Hiku is no battle charger, but he is sure-footed in the hills. Take him, Kerovan, with the Fourth Blessing." Once more he reached for his wand and with it he tapped me lightly on the forehead, right shoulder, left, and over my heart.
Straightway then, I had the feeling my decision had pleased him. Yet I also knew that I could not be sure it was the right one. For it was laid upon me that I must choose my own road for myself and not by the advice of another.
I had forgotten my bare hoofs, but now as I moved toward the pony, my boots swung against my thigh and I caught at them. As I loosened them from my belt, ready to draw them on again, I had a strong revulsion against hiding my deformity—or was it so? It was a difference, yes, but what I had seen in my father's chamber this night, a deformity of spirit, seemed to me the greater evil.
No, I was done with hiding. If Joisan and her kind turned from me in disgust, then I was free of them. I hurled the boots from me, holding to that sense of freedom I had had since I shed them.
"Well done," Neevor said. "Be yourself, Kerovan, not ruled by the belief that one man must be like another. I have hopes for you after all."
Deliberately he urged the second pony on a few paces and then, standing with one hand on its shoulder, he drew a circle on the earth around the animal and himself. Following the point of his wand there sprang up a thin, bluish haze. As he completed the encirclement, it thickened to hide both man and beast. As I watched, it faded, but I was not surprised to see that its going disclosed emptiness—that the trader and his mount were gone.
I had guessed that he was one of the Old Ones. And that he had come to me was not by any chance. But he left me much to think on. Half-blood was I then, having kinship to the mysterious forelords in this land? I was a tool of my mother's desiring, though not to her purpose—yes, so much he told me fitted with the facts I knew and answered many questions.
I clung to the human part of me now: the fact that I was in truth Ulric's son, no matter what that sorceress had done to set it awry. That thought I cherished. For in death my father had come closer and dearer to me than ever he had been in life. Ulmsdale had been his. Therefore I would do what I could to see it safe, which meant I must ride for Ithkrypt.
I could not even be sure that the pony was of the natural order of beasts, seeing by whose hand he had come to me. But he seemed to be exactly like any other of his breed. He was sure-footed. Still there were stretches where I must go afoot and lead him.
By dawn I was well into the heights. As I made a rough camp, I lifted off my gift-mount something Neevor had not removed along with the pack, a stout hide bag. In it was a water bottle of lamantine wood. But it did not contain water, rather a white drink that was more refreshing and warming than any wine I had ever tasted. There was also a round box of the same wood with a tightly fitting cover. I worked this off, to find inside journey cakes that had been so protected by their container they seemed fresh from the griddle. Nor were they the common sort, but had embedded in them bits of dried fruit and cured meat. One of them satisfied my hunger, as great as that now was, and the rest I kept for the future.
Though sleep tugged at my eyelids and my body demanded rest, I sat for a while in a niche between two rock teeth, thinking on all that had happened to me this night. My hoofed feet stretched before me. I studied them, trying to put myself in the place of one sighting them without warning. Perhaps I had been foolish to cast away my boots. No, in the same instant as that thought entered my mind, I rejected it. This I would do and so I would go—Joisan and her people must see me as I was and accept or deny me. There must be between us no untruths or half-truths, such as had filled my father's house with a web of dark deceit and clung there now as a foul shadow.
I unlatched my belt-purse and for the first time in months brought out that case, deliberately opened it, and slid into my hand Joisan's picture. A girl's face, and one painted nearly two years ago. In that time we had both grown older, changed. What was she like, this maid with large eyes and hair the color of autumn leaves? Was she some subdued daughter-of-the-house, well-lessoned in the ways of women but ignorant of the world outside Ithkrypt's stout walls? For the first time I began to think of her as a person, apart from the fact that by custom she was as much my possession as the sword at my belt, the mail on my back.
My knowledge of women was small. In the south I had listened to the boasting tales men tell around the campfires of any army. But I had added no experience of my own. I thought now that perhaps my mixed blood, my inheritance from the Old Ones, had marked me with more than hoofs; that it had set upon my needs and desires some barrier against the dale maids. If that was so, what would become of my union with Joisan?
I could break bride-oath, but to do such would be to lay a stain on her, and such a trick would be as evil as if I stood up before her assembled House and defamed her. That I could not do. But perhaps when we at last met face to face she would look upon me and make such a repudiation, and I would not gainsay it. Nor would I allow thereafter any dale-feud to come from my dismissal.
Yet at this moment when I looked upon her face in the dawn light, I wanted to see her, and I did not relish her breaking bride-oath with me. Why had I sent her the englobed gryphon? Almost during the past months I had forgotten that—but my interrupted journey to see Riwal brought it back to mind. What had lain so heavily on me then that I had sent that wonder to her, as if such a gift was necessary? I tried to picture it in my mind now—the crystal ball with its gryphon within, a warning claw raised—
But—
I was no longer looking at the dale below. I could no longer see the pony grazing. Rather I saw—her!
She was before me so clearly at that moment that I might have reached out and touched her skirt where she crouched. Her russet hair was in wild disorder over her shoulders, and through its straying strands I could see the gleam of mail. On her breast hung the gryphon, and it blazed with light. Her face was bruised, and there was fear in her eyes. Against her knee rested the head of a young man. His eyes were closed, and across his lips bubbled the froth of blood that marked a wound from which there was no healing. Her hand touched his forehead gently, and she watched him with a tenseness that meant his life or death had meaning for her.
Perhaps this was farseeing, though I had only once had such a gift or curse set upon me before. I knew that the face of the dying man was not mine, and her sorrow was for another. Perhaps therein lay my answer. Nor could I fault her if it was. For we were naught to each other but names. I had not even sent her the picture that had been her own asking from me.
That the gryphon blazed so clearly puzzled me a little after I had schooled myself to accept the meaning of what I saw. It was as if life had poured into that globe. So—perhaps now I knew the reason why I had been so strangely moved to send it to her. Though I had been the finder and had treasured it, it was not mine to have and hold, but was meant rather to lie where it now rested, and was truly hers and not mine. I must accept that also.
How long did that farsight or vision last? I did not know. I knew only that it was true. The strange youth was dying, or would die, and she would mourn him thereafter.
But such a death argued that Ithkrypt was not the refuge I had looked to find. It was not usual for a daleswoman to wear mail like a warrior, but we did not live in ordinary times. She was armored, and her comrade was dying; to that there could be only one explanation. Ithkrypt was either attacked or soon would be. Still that knowledge did not deter me. Rather it drew me—for I had a duty also to Joisan, whether or not she would ever now turn to me happily. If she were in danger, there was even more reason I should cross the ridges to her.
Ulmsdale, once my father's and now under the hands of those I knew to be of ill intent; Ithkrypt perhaps overrun—I was traveling from one danger to another. Death was surely sniffing at my heels, ready to lay claw-hand on my shoulder. But this road was mine, and I could take no other.
The vision was gone, and with its going my weariness settled so heavily upon me that I could not fight it. I slept the day away in my hole, for when I roused it was already dusk. I wakened to die pony nudging against my shoulder, as if the beast were a sentry on guard.
Dusk—yes—and more. There was a gathering of thick clouds, such as I had seldom seen. So dark and heavy was that massing that I could not now sight the Giant's Fist! And the pony crowded in against me as I scrambled to my feet.
The beast was sweating; the smell of it was rank. He pushed his head against my shoulder, and I gentled him with neck-stroking. This was fear the like of which I had seldom seen before in any animal. Emotion gripped me also: a vast apprehension, as if some force beyond understanding gathered, a force that was inimical to all my kind and could, if it would, sweep us like grains of dust from its path.
I backed against the rock wall, my hands still on the pony, waiting. I did not know for what, except I feared it as I never had feared anything before in my life.
There was no wind, no sound. That terrible stillness added to my fear. The dale, the world, cowered and waited.
From the east there was a sudden flash of light. Not the usual lightning, but rather a wide swath across all the heavens. Eastward—over the sea—
Power of wind and wave they had spoken of—were they about to summon that? Then the invaders' ship must lie near to Ulmsport, and they had had little time to ready their plans. What would happen?
The pony uttered a strange sound such as I had never heard from any mount before. It was almost a whimper. And that oppression increased until it seemed that the very air about us was kept from our lungs and we could not breathe freely. Still there came no wind, but sheet lightning flashed seaward. Now came a long roll—as if a thousand war drums beat together.
Above the clouds was a night of such darkness I could see no more than if I were blindfolded. Surely this was no ordinary storm, at least like none I had seen before in my lifetime. My lifetime. Deep in me a thread of memory stirred—but it could not be memory—for it was not of this life but another.
But that was foolishness! A man had but one lifetime and the memories of that—one lifetime—
My skin, where it was exposed to the air, itched and burned as if the atmosphere were poisoned. Then I saw light—but not in the sky—rather auras about rocks as if they were palely burning lanterns, their light a foggy discharge.
For the third time, sheet lightning blanketed the east, and after it came the drum roll. Then followed the wind—
Wind, but such wind as I swear the dale had never felt before. I crouched between the rocks, my face buried in the trembling pony's rough mane, the smell of the beast's sweat in my nostrils. He was steaming wet under my hands. There was no way I could shut out the sound of that wind. And surely we would be scooped out of our small refuge by its force, whirled out to be beaten to death in the open.
I braced my hoofs deep in the ground, used the rock at my back and side as best I could to anchor me, and felt the pony, iron-tense in my hold, doing likewise. If the poor beast whimpered now, I could no longer hear him, for the sound of the elements was deafening. The drum beat had become a roar to which there was no end.
I could not think; I could only cower in dull hope of escaping the full fury. But as it continued I grew somewhat accustomed to it, as one can when the first sharp edge of any fear is dulled by a continuation of its source. I realized then that the wind blew from east to west, and its power must be directed from the sea upon Ulmsport.
What such a storm might do along the coast I could not imagine, save that it would utterly devastate everything within its hammer blows. If there had been an enemy fleet drawing to port, that must be completely overwhelmed. But the innocent would suffer with the invader. What of the port and those who dwelt there? If this storm was born of the Power those in the keep thought to summon, then they had either lost control of it or had indeed drawn hither something greater than they had planned.
How long did it last? I lost track of time. There was no night, no day—only black dark and the roaring—and the fear of something that was not of normal nature. What of the keep? It seemed to me that this fury could well shake even those great stones one from the other, splitting open the firm old building as if it were a ripe fruit.
There was no slacking off as would occur in a true storm. One moment the deafening roar, the fury—then silence, complete, dead. I thought at first that the continued noise, the pressure, had deafened me. Then I heard a soft sound from the pony. He pushed against me, backing into the open.
Above, it was once more dawn. The dark clouds, tattered as my father's death banner, faded into nothingness. Had it been so long we had been pent there? I stumbled after the pony into the quiet open.
The air no longer held that acridness which had tortured our breathing, but was fresh and cool. And there was a curious—I could only define it as emptiness—in it.
I must see what had happened below. That thought drove me. Leading Hiku along the narrow rim of the dale, I headed back toward the Giant's Fist. These heights had been scoured. Vast areas of trees and brush had been simply torn away, leaving scars in the earth to mark their former rooting.
So obvious were these signs of destruction, I was prepared in part for what I did sight at the foot of the heights. Yet it was far worse than I expected.
Part of the keep still stood, though its outline was not that of a complete building any longer. About it was water—a great sheet of water on the surface of which floated a covering of wreckage, perhaps part of it ships, part the houses of Ulmsport, but too tangled to be identified with any surety. And that water came from the east—the sea had claimed most of Ulmsdale.
Had those below escaped? I could see no signs of life. The village was under water save for a roof or two. So the disaster those below had wantonly summoned had fallen.
Were they caught up in the maelstrom of the force they could not control? That I hoped. But that Ulmsdale as I had known it was dead, was manifest. No man could have a future here. For I believed that what the sea had won, it would not surrender. If the invaders thought to use this as a foothold, they were defeated.
I turned my face from, that lump which had been the keep, and so from the past. In a way, I still had a duty laid upon me—I must learn how it fared with Joisan. And then—there lay the south and the long, long battles to come.
Thus I tramped away from the Fist with no desire to look again at the ruin in the dale, and my heart was sore, not for any loss of mine, for I had never truly felt that it was my holding, but for the wreckage of all my father had cherished and sought by every means he knew to protect. And I think I cursed as I went, though silently, those who I had done this thing.
12
Joisan
As we stood under the moon in that secret place of stone, the gryphon blazing on my breast, Toross slipped from my hold to the ground. I knelt beside him, drawing the garment from his chest that I might see his hurt. His head lay against my knee, and from his lips a stream of blood trickled. That he had come this far was a thing hardly to be believed when I saw the wound exposed. Enough of Dame Math's healing knowledge was mine to know that it was indeed a death blow, though I slit my underlinen with his knife and made a pad to halt the seepage.
I gentled his head against me. In so very little could I ease the passing of this man who had given his life that I might live. In the light of the gryphon and the moon I could well see his face.
What twisting of fate had brought us two together? Had I allowed myself, I might have wanted to joyfully welcome Toross for my lord. Why had I not?
In the library of the Abbey I had found many curious pieces of lore not generally taught, perhaps considered mysteries of the Flame. And one such roll of runes I now remembered—that a man—or woman—does not live a single life, but rather returns to this world at another time for the purpose of paying some debt that he or she owes to another. Therefore in each life one is bound to some other by ties that are not of this life and time, but reach far back into a past no seer can delve. Toross had been drawn to me from the first, so much so that he had nearly dimmed the honor of his House to seek me out, to urge upon me a similar feeling.
Though I had held fast against him, yet he had come here to die in my arms, because my life meant more to him than his own. What debt had he owed me, if that old belief were true? Or had he now laid some debt upon me that must be paid in turn?
His head moved in my hold. I leaned close to hear his whisper:
"Water—"
Water! I had none. To my knowledge the closest lay in the river a long distance from us. I took up my skirt, still heavily dampened, and wiped his face, wishing bitterly that I could give him such a little thing. Then I saw in the dead-white radiance, which seemed so intense in this place, that plants grew about the paved space. Tall as my shoulder they stood, with great, fleshy leaves outspread in the moonlight. And on those were drops of silver. I recognized a plant Dame Math had used. Yet hers had been very small compared to these. These plants had the art of condensing water on their leaves with the coming of night's cool.
Gently I laid Toross down and went to gather this unexpected boon, tearing off the largest leaves with care, lest I spill their precious cargo. And I brought them to wet his lips, eased a few drops into his mouth. So little it was that I despaired, but perhaps the leaves had some healing quality they imparted to those droplets, for these seemed to satisfy his thirst.
I took him up again, and as I settled his head against me, his eyes opened and he knew me. He smiled.
"My—lady—"
I would have hushed him, not for what he said, but that it wasted his strength, and of that he had so little left. But he would not have it so.
"I knew—my—lady—from—the—first—I saw you." His voice grew stronger as he talked, instead of weaker. "You are very fair, Joisan, very wise, very—desirable. But it—" He coughed, and more blood came, which I wiped away quickly with the wet leaves. "Not for me," he ended clearly.
He did not try to speak again for a while, and then,
"Not for the lordship, not ever, Joisan. You—must—believe that. I—would—have—come wooing if you had no dowry at all. Not the lordship—though they said it was the way to make sure of that. I wanted—you!"
"I know," I assured him. That was true. His kin might have urged him to wed with me for Ithkrypt, but Toross had wanted me more than any keep. The great pity of it was that all I could feel for him was friendship, and such love as one might give a brother—nothing more.
"Had you not wedded—" He gasped and choked. Now speech was beyond him.
At the last I gave what I had to offer to ease him—a lie that I spoke with all the ring of truth I could muster.
"I would have welcomed you, Toross."
He smiled then, such a smile as was a crossbow's bolt in my heart. And I knew that my lie had been well said. Then he turned his head a little, resting his stained lips against my breast, and his eyes closed as if he would sleep. But it was not sleep that came as I held him so. After a space I laid him down and wavered to my feet, looking about me, unable for that moment to look upon him.
I set myself rather to view this place. That we had come to some site of the Old Ones I had realized. But then its shape had been of no importance, merely that it was the end to which I could bring Toross. Now, sharply defined in the moonlight, I could see all of it.
There were no walls, no remains of such, just the pavement, dazzling in the moonlight. For the first time I was aware that some of the light came from the ancient stones themselves, similar to the glow of the globe.
Still those stones, in spite of their gleam, appeared to be little different from the rocks that formed the walls of Ithkrypt. Only the light pulsated a little as if it came and went like the breathing of some great animal.
Not only the glow but the shape of the pavement astonished me. It was laid in the form of a five-pointed star. As I stood there, swaying a little, it seemed to force its form upon my eyes as if it had a meaning that was necessary for me to see and understand. But my knowledge of the Old Ones and their ways was so fragmentary that I could not guess into what we had intruded, save it had never been fashioned to serve a Dark Power, but Light, and that it had indeed been a place of forces, some remnants of which still clung.
Had I only known how to use those! Perhaps I could have saved Toross, saved the dalespeople who would now look to me for leadership. If I only knew more! I think I cried out then in my desolation of spirit, for the loss of something I had never had, but which might have meant so much.
There was something here— Suddenly I threw back my head and gazed upward, stretched wide my arms. It was as if I were trying to open some long-closed door within me, to welcome into starved darkness a filling of light. There was a need in me, and if I asked I would be given. Yet I did not know what I was to ask for, and so in the end my arms fell to my sides and I was still empty. I was gnawed also by the knowing that I had been offered something wondrous which I was too ignorant to take. The thought of my own failure was the bitterest of all.
With this loss still upon me I turned about to face Toross. He lay as if asleep, and the glow of the stone was all about him. There was no way I could entomb him after the manner of the dales, with his armor upon him, his hands folded on the hilt of his sword to show that he had fallen valiantly in battle. Even this I could not do for his honor. Yet in this place such seemed unnecessary, for he rested in such glory and peace as I did not think any of our tomb chambers held. And he slept.
So I knelt and took up his hands, crossing them, though not on any sword hilt. And, last of all, I kissed him as he slept, for he had desired and served me to the utmost, even if I could not be to him as he wished.
Then I went forth from the star place and I broke off ferns and sweet-smelling herbs which grew here as if in a Wisewoman's garden. These I brought, and with them I covered Toross, save for his face, which I left open to the night. And I petitioned whatever Power lingered in this place that he indeed rest in peace. Then I turned and left him, knowing within my heart that with Toross now all was well, no matter what lay elsewhere in this war-riven and tormented land.
Beyond the edge of the star I hesitated. Should I retrace my way or strive to travel on, using the wood for cover, hoping beyond that to find some trail my people had taken? In the end I chose the latter.
Here the trees stood thicker and there was no path, nor could I be sure that I headed straight. I was no woodsman and I might be wandering. But I did my best.
When I came at last to that screen of thick brush which was the outer ring of the wood, my mouth was dry with thirst. I wavered as I walked from weariness, being faint with hunger. But before me was the narrowing end of the dale and the heights over which the refugees from Ithkrypt must have fled.
The light of pre-dawn was in the sky, my only lamp, for the glow had gone from the globe. It was dead, and I was alone, and the burden of a heavy heart weighed upon me as much as my weariness.
I reached a spur of rock behind which there was a hollow, and I knew I could go no farther. Around it grew sparse patches of berries, some of the fruit ripe. It was tart, mouth-twisting, what one would not usually eat without meat, for which it would be a relish. But it was food, and I stripped the ground-hugging bushes quickly, stuffing the fruit into my mouth as ravenously as anyone who knows bitter hunger.
That I could go on without rest I doubted; nor did I think I could find a better hiding place than this hollow. But before I crept within, I used Toross' knife to change my clothing for this wilderness scrambling. My skirt was divided for riding, but the folds were so thick and long they had nearly proved my undoing. Now I slashed away, tearing off long strips. These I used to bind down the "legs" of my shortened skirt, narrowing the folds and anchoring them as tightly as I could above my boots. The garment was far more bulky than a man's breeches, but I had greater freedom of movement than before.
Having done this, I huddled back into the hollow, sure my whirling thoughts would not let me sleep, no matter how deep my fatigue. My hands went to my breast, closing about the globe, without my willing.
It had no warmth now, yet there was something about the smooth feel of it that was comforting. And so, clasping it, I did fall asleep.
All men dream, and usually upon waking one remembers such dreams only in fragments—which may be of terror and darkness or, at long intervals, of such pleasure that one longs to hold on to them even as they fast fade. Yet what I experienced now was unlike any dream I had ever known.
I was in a small place, and outside swept storm winds—but winds of far more than normal fury. There was someone with me in that place. I caught a suggestion of a shoulder outline, a head turned from me, and there was a strong need that I know who this was. But I could not see a face or name a name. I could only cower as those racking winds beat by the opening of the crack in which we sheltered. As it had been in the place of the star, so was it here, the knowledge that had I only the gift, the ability, I could gain what I needed and that good would come of it. Yet I had it not, and the dream was gone—or else I could not remember more of it then or ever.
When I roused, the sun was almost down, and the shadows long about me. I sat up, still weary, still thirsty and longing for even as much water as I had shaken from the leaves in the wood. There was a dull ache in my middle, perhaps from the berries, perhaps from lack of food. I got to my knees and peered down-slope for any sign of the enemy.
Thus it was I spied those two making their way along as scouts do. My hand was at knife hilt in an instant. But in a moment I saw these were dalesmen. I whistled softly that call that we had learned for just such a use as this.
They flattened themselves instantly to the earth, then their heads rose at my second whistle. Seeing me, it took them only a few moments to join me, and I knew them for Toross' armsmen.
"Rudo, Angarl!" They could have been my brother-kin, so rejoiced was I to see them.
"My Lady! Then the Lord Toross brought you forth!" Rudo exclaimed.
"He did indeed. Great honor he cast upon his House."
The armsman looked beyond me into the hollow, and I saw that he guessed what dire report I must make now.
"The invaders have a weapon that can slay from a distance. As we ran, the Lord Toross was struck. He died in the safety to which he brought me. Honor to his name forever!" Did it help that at this moment I could use the formal words of a warrior's last farewell?
Both these men were well past middle age. What Toross might have been to them, or what ties—perhaps of almost kin-friendship—he might have had with them, I did not know. They bowed their heads at my words and repeated harshly after me, "Honor to his name forever!"
Then Angarl spoke. "Where is he, Lady? We must see to him—"
"He lies in a holy place of the Old Ones. To that we were guided, and there he died. And the peace of that place shall be his forever."
They glanced from one to the other. I could see their sense of custom warred in them with awe. And I added, "That which abides there welcomed him, yes, and gave him to drink in his final hour, and offered sweet herbs for his bedding. He rests as becomes a proud warrior, and on this you have my oath."
That they believed. For we all know that while there are places of the Dark Power to be shunned, there are others that offer peace and comfort, even to interlopers. And if such a place welcomed and held Toross now, he was indeed laid to rest with honor.
"It is well, Lady," Rudo answered me heavily, and I could see that indeed Toross had meant much to these two.
"You have come from the dalespeople?" I asked in turn. "And have you aught to eat—or drink?" My pride departed, and I wanted badly what they might carry.
"Oh—of a certainty, Lady." Angarl used his good hand to unstrap a bag from his belt, and in it was a bottle of water and tough journey cakes. I had to use all my control to drink sparingly and eat in small bites, lest my stomach rebel.
"We are of the band that went with the Forester Borsal. My lady and her daughter were also with us. But they turned back to see Lord Toross. We have been hunting him, since he did not join us by moonrise—"
"You are on this side of the river then—"
"Those demons hunt over the dale. If our lord had lived, this would have been the only free way," Rudo said simply.
"They are in all the dale now?"
Angarl nodded. "Yes. Two bands of our people were captured because they moved too slowly. Also, few of the flocks and herds got away. The beasts refused the climb to the pass, and the herders and shepherds could not force them to it. Those who tried too long—" He made a small gesture to signal their fate.
"You can find your way back?"
"Yes, Lady. But we had best be on our way quickly. There are parts that are hard going in the dark. Were it not summer and the dusk later in coming, we could not do it."
Their food heartened me, and their company even more. Also I found that the precautions I had taken to turn my skirt into breeches aided my going, so I was able to set out at a pace I could not have held the night before. Before I went, I returned the gryphon into hiding under my mail, for to me it was a private thing.
Our way was rough, and even my guides had to pause and cast around at times to find landmarks by which they could sight their way, for here there was not even a sheep track for a road. We climbed as the night grew darker. It was colder here, and I shivered when the wind struck full on. We talked very little, they no more than giving me a word of guidance when the occasion arose. My weariness was returning. But I made no complaint and did the best I could, asking nothing from them. In this hour their companionship was enough.
We could not take the last climb through the pass in the dead of night, so once more I sheltered among rocks, this time with Rudo on my right, Angarl on my left. I must have slept, for I do not remember anything after our settling there until Rudo stirred and spoke.
"Best be on now, Lady Joisan. There is the dawn, and we do not know how high those murderers range in their search for blood on their swords."
The light was gray, hardly better than twilight. I sighted gathering clouds. Perhaps we were to face rain—though we should welcome what washed away our tracks.
It did begin to rain with steady persistence. There was not even tree cover as we slipped and slid down from the pass into the valley beyond. I knew this country only vaguely. At the level of the dale, if one traveled through the lower pass, mere was a road—heading toward Norstead. Though the lords cared for it after a fashion (mainly by chopping back any undergrowth for three spear lengths on either side so mat the opening might deter ambushes by outlaws), it was not a smooth track.
The river was too wide and shallow hereabouts to provide for any craft save when the spring floods were in spate. And this part of the country lacked settlers. It was grazing ground, and in winter the grasses on the lower lands helped to feed the stock. But no one lived here, save seasonally.
Our people are few in the western dales. And dalesmen cling to company. Those who do turn hunter, forester, or trader are misfits who do not rub well against their fellows and are usually looked upon as being but a grade or two above the outlaws, since they are wandering, rootless men of whom anything may be expected. Thus we largely keep to the richer lands and within arrow flight of our dales. Our people dales are scattered. Norsdale, perhaps five days' journey westward by horse, more on foot, was the nearest settlement of which I knew.
But we did not descend to the road to Norsdale, being warned by fire smoke on the valley floor. Our people would not light such. Again we kept to the upper slopes, angling south. So we came near to being the targets of crossbow bolts from our own kin.
There was a sharp challenge from a screen of bush; then a woman came into the open to face us. I knew her for Nalda, whose husband had been miller at Ithkrypt, a tall woman of great strength in which she took pride, sometimes in her way seeming more man than woman when compared to the chattering gossips in the village. She held her bow at ready, the bolt laid on, and did not lower it as we came up. But on seeing me, her rough face lightened.
"My Lady, well come, oh, well come!" Her greeting warmed my long-chilled heart.
"Well come in truth, Nalda. Who is with you?"
She reached forth to touch my arm, as if she needed that to assure herself I did indeed stand there.
"There be ten of us—the Lady Islaugha and the Lady Yngilda, my dad Timon and—but, Lady Joisan, what of Stark, my man?"
I remembered that red slaughter by the river. She must have read it in my face. Her own grew hard and fierce in an instant.
"So be it," she said then, "so be it! He was a good man, Lady, and he died well—"
"He died well," I assured her speedily. I would never tell any dales woman the manner of dying I had seen. For the dead were our heroes, and that is all we needed to know to hold them in honor.
"But what do I think of! Come quickly—those demons are in the valley below. We would have moved on, but the Lady Islaugha, she would not go, and we could not leave her. She waits for the Lord Toross."
"Who will not come now. And if the invaders are already hereabouts we must move on quickly. Ten of you—what men?"
"Rudo and Angarl." She nodded to my companions. "Insfar, who was shepherd in the Fourth Section. He escaped over the rocks with a hole in his shoulder, for these thrice-damned hunters of honest men have that which kills at a distance. The rest are women and two children. We have four crossbows, two long bows, our belt knives and Insfar's wolf spear. And among us food for mayhap three days, if we eat light and make one mouthful do the work of three."
And Norsdale was far away—
"Mounts?"
"None, my Lady. We took to the upper pass and could not bring them. That was where we lost the rest Borstal guided—they went ahead in the night. There are sheep, perhaps a cow or two running wild—but whether we can hunt them—" She shrugged.
So much for all our plans of escape. I only hoped that some other parties of our people had gotten away sooner, been better equipped, and could get through to Norsdale. Whether they could rouse any there to come to our rescue, I doubted. In spite of my own need I could understand that those there would think a second time before venturing forth on such a search. They would be better occupied making ready their defenses against the invaders.
Thus escorted by Nalda, who seemed to have taken command of this small band, I came into their camp, though there was little about it of any camp. Seeing me, the Lady Islaugha was on her feet instantly.
"Toross?" Her cry was a demand. In her pale face her eyes glowed as the gryphon globe had glowed, as if within her was a fire.
My control was shaken. As I tried to find the right words, she came to me, her hands on my shoulders, and she shook me as if so to bring my answer.
"Where is Toross?"
"He—he was slain—" How could I clothe it in any soothing words? She wanted only the truth, and no one could soften that for her.
"Dead—dead!" She dropped her hold on me and stepped back. Now first there was horror in her expression, as if in me she saw one of the invaders bloody-handed from slaughter, and then a hardening of feature, a mask of hate so bitter it was a blow.
"He died for you—who would not look to him. Would not look to him who could have had any maid, yes, and wife, too, if he only lifted his finger and beckoned once! What had you to catch his eye, hold him? If he gained Ithkrypt with you, yes, that I could accept. But to die—and you stand here alive—"
I had no words. I could only face her storm. For in her twisted way of thinking she was right. That I had given Toross no encouragement meant nothing to her. What mattered was that he had wanted me and I had stood aloof, and he had died to save me.
She paused, and now her mouth worked and she spat, the spittle landing at my feet.
"Very well, take my curse also. And with it the oath of bearing and forebearing, for that you owe to me—and to Yngilda also. You have taken our kin-lord—therefore you stand in his place."
She invoked the old custom of our people, laying upon me the burden of her life as a blood-price, which in her eyes it was. From this time forth I must care for her—and Yngilda—protect them and smooth their way as best I could, even as if I were Toross himself.
13
Kerovan
Once more I stood on heights and looked down to death and destruction. Wind and wave had brought death to Ulmsdale, but here destruction had been wrought by the malice of men. It had taken me ten days to reach the point from which I spied on Ithkrypt, or what remained of it. One whole day of that time had been spent in reaching this pinnacle from which I could see a keep battered into dust.
Oddly enough there were no signs of the crawling monsters that breached walls in this fashion. Yet there were few stones of the keep still stacked one upon the other. And it was plain an enemy force was encamped here.
They had come upriver by boats, and these were drawn up on the opposite shore of the river.
My duty was divided. This landing must be reported, yet Joisan was much on my mind. No wonder I had sighted her in that vision clad in mail and gentle with a dying man.
Was she captive or dead? Back in the trackless wilderness through which I had come, I had crossed trails of small groups moving westward, refugees by all evidence. Perhaps she had so fled. But where in all those leagues of wild land could I find her?
Lord Imgry had set me a duty that was plain. Once more I was torn between two demands, and I had one thin hope. There were signal posts in the heights. Messages could be flashed from peak to peak—in the sun by reflection from well-burnished shields; at night by torches held before the same backing. I knew the northern one of those, which could not be too far from Ithkrypt. If that had not been overrun and taken, Imgry would have his warning, and I would be free to hunt for Joisan.
Using scout skill I slipped away from the vantage point that had shown me Ithkrypt. There were parties of invaders out, and they went brashly, with the arrogance of conquerors who had nothing to fear. Some drove footsore cattle and bleating sheep back to slaughter in their camp; others worked to the west, seeking fugitives perhaps, or making out trails to lead an army inland even as Imgry had feared, that they might come down on us to crush our beleaguered force between two of theirs.
I found Hiku an excellent mount for my purposes. The pony seemed to follow by his own choice that country into which he could merge so that only one alerted to our presence there could have been aware of our passing. Also he appeared tireless, able to keep going when a stable-bred mount might have given out, footsore and blowing.
That these dales had their natural landmarks was a boon, for I could so check my direction. But I came across evidence that the invaders were spreading out from Ithkrypt. And I knew that I would not be a moment too soon. Perhaps I was already a day or so too late in sounding the alarm.
I found the crest on which the signal niche was located and there studied with dismay the traces of those come before me. For it was the order of such posts that there be no trail pointing to them; yet here men, more than two or three, had passed openly, taking no trouble to cover their tracks.
With bare steel in my fist, my war hood laced, I climbed to the space where I should have found a three-man squad of signal-men. But death had been before me, as the splashes of clotted blood testified. There was the socket in which the shield had been set so it could be readied to either catch the sun or frame the light of a torch—there was even a broken torch flung to earth and trampled. I looked south, able to make out the next peak on which one of our outposts was stationed. Had those attacked here been able to give the alarm?
Now I applied my forest knowledge to the evidence and decided that battle had been done in mid-morning. It was now mid-afternoon. Had they wings, perhaps the invaders could have flown from this height to the one I could see, but men, mounted or on foot, could not have already won to that second point in such time. If no warning had gone forth, I must in some manner give it.
The shield had been wrenched away. I carried none myself, for my activities as scout had made me discard all such equipment. Signal—how could I signal without the means?
I chewed a knuckle and tried to think. I had a sword, a long forester's knife, and a rope worn as a belt about my middle. My mail was not shining, but coated over on purpose with a greenish sap which helped to conceal me.
Going forth from the spy niche I looked around—hoping against hope that the shield might have been tossed away. But that was rating the enemy too low. There was only one thing I might do, and in the doing I could bring them back on me as if I had purposefully lashed a nest of Anda wasps—set a fire. The smoke would not convey any precise message, as did the wink of reflection from the shield, but it would warn those ahead.
I searched the ground for wood, carrying it back to the signal post. My last armful was not culled from the ground under the gnarled slope trees, but selected from leaves on those same trees.
I used my strike light and the fire caught, held well. Into it then I fed, handful by handful, the leaves I had gathered. Billows of yellowish smoke appeared, along with fumes that drove me coughing from the fireside, my eyes streaming tears. But as those cleared I could see a column of smoke reaching well into the sky, such a mark as no one could miss.
There was little or no wind, even here on the upper slopes, and the smoke was a well-rounded pillar. This gave me another idea—to interrupt the smoke and then let it flow again would add emphasis to the warning. I unstrapped my cloak, and with it in hand went back to the fire.
It was a chancy business, but, though I was awkward, I managed to so break the column that no one could mistake it was meant for a signal. A moment or two later there came a flash from the other peak that I could read. They had accepted the warning and would now swing their shield and pass it on. Lord Imgry might not have exact news from the north, but he would know that the enemy had reached this point.
My duty done, I must be on my way with all speed—heading westward. To seek Joisan I could do no more than try to find the trail of one of those bands of refugees and discover from them what might have happened to my lady.
If so far fortune had favored me, now that was not true. For I learned shortly that the pursuit was up, and such pursuit as made my heart beat fast and dried my mouth as I urged Hiku on. They had out their hounds!
We speak of those of Alizon as Hounds, naming them so for their hunters on four legs, which are unlike our own dogs of the chase. They have gray-white coats, and they are very thin, though large and long of leg; their heads are narrow, moving with the fluid ease of a serpent, their eyes yellow. Few of them were with the invaders, but those we had seen in the south were deadly, trained to hunt and kill, and with something about them wholly evil.
As I rode from where that smoke still curled, I heard the sound of a horn as I had heard it twice before in the south. It was the summoning of a hound master. I knew that once set on my trail, those gray-white, flitting ghosts, which had no true kinship with our guardian dogs of the dale keeps, would ferret me out.
So I set to every trick I knew for covering and confusing my trail. Yet after each attempt at concealment, a distant yapping told me I had not escaped. At last Hiku, again of his own accord, as if no beast brain lay within his skull but some other more powerful one, tugged loose from my controlling rein and struck to the north. He half slid, half leaped down a crumbling bank into a stream, and up that, against the current, he splashed his way.
I loosed all rein hold, letting him pick his own path, for he had found a road which could lead us to freedom. It was plain he knew exactly what he was doing.
The stream was not river-sized, but perhaps a tributary to the river that ran through Ithkrypt. The water was very clear. One could look down to see not only the stones and gravel that floored it, but also those finned and crawling things to which it was home.
Suddenly Hiku came to a full stop, the water washing about his knees. So sudden was that halt that I was nearly shaken from my seat. The pony swung his head back and forth, lowering it to the water's edge. Then he whinnied and turned his head as if addressing me in his own language.
So odd were his actions I knew this was no light matter. As he lowered his head once more to the stream I believed he was striving to call my attention to something therein and was growing impatient at my not understanding some plain message.
I leaned forward to peer into the water ahead. It was impressed upon me now that Hiku's actions were not unlike those of someone facing an unsprung trap. Was there some water dweller formidable enough to threaten the pony?
Easing sword out of sheath, I made ready for attack. The pony held his head stiffly, as if to guide my attention to a certain point. I took my bearing from him in my search.
Stones, gravel—then—yes! There was something, hardly to be distinguished from the natural objects among which it lay.
I dropped from Hiku's back, planting my hoofs securely in the stream bed against the wash of the current. Then I worked forward until I could see better.
There was a loop, not of stone, or at least of any stone I had ever seen, blue-green in color. And it stood upright, seemingly wedged between two rocks. With infinite care I lowered sword point into the flood, worked it within that loop, and then raised it.
Though it had looked firmly wedged, it gave to my pull so easily I was near overbalanced by the release. And I snapped the blade up in reflex, so that what it held could not slide back into the water.
Instead it slipped down the length of the blade to clang against the hilt, touch my fingers. I almost dropped it, or even flung it from me, for there was an instant flow of energy from the loop into my flesh.
Gingerly I shook it a little down my sword, away from my fingers, and then held it closer to my eyes. What ringed my steel was a wristlet, or even an archer's bow guard, about two fingers's wide in span. The material was perhaps metal, though like no metal I knew. Out of the water it glowed, to draw the eye. Though the overall color was blue-green, now that I saw it close, I made out a very intricate pattern woven and interwoven of threads of red-gold. And some of these, I was sure, formed runes.
That this had belonged to the Old Ones I had not the least doubt, and that it was a thing of power I was sure from Hiku's action. For we dalesmen know that the instincts of beasts about some of the ancient remains are more to be relied upon than our own. Yet when I brought the armlet closer to the pony, he did not display any of the signs of alarm that I knew would come if this was the thing of a Dark One. Rather he stretched forth his head as if he sniffed some pleasing odor rising from it.
Emboldened by his reaction, I touched it with fingertip. Again I felt that surge of power. At length I conquered my awe and closed my fingers on it, drawing it off the blade.
Either the flow of energy lessened, or I had become accustomed to it. Now it was no more than a gentle warmth. And I was reminded of that other relic—the crystal-enclosed gryphon.
Without thinking I slipped the band over my hand, and it settled and clung about my wrist snugly, as if it had been made for my wearing alone. As I held it at eye level, the entwined design appeared to flow, to move. Quickly I dropped my hand. For the space of a breath or two I had seen—what? Now that I no longer looked, I could not say—save that it was very strange, and I was more than a little in awe of it. Yet I had no desire to take off this find. In fact, when I glanced at my wrist as I remounted, I had an odd thought that sometime—somewhere—I had worn such before. But how could that be? For I would take blood oath I had never seen its like. But then who can untangle the mysteries of the Old Ones?
Hiku went on briskly, and I listened ever for the sound of the hound horn, the yap of the pack. Once I thought I did hear it, very faint and far away, and that heartened me. It would seem that Hiku had indeed chosen his road well.
As yet he made no move to leave the stream, but continued to plow through the water steady-footed. I did not urge him, willing to leave such a choice to him.
The stream curved, and a screen of well-leafed bushes gave way to show me what lay ahead. Hiku now sought the bank of a sand bar to the right. But I gazed ahead in amazement. Here was a lake not uncommon in the dales, but it was what man—or thinking beings—had set upon it that surprised me.
The water was bridged completely across the widest part. However, that crossway was not meant for a roadway, but rather gave access to a building, not unlike a small keep, erected in the middle of the water. It presented no windows on the bridge level, but the next story and two towers, each forming a gate to the bridge, had narrow slits on the lower levels, wider as they rose in the towers.
From the shore where we were, the whole structure seemed untouched by time. But the far end of the bridge, reaching toward the opposite side of the lake, was gone. The other bridgehead was not far from us and, strange as the keep appeared (it was clearly of the Old Ones' time), I thought it offered the best shelter for the coming night.
Hiku was in no way reluctant to venture onto the bridge, but went on bravely enough, the ring of his hoofs sounding a hollow beat-beat which somehow set me to listening, as if I expected a response from the building ahead.
I saw that I had chosen well, for there was a section of the bridge meant to be pulled back toward the tower, leaving a defensive gap between land and keep. Whether that could still be moved was my first concern. Having crossed, I made fast the end of my rope to rings there provided.
Hiku pulled valiantly and, dismounted, I lent my strength to his. At first I thought the movable bridge section too deeply rooted by time to yield. But after picking with sword point, digging at free soil and wind-blown leaves, I tried again. This time, with a shudder, it gave, not to the extent its makers had intended, but enough to leave a sizable gap between bridge and tower.
The gate of the keep yawned before me darkly. I blamed myself for not bringing the makings of a torch with me. Once more I relied on the pony's senses. When I released him from the drag rope, he gave a great sigh and footed slowly forward, unled or urged, his head hanging a little. I followed after, sure we had come to a place where old dreams might cluster, but that was empty of threat for us now.
Over us arched the bulk of the tower, but there was light beyond, and we came into a courtyard into which opened the main rooms of the structure. If it had been built on a natural island, there was no trace of that, for the walls went straight down on the outer side to the water. In the courtyard a balcony, reached by a flight of stairs on either side, ran from one gate tower to the other, both right and left.
In the center of the open space there were growing things. Grass, bushes, even a couple of small trees, shared crowded space. Hiku fell to grazing as if he had known all along that this particular pasture awaited him. I wondered if he did; if Neevor had come this way.
I dropped my journey bag and went on, passing through the other tower gate out to the matching bridge. It was firm, uneroded. I thought of the wide differences among the Old Ones' ruins. For some may be as ill-treated by time as those Riwal and I had found along the Waste road, and others stand as sturdy as if their makers had moved out only yesterday.
When I came to the cutoff end of the bridge, I found—not as I had half-expected now, a section pulled back—that the bridge material was fused into glassy slag. I stretched my hand to touch that surface and felt a sharp throb of pain. On my wrist the band was glowing, and I accepted what I believed to be a warning. I retreated to the courtyard.
Wood I found in the garden, if garden it had been. But I did not hasten to make a torch. I had no desire now to enter the balcony rooms to explore in the upper reaches of the towers. Instead I scraped up dried grass of an earlier season, and with my cloak, which still reeked of the signal smoke, I made a bed. In my exploration I found water running from a pipe that ended in a curious head, the stream pouring from both mouth and eyes into a trough and then away along a runnel. Hiku drank there without hesitation, and I washed my smut-streaked face and hands and drank my fill.
I ate one of my cakes, crumbled another, and spread it on wide leaves for Hiku. He relished that and only went back to grazing when he had caught up the last possible crumb with his tongue. Settling back on my cloak bed, my battle hood unlaced, and as comfortable as any scout can be in the field, I lay looking up at the stars as the night closed down.
One could hear the wash of water outside the walls, the buzz of an insect, and, a little later, the call of some night hunter on wings. The upper reaches of the tower could well house both owls and nighthawks. But for the rest there was a great quiet that matched an emptiness in this place.
I was heartened by what seemed to be the good fortune of this day—the fact that my signal had been read, the finding of the talisman—
Talisman? Why had my thoughts so named the armlet? I sought it now with the fingers of my other hand. It was slightly warm to the touch; it fitted my wrist so snugly, it did not turn as I rubbed it, yet I was aware of no punishing constriction. I felt, under my fingertips, that the designs upon it were in slight relief, and I found that I was trying to follow this line or that by touch alone. I was still doing it as sleep overcame me.
That sleep was deep, dreamless, and I awoke from it refreshed and with confidence. It seemed to me that I could face without fears all this day might bring, and I was eager to be gone.
Hiku stood by the trough, shaking his head, the water flying in drops from his muzzle. I hailed him happily as if he could answer me in human speech. He nickered as though he found this a morning to make one feel joyfully alive.
Even though I had daylight as an aid, I had no wish to explore. The driving need to know what had happened to Joisan was part of me. I waited only to eat, and then I readied to leave.
Whether the portion of the bridge that had moved at our urging last night could now be replaced, I began to wonder. When we came to the portion lying on top of the other surface, I examined it with care. In the bright light of day I saw, jutting up on the north side of the parapet, a rod as thick as my forearm.
This was too short to have been a support for anything overhead, but it must have a purpose, and I hoped it dealt with the controls on the bridge. In test I bore down on it with all my strength, and nothing happened. From steady pressure I turned to quick, sharp jerks. There was a hard grating, it loosened, and once more I applied pressure.
The bridge section we had worked with such infinite labor to drag back trembled and began, with screeches of protest, to edge forward. It did not quite complete the span again, but lacked only perhaps a foot of locking together. The gap was not enough to prison us.
Back on shore, before I mounted Hiku, I gazed back at the lake keep. It was so strongly built a fortress, so easily defended, that I marked it down to serve at some future time. With the bridge drawn back, even the crawling monsters of the invaders could not reach it. And its lower walls without breaks could safely hide a third of the army in the south. Yes, this was a fortress that we might make good use of.
Now as I turned Hiku north, planning to cut across refugee trails heading west, I saw that the land about this portion of the lake must once have been under plow. There were even patches of stunted grain still growing. I passed an orchard of trees with ripening fruit. This land must have fed the lake dwellers once. I would have liked to explore, but Joisan's plight did not allow that.
A day it took me to cross that countryside to the next rise of hills. I saw animals in plenty, deer grazing, which meant no hunters. Among them, as I neared the hills, were some gaunt and wild-eyed cattle which I believed had been lost from some herd harried by the invaders. Those sighting me snorted and galloped away clumsily.
As I re-entered the hills, I found the cattle's trail marked by hoof prints and droppings. It angled through a rift, and I followed it wearily, hoping for an easy passage, but also aware that the cattle might be hunted.
Yet I met no enemy. At length, a day later, I chanced directly on what I sought, tracks left by a small band who were not forest-trained enough to hide their going. There were only three horses, and most of the traces had been left by women and children. These must be fugitives from Ithdale, and though there was one chance in perhaps a thousand of Joisan being among them, I might learn something of her.
The tracks were several days old. They tried to head west, but the nature of the rough ground kept pushing them south instead. And this was wild country.
On the morning of the fourth day of trailing, I came to the top of a ridge and, smelling smoke, I crept up to make sure this was the party I sought and not a band of enemy scouts.
The valley was wider here, with a stream in its middle. By the banks of that were shelters of hacked branches covered with other branches and grass. A woman bent over a fire, feeding it one stick at a time. As I watched a second figure crept from one of those lean-tos and straightened to full height.
Morning light caught the glitter of mail that the newcomer was now pulling on. Her head was bare, her hair tied back in a red-brown rope falling between her shoulders. Fortune had favored me once again—that this was indeed Joisan, though I was too far to see her face, I was somehow sure.
My purpose was now clear. I must front her as soon as possible. And when she moved purposefully away from the fire and set off along the river, I was glad. I wanted to meet her alone, not under the staring eyes of her people.
If she were to turn from me in disgust at the sight of my hoofs, any relationship would end before it was begun. I must know that without witnesses. I slipped down-slope to intercept her, using the same caution I would have had she been the enemy.
14
Joisan
In our struggle westward during the flight from Ithdale, we had had a little luck in the finding of three strayed ponies, upon the backs of which our weaker ones could ride in turn. This I ordered—that all was share and share alike, with no deference to rank. Yngilda glowered at me, though the Lady Islaugha, after her first violent outburst, went silent as if I did not exist. I thankfully accepted that.
That there was no easy road to Norsdale, we learned within the second day of our journey. The invaders, either in pursuit of fugitives or animals, quartered the land, and we were driven far off our course.
Food was our great need, for luckily, this being summer, we could shelter in the open. But though our animals could graze, we could not live on grasses. And the distance we covered in a day became less and less, since we must also hunt to fill our bellies.
Insfar, who had been a shepherd and had knowledge of wild berries and edible plants, was our guide here. There were mushrooms, and every stream or pool was an invitation to try fishing.
Our arrows and bolts were too few to waste on hunting unless we could make very sure of the shot, and I forbade their being thrown away. Rudo, in spite of being one-eyed, had luck with a slingshot and carried it ever with him, along with a store of pebbles. Four times he added rabbits to the pot. But there was less than a mouthful of meat apiece when that was served out.
We had a second problem, one which had slowed this band of fugitives from the first. Martine, who had been wedded only last fall to the son of the village headman, was heavy with child, her time near upon her. I knew we must find, and shortly, a place wherein we could not only camp for a space, but also have food. Yet nowhere in this rugged land did there seem any welcome.
On the fifth day of our frighteningly slow travel, Rudo and Timon, scouting ahead, returned with brighter faces. We had not cut across any invader trail now for more than a day and so we had a faint ray of hope we had gotten beyond their ranging. What our scouts offered us was a camping ground. And none too soon, I believed, for Nalda, who had kept an eye on Martine, looked very sober.
If we turned a little south, Rudo reported, we would find a valley with not only water but game. He had also discovered a thicket of pla-plums fully ripe. And there was no sign of any visitors.
"Best foot in there, Lady Joisan." Nalda spoke with her usual frankness. "That one"—she nodded at Martine, who sat on the nearest pony, her head dropping, her hands pressed to her swelling belly—"is nigh her time. I do not think she is going to get through this day before her pains come."
We came into the valley. As Rudo promised, it had many advantages. And the men, though Insfar could use only one arm and Angarl one hand, set about hacking down saplings and setting up lean-tos—the first of which Nalda took for Martine.
She had foreseen rightly. By moonrise our party had gathered a new member, squalling lustily, and named Alwin for his dead father. Thus also our staying here for some time was ordered.
It was the next morning I set my will against Yngilda's. If we were to survive, we must gather all the food we might find, keeping ourselves on spare rations while we dried or otherwise prepared the rest for the trail ahead.
I was learned in the provisioning of a keep, but here where there was no salt, no utensils with which to work—nothing but hands, my memory, and what I could improvise—it seemed I faced an impossible task. Yet it was one I must master.
The village women made no murmur, and even the two children did as they were bid at their mother's side. It made me hot with anger when Yngilda did not bestir herself from the lean-to or make any move to join our foraging party.
I went to her, a bag roughly woven from grass and small vines in my hand. Coaxing would not stir her, that I was sure. This was a case for the rough of one's tongue, and that, exasperated and driven as I was, I could easily give.
"On your feet, girl! You will go with Nalda and take heed of what she says—"
She looked at me stony-eyed. "You are bondswoman to us, Joisan. If you would grub in the dirt with fieldwomen, that is your choice. I do not forget my blood—"
"Then live upon it!" I bade her. "Who hunts not food does not eat by another's labor. And I am no bondswoman."
I threw the bag to her, and she spurned it with her foot. So I turned and tramped away to join the others. But I swore that I would hold to my promise. She was able-bodied and young—I would share with the Lady Islaugha, but not with her.
Of the Lady Islaugha I thought now impatiently. She had sunk into herself: for no better way could I describe her appearance since I had reported Toross' death. As with Dame Math at the last, age had settled upon her in a single day; so, though she was still in middle years by reckoning, she was to all eyes an aged woman.
She had retreated into her own thoughts, and sometimes we could not rouse her, even to eat what was put into her hand, without a great effort. Now and then she muttered in whispers of which I could not catch more than a word or so, and from these I guessed that she spoke with those I could not see and who, perhaps, were long gone from this world.
I hoped that this was a temporary state born of shock and that in time she would be herself. But of that I could not be sure. If I could only get her to Norstead Abbey where the Dames were learned in nursing, perhaps she might be brought back to the world. But Norsdale seemed farther from us each day.
Yngilda had no such excuse, and she must take upon herself a share of our hardships. The sooner she learned that fact, the better! It was with no pleasant feelings that I went out to hunt.
I had a long bow and three arrows. At Ithkrypt in practice shooting I had proven myself marksman. But shooting at a target and at living prey were, I knew, two different matters, and I must not waste any more of those arrows. So my greater hope this morning was fixed on the river.
With patience and care I had worked at the edge of my mail shirt and broken off a couple of links, shaping them roughly into hooks, raveling my cloak hem and twisting together fibers for a cord. It was poor equipment for a fisherman, but the best I had. And as the foragers separated, the men heading for the grassland where rabbits might be found, the women for the plum thicket, I kept on along the river bank.
Only necessity made it possible for me to bait the first hook with a living insect. I had always shrunk from hurting any creature, and this use of a small life was to me another horror to be added to those of the immediate past.
I found a place where I could wade out to a square rock around which the water washed. There were trees here, and it was cool, shadowed from the sun. But it was still so warm that I shed my mail and the padded jacket under that, keeping on only my undershift; but wishing I might drop that also and slide into the water to wash clean, not only from the dust and sweat of our journeying, but from memory also.
The gryphon swung free, but it held none of the life it had shown the night when Toross and I fled together. I studied it now. It was marvelously wrought. Where had it come from? Overseas—a fairing bought from some Sulcar trader? Or—was it a talisman of the Old Ones?
Talisman—my mind played with that thought. Had it served us as a guide on our flight from Ithkrypt to the star place in the wood? That had been of the Old Ones, and this—could it be, I speculated, that such baubles as this had connection with remains of the Old Ones?
It was an interesting thought, but not one to produce food. I had best attend to the reason for my being here. I dropped my baited line into the water.
Twice I had a strike, but the fish got away. And the second time it took my hook. I had never possessed great patience, but that morning I forced myself to cultivate it as I never had before.
I gained two fish with the second hook. But neither was large. And I feared that unless luck changed, this was no way to replenish our supplies. Leaving my rock, I trudged farther along the stream and, to my joy, found in a side eddy a bed of watercress I plundered.
As the sun turned westward, I turned back to camp. I had eaten some berries and chewed on a handful of my watercress. But I ached with hunger as I went, hoping that the rest had had better luck. When I struck away from the river, I came across the first piece of real fortune I had had all day.
There was a snarl and a deeper answer. Dropping my bag of fish and watercress, I put arrow to bow string and stole forward past a screen of brush.
On the body of a fresh-killed cow crouched a half-grown snow cat, its ears flattened to its skull, its teeth bared in a death-promising grin. Facing it was a broc-boar.
These grim scavengers were meat eaters, but this one must have either been wild with some private fury, or ravening with hunger, or it would not have challenged the cat over its own kill. And it would seem that the cat was wary of the boar, as if it sensed that the other's challenge had a double element of danger.
The boar was digging its tusks into the earth already softened by its pawing forefeet, tossing bits of sod into the air and squealing in a rising crescendo of sound.
Side by side on the ground the boar would outweigh the cat, I thought. I had seen only two broc-boars in my life, and both had been well under the weight and shoulder height of this monster.
The cat screamed in fury as it sprang, not at the boar, but back from the prey it had cut down. And the boar moved after it with a nimbleness I would not have guessed possible. With another protest of feline rage, the snow cat leaped to a crag and up, soon gaining the heights. From there I could hear its hissing and growling growing fainter as it left the field to the boar who stood, its head cocked, listening.
Almost without planning, I moved then. It was dangerous. Wound that tight package of porcine fury, and I might be horribly dead. But as yet the boar had not winded me, and I saw in it such a promise of food as I could not, in my hunger, resist. Also it was standing now in just the position where I could get a telling shot.
I loosed my arrow and a second later threw myself backward into such hiding as the brush gave me. I heard a terrible squeal and a thudding, but I dared not wait. If I had failed, that four-footed death would be after me. So I ran.
Before I reached camp I sighted Rudo and Insfar and gasped out my story.
"If the boar did not follow you, Lady, it was because it could not," Insfar said. "They are devils for attack. But it may well be your shot was lucky—"
"It was folly," Rudo commented sourly and directly. "It might well have slain you."
He had the truth of it. My hunger had betrayed me into the rankest folly. I accepted his words humbly, knowing that I might now be lying dead.
We returned together, scouting the terrain as if we expected an attack from ambush. We had circled, going upslope. When we finally reached the scene, there lay the cow and, beyond, on bloodstained ground torn by hoofs and tusks, the boar also. My arrow had sunk behind its shoulders and into the heart.
I found this stroke of fortune earned me awe from the rest of the party. Such a happening was so rare that it might be deemed an act dictated by the Power. I believe that from that hour my people held that some of Dame Math's knowledge and skills were also mine. Though they did not say it to my face. I saw them send favor signs in my direction, and they paid heed to all I said, as if what I uttered were farseeings.
Yngilda remained my thorn-in-the-flesh. I kept to my resolve that first night, and when the flesh was roasting on spits above the fire, so that the savor of it brought juices flowing into the mouth, I spoke aloud so all could hear.
The able-bodied who did not labor equally to supply us all would not share in the fruits of our seeking. So I said, after I had given full praise for the results of that day's harvesting. I saw that all shared that night—save Yngilda. But her I refused openly, that all might note I did not accept rank as an excuse for idleness.
She flung at me that I was under blood-curse to her family. But I said as firmly that I accepted the Lady Islaugha as my charge, and her I would serve. To that these assembled could bear witness. However, Yngilda was young, of strong body, and therefore she would find none here to wait upon her—it would be equal sharing.
I think she would have liked to fly at me, to rake my face and eyes with her fingers. But in that company she stood alone, measured for what she was, and she knew it. So at last she turned from us and crawled back into her lean-to, and I heard her crying, but such weeping as comes from anger and not from sorrow. I had no pity for her. But I also realized that I had made an enemy who would remain an unfriend.
However, it seemed as one day followed another Yngilda had reconsidered her position and thought the better of her obstinacy. She did not do her share of the work graciously, but work she did, even to the odorous business of helping to spread the strips of beef to sun-dry after we butchered the cow that had fallen to the snow cat.
We were frugal, even making use of the bones of both slaughtered animals, their hides (though these could only be rough-cured), and the tusks of the boar. Martine regained her strength, so I had hopes that before the warm weather was past we could fight our way to Norsdale and I could at last lay down my burden of leadership.
Lady Islaugha took to wandering away, in search, perhaps, of Toross. One of our number had to guard her ever, since while so driven to this wandering, her strength seemed the greater and she would set off briskly, often struggling with her guardian if he tried to thwart her, only to falter later when she tired. Then her guardian would lead her back.
Timon fashioned some better fishhooks, and I continued to try my luck along the river. I think out of stubbornness, determined to win a victory here as I had with the boar. But, judging by my continued failures, my luck did not hold in water as it had on land. So clear was that water that ofttimes I sighted the shadows of what were indeed giants compared to the unwary fish I managed to pull out. But either there was some trick of baiting I did not understand, or else these were warier than most fish.
It was during the third day when I followed the river that there came upon me the strong sensation of being watched. So acute did this grow that my hand went to Toross' knife in my belt. From time to time I halted to look around, certain that if I turned quickly enough I could sight some face framed in the grass or in a bush.
I grew so uneasy that I decided to return to camp and alert my people. Some scout of the invaders might have found our backtrail. If so, we might be already doomed, unless we could find and slay him before he reported to his force.
As I turned, the bush parted and one stepped into the open. I had drawn steel in the same instant, ready to defend myself.
He held up empty hands as if he knew what passed in my mind. At the same time, seeing him in full, I knew he was no invader. His battle hood was loosened to lie back on his shoulders. And he wore no over-jerkin or tabard with arms emblazoned on it. Rather did his mail and leather look dulled and dingy, as if purposefully darkened.
But—as my eyes swept down his slim body I stiffened. He wore no boots, his leather breeches were in-fastened at his ankles with straps and his feet—but he had no feet! He stood upon hoofs like one of the cows.
From that impossibility I swiftly looked to his face again, half-expecting to find it monstrous also in some way. But it was not. A man's face truly, browned by sun and wind, the cheeks a little hollowed, the mouth firm-set. He was not as handsome as Toross and—my eyes met his, and in spite of my control I took a step or so back. For those eyes, like the hoofed feet, were not of human man.
They were the color of that amber known to us as "butter," a deep yellow, and in them the pupil was more slit than circle. Not a man's eyes—
When I drew back there was a change in his face, or did I only imagine that? And now I remembered Dame Math's teaching (had she ever in the past, before she had joined Norsdale, met such a one?), so between us in the air I drew a certain sign.
He smiled, but it seemed that smile was a wry, almost twisted one, as if in some manner he regretted that I knew him for what he was—one of the Old Ones. For the first time he spoke:
"Greetings, Lady."
"And to you—" I hesitated, for by what honorific should an Old One be courteously addressed? That had not been in my training. Thus I gave him what I would grant one of my rank in the dales. "Lord, greeting."
"I do not hear you add 'fair meeting,'" he said then. "Do you deem me without testing unfriend?"
"I deem you beyond my measuring," I answered frankly, for I believed that to be true, and perhaps he could read my mind. Such was child's play for one of them.
He looked puzzled. "Who think you that I am, Lady?"
"One of those who held dominion here before my blood came to High Hallack."
"An Old One—but—" His wry smile came again. "So be it, Lady. I shall say you neither aye nor nay, since you have named me so. But you and your folk yonder seem in a sorry case. It may be that I can be of some assistance to you."
I knew so little—there were those among the Old Ones who were said to be favorable to men, who had on occasion given them assistance. There were others of the Dark whose malice meant great peril. Trust is a precious gift. If I chose wrong now, we would all suffer. Yet there was that about him which argued that he was not of the Dark.
"What have you to offer? We would reach Norsdale if we can, but the way—"
He interrupted me. "If you seek to go westward there are many perils. But I can bring you to a shelter that will serve you better than here. There is fruit and game there also—"
I gazed into those golden eyes, troubled. When he spoke so, I wanted to believe. But I was not alone; there were these people of mine. And to trust an Old One—
His smile went as I hesitated. There was a coldness in his face, as if he had held out his hand and been rebuffed. My unease grew. Perhaps he was one disposed to aid my kind, but would take offense if that aid were refused, thus bringing on us his displeasure.
"You must forgive me." I sought for words to assuage the ire I feared might be rising in him. "I have had no meeting with—with your people heretofore. If I do not comport myself as I should, it comes from ignorance alone and not from any wish to offend. Among the dales you appear only in our legends. Some of those are favorable; some deal with the Dark Ones who give us hate instead of friendship. Thus we walk warily in your presence."
"Because the Old Ones have what you call the Power," he said. "Well, that may be so. But I mean you nothing but good, Lady. Look upon what you wear there on your breast—hold it out that I may touch it with my fingertip—you will see that this is so."
I looked to the crystal gryphon. Though there was bright sun on us, not moonlight, I could perceive that it was glowing; almost it appeared as if the creature within the ball was about to give tongue and speak for this stranger, so oddly knowing did the carving look. I did then as he suggested, took the chain from about my neck and held the globe in my hand, stretching it forth to him.
He touched it with fingertip only, and the globe flashed into such radiant glory that I near dropped it. In that moment I knew that all he said was the truth, and that here had come one out of the unknown past of this land to do us service. So my heart lightened, though my awe of him was greater.
"Lord." I bowed my head before him, giving homage to that which he was. "In all things behold one whom you can bid—"
Again he interrupted me, this time speaking with such sharpness that it was a rebuke.
"I am no lord of yours, Lady, nor are you under my bidding. You are to make your choice freely. I can offer you and your people a strong shelter and what aid a single man can give—and I am not ignorant of the ways of war. But more than your friendship—if you can find it in you sometime when we are better acquainted to offer me that—I do not claim!" There was such authority in his tone as he spoke, as if it were very necessary that he make this plain to me (though I was not sure of his meaning), that I was abashed. But I knew I must abide by his wishes.
He came back with me to my people, and they were also in awe of him, shrinking away. I watched him and saw the bitterness in his face, and there came into my mind that he was hurt at this effect he had upon people, so in turn I felt something of his pain—though how I knew all this I could not tell.
But he being who he was, his orders were obeyed without question. He whistled, and there came down from the hills in answer a mountain pony sure-footed and steady. And on this he mounted Martine. On our other horses we packed what we had harvested here and we went under his guidance.
In the end he brought us to a place that was indeed a greater wonder in its way than any keep of the dales, for it was built upon lake. To it led two bridges, though one was broken off and useless. But the other, as he showed us, could have a section moved in it, leaving a defensive gap across which no foeman could come.
Best of all, this land had once been under cultivation. There was fruit and stunted, wild-growing grain to be harvested.
With this promise against famine we knew we could remain here for as long as we needed to give our people strength and gather more food for the journey. And my trust in him was secure.
He gave us no name. Perhaps with him it was as it is with the Wisewomen who believe that if one knows their name one can then establish domination over them. In my mind I called him Lord Amber, because of his eyes.
Five days he stayed with us, seeing all was well. Then he said that he would go on scout, making sure those of Alizon had not struck inland far enough to trouble us.
"You speak as if the Hounds are also your enemies," I said, curious. "Yet your people have not been attacked—it is my kin they sweep away."
"The land is mine," he answered. "I have already fought these elsewhere. I shall fight again, until they are driven back into the sea from which they came."
I knew a thrill of excitement. What would it mean to my poor, beleaguered people if there were others such as he, possessors of the Power who would aid us in this death struggle? I longed to ask him if there were, and again it was as if he read my thoughts.
"You think the Power could be used to vanquish these?" There was brooding sadness in his face. "Be not wishful for that, Lady Joisan. He or she who summons such cannot always control it. Best not to call so. But this I will tell you: I believe this place to be as safe as any now in the hills. If you choose well, you will abide here until I return."
I nodded eagerly. "Be assured we shall, Lord." In that moment I had a queer desire to stretch forth my hand, perhaps touch his arm—as if some touch of mine could lighten that burden I thought rested ever on him. Such a fancy had no meaning in it, and of course I did not follow that odd whim.
15
Kerovan
I saw the shrinking in her eyes and knew that there could be naught between us. But it was only in my losing that I came to learn how much I had held in my heart the thought that, to at least one, I would be no monster but a man. How right had been that instinct that had kept me from obeying her wish and sending her my portrait. For she would never know now that I was Kerovan.
That Joisan took me to be one of the Old Ones was a two-way matter. On the one hand it kept her from asking questions that I must either struggle to evade or to which I must give answers. On the other hand she would expect from me some evidences of the "Power." For that lack, I would also need to find excuses. But for a few days I could set actions before words.
The poor camp was a refuge for a pitiful band. There were but four who might be termed—very loosely indeed—fighting men. Two were past middle age, one lacked a hand, the other an eye. There was a green boy who I suspected had never drawn steel, and a hill shepherd who was wounded. The rest were women and children, though of that number there were some who could stand shoulder to shoulder with any armsman if there be need.
Mostly they were villagers, but there were two other ladies—one old, one young. The older was in a state of shock and had to be watched lest she wander off. Joisan confided in me that her son had been slain and that she now refused to accept that, but wished to search for him.
Of Ithdale itself she had a strange tale, almost as strange as the blasting of Ulmskeep. It seemed that one of her own House had called down some manifestation of the Power on the keep, catching thereby a goodly number of the Hounds. But before that there had been some warning, and she hoped many of her people had managed to flee westward. Their goal was Norsdale, but now they had no guide. Also there was one of the women who had recently given birth and could not travel long or hard.
It was then I thought of that fortress in the lake and how it could be a shelter for this band until they recovered strength. That much I could do for my lady, bring her and hers to a roof over their heads and a small measure of safety.
That she wore my gift was no matter of pride for me, for I was certain she did not wear it because of any attachment for its giver, but rather because she delighted in it for itself. Now and then I saw her hand seek and caress it, almost as if from such fondling she received strength.
The younger of the two ladies, Yngilda, who was kinsman to Joisan, I did not like. She watched my lady from under downcast lids with a sly hatred, though Joisan showed no ill will toward her. What lay between them in the past I had no inkling, but this Yngilda I would noways trust.
Of Joisan herself—but those thoughts I battled, summoning always in answer to them the memory of her expression when she had seen my unbooted feet. How wise I had been in my choice to bare that deformity to the world. Had I continued to hide it, made myself known to Joisan, found her welcoming and then— No, it was far better to eat bitterness early than to have it twist one doubly by following on the sweet.
That Joisan was such a lady as one might treasure, that I learned in those few days when, under my guidance, they made their way to the keep in the lake. Though many times she must have been weary and downhearted, yet she was ready to shoulder duty's burdens without complaint. And her courage was as great as her heart. Had I only been as other men—
Now I chewed upon the sourness of that memory-vision in which she had supported a dying man in grief. Did that lie in the past or the future? I had no right to question her, for I had surrendered that with my claim upon her.
Somehow, if it was in my power, I would get them to what safety Norsdale promised. Then—well, I was now a landless man. It would be easy to drop out of reckoning. I could join some lord's menie. Or I could go into the Waste, where those who were outcast from the dales carried their shame or despair. However, I would see Joisan safe before I said farewell.
When those from Ithdale had settled into the keep and had mastered the defensive use of the sliding bridge, I sought Joisan and told her that I must scout for the invaders. That was in part the truth, but with that need there was another, that I go forth and do battle with myself, for there were times when she seemed to reach to me. Not with hand, or invitation of voice. But she would look at me when she thought my attention elsewhere, and there was a vague questioning in her face, as if she sensed that there was a bond between us.
Because there was a weakness in me that yearned toward making myself known, trading on any familiarity that had grown, I was determined that I must get away until I could build such an inner control as would never yield. In her eyes at first sighting I had been a monster. Because she now believed me an Old One with whom the human standards of form did not hold, she accepted me fully. But as a man—I was flawed.
On leaving the lake keep, I circled out to the north and west. This was wild country, though it did not have the desolation of the Waste. Nor did I come upon any other Old Ones' ruins, as I expected to with the impressive lake structure so near.
Three days I quested in the direction I thought we must follow to Norsdale. Though I did not know any trail, I had a good idea of the general direction. The way was rough; the wider dalelands of the east changed here to a series of knife-narrow valleys walled by sharp ridges, weary path for foot travelers. Also they would not be able to go fast. And I began to wonder as I struck more westerly whether they would not be better placed to wait within the lake keep for spring.
On the fourth day I cut across a fresh trail. It was of a small party, not more than four. They were mounted, but their animals were plainly not the heavier beasts the Hounds favored, the tracks being those of unshod hill ponies. More fugitives? Perhaps—but in times of war no man accepts an untested assumption.
Joisan had told me her people had been widely scattered when they fled. These could be more from Ithdale. I had a duty to make sure, to guide them if they were.
Yet I took no chances and followed with a scout's wiles. The trail was, I believed, several days old. Twice I came to where they had camped, or rather sheltered, for there were no signs of fires. Also—their going spoke of more than flight—it was as if they knew exactly where they were going, driven more by purpose than fear.
Their direction, making allowances for natural obstacles, led directly to the lake keep. Noting that, I was more than a little uneasy. Four men did not make a formidable force, but four men, armed and ready, could take Joisan's people by surprise. And these might be outlaws drawn out of the Waste by the bait of loot.
I might have held their trail had it not been for the storm. It came with the evening of the same day. Though it was a summer shower compared to the fury I had seen unleashed in Ulmsdale, yet it was not an easy torrent of rain or force of wind to face. With darkness added, I was made to seek cover and wait it out.
As I waited, my mind fastened on one evil chance after another. I could not rid myself of the belief that these I followed were unfriends. And I had lived on the edge of the Waste too many years not to know what outlaws would do to the helpless. Now I could only hold to self-control, try to believe that Joisan had followed my last orders and taken up the bridge section and that they would not admit strangers to the keep. Ithkrypt had not known such raiders. They could accept any dalesman as friend.
By morning the storm had cleared, but it had washed away the trail. I was too concerned to cast about hunting it. The need to return to the lake, discover what chanced there, rode me strong.
But I had two days yet on the way, even though I pushed myself and Hiku to the limit of endurance. When I came into the lake valley and saw the keep, I had worked myself into such a state of foreboding as prepared me to find a bloody massacre.
There was a hail from one of the largely overgrown fields that brought me up short. Nalda and two of the other women waved from a peaceful scene. They were engaged in harvesting the thin stand of stunted grain, gleaning every stalk and heaping it on two outspread cloaks.
"Fair news, my Lord!" Nalda's voice rang loud in my ears as she crossed the field to reach me. "My Lady's lord has come at last! He heard of her distress and rode to her service!"
I stared at her unbelieving. Then common sense made it all clear—she did not mean Joisan, of course. She spoke of the lord wed to Yngilda—though I had thought he was supposed to have died when his keep was overrun in the south.
"The Lady Yngilda must be giving praise to Gunnora at this hour," I had wit enough to answer.
Nalda stared at me as strangely as perhaps I had at her a moment earlier.
"That one—she is widowed, not wed! No, it is the Lord Kerovan, he who has been so long wed to our dear lady! He rode in three days since—to rejoice our hearts. Lord, my Lady has asked all to watch for you, to urge you to hasten to the keep—"
"Be sure that I shall!" I said between set teeth. Who this false Kerovan was, I had no idea, but that I must see him, must save Joisan from any danger. Someone who knew of our marriage, who perhaps thought me dead, was taking advantage of the situation. And the thought that he could be with Joisan now was like a sword through me. That she might in time turn to another I had tried to endure, but that another had come to her in false guise—that I would not countenance.
For now I could call on my repute as an Old One of unknown but awesome powers. I could proclaim Kerovan dead, unmask this intruder, and she would believe me. Thus I had only to reach the keep, confront the imposter.
I urged my weary Hiku to a trot, though I longed to fling myself from his back, go pounding ahead into the keep, to call out this thief of another man's name and slay him out of hand—not because he had taken my name, but because he had boldly used it as a cloak to reach Joisan. In that moment how I wished I was truly what she thought me, one who could summon forces beyond the understanding of men.
Angarl, the one-handed, was on sentry duty and gave me a greeting I forced myself to answer. Very shortly I was in the courtyard. The deserted emptiness which had restrained me on my first visit here from too detailed exploration was gone. Life had returned to the hold, and it was now a place for humans.
Two men loitered by the water-trough, chaffing with one of the village girls, their deeper laughter banishing even more the atmosphere of the alien. They wore House badges adorned with my gryphon on their over-jerkins.
Before they looked up, I studied them. Neither was familiar—I had begun to wonder if survivors from Ulmsdale might have been drawn into this. However, the fact I did not recognize them meant little, for I had been away from Ulmskeep for months before my father's death, and he might have hired newcomers to augment his forces, taking the places of those who rode south with me.
That they wore such badges meant this was no hastily improvised scheme on the part of someone who had heard a rumor or two. Here was careful preparation—but why? Had Joisan still had Ithkrypt with men and arms, then I could have understood such a move. The false Kerovan would have ruled in Ithdale. But she was a landless, homeless fugitive. Why then?
One of the men glanced up, saw me, and nudged his fellow. Their laughter was gone; they eyed me warily. But I did not approach them. Rather I slid from Hiku's back and walked, stiff with the weariness of the trail, toward that tower room Joisan had made her own.
"You—!" The call was harsh, arrogant.
I swung around to see the two armsmen striding toward me. It was not until they fronted me that they seemed to realize I differed from their kind. I faced them calmly, bringing into my bearing the stiffness of one who has been approached by those beneath him in rank in a manner highly unbecoming.
"You—" the leader began again, but he was now uncertain. I saw his comrade nudge him in the ribs. It was the second man who now pushed a little to the fore.
"Your pardon, Lord," he said, his eyes searching me up and down. "Whom do you seek?"
His assumption of a steward's duties here fed my anger.
"Not you, fellow." I turned away.
Perhaps they would have liked to have intercepted me, but they did not quite dare. And I did not look to them again as I came to the tower doorway, now curtained with a horse blanket.
"Good fortune to the house!" I raised my voice.
"Lord Amber!" The blanket was thrust aside and Joisan stood before me.
There was that in her face at that moment which hurt. So he had won this already, this radiance! So ill had I played my part I had thrown away all. All? Another part of me questioned it. I had already decided that this was not for me. How could I then question her happiness if one she believed to be her lord had come to serve her in the depths of her need? That he was an imposter was all I must think on, that she must not be deceived.
"Lord Amber, you have come!" She put forth her hand, but did not quite touch mine, which I had raised against my will. Having spoken so, she stood looking at me. I could not understand—"
"Who comes, my fair one?"
I knew that voice from the dusk of the room behind her. Knowing it, my hate near broke bounds so that I thirsted to draw steel and press him into combat. Rogear here—but why?
"Lord Amber, have you heard? My lord has come—hearing of our troubles he has come—"
She spoke hurriedly and there was that in her voice which made me watch her closely. I had seen Joisan afraid. I had seen her rise above fear and pain of heart and mind, be strong for others to lean upon. But at this moment I thought that it was not joy that colored her tone. Outwardly she might present this smiling face, inwardly—no—
Her lord had not brought her happiness! Excitement stirred in me at that which I thought no guess but honest truth. She had not found in Rogear what she sought.
She retreated a step or two, though she had not answered his question. I followed, to stand facing my mother's kin. He wore a war tabard over his mail, with that gryphon to which he had no right worked on it. Above that his face was handsome, his mouth curved in that small secret smile, until he saw me—
In that instant the smile was wiped from his lips. His eyes narrowed, and there was about him watchfulness as if we both held swords in hand and were set against one another.
"My Lord." Joisan spoke hurriedly, as if she sensed what lay between us and wished to avoid battle. But she addressed me first as the higher in rank. "This is my promised Lord, Kerovan who is heir to Ulmsdale."
"Lord—Kerovan?" I made a question of that. I could denounce Rogear at once. But so could he me. Or could he? That which had been my bane, my deformity—would not Joisan continue to think it proved me alien? At any rate, Rogear must not be allowed to play any dark game here—whether it meant Joisan turning from me in disgust or not.
"I think not!" My words fell into the silence like blows.
In that moment Rogear's hand came up, something flashing in it—the crystal gryphon. From it struck a ray of light straight at my head. Pain burst behind my eyes, I was both blind and in such agony that I could not think, only feel. I staggered back against the wall, fighting to keep my feet. My arm was upflung in a futile effort to counter this stroke I was unprepared to face. I heard Joisan scream, and hard upon that cry another of rage and pain. Still blind, I was thrust aside and fell to the floor.
Joisan screamed again, and I heard sounds of a struggle. But I could not see! Not trying to rise, I threw myself toward the sounds.
"No! No!" Joisan's voice. "Loose me!"
Rogear had Joisan! A foot stamped upon my hand, giving me a second thrust of agony so great I could not stand it, yet I must! If he had Joisan, could drag her out—
I flung out my sound arm, touched a body, embraced kicking legs, and threw the weight of my shoulder against him, bearing him under me to the floor.
"Joisan—run!" I cried. I could not fight when I could not see; I could only hold on, taking his blows, trying to keep him so she could escape.
"No!" Her voice again, and with a cold note in it that I had never heard before. "Lie you very still, my Lord."
"Lord Amber," she said now, "I hold a knife blade to his throat. You may loose him."
He did indeed lie as one who would offer no more fight. I backed away a little.
"You say," she continued in that same tone, "this is not Kerovan. Why, my Lord?"
I made my choice. "Kerovan is dead, my Lady. Dead in an ambush laid by this Rogear above his father's keep. This Rogear has knowledge of the Old Ones—from the Dark Path—"
I heard a quickly drawn breath. "Dead? And this one dares to wear my lord's name to deceive me?"
Rogear spoke up then. "Tell her your name—"
"As you know, we do not give our names to mankind," I improvised.
"Mankind? And what are—"
"Lord Kerovan." My head jerked toward that new voice. "What do you—" It was one of the armsmen from the courtyard.
"Lord Kerovan does nothing," Joisan answered. "As for this one, take him and ride."
"Shall we take her, Lord?" asked the armsman.
I had gotten to my feet, faced toward that voice, though I could not see.
"Let the wench go. She is of no importance now." By his tone Rogear had regained his full confidence.
"And him, Lord?" Someone was moving toward me. My crushed hand hung useless. In any event I could not see him.
"No!" Rogear's answer was the exact opposite of what I expected to hear. In that moment a single thrust from a sword would have finished it all in his favor, and he could have had his will of Joisan. "Touch him on your peril!"
"We ride," he added. "I have what I came for—"
"No! Not that! Give me the gryphon!" Joisan's cry ended in the thud of a blow, and her slight body struck against mine. She would have slipped to the floor, but I flung my arm about her. They were gone, though I cried out for any in the courtyard to stop them.
"Joisan!" I held her close against me. She was a slack weight—if I could only see! What had that devil done to her? "Joisan!" Had he killed her?
But Joisan was not dead, only struck senseless, as those who came running told me. As for Rogear and his men, they were away. I sat by Joisan's bed, holding her hand in mine. About my useless eyes was bound a cloth wet in water in which herbs had been steeped. Only in that hour did I begin to face the fact that perhaps my sight was gone from me. Just as I had not been able to save Joisan from that last blow, so I would never again be able to step between her and any other harm. That was the black hour in which I learned how much she had come to be a part of me. The pain I had known earlier as I stood aside from making myself known was as nothing to what I now felt.
"Lord—" Joisan's voice weak and thin, but still her voice.
"Joisan!"
"He took—he took my lord's gift—the gryphon." She was sobbing now.
Fumbling, I drew her into my arms, so she wept upon my shoulder.
"It was the truth you spoke; he is not Kerovan?"
"The truth. It is as I said. Kerovan died in ambush at Ulmsdale. Rogear is betrothed to Kerovan's sister."
"And I never saw my lord. But his gift—that one shall not have it! By the Nine Words of Min, he shall not! It is a wondrous thing, and his hands besmirch it. And he used it as a weapon, Lord—he used it to burn your eyes!"
That flash from the globe—
"But also, Lord, your own power answered, from this band on your wrist. If you had only held that sooner as a shield.'' Her fingers were feather-light on my arm above the armlet. "Lord, they say those of you people are mighty in healcraft. If you have not that talent yourself, can we not take you to them? It is in my service, this grievous hurt came to you. I owe you a blood-debt—"
"No. There is no debt between us," I denied quickly.
"This Rogear has always been my unfriend. Had we met anywhere he would have sought to kill me." And, I thought bleakly then, perhaps I would be better dead now of a wound, than alive with this cloth about my head marking my loss.
"I have something of healcraft, and so has Nalda. Perhaps the sight will come again. Oh, my Lord, I do not know why he sought me here. I have no longer lands or fortune—save that which he took with him. Know you of this gryphon? It was sent me by my lord. Was it then such a great treasure of his House that this Rogear would risk so much to get it into his hands?"
Her query drew my thoughts away from my own darkness to consider why Rogear had come. The crystal gryphon—that it had strange powers was entirely possible. He was learned in the lore of the Old Ones—the Dark Old Ones. I had heard enough from Riwal to know that when one went some distance down the path of alien knowledge, things of power, both light and dark, could make themselves known to the initiate.
I had been with Riwal when I first found it. Neevor had said Riwal was dead, but he had evaded giving a description of how my friend had died. Supposing Rogear, already practiced in the Old Ones' learning, had somehow ferreted out Riwal, learned from him about the gryphon, traced it thereafter to my lady? That would mean it was such a talisman as could cause great troubles. In Rogear's hand its use would be a danger to the world I knew. Joisan was right, we must strive to get it back. But how—? I put my hand to my bandaged eyes with a sigh. Could it ever be done?
16
Joisan
I was in the eastern watchtower when my lord came to us. Though there was no alarm gong hung there, yet in the uppermost room we had found a metal plate set into the wall. A sword hilt laid vigorously to that brought forth a carrying sound. After Lord Amber rode out, we kept a sentry there by day—secure at night with the bridge drawn against any shoreside visitor.
So was I alert to the riders and beat out an alarm before I saw that they rode at a quiet amble and that Timon came afoot with them. He was under no restraint, but spoke with their leader in a friendly fashion. For a moment or two I was excited by the belief that some of our men might have escaped the slaughter by the river and been able to trace us. Then I saw that their battle tabards were not red but green.
They could be scouts of some other dale, and from them we could claim escort to Norsdale, though that thought made me a little unhappy. I wanted to reach Norsdale with my people. There the Lady Islaugha would find the proper tending, and the others for whom I was responsible would find new homes. Though the Lord Amber had indeed provided us with such a refuge as I had never hoped to find in this wilderness, we could not stay here forever.
I did not know why I had been so content here, as I had not been since that long-ago time before Lord Cyart had ridden south at the beginning of our time of trouble.
Now, even as the sound died away, I sped down the stair, eager to see who these newcomers might be. Yet I ordered my pace as I reached the courtyard, for chance had made me ruler in this place, and I must carry myself with proper dignity.
When I saw what was emblazoned on the tabard of he who led that very small company I was shaken. So well did I know that rearing gryphon! These were my lord's people. Or even—perhaps—
"My dear Lady!" He swung from his saddle and held out his hands to me in greeting, his voice warm.
Though I wore no skirt now, I swept him the curtsy of greeting. However, I was glad he attempted no wanner salute than voice and hand.
"My Lord Kerovan?" I made that more question than recognition.
"I so stand before you." He continued to smile. Yet—
So this was my wedded lord? Well, he had not Toross' regularity of feature, but neither was he unpleasing to look upon. For a dalesman his hair was very dark—a ruddy darkness—and his face less broad across the jaw—more oval. Despite those ugly rumors that Yngilda had first loosed for me, there was certainly nothing uncanny in him. Now Lord Amber was plainly of alien stock, but my Lord was as any dalesman.
That was our first meeting, and it was a constrained and uneasy one with many eyes watching. But how else could it be in this time and place when we two so bound together were strangers to one another?
I was grateful that he treated me with courtesy and deference, as one yet to be won, not a possession already in his hand. And he was both kind and gentle in his manner. Still—
Why did I feel that I wanted nothing more of him, that I regretted that he had come? He spoke me fair, always in that voice which was so pleasing, telling me that he, too, was now homeless—that Ulmskeep had fallen to the invaders. He and his men had been in flight from there, striving to reach Ithdale, when he had crossed the path of some of our people. Learning how it had been with us, he then pushed on to search for me.
"We were told you were with the army in the south, my Lord," I said, more to make conversation than to demand any explanation.
"That I was, for my father. But when he became ill he sent for me. Alas, I arrived too late. Lord Ulric was already dead, and the invaders so close to our gates that our last battle was forced upon me in great haste. But we were favored in that there was a storm from the sea, and so those of Alizon took nothing. In the end they were totally destroyed."
"But you said that your keep had fallen."
"Not to the Hounds. It was the sea that brought down our walls: wind and wave swept in, taking the land. Ulmsport"—he gestured—"now all lies under water.
"However," he continued briskly, "the lad who brought us here says you seek Norsdale—"
I told him our tale and of how I was curse-bound to the Lady Islaugha and must see her to safety.
He was grave-faced now, hearing me through without question, nodding now and then in agreement to something I said.
"You are well south of your proper trail," he replied, with the authority of one who knew exactly where he was. "And your fortune has been great to find such shelter as this."
"The finding was not ours, but Lord Amber's."
"Lord Amber? Who bears such an unusual name?"
I flushed. "He does not speak his name—that is mine for him, since every man must have a name. He—he is one of the Old Ones—one well disposed toward us."
I was not too ill at ease at that moment to miss noting his reaction to my words. He stiffened, his face now a mask behind which thoughts might race without sign. I have seen a fox stand so still as it listened for a distant hunter. A moment later that alert was gone, or else he concealed it.
"One of the Old Ones, my Lady? But they have been long gone. Perhaps this fellow seeks to deceive you for some purpose. How can you be sure he is such? Did he proclaim it so?"
"He had no need, as you will see when he returns. No man would be like him." I was a little irked at the note in his voice, as if he believed me some silly maid to whom any tale can be told. And since I had led my people I had become one to make decisions, to stand firm. I had no liking to be pressed again into the old position of a daleswoman, that only my lord or another man could see the truth in any question and then decide what was best for me. Before I was Kerovan's lady I was myself!
"So he returns? Where is he now?"
"He went some days ago to scout for the invaders," I replied shortly. "Yes, he will return."
"Well enough." My lord nodded again. "But there are different kinds of Old Ones, as our people learned long ago, to their sorrow. Some would be friends, after a distant fashion; some ignore us unless we infringe upon their secrets; and others follow the Dark Path."
"As well I know," I answered. "But Amber can touch your gift and it blazes as it did when it saved me from the Hounds."
"My gift—?"
Did I hear aright now? Had there been surprise in his tone? But no, I schooled my thoughts, I must not react so to everything he said as if we were unfriends and not life comrades, as we must learn to be.
He was smiling once more. "Yes, my gift. Then it has been of good use to you, dear one?"
"The best!" My hand went to my breast, where the gryphon lay safe as I ever kept it so. "My Lord, I am not wrong then in believing it is a treasure of the Old Ones?"
He leaned forward across the stone-slab table between us, where he had broken his fast on the best we had to offer. There was eagerness in his eyes, even if emotion did not show elsewhere on his countenance.
"You are not wrong! And since it has served you so well, I am doubly glad that it was in your hands. Let me look upon it once again."
I loosed the lacing of my jerkin, pulled at the chain to draw it forth. Yet something kept me from slipping it off and putting it into the hand he laid palm up on the table as if to receive it.
"It does not leave me, as you see, night or day." I made excuse. "Can you say the same for my return gift to you, my lord?" I tried to make that question light, the teasing of a maid with her wooer.
"Of course." Had he touched his own breast swiftly? Yet he made no move to show me the picture I had sent him.
"Still you did not follow my wishes sent with it." I pursued the matter. I do not know why, but my uneasiness was growing. There was nothing I had seen in him or his manner to make me unhappy yet, but there was a shadow here that I could sense and that vaguely alarmed me. I found that I was restoring the gryphon to its hiding place hastily, almost as if I feared he might take his own gift from me by force. What lay between us that I should feel this way?
"There was no time, nor trusted messenger," he was saying. However, I was sure he watched the globe as long as it was in sight with greater interest than he had for my face above it.
"You are forgiven, Lord." I kept to the light voice, molding my manner on that of the maids I had seen in Trevamper in that long-ago time before all this evil had broken upon the dales. "And now—I must be back to my duties. You and your men must have lodging, though you will find it somewhat bare. None of us sleep soft here."
"You sleep safe, which in these days is much to say." He arose with me. "Where do you quarter us?"
"In the west tower," I answered, and drew a breath of relief that he had not asserted his right and demanded my compliance. What was the matter with me? In all ways he had shown consideration and courtesy, and in the days that followed it was the same.
He spoke of our efforts to harvest the stunted grain from the old fields and pick and dry the fruit. He praised our forelooking. He was quick to say that such a band as ours—of the old, the very young, and the ailing—would be extra long on the road and would need all the food we could gather. His armsmen took over sentry duty, leaving us free to work in field and orchard.
Nor did he press for my company, which eased my feeling of constraint. Only I was haunted by an uneasiness I could not account for. He was kind; he strove to make himself pleasing to me in thought and word. And yet—the longer he stayed, the more unhappy I was. Also I was a little frightened at the prospect to come of the time when he would be my lord in truth.
Sometimes he rode forth to scout along the hills, assuring us added protection. I would not see him from dawn until sunset, when we drew in that movable part of the bridge. However, twice in the dusk I saw him in conversation with Yngilda, whom he treated with the same courtesy he showed to me and Lady Islaugha, though the latter seemed hardly aware of his presence.
I do not believe he sought out my kinswoman. Their meetings must have been of her contrivance. She watched him hungrily. At the time I believed I knew what lay in her mind—her lord was dead; she had naught to look forward to save a dragging life at Norsdale. Yet I, whom she hated—yes, hated, for I no longer disguised from myself that her dislike of me had crystallized through envy to something deeper and blacker—had Kerovan, eager to serve me. That she might make trouble between us, well, that, too, was possible. But I did not think so, for I could see no way that even a spiteful tongue could disturb the relationship we now had, it being so shallow a one.
Had my heart followed my hand, that would have been a different matter. Undoubtedly I would have resented meddling. As it was, no action of hers disturbed me.
On the fifth day after his arrival, Kerovan did not stay so late in the hills, but came back in mid-afternoon. I had been in the fields, for another pair of hands was needed. One of the children had run a thorn into her foot, so I brought her back to wash and bind it with healing herbs we had found.
Her tears soothed, her hurt tended, she went running back to her mother, leaving me to put away our small supply of medicants. While I was so busied, my lord came to me.
"My fair one, will you give into my hand that gift I sent you? There is perhaps the chance that it can serve you even better than it has, for I have some of the old learning and have added to it since I sent you this. There are things of power which, rightfully used, are better than any weapons known to our kind. If we indeed have such a one, we can count on an easier journey to Norsdale."
My hand went to what I wore. In no way did I wish to yield to his request. Still, I had no excuse. I did not understand myself, why I shrank from letting it go out of my hands. Most reluctantly I loosed lacings and drew it forth. Still I held it cupped for a long moment while his hand was held forth to take it. He smiled at me as one who would encourage a timid child.
At length, with an inner sigh, I gave it to him. He took a step into the light of the slit window, held it at eye level as if he were fronting the tiny gryphon in some silent communication.
It was at that moment that I heard from without the traveler's greeting. "Good fortune to his House."
I did not have to see the speaker. I knew at once and was past Kerovan into the open.
"Lord Amber!" I did not understand the emotion awakened in me at the sound of his voice. It was as if all the uneasiness of the past few days was gone, that safety stood here on hoofed feet, gazing at me with golden eyes.
"You have come!" I put forth my hand, eager to clasp his. But that reserve which was always a part of him kept me from completing that gesture.
"Who comes, my fair one?" Kerovan's voice broke through the wonderful feeling of release, of security. I had to find other words.
"Lord Amber, have you heard? My lord has come—hearing of our troubles he has come."
I retreated a little, chilled by a loss I could not define. The Old One followed. I did not look to my lord, only still into those golden eyes.
"My Lord, this is my promised lord, Kerovan, who is heir to Ulmsdale.''
His face was secret, close, as I had seen it other times.
"Lord Kerovan?" He repeated it as if he asked a question. And then he added with the force of one bringing down a sword in a swift, lashing stroke, "I think not!"
Kerovan's hand came up, the globe clasped tight in his fingers. From that shot a piercing ray of light, striking full into Lord Amber's eyes. He flung up one hand as he staggered back. On the wrist of that hand I saw an answering glow, as if a blue mist spread to raise a curtain about him.
I screamed and struck at Kerovan, striving to snatch the crystal from him. But he beat me off, and in his face I read that which changed my uneasiness to fear.
Kerovan caught me in a strong hold and dragged me toward the door. Lord Amber, one hand to his eyes, was on his hands and knees, his head moving as if he strove to find us by sound alone, as if he were blind!
I beat at Kerovan, twisting in his hold. "No! Loose me!"
Lord Amber lurched toward us. I saw Kerovan raise his boot, stamp, and grind one of those groping hands against the floor. But with his other arm Lord Amber caught Kerovan about the knees and bore him down by the weight of his body.
"Joisan, run!" he cried.
I was free of Kerovan, but run I would not. Not while Lord Amber took the vicious blows Kerovan was dealing.
"No!" My belt knife, Toross' legacy, was in my hands. I crouched above the struggling men, caught Kerovan's hair, and jerked his head back, putting the edge of that blade to his throat. "Lie you very still, my lord," I ordered. He must have read my purpose in my face, for he obeyed.
Not taking my eyes from him, I said then, "Lord Amber, I hold knife-edge to his throat. Loose him."
He believed me and edged away.
"You say," I continued, "this is not Kerovan. Why?"
He was rising to his feet one hand still to his eyes. "Kerovan is dead, my Lady." His voice sounded very weary. "Dead in an ambush laid by this Rogear above his father's keep. This Rogear has knowledge of the Old Ones—from the Dark side."
My breath hissed between my teeth. So much was made plain to me now. "Dead? And this one dared to wear my lord's name to deceive me?"
Rogear spoke up then. "Tell her your name—"
Lord Amber answered him. "As you know, we give not our names to mankind."
"Mankind? And what are—"
"Lord Kerovan!" I was so startled by that voice from the door I jerked away from my guard post. "What do you—?" One of his armsmen stood there.
I spoke quickly. "Lord Kerovan does nothing! As for this one—take him and ride!"
There was a second man behind him, with the bolt on his crossbow, which was aimed not at me but at Lord Amber. And in his face a horrible eagerness, as if he would joy in loosing death.
"Shall we take her, Lord?" asked the first man.
Lord Amber was moving toward him, his hands empty. And the man held ready steel.
Rogear had rolled away from me. "Let the wench go. She is of no importance now."
"And him, Lord?"
"No! Touch him not, on your peril!"
I had thought he would order Lord Amber's death, if one of the Old Ones can be so killed.
"We ride," he added. "I have what I came for." He was putting the gryphon into the inner pocket of his tabard.
That roused me to action. "No. Not that!" I sprang at him. "Give me the gryphon!"
He aimed a blow at my head with his other hand, and I did not dodge in time. A burst of pain drove me into darkness.
When I awoke I lay on my bed-place, and the dusk was deep. But I saw that Lord Amber was beside me, and my hand lay in his. There was a bandage bound about his head, covering his eyes.
"Lord—"
He turned his head instantly to me. "Joisan!"
"He took the gryphon!" For I had brought out of the dark that memory, strong enough to urge me into action.
Lord Amber drew me gently to him, and I wept as I had not in all those days of danger and sorrow that lay behind me. Between my sobs I asked, "It was the truth you spoke? He was not Kerovan?"
"It was the truth. It is as I said, Kerovan died in Ulmsdale. Rogear, who is betrothed to Kerovan's sister, arranged the ambush."
"And I never saw my lord," I said then in sad wonder. "But his gift, that one shall not have it!" Anger brought me strength. "By the Nine Words of Min, he shall not! It is a wondrous thing and his hands besmirch it! He used it as a weapon, Lord—he used it to burn your eyes. It was what rested on your wrist that defeated him. If you had only used it sooner as a shield!" I put my fingers to his wrist a little above that armlet. "Lord," I continued, "they say those of your people are mighty in healcraft. If you cannot aid yourself, can we not take you to them? It is in my service that this grievous hurt was done you, I owe this as a blood-debt—"
But he denied that with force and quickly. "No! There is no debt between us. Had we met elsewhere he would have sought to kill me."
"I have something of healcraft," I said then, "and Nalda.'' But in my heart I knew how limited we were, and that gave birth to fear. "Perhaps the sight will return. Oh, my Lord, I do not know why he sought me here—I have no longer lands nor fortune—save what he took with him. Know you of the gryphon? It was sent to me by Kerovan. Was it then such a great treasure of his House that this Rogear would risk much to get it into his hands?"
"No. It is no treasure out of Ulmsdale. Kerovan himself found it. But it is a thing of power and Rogear has enough of the Dark knowledge to use such. To leave it in his hands now—"
I could reach his thoughts as well as if he put them into words—to leave such a weapon with one of the Dark Ones was something we.were bound in honor not to allow. But Rogear—not only did he ride with armsmen prepared to slay, but he had already shown he could harness the gryphon to his service too.
"My Lord, what can we do then to gain it once more?" I asked simply. For in this man (if man one might call him), I centered now all my trust.
"For the present"—weariness was deep in his voice—"I fear very little. Perhaps Rudo or Angarl can follow his trail a little, mark his path from here. But we cannot follow—yet—"
Again I believed that I knew his thoughts. He must nurse some hope that his sight would return. Or else he had some power of his own he could summon to aid. In this thing he must ride as marshal, I as an armsman. For I knew that the quest, or coming battle, was as much mine as his. It was my folly that had delivered the gryphon to Rogear. Thus my hand must have a part in its return.
My head ached cruelly, and Nalda brought a bowl of herb tea that she said I must drink. I suspected that it would make me sleep, and I would have refused. But Lord Amber urged me to it, and I could not set my will against his.
Then Nalda said she had a new ointment for his eyes, something she had used on burns, and that she would dress them again. I do not think he believed it would help, but he allowed her to take his hand and guide him forth.
I was only on the edge of drowsiness when Yngilda came to me, standing above my bed and staring down as if I had, in the space of hours, taken on a new face.
"So your Lord is dead, Joisan," she said. I detected satisfaction in her words. That I did not prosper over her meant much.
"He is dead." I felt nothing. Kerovan had been a name for eight years—little more. To me he was still a name. How can one sorrow for a name? Instead it was a matter for rejoicing that I had that strange, instinctive dislike for the imposter. Rogear was not my lord; I need feel no discomfort or guilt because I did not like or trust him. My lord was dead, having never really lived for me.
"You do not weep." She watched me with that sly malice with which she so often favored me.
"How can I weep for one I never knew?" I asked.
She shrugged. "One shows proper feeling—" she accused.
We were no longer by keep custom, not here, not with our world swept away by the red tide of war. Were I back at Ithkrypt, yes, I would have kept the terms of conventional mourning as would be expected of me. Here there was no reason for form alone. I was sorry that a good man had died, and by the treachery of his kin, but mourn more than that I could not.
She drew from an inner pocket a strip of cloth made into a bag. I caught a whiff of scent from it and knew it for one of the herb bags put under pillows for those with aching heads.
"My mother's, but she does not need it this night." Yngilda spoke brusquely, as if she believed I might refuse her offering.
I was surprised, yes, but not unduly so. Perhaps now that we were equal before the world, Yngilda would no longer think me the more fortunate. So I thanked her and allowed her to slip the scented bag beneath my head where the warmth of my body could release the odor to soothe me.
The herb broth was doing its work also. I found it hard to keep my eyes open. I remember seeing Yngilda turn away toward the door, and then—I must have slept.
17
Kerovan
"It is Nalda?" I turned my head, though I could not see.
"Yes, Lord." She spoke briskly, and I silently thanked her for her way toward me. There was no pity in her manner, only the confidence of one who had nursed hurts and expected healing to come from her service.
"Lady Joisan?"
"She sleeps, Lord. And truly she has taken little hurt, save that the blow was a hard one. But there be no bones broken or other injury."
"Have the men reported in yet?"
"No, you shall see them at once when they come. Now here is some soup, Lord. A man must keep his belly filled if he would hold his strength. Open your mouth—"
She spooned it into me as if I were a baby. Nor could I then say her "no." But in me was a rage against what had happened and a dark feeling of misery that there was naught I could do for myself.
Nalda guided me to my bed, and I stretched thereupon. But sleep, even rest, was far from me, wearied from the trail as I was. I lay rigid, as one who expects any moment to be called to arms, though I might never be again.
I thought of Joisan—of her need to regain the gryphon. I knew that she was right; that it must be taken from Rogear. He had not been caught in the doom of wind and wave. Had then those others escaped also—Hlymer who was no true brother, the Lady Tephana, Lisana—?
Now I raised my hands to explore the bandage over my eyes. It was still damp, and I was sure it was of no use to me.
Rogear—if he had come after the gryphon—how could he have known of it save through Riwal, and from Jago, that it had passed to Joisan? What was it that he had come to seize? I knew so little at a time when knowledge was so essential.
I rested my arm across my forehead, the back of my wrist upon that bandage. How long was it before my thoughts were shaken out of the dreary path they followed, and I realized something was in progress?
The wrist band! Joisan had said it defeated the ray from the crystal. From it now—I sat up and tore away the bandage. A warmth spread from the wristlet where it touched my flesh. Perhaps instinct, perhaps "memory" guided me now, for I held that band of strange metal first against my right eye and then against my left, pressing upon the closed lids. I did not try to see as yet. What I did was simple. Why I did it I did not know, save there came from that act a sense of well-being, a renewed confidence in life.
I dropped my hand and opened my eyes.
Dark! I could have cried my vast despair aloud. I had thought—hoped—
Then I turned my head a little and—light! Limited—but there. And I realized that I sat within a darkened room with light marking the doorway. Hastily I arose and went to it.
Night, yes, but no darker than any night I had seen before. When I had raised my head to look heavenward—stars! Stars glittering more brightly than I could remember. I could see!
Joisan! Instantly in my joy I knew I must share this with her. And that was mainly instinct too. I looked around the courtyard to get my bearings and headed to her room.
The doorway curtain was down and made me pause. Nalda had said she had given her lady a sleeping draught and that she would rest until morning. But even if I could not tell her of this miracle, I could at least look upon her dear face. There was a faint glimmer of light behind the curtain. They must have left one of the rush lamps with her.
So I entered, wishing to shout aloud my tidings, yet walking softly, trying to control even my breathing lest I disturb her rest.
Only, there was no one on her bed! The light cloak that must have been her night covering was tossed aside; the couch of leaves, grass, and brush was empty.
Empty save for something that lay in the hollow where her head must have rested. It caught my eye and I scooped it up. I held a bag, lumpily stuffed with herbs, which gave forth a strong odor. Among the leaves and roots there was a harder knot.
I gasped and the bag fell to the floor. About the band on my wrist coiled a thin blue haze, as if the metal had given forth a puff of smoke. I needed no instruction as to the nature of what was in that bag. Black evil shouted aloud in my mind.
Stooping, I caught up the bag by the point of my knife and dropped it on the stone table, close to the rush light. Without laying finger on it I slit the cloth, using the knife point to dig and probe until I brought into sight a thing about the size and shape of a Sulcar trading coin. It was dull black, yet also veined in red, and those veins—No, they were not veins after all, but some runic pattern as involved as those on my wristlet.
This was a thing of Power, that I knew. But of the Dark Power. Anyone touching it—
Joisan! How had this evil thing come into her bed? In that moment such a fear rent me that I shouted, calling on Nalda who should certainly be near at hand. The fury of my voice echoed hollowly out into the courtyard. I called again, heard other voices upraised—
"My Lord," Nalda stood in the doorway, "what—"
I pointed to the bed. "Where is my lady?"
She exclaimed, hurried forward, stark surprise on her face.
"But—where else could she be, Lord? She was sleeping, as the drink would make her. I would take Gunnora's Three Oaths that she could not stir until morning—"
"Did you leave this in her bed?" I had controlled my fear, outwardly at least. Now I used the knife to indicate the torn bag and its contents.
She leaned close, sniffing. "My lord, this is a soothing bag such as we make for the Lady Islaugha when she is restless. One of these beneath her pillow, and she is not so led by her fancies. It is of good herbs—"
"Do you also add this?" My knife point was close to the evil symbol.
She bent her head again. When she raised it and her eyes met mine, she looked stricken.
"Lord, I know not what that thing is, but—it is wrong!" Then something else burst upon her. "Lord—your eyes—you can see!"
I brushed that aside. Once that relief had filled my world, but of greater concern now was what had happened to Joisan. That she had been exposed to this thing of the Dark was agony to think on.
"Yes," I answered shortly. "But my lady slept with this by her, and she is gone. I know not what deviltry has been wrought here—but we must find her soon!"
So the aroused company searched from sentry towers to bridge ends. As the one that worked was pulled up for the night, I could see no way Joisan could have gone ashore. Yet it was plain she was not hidden in any of the rooms we explored.
In the end I had to accept that Joisan was nowhere in the keep. There remained—the lake! I stood at the bridge gap looking into the water, holding my torch to be reflected from its surface. Rogear—there was only one who would have done this thing! But he had been well away when Joisan had been laid on her bed. Someone in this place had been his servant in the matter. And from that servant I would have the truth!
I summoned them all, men, women, children, into the courtyard, and on a stone there I placed that ominous thing which had been a weapon aimed at my lady. I no longer felt the heat of first anger in me. For there had crept a cold along my bones, and my mind fastened on one thing alone—there would be such a blood-price for Joisan as these dales had never seen.
"Your lady has been taken from you by treachery." I spoke slowly, simply, so that the youngest there might understand. "While she was weak of body this evil thing was put into her bed, and so she was driven forth, perhaps to her death." Now I ventured onto ground of which I was not sure, depending heavily on what I had learned from Riwal. "Those who meddle with such a thing as this carry the taint of it upon them. For it is an essence of evil as to soil beyond cleansing. Therefore you shall each and every one of you display your hands and—"
There was a swirl among the women, a cry. Nalda had seized upon one who stood beside her, held fast a screaming girl. I was with them in an instant.
The Lady Yngilda—I might have expected it.
I spoke to Nalda. "Bring her! Do you need help?"
"Not so!" She was a strong wench and she held the whimpering girl easily.
I spoke then to the others. "I shall settle this matter. And I lay upon you—let no one touch, for his spirit's sake, what lies here."
They did not move from the courtyard, and none followed us as we returned to Joisan's chamber.
I thrust my torch into one of the wall rings, thus giving us more light. Nalda had twisted Yngilda's arms behind her back, prisoning her wrists in a grip I think even few men could have broken. She swung her captive around to face me. The girl was blubbering, still jerking futilely to loose herself.
Catching her by the chin, I forced her head up to meet me eye to eye.
"This was of your doing." I made that an accusation, no question.
She wailed, looking half out of her wits. But she could not escape me so.
"Who set you to this? Rogear?"
She wailed again, and Nalda gave her a vigorous shake. "Answer!" she hissed into her ear.
Yngilda gulped. "Her lord—he said she must come to him—that would bring her—"
I believed that she spoke the truth of what Rogear had told her. But that Yngilda had been moved by any goodwill toward Joisan in the doing of this I knew was not so. That Rogear had left such a trap out of malice I could also believe.
"Bring her to her death," I said softly. "You stand there with her blood on your hands, Yngilda, as surely as if you had used your knife!"
"No!" she cried. "She is not dead, not dead! I tell you she went—"
"Into the lake," I finished grimly.
"Yes, but she swam—I watched—I did, I tell you!"
Again I believed she spoke truthfully, and that cold ice in me cracked a little. If Joisan had gone ashore, if she were under some ensorcellment—then I still had a chance to save her.
"It is a long swim—"
"She climbed ashore; I saw!" She screamed back at me in a frenzy of terror, as if what she read in my face near broke her wits.
I turned to the door. "Insfar, Angarl." I summoned those two who had proven best at tracking. "Go ashore and look for any sign that someone came out of the lake!"
They were on their way at once. I came back to Nalda and her charge.
"I can do no more for you and your people now," I told Nalda. "If my lady has been ensorcelled—"
"She is bespelled," Nalda broke in. "Lord, bring her back safe from that!"
"What I can do, be sure that I will." I said that as solemnly as any oath one could make with blood before kinsmen. "I must follow my lady. You will be safe here—at least for a time."
"My Lord, think not of us. But rather fasten your thoughts upon my lady. We shall be safe. Now—what of this one?" She looked to Yngilda, who was weeping noisily.
I shrugged. Now that I had what I wanted from her, the girl was nothing to me. "Do as you will. But I lay upon you that she should be well watched. She has dealt with a Dark One and obeyed him. Through her more evil may come."
"We shall see to her." There was such a promise in Nalda's voice that I thought Yngilda might well shiver.
I went back to the courtyard and took up the coin of evil on the point of my knife and carried it into the water. I would not bury it in the ground lest the unknowing chance upon it.
Dawn was breaking when I rode forth on Hiku with fresh provisions for the trail. Yngilda had spoken the truth; a swimmer had come ashore, crushing lake reeds and leaving a trace that could not be mistaken. Beginning there I must follow my lady.
What manner of sorcery had been used on her I did not know, but that she was drawn against her will I had no doubts. I tracked her to the valley rim. There she was met by those who were mounted, and I knew that Rogear and his armsmen had lurked there waiting for her.
Four they were, and with perhaps such weapons as I could not imagine, my lady probably well bound so I could not entice her in any way from their company. I might only follow, trusting fortune to give me a chance, ready to help fortune when it did.
The trail led west and north, as I thought it might. It was my belief that Rogear intended to return to his own keep. He had come to Ulmsdale to obtain power. Perhaps now with the gryphon he had it.
They did not often halt, and for all my pushing they kept ever ahead. On the second day I found traces that told me their party had been augmented by three more riders. Also there were led horses, so that they could change mounts when theirs wearied. Whereas I had only Hiku, who was already worn.
Still the rough-coated pony never failed me, and I thought that any mount supplied by Neevor might be more than he seemed in outward appearance. It was after I snatched the rest I must have on the third night and headed on in the morning that I realized we were skirting lands I knew, coming into the forested fringe which had been my boyhood roving place.
There could be only one goal for those I followed. They were heading for the Waste. Well, what else could I expect—they dabbled in forbidden knowledge; surely they would turn to some possible source of the Power they wooed. But why had they taken my lady? To spite me? No, Rogear would have no interest in that. To his mind I was maimed, not to be considered an enemy any longer. And he had the gryphon—why must he have Joisan also? I kept thinking of this as I went, trying this explanation and that, yet none seemed to fit.
On the morning of the fifth day I reached the edge of the Waste, near, I realized upon checking landmarks, to that road which ran to the naked cliff. And I was not greatly surprised to discover the trail I followed led in that direction.
Once more I rode on that ancient pavement. But it was difficult to remember that time, as if what had happened to me before had been the actions of another Kerovan who was not I, or even close kin. How I wished now for Riwal. He would have known so much more, though he was no Old One. But the safeguards he had had were not mine, and those I trailed were far more learned, I feared, than Riwal.
One night I camped along the road, scanting my rest, on my way before dawn. Here were the hills where those carvings stood out on the cliff faces. I found in my going curling runes resembling those on my wristlet. From time to time, viewing them, I felt a quickening excitement, as if I were on the very verge of understanding their meaning, yet I never did.
As before, I believed I was dogged by something that spied upon me. Though it might not have been dangerous in itself, what it might serve was another matter. I reached again the place of the great face. And before it I found evidence of those I hunted.
Set out on a rock before that great countenance was a bowl and, flanking it, two holders of incense. The bowl still held a film of oily liquid, and the incense holders had been recently used. All were of a black metal or stone I did not know. But I would not have set fingertip to them for my life's sake. Around my wrist once more that blue warning arose. What I did I was moved to by revulsion. I hunted about for stones and, with the largest of these, I smashed all that was set out. There came a shrill noise as they were powdered into fragments. Almost one could believe that the things had life of their own. But I did not leave them behind me as a ready focus for any remnant of the Dark that might linger here.
When I came to that great star, which had so awed Riwal, I found no similar signs of any ceremony, only marks in the earth to show that here they had left the narrowed road on the far side, squeezed by as if they wished to be as distant from that carving in passing as they could get. This, then, they feared. I paused for a moment to study it. But it held no heartening message save that—they had feared it.
Ahead lay only the cliff wall; they could go no farther. I had come to my journey's end, and I had no better plan in my head than to front boldly what waited me there. So I dismounted and spoke to Hiku:
"Friend, you have served me well; return now to him who gave you." I stripped away bridle and riding pad, dropping them to the road, because I believed that what lay before me was death. It would not be their choice of death, however, for my lady and me, but mine. If need be she would die by my hand, clean of the evil they might try to lay on her.
My fingers went to the band on my wrist, seeking the pattern there. It was a thing of power, I knew. Only I had not the key of its use. However, touching it so, I stared upon the star and longed to know what would defeat the Dark Ones ahead.
It was at that moment Neevor's words returned to me:
"You shall seek and you shall find. Your own heritage shall be yours. The discovery of what you are and can be you must make for yourself."
Brave words—said only to hearten? Or were they prophecy? Riwal said that to call upon a name in this place would unlock some force. But I knew no names; I was only human—of mixed blood perhaps—but human—
It seemed to me in that moment that I had spoken that word aloud; that it echoed back to me from the walls.
I flung up my arm before the star, and I made my plea, but not aloud. If there was any power here that might be drawn upon, let it come to me. Even if it blasted me, let me hold it long enough to free my lady, to deal with Rogear who sought to bring to this land that which was better lost. Let—it—fill—me.
It was as if something within me moved, slowly, grudgingly, as might a long-locked door. There was a flow from behind that door, one I did not understand. With it came such a maze of shadow memories as nearly overbore me. But I fought to remember who I was and why I stood there. And the memories were but shadows after all; my will was the sun to banish them.
But I knew! The shadows left behind that much. I had a weapon. Whether it would stand against what those others might marshal I could not tell until I put it to the final test. And the time was now!
I trotted ahead, urgency driving me. A sound broke the silence, a chanting that rose and fell as waves pound a coast. I rounded the bend and came upon those I sought. But of me they took no notice. They were too intent on what they would do here.
Upon the ground was a star enclosed in a circle. And that circle had been drawn in blood, blood that smoked and stank and had been drained from the armsmen who lay dead at one side like so much refuse.
On each point of the star was a spear of darkness, of oily smoke, that struck up into the sky, adding its stench to that of the blood. And before each of the points stood one of their party, four facing inward, the fifth placed before the wall to stare blank-eyed at it.
Hlymer, Rogear, Lisana, the Lady Tephana and, with her face to the wall, my Joisan. The four chanted, but she stood as one who walked through nightmares and could not help herself. Her hands were at her breast, and between them she held the gryphon.
It was if they shouted their purpose aloud, for I knew it. They were before a door, and Joisan held the key. By some fate she alone could use it, and so they had brought her for that task. What lay behind that door to which this road ran—who knew. But that I would let them open it—no!
Still they did not see me, for they were so intent upon what they did that the world beyond their star-in-circle had ceased to have real existence for them. Now I perceived something else around that line of smoking: blood-edged creatures as wispy as shadows. Now and then some dreadful snout sniffed at that barrier or dabbled in it. Fresh blood drew these remnants of ancient evil, but they were worn by centuries to such poor things they were shadows only. Of them I had no fear.
Some sighted me and came padding in my direction, their eyes glinting like bits of devilish fire. Without my willing it consciously, my arm swung up and they cowered away, their eyes upon my wrist band. So I came to the circle of blood. There the smoke made me sick to the center of my being, but against that body weakness I held firm.
Now I raised my voice and I named names, slowly, distinctly. And my words cut through the spell their chanting raised.
"Tephana, Rogear, Lisana, Hlymer—" As I spoke each, I faced a little toward the one I so named. There was a shadow flicker in my mind. Yes, this was the right of it! This had I done once before in another place and time.
All four of them started as if they had been quick-awakened from sleep. Their eyes no longer centered on Joisan's back; they turned to me. I saw black rage flare in Rogear's, and perhaps in those of Lisana and Hlymer. But the Lady Tephana smiled.
"Welcome, Kerovan. So, after all, you prove the blood runs true." Her voice was sweeter than I had ever heard it, as she counterfeited what should have bound us together and never would. But if she thought me so poor a thing that I could be so deceived, she reckoned little of what she had once wrought.
Again that shadow knowledge moved in my mind, and I made her no answer. Instead I raised my hand, and from my wristlet a beam of blue light shot to touch the back of Joisan's head.
I saw her sway, and she gave a piteous cry. Still that which controlled me kept me to the attack, if attack it was. Slowly she turned around, seeming to shrink under a blow she could not ward off. Now she was away from the wall, facing me across the star-in-circle. Her eyes were no longer empty of what was Joisan. There was intelligence and life in them again, as she looked about her.
I heard a beast's growl from Hlymer. He would have leaped for my throat, but the Lady Tephana gestured, and he was silent and quiet in his place. Her hands moved back and forth in an odd manner as if she wove something between them. But I had little time to watch, for Rogear had moved also. He had Joisan in his hold, keeping her between us as a shield.
"The game is still ours, Kerovan, and it is to the death," he said. We might have been facing each other across a gaming board in a keep hall.
"To the death—but to yours, not mine, Rogear." With my upheld hand I sketched a sign, a star without a circle. Between us in the air that star not only glowed blue-green, but it traveled through the space between us until it was close to him at face level.
I saw his face go gaunt, old. But he did not lose his belief in himself. Only he dropped his hold on Joisan and stepped forward saying, "So be it!"
"No!" The Lady Tephana raised her eyes from what she wove without substance. "There is no need. He is—"
"There is every need," Rogear told her. "He is much more than we deemed him. He must be finished, or we shall be finished too. Spin no more small spells, Lady. You had the fashioning of him flesh and bone, if not spirit. Lend me your full will now."
I saw for the first time uncertainty in her face. She glanced at me and then away swiftly, as if she could not bear to look upon me.
"Tell me," Rogear pressed, "do you stand with me in this? Those two"—he motioned to Hlymer and Lisana—"can be counted as nothing now. It is us against what you sought to make and failed in the doing."
"I—" she began, and then hesitated. But at last the agreement he wished came from her. "I stand with you, Rogear."
And I thought—so be it. From this last battle there would be no escape, nor did I wish it.
18
Joisan
I dreamed and could not wake, and the dream was dark with fear at its core. For me there was no escape, for in this dream I walked as one without will of my own. He who gave the order was Rogear.
First there was the calling, a need so laid upon me that I left the keep, trusted myself to the waters about it, swam for the shore. Then I must have traveled yet farther across those deserted fields until Rogear was there and he horsed me before him to ride.
There were parts I could not remember. Food was put in my hands and I ate, yet I tasted nothing. I drank and was aware of neither thirst nor the quenching of it. We were joined by others, and I saw them only as shadows.
On we rode into strange places, but these were the places of dreams, never clearly seen. At last we came to the end of that journey. There was—no! I do not wish to remember that part of the dream. But afterward, I held my lord's gift in my hands and it was laid upon me, as much as if I were in bonds, that I must stand, and when orders came I must obey. But what I was to do—and why—?
Before me was a cliff rising up and up, and behind me I heard a sound, a sound that lashed at me. I wanted to run—yet as in all ill dreams I could not move, only stand and look upon the rock and wait—
Then—
There was pain bursting in my head, like fire come to devour my mind, burn out all thought. But what vanished in those flames was that which held me prisoner to another's will. Weakly I turned away from the cliff to look upon those who held me captive.
Lord Amber!
Not as I had seen him last with bandaged eyes, fumbling in blindness, but as a warrior now, ready for battle, though his sword was sheathed and he had no knife-of-honor ready. Still, that he warred in another way, I knew.
There were four others. And I saw then there was a star drawn in the earth and that I stood in the point that fronted the cliff, those others to my right and left in the other points.
One was Rogear, two were women, the fourth another man. He made a move in the direction of Lord Amber, but the woman to my right stayed him with a gesture. Rogear sprang before I could move and held me like a battle shield.
"The game is still ours, Kerovan," he said, "and it is to the death."
Kerovan! What did he mean? My lord was dead.
Lord Amber—it was Lord Amber who answered him. "To the death, but to yours, not mine, Rogear." I saw him draw a sign in the air, and there was a blue star that traveled to hang before Rogear's eyes.
He loosened me and stepped away, saying, "So be it."
"No!" The woman to my right spoke. "There is no need. He is—"
Rogear interrupted her. "There is every need. He is much more than we deemed him. And he must be finished, or we shall be finished too. Spin no more small spells, Lady. You had the fashioning of him, flesh and bone, if not spirit. Lend me your full will now!"
She glanced swiftly then at Lord Amber; then away. I saw her lips tighten. In that moment she was far older than she had seemed earlier, as if age settled on her with the thoughts in her mind.
"Tell me," Rogear continued, "do you stand with me in this? Those two"—he motioned toward the other man, the girl—"can be counted as nothing now. It is us against what you sought to make and failed in the doing."
I saw her bite her lip. It was plain she was in two minds. But at last she gave him what he desired. "I stand with you, Rogear."
"Kerovan," Rogear had called him, this man I would have taken blood-oath was one of the Old Ones. At that moment, all those sly whispers and rumors flooded back in my mind—that my lord was of tainted blood, becursed, that his own mother could not bear to look upon him. His own mother! Could it be—? Rogear said this woman had the fashioning of him, flesh and bone, but not spirit. Not spirit!
I looked upon Lord Amber and knew the truth, several truths. But this was not the time for the speaking of truth, nor the asking of whys and wherefores. He faced those who were deadly enemies, for there be no more deadly enemies than those of close blood-kin when evil works. And they were four against his one!
His one—! I looked about me wildly. I had no weapons—not even Toross' knife. But a stone—even my bare hands if need be— Only this was not fighting as I had known it. This was a matter of Power—Power such as Math had loosed at her death hour. And I had no gift of such. I tried to clench my fist. A chain looped about my fingers and cut my flesh. The gryphon—I still had the gryphon! I remember how Rogear had used it before—could not Lord Amber do likewise? If I could throw it to him—But Rogear was between; he need only turn, wrest it from me, use it as he had before—
With this in mind, I wrapped my two hands tight about the globe, saying to myself that Rogear would not take it from me to use against my lord, not while I had life to defend it!
My lord—Kerovan? I did not know the rights of that—whether Lord Amber had lied to me. But had he, my heart told me, then it had been with good reason. For just as I had shrunk from Rogear when he played Kerovan to entrap me, so did I now range myself with this other in time of battle. Old One or no, Kerovan or no, whether he wished it or no, in that instant of time I knew that we were tied together in such a way no axe bond or Cup and Flame ceremony could add to. That I welcomed this I could not have said, only it was as inevitable as death itself.
This being so, I must stand to his aid. Though how I could—
Almost I cried aloud with pain. My hands—! I looked down. My shrinking flesh could not hide the glow I held. The gryphon was coming to life, growing hotter and hotter. Might I then use it as Rogear had—to strike out in flame? But I could not hold it—the pain was too intense now.
If I grasped it by the chain alone—? I loosed it a little to dangle. It was as if all the lamps that had once burned in Ithkrypt's shattered hall were gathered into one!
"Look at her!" The girl on my left leaped at me, her hand outstretched to strike the gryphon from my hold.
By its chain I swung it at her and she cowered away, her hands to her face, falling to the ground with a scream.
So I had learned how to use what I held! Having so learned, I prepared to put it into further practice. A small black ball fell at my feet, thrown by the other woman. It broke, and from it curled an oily black snake, to wreathe about my ankles with the speed of a striking serpent, holding me as fast as if those coils were chains of steel.
I had been so occupied by my discovery concerning the gryphon that I had not seen what chanced with my lord. But now, fast captive, unable to swing my globe far enough, I watched in despair.
The other man held forth his right hand, and Rogear clasped it. Just so was he hand-linked to the woman, and the three faced my lord as one. Now the woman took into her other hand, from where it was set in her girdle as a sword might be, a rod of black along which red lines moved as if they were crawling things. She pointed this at my lord and began to chant, outlining his body with her wand—head to loins and up again to head.
I saw him tremble, waver, as if a rain of blows battered him. He held his arm ever before him, striving to move it so that the blue band about his wrist was before the point of the rod. Yet that he was hard set it was plain to see, and I wrestled with the smoky tangle about my feet, striving to reach those evil three with the globe.
"Unmade, I will it!" The woman's voice rolled like thunder. "As I made, so shall it be unmade!"
My lord—by the Flame, I swear it! I saw his body shaken, thin, becoming more shadow than substance. And out of nowhere came a wind to whirl and buffet that shadow, tearing at it.
I feared to loose the gryphon, but this must be stopped—the wind, that roll of chant-thunder—the rod that moved, erasing my lord as if he had never been! Shadow though he was, torn as he was, still he stood, and it seemed to me the black rod moved more slowly. Was she tiring?
I saw Rogear's face. His eyes were closed, and there was such a look of intense concentration there I guessed his will was backing hers. Did I dare loose my only weapon now?
Hoping I had not made the wrong choice, I hurled the gryphon at Rogear. It struck his shoulder, fell to the ground, rolled across the point of the star, stopped just within the circle. But the hand with which Rogear had gripped that of the woman fell from her grasp, limply to his side. He went to his knees, dragging with him the other man, who fell forward and lay still. While along Rogear's body, spreading outward from where the gryphon had struck, played lines of blue like small hungry flames, and he rocked back and forth, jerking with his other hand to free himself from the hold on the prone man. Yet it appeared he could not loose that finger locking.
The lines of fire ran down his arm swiftly, crossing to the body of the other man. Now Rogear did not strive to break that hold, and I guessed that he was willing the fire to pass from him into the other, who was now writhing feebly and moaning.
While he fought thus with his will, the woman stood alone. And her wand was held in a hand plainly failing. My lord was no longer a shadow, and the wind was dying. He looked to the woman steadily and without fear. In his eyes was something I could not read. Now he did not trouble to move his hand to ward off the rod. Rather, he held the wristlet level between them at heart height and he spoke, his words cutting through her chant.
"Do you know me at last, Tephana. I am—" He uttered a sound which might be a name, yet was unlike any name I had ever heard.
She raised her rod like a lash, as if she would beat him across the face in a rage too great to be borne.
"No!"
"Yes and yes and yes! I am awake—at long last!"
She twirled the rod at shoulder height, as I have seen a man ready a throwing spear. And throw it she did, as if she believed its point would reach his heart.
But, though he stood so close, it did not touch him, passing over his shoulder to strike against a rock and shatter with a ringing sound.
Her hands went to her ears, as if that sound were more than she could bear to hear. She wavered, but she did not fall. Now Rogear dragged himself up to his feet, moved beside her. His one hand still hung limply by his side, the other he raised swiftly, and let it fall on her shoulder. His face was white, stricken, yet I saw his eyes and knew that his will and his hate were blazingly alive.
"Tool!" His lips moved as if his face had stiffened into a mask. "Fight! You have the Power. Would you let that which you marred in the making triumph over you now?"
Lord Amber laughed! It was joyful laughter, as if he had no cares in the world.
"Ah, Rogear, you would-be opener of gates, ambitious for what, if you knew all, you would not dare to face. Do you not yet understand the truth? You seek to reach that which is beyond you: not only to reach it, but to put to use that which is not for your small mind—to Dark use—"
It was as if each word was a lash laid across Rogear's face, and I saw such anger mirrored there as I thought no human features could contain. His mouth worked, and there was spittle on his lips. Then he spoke.
I cannot put into words what rang then in my head. I know that I sank to the ground, as though a great hand were pressing me flat. Above Rogear's head stood a column of black flame, not red like honest fire but—black! Its tip inclined toward Lord Amber. But he did not start away. He stood watching as if this did not concern him.
Though I cried a warning, I did not hear my own words. The flame leaned and leaned, out across the star, the circle which enclosed it, poised over Lord Amber's head. Yet he did not even raise his eyes to see its threat, only watched Rogear.
About Rogear and the woman he held to him, the flame leaped and thickened as if it fed upon their bodies. It grew darker than ever, until they were hidden. And the tip of the flame moved as if trying to reach Lord Amber. Still it did not.
Slowly it began to die, fall back upon itself, growing less and less. And as it went it did not disclose Rogear or the woman. Finally it was but a glowering spark on the pavement—and nothing! They, too, were gone.
I put my hands to my eyes. To see that ending—it gave me such fear as I had never known, even though it did not threaten me. Then—there was silence!
I waited for my lord to speak—opening my eyes when he did not. And I cried out, forgetting all else. For no longer did he stand confidently upright to face his foes. He lay as crumpled beyond the circle as those who left within it—and as still.
About my feet the serpent no longer coiled. I staggered toward him, stopping only to pick up the gryphon. That was plain crystal again, its warmth and life gone.
As I had once held Toross against the coming of death, so did I now cradle my lord's head against me. His eyes were closed. I could not see their strange yellow fires. At first I thought he was dead. But under my questing fingers his heart still beat, if slowly. In so much he had won, he was still alive. And if I could only keep him so—
"He will live."
I turned my head, startled, fierce in my protection of the one I held. From whence had this one come? He stood with his back to the wall of the cliff, leaning a little on a staff carven with runes. His face seemed to shift queerly when I looked upon him, now appearing that of a man in late middle life, again that of a young warrior. But his clothing was gray as the stone behind him and could have been that of a trader.
"Who—?" I began.
He shook his head, looking at me gently as one who soothes a child. "What is a name? Well, you may call me Neevor, which is as good a name as any and once of some service to me—and others."
Now he stood away from the cliff and came into the circle. But as he came he used his staff to gesture right and left. The evil outer circle was gone; the star also. Then he pointed to the girl, the other man, to all other evidences of those who had striven to call the Power here. And they were also gone, as if they were part of a dream from which I had now awakened.
At last he neared me and my lord, and he was smiling. Putting out the staff again, he touched my forehead and, secondly, touched my lord on the breast. I was no longer afraid, but filled with a vast happiness and courage, so that in that moment I could have stood even against the full army of invaders. Yet this was better than battle courage, for it reached for life and not death.
Neevor nodded to me. "Just so," he said, as if pleased. "Look to your key, Joisan, for it will turn only for you, as that one who dabbled in what was far beyond him knew."
"Key?" I was bemused by his order.
"Ah, child, what wear you now upon your breast? Freely given to you it was with goodwill, by one who found the lost—and not by chance. Patterns are set in one time, to be followed to the end of all years to come. Woven in, woven out—"
The tip of his staff moved across the ground back and form. I watched it, feeling that I could understand its meaning if I only made some effort, knew more.
I heard him laugh. "You shall, Joisan, you shall—all in good time."
My lord opened his eyes, and there was life and recognition in his expression, but also puzzlement. He stirred as if to leave my hold, but I tightened that.
"I am—" he said slowly.
Neevor stood beside us, regarding us with the warmth of a smile.
"In this time and place you are Kerovan. Perhaps a little less than you once were, but with the way before you to return if you wish. Did I not name you 'kinsman'?"
"But I—I was—"
Neevor's staff touched him once more on the forehead. "You were a part, not the whole. As you now are, you could not long contain what came to remind you of what you were and can be. Just as those poor fools could not contain the evil they called down, which consumed them in the end. Be content, Kerovan, yet seek—for those who seek find." He turned a little and pointed with his staff to the blankness of the cliff. "There lies the gate; open it when you wish, there is much beyond to interest you."
With that he was gone!
"My Lord!"
He struggled up, breaking my hold. Not to put me aside as I feared he now would, but rather to take me in his arms.
"Joisan!" He said only my name, but that was enough. This was the oneness I had ever sought, without knowing, and finding it was all the riches of the world spread before me for the taking.
19
Kerovan
I held Joisan in my arms. I was Kerovan, surely I was Kerovan. Still—
Because that memory of the other one, the one who had worn my body for a space as I have worn mail, clung, so did I also cling to Joisan, who was human, who was living as I knew life and not—
Then the full sense of who and what I was as Kerovan returned. Gently I loosed my hold of her. Standing, I drew her to her feet. Then I was aware that the happiness of her face was fading and she watched me with troubled eyes.
"You—you are going away!" She clutched my forearms with her hands, would not let me turn from her. "I can feel it—you are going from me because you wish to!" Now her words had an angry ring.
I could remember our first meeting and how she had looked upon me then—I who was not wholly human, part something else that I did not yet know or understand.
"I am not an Old One," I told her straightly. "I am indeed Kerovan, who was born thus!" I shook off her hold to step back and show her one of those hoofed feet, thrust stiffly out that she might see it plain. "I was born by sorcery, to become a tool for one who aspired to the Power. You watched her try to destroy what she had created, and instead she herself was destroyed."
Joisan glanced to where the flame had eaten up those two.
"Twice-cursed was I from the beginning: by my father's line and by my mother's desire. Do you understand? No fit mate for any human woman am I. I have said—Kerovan is dead. That is the truth, just as Ulmsdale is destroyed, and with it all the House of Ulm. ..."
"You are my promised, wedded lord, call yourself what you desire."
How could I break that tie she pulled so tight between us? Half of me, no, more than half, wanted to yield, to be as other men. But the fact that I had been a vessel into which something else had poured, even though that was gone again—How could I be sure it would not return, with force enough to reach out to Joisan? I could not—tainted, cursed, deformed—give me that name best-suited. I was no lord for her.
Once more I retreated, edging away, lest her hand meet mine once again and I could not control the desire of that part of me human-rooted. Yet I could not turn and go from her, leave her alone in the Waste. And if I went with her, back to her people, could I continue to hold to my resolve?
"Did you not listen to Neevor?" She did not follow me; rather she stood, her hands clasped on her breast over the englobed gryphon. "Did you not listen then?" Still there was anger in her voice, and she regarded me as one who is exasperated by stupidity.
"He called you kinsman—therefore you are more than you deem yourself. You are you—no tool for any one, Kerovan. And you are my dear Lord. If you strive to say me 'no,' then you shall discover I have no pride. I shall follow wherever you go, and in the face of all shall I claim you. Do you believe me?"
I did, and believing could see that now I could not deal otherwise than seem to agree.
"Yes," I made simple answer.
"Good enough. And if in the future you try to walk away from me again, never shall you find that easy." It was not a warning of a promise, but a statement of fact. Having seemingly settled that to her satisfaction, she looked once more to the cliff.
"Neevor spoke of a door and a key which I hold. Someday I shall put that to the test."
"Someday?" Yes, I could remember Neevor's words better now that I had my emotions under control.
"Yes. We—we are not ready—I think—I feel—" Joisan nodded. "It is something we must do together, remember that, Kerovan—together!"
"Where to, then? Back to your people?" I felt rootless, lost in these dales. I would leave the choice to her, since all my ties were gone save one.
"That is best," she answered briskly. "I have promised them what measure of safety can be found nowadays. When they have won to that, then we shall be free!"
Joisan flung wide her arms, as if the taste of that freedom was already hers. But could it be freedom if she held to that other tie? I would walk her road for now because I had no choice. But never would I let her be the loser because she looked at me and saw a Kerovan to whom she was oath-tied.
20
Joisan
My poor lord, how bitter must have been his hurts in the past! I wish that I could run back down the years and rub out the memory of each, one by one. He has been named monster until he believes it—but if he could only look upon himself through my eyes—
We shall walk together, and I shall build a mirror that he may see himself as he is, and, so seeing, be free from all the sorrow the Dark Ones laid upon him. Yes, we shall return to my people—though they are not truly mine anymore—for I feel as one who has taken another road and can look back only a little way. We shall make sure that they reach Norsdale. For the rest—
So I thought in that hour, and wise was that thought. For sometimes wisdom comes not altogether through age and experience, but suddenly like an arrow flight. I nursed my key within my hand—that bride gift which had been first my bane and then my salvation. I put my other hand within my lord's, so we went together, turning away for a space from the gate Neevor promised us, knowing within my heart that we would return and that it would open upon—But what mattered what lay beyond if we went together to see?
All rights reserved, including without limitation the right to reproduce this ebook or any portion thereof in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.
This is a work of fiction. Names, characters, places, events, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.
Copyright © 1972 by Andre Norton
ISBN: 978-1-4976-5616-1
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{"url":"https:\/\/en.wikipedia.org\/wiki\/Dictionary_learning","text":"# Sparse dictionary learning\n\n(Redirected from Dictionary learning)\n\nSparse coding is a representation learning method which aims at finding a sparse representation of the input data (also known as sparse coding) in the form of a linear combination of basic elements as well as those basic elements themselves. These elements are called atoms and they compose a dictionary. Atoms in the dictionary are not required to be orthogonal, and they may be an over-complete spanning set. This problem setup also allows the dimensionality of the signals being represented to be higher than the one of the signals being observed. The above two properties lead to having seemingly redundant atoms that allow multiple representations of the same signal but also provide an improvement in sparsity and flexibility of the representation.\n\nOne of the most important applications of sparse dictionary learning is in the field of compressed sensing or signal recovery. In compressed sensing, a high-dimensional signal can be recovered with only a few linear measurements provided that the signal is sparse or nearly sparse. Since not all signals satisfy this sparsity condition, it is of great importance to find a sparse representation of that signal such as the wavelet transform or the directional gradient of a rasterized matrix. Once a matrix or a high dimensional vector is transferred to a sparse space, different recovery algorithms like basis pursuit, CoSaMP[1] or fast non-iterative algorithms[2] can be used to recover the signal.\n\nOne of the key principles of dictionary learning is that the dictionary has to be inferred from the input data. The emergence of sparse dictionary learning methods was stimulated by the fact that in signal processing one typically wants to represent the input data using as few components as possible. Before this approach the general practice was to use predefined dictionaries (such as Fourier or wavelet transforms). However, in certain cases a dictionary that is trained to fit the input data can significantly improve the sparsity, which has applications in data decomposition, compression and analysis and has been used in the fields of image denoising and classification, video and audio processing. Sparsity and overcomplete dictionaries have immense applications in image compression, image fusion and inpainting.\n\nImage Denoising by Dictionary Learning\n\n## Problem statement\n\nGiven the input dataset ${\\displaystyle X=[x_{1},...,x_{K}],x_{i}\\in \\mathbb {R} ^{d}}$ we wish to find a dictionary ${\\displaystyle \\mathbf {D} \\in \\mathbb {R} ^{d\\times n}:D=[d_{1},...,d_{n}]}$ and a representation ${\\displaystyle R=[r_{1},...,r_{K}],r_{i}\\in \\mathbb {R} ^{n}}$ such that both ${\\displaystyle \\|X-\\mathbf {D} R\\|_{F}^{2}}$ is minimized and the representations ${\\displaystyle r_{i}}$ are sparse enough. This can be formulated as the following optimization problem:\n\n${\\displaystyle {\\underset {\\mathbf {D} \\in {\\mathcal {C}},r_{i}\\in \\mathbb {R} ^{n}}{\\text{argmin}}}\\sum _{i=1}^{K}\\|x_{i}-\\mathbf {D} r_{i}\\|_{2}^{2}+\\lambda \\|r_{i}\\|_{0}}$, where ${\\displaystyle {\\mathcal {C}}\\equiv \\{\\mathbf {D} \\in \\mathbb {R} ^{d\\times n}:\\|d_{i}\\|_{2}\\leq 1\\,\\,\\forall i=1,...,n\\}}$, ${\\displaystyle \\lambda >0}$\n\n${\\displaystyle {\\mathcal {C}}}$ is required to constrain ${\\displaystyle \\mathbf {D} }$ so that its atoms would not reach arbitrarily high values allowing for arbitrarily low (but non-zero) values of ${\\displaystyle r_{i}}$.${\\displaystyle \\lambda }$ controls the trade off between the sparsity and the minimization error.\n\nThe minimization problem above is not convex because of the 0-\"norm\" and solving this problem is NP-hard.[3] In some cases L1-norm is known to ensure sparsity[4] and so the above becomes a convex optimization problem with respect to each of the variables ${\\displaystyle \\mathbf {D} }$ and ${\\displaystyle \\mathbf {R} }$ when the other one is fixed, but it is not jointly convex in ${\\displaystyle (\\mathbf {D} ,\\mathbf {R} )}$.\n\n### Properties of the dictionary\n\nThe dictionary ${\\displaystyle \\mathbf {D} }$ defined above can be \"undercomplete\" if ${\\displaystyle n or \"overcomplete\" in case ${\\displaystyle n>d}$ with the latter being a typical assumption for a sparse dictionary learning problem. The case of a complete dictionary does not provide any improvement from a representational point of view and thus isn't considered.\n\nUndercomplete dictionaries represent the setup in which the actual input data lies in a lower-dimensional space. This case is strongly related to dimensionality reduction and techniques like principal component analysis which require atoms ${\\displaystyle d_{1},...,d_{n}}$ to be orthogonal. The choice of these subspaces is crucial for efficient dimensionality reduction, but it is not trivial. And dimensionality reduction based on dictionary representation can be extended to address specific tasks such as data analysis or classification. However, their main downside is limiting the choice of atoms.\n\nOvercomplete dictionaries, however, do not require the atoms to be orthogonal (they will never be a basis anyway) thus allowing for more flexible dictionaries and richer data representations.\n\nAn overcomplete dictionary which allows for sparse representation of signal can be a famous transform matrix (wavelets transform, fourier transform) or it can be formulated so that its elements are changed in such a way that it sparsely represents the given signal in a best way. Learned dictionaries are capable of giving sparser solutions as compared to predefined transform matrices.\n\n## Algorithms\n\nAs the optimization problem described above can be solved as a convex problem with respect to either dictionary or sparse coding while the other one of the two is fixed, most of the algorithms are based on the idea of iteratively updating one and then the other.\n\nThe problem of finding an optimal sparse coding ${\\displaystyle R}$ with a given dictionary ${\\displaystyle \\mathbf {D} }$ is known as sparse approximation (or sometimes just sparse coding problem). A number of algorithms have been developed to solve it (such as matching pursuit and LASSO) and are incorporated in the algorithms described below.\n\n### Method of optimal directions (MOD)\n\nThe method of optimal directions (or MOD) was one of the first methods introduced to tackle the sparse dictionary learning problem.[5] The core idea of it is to solve the minimization problem subject to the limited number of non-zero components of the representation vector:\n\n${\\displaystyle \\min _{\\mathbf {D} ,R}\\{\\|X-\\mathbf {D} R\\|_{F}^{2}\\}\\,\\,{\\text{s.t.}}\\,\\,\\forall i\\,\\,\\|r_{i}\\|_{0}\\leq T}$\n\nHere, ${\\displaystyle F}$ denotes the Frobenius norm. MOD alternates between getting the sparse coding using a method such as matching pursuit and updating the dictionary by computing the analytical solution of the problem given by ${\\displaystyle \\mathbf {D} =XR^{+}}$ where ${\\displaystyle R^{+}}$ is a Moore-Penrose pseudoinverse. After this update ${\\displaystyle \\mathbf {D} }$ is renormalized to fit the constraints and the new sparse coding is obtained again. The process is repeated until convergence (or until a sufficiently small residue).\n\nMOD has proved to be a very efficient method for low-dimensional input data ${\\displaystyle X}$ requiring just a few iterations to converge. However, due to the high complexity of the matrix-inversion operation, computing the pseudoinverse in high-dimensional cases is in many cases intractable. This shortcoming has inspired the development of other dictionary learning methods.\n\n### K-SVD\n\nK-SVD is an algorithm that performs SVD at its core to update the atoms of the dictionary one by one and basically is a generalization of K-means. It enforces that each element of the input data ${\\displaystyle x_{i}}$ is encoded by a linear combination of not more than ${\\displaystyle T_{0}}$ elements in a way identical to the MOD approach:\n\n${\\displaystyle \\min _{\\mathbf {D} ,R}\\{\\|X-\\mathbf {D} R\\|_{F}^{2}\\}\\,\\,{\\text{s.t.}}\\,\\,\\forall i\\,\\,\\|r_{i}\\|_{0}\\leq T_{0}}$\n\nThis algorithm's essence is to first fix the dictionary, find the best possible ${\\displaystyle R}$ under the above constraint (using Orthogonal Matching Pursuit) and then iteratively update the atoms of dictionary ${\\displaystyle \\mathbf {D} }$ in the following manner:\n\n${\\displaystyle \\|X-\\mathbf {D} R\\|_{F}^{2}=\\left|X-\\sum _{i=1}^{K}d_{i}x_{T}^{i}\\right|_{F}^{2}=\\|E_{k}-d_{k}x_{T}^{k}\\|_{F}^{2}}$\n\nThe next steps of the algorithm include rank-1 approximation of the residual matrix ${\\displaystyle E_{k}}$, updating ${\\displaystyle d_{k}}$ and enforcing the sparsity of ${\\displaystyle x_{k}}$ after the update. This algorithm is considered to be standard for dictionary learning and is used in a variety of applications. However, it shares weaknesses with MOD being efficient only for signals with relatively low dimensionality and having the possibility for being stuck at local minima.\n\nOne can also apply a widespread stochastic gradient descent method with iterative projection to solve this problem.[6][7] The idea of this method is to update the dictionary using the first order stochastic gradient and project it on the constraint set ${\\displaystyle {\\mathcal {C}}}$. The step that occurs at i-th iteration is described by this expression:\n\n${\\displaystyle \\mathbf {D} _{i}={\\text{proj}}_{\\mathcal {C}}\\left\\{\\mathbf {D} _{i-1}-\\delta _{i}\\nabla _{\\mathbf {D} }\\sum _{i\\in S}\\|x_{i}-\\mathbf {D} r_{i}\\|_{2}^{2}+\\lambda \\|r_{i}\\|_{1}\\right\\}}$, where ${\\displaystyle S}$ is a random subset of ${\\displaystyle \\{1...K\\}}$ and ${\\displaystyle \\delta _{i}}$ is a gradient step.\n\n### Lagrange dual method\n\nAn algorithm based on solving a dual Lagrangian problem provides an efficient way to solve for the dictionary having no complications induced by the sparsity function.[8] Consider the following Lagrangian:\n\n${\\displaystyle {\\mathcal {L}}(\\mathbf {D} ,\\Lambda )={\\text{tr}}\\left((X-\\mathbf {D} R)^{T}(X-\\mathbf {D} R)\\right)+\\sum _{j=1}^{n}\\lambda _{j}\\left({\\sum _{i=1}^{d}\\mathbf {D} _{ij}^{2}-c}\\right)}$, where ${\\displaystyle c}$ is a constraint on the norm of the atoms and ${\\displaystyle \\lambda _{i}}$ are the so-called dual variables forming the diagonal matrix ${\\displaystyle \\Lambda }$.\n\nWe can then provide an analytical expression for the Lagrange dual after minimization over ${\\displaystyle \\mathbf {D} }$:\n\n${\\displaystyle {\\mathcal {D}}(\\Lambda )=\\min _{\\mathbf {D} }{\\mathcal {L}}(\\mathbf {D} ,\\Lambda )={\\text{tr}}(X^{T}X-XR^{T}(RR^{T}+\\Lambda )^{-1}(XR^{T})^{T}-c\\Lambda )}$.\n\nAfter applying one of the optimization methods to the value of the dual (such as Newton's method or conjugate gradient) we get the value of ${\\displaystyle \\mathbf {D} }$:\n\n${\\displaystyle \\mathbf {D} ^{T}=(RR^{T}+\\Lambda )^{-1}(XR^{T})^{T}}$\n\nSolving this problem is less computational hard because the amount of dual variables ${\\displaystyle n}$ is a lot of times much less than the amount of variables in the primal problem.\n\n### LASSO\n\nIn this approach, the optimization problem is formulated as:\n\n${\\displaystyle \\min _{r\\in \\mathbb {R} ^{n}}\\{\\,\\,\\|r\\|_{1}\\}\\,\\,{\\text{subject to}}\\,\\,\\|X-\\mathbf {D} R\\|_{F}^{2}<\\epsilon }$, where ${\\displaystyle \\epsilon }$ is the permitted error in the reconstruction LASSO.\n\nIt finds an estimate of ${\\displaystyle r_{i}}$ by minimizing the least square error subject to a L1-norm constraint in the solution vector, formulated as:\n\n${\\displaystyle \\min _{r\\in \\mathbb {R} ^{n}}\\,\\,{\\dfrac {1}{2}}\\,\\,\\|X-\\mathbf {D} r\\|_{F}^{2}+\\lambda \\,\\,\\|r\\|_{1}}$, where ${\\displaystyle \\lambda >0}$ controls the trade-off between sparsity and the reconstruction error. This gives the global optimal solution.[9] See also Online dictionary learning for Sparse coding\n\n### Parametric training methods\n\nParametric training methods are aimed to incorporate the best of both worlds \u2014 the realm of analytically constructed dictionaries and the learned ones.[10] This allows to construct more powerful generalized dictionaries that can potentially be applied to the cases of arbitrary-sized signals. Notable approaches include:\n\n\u2022 Translation-invariant dictionaries.[11] These dictionaries are composed by the translations of the atoms originating from the dictionary constructed for a finite-size signal patch. This allows the resulting dictionary to provide a representation for the arbitrary-sized signal.\n\u2022 Multiscale dictionaries.[12] This method focuses on constructing a dictionary that is composed of differently scaled dictionaries to improve sparsity.\n\u2022 Sparse dictionaries.[13] This method focuses on not only providing a sparse representation but also constructing a sparse dictionary which is enforced by the expression ${\\displaystyle \\mathbf {D} =\\mathbf {B} \\mathbf {A} }$ where ${\\displaystyle \\mathbf {B} }$ is some pre-defined analytical dictionary with desirable properties such as fast computation and ${\\displaystyle \\mathbf {A} }$ is a sparse matrix. Such formulation allows to directly combine the fast implementation of analytical dictionaries with the flexibility of sparse approaches.\n\n### Online dictionary learning (LASSO approach)\n\nMany common approaches to sparse dictionary learning rely on the fact that the whole input data ${\\displaystyle X}$ (or at least a large enough training dataset) is available for the algorithm. However, this might not be the case in the real-world scenario as the size of the input data might be too big to fit it into memory. The other case where this assumption can not be made is when the input data comes in a form of a stream. Such cases lie in the field of study of online learning which essentially suggests iteratively updating the model upon the new data points ${\\displaystyle x}$ becoming available.\n\nA dictionary can be learned in an online manner the following way:[14]\n\n1. For ${\\displaystyle t=1...T:}$\n2. Draw a new sample ${\\displaystyle x_{t}}$\n3. Find a sparse coding using LARS: ${\\displaystyle r_{t}={\\underset {r\\in \\mathbb {R} ^{n}}{\\text{argmin}}}\\left({\\frac {1}{2}}\\|x_{t}-\\mathbf {D} _{t-1}r\\|+\\lambda \\|r\\|_{1}\\right)}$\n4. Update dictionary using block-coordinate approach: ${\\displaystyle \\mathbf {D} _{t}={\\underset {\\mathbf {D} \\in {\\mathcal {C}}}{\\text{argmin}}}{\\frac {1}{t}}\\sum _{i=1}^{t}\\left({\\frac {1}{2}}\\|x_{i}-\\mathbf {D} r_{i}\\|_{2}^{2}+\\lambda \\|r_{i}\\|_{1}\\right)}$\n\nThis method allows us to gradually update the dictionary as new data becomes available for sparse representation learning and helps drastically reduce the amount of memory needed to store the dataset (which often has a huge size).\n\n## Applications\n\nThe dictionary learning framework, namely the linear decomposition of an input signal using a few basis elements learned from data itself, has led to state-of-art results in various image and video processing tasks. This technique can be applied to classification problems in a way that if we have built specific dictionaries for each class, the input signal can be classified by finding the dictionary corresponding to the sparsest representation.\n\nIt also has properties that are useful for signal denoising since usually one can learn a dictionary to represent the meaningful part of the input signal in a sparse way but the noise in the input will have a much less sparse representation.[15]\n\nSparse dictionary learning has been successfully applied to various image, video and audio processing tasks as well as to texture synthesis[16] and unsupervised clustering.[17] In evaluations with the Bag-of-Words model,[18][19] sparse coding was found empirically to outperform other coding approaches on the object category recognition tasks.\n\nDictionary learning is used to analyse medical signals in detail. Such medical signals include those from electroencephalography (EEG), electrocardiography (ECG), magnetic resonance imaging (MRI), functional MRI (fMRI), continuous glucose monitors [20] and ultrasound computer tomography (USCT), where different assumptions are used to analyze each signal.\n\n## References\n\n1. ^ Needell, D.; Tropp, J.A. (2009). \"CoSaMP: Iterative signal recovery from incomplete and inaccurate samples\". Applied and Computational Harmonic Analysis. 26 (3): 301\u2013321. arXiv:0803.2392. doi:10.1016\/j.acha.2008.07.002.\n2. ^ Lotfi, M.; Vidyasagar, M.\"A Fast Non-iterative Algorithm for Compressive Sensing Using Binary Measurement Matrices\"\n3. ^ A. M. Tillmann, \"On the Computational Intractability of Exact and Approximate Dictionary Learning\", IEEE Signal Processing Letters 22(1), 2015: 45\u201349.\n4. ^ Donoho, David L. (2006-06-01). \"For most large underdetermined systems of linear equations the minimal \ud835\udcc11-norm solution is also the sparsest solution\". Communications on Pure and Applied Mathematics. 59 (6): 797\u2013829. doi:10.1002\/cpa.20132. ISSN\u00a01097-0312. S2CID\u00a08510060.\n5. ^ Engan, K.; Aase, S.O.; Hakon Husoy, J. (1999-01-01). Method of optimal directions for frame design. 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing, 1999. Proceedings. Vol.\u00a05. pp.\u00a02443\u20132446 vol.5. doi:10.1109\/ICASSP.1999.760624. ISBN\u00a0978-0-7803-5041-0. S2CID\u00a033097614.\n6. ^ Aharon, Michal; Elad, Michael (2008). \"Sparse and Redundant Modeling of Image Content Using an Image-Signature-Dictionary\". SIAM Journal on Imaging Sciences. 1 (3): 228\u2013247. CiteSeerX\u00a010.1.1.298.6982. doi:10.1137\/07070156x.\n7. ^ Pint\u00e9r, J\u00e1nos D. (2000-01-01). Yair Censor and Stavros A. Zenios, Parallel Optimization \u2014 Theory, Algorithms, and Applications. Oxford University Press, New York\/Oxford, 1997, xxviii+539 pages. (US \\$ 85.00). Journal of Global Optimization. Vol.\u00a016. pp.\u00a0107\u2013108. doi:10.1023\/A:1008311628080. ISBN\u00a0978-0-19-510062-4. ISSN\u00a00925-5001. S2CID\u00a022475558.\n8. ^ Lee, Honglak, et al. \"Efficient sparse coding algorithms.\" Advances in neural information processing systems. 2006.\n9. ^ Kumar, Abhay; Kataria, Saurabh. \"Dictionary Learning Based Applications in Image Processing using Convex Optimisation\" (PDF).\n10. ^ Rubinstein, R.; Bruckstein, A.M.; Elad, M. (2010-06-01). \"Dictionaries for Sparse Representation Modeling\". Proceedings of the IEEE. 98 (6): 1045\u20131057. CiteSeerX\u00a010.1.1.160.527. doi:10.1109\/JPROC.2010.2040551. ISSN\u00a00018-9219. S2CID\u00a02176046.\n11. ^ Engan, Kjersti; Skretting, Karl; Hus\u00f8y, John H\\a akon (2007-01-01). \"Family of Iterative LS-based Dictionary Learning Algorithms, ILS-DLA, for Sparse Signal Representation\". Digit. Signal Process. 17 (1): 32\u201349. doi:10.1016\/j.dsp.2006.02.002. ISSN\u00a01051-2004.\n12. ^ Mairal, J.; Sapiro, G.; Elad, M. (2008-01-01). \"Learning Multiscale Sparse Representations for Image and Video Restoration\". Multiscale Modeling & Simulation. 7 (1): 214\u2013241. CiteSeerX\u00a010.1.1.95.6239. doi:10.1137\/070697653. ISSN\u00a01540-3459.\n13. ^ Rubinstein, R.; Zibulevsky, M.; Elad, M. (2010-03-01). \"Double Sparsity: Learning Sparse Dictionaries for Sparse Signal Approximation\". IEEE Transactions on Signal Processing. 58 (3): 1553\u20131564. Bibcode:2010ITSP...58.1553R. CiteSeerX\u00a010.1.1.183.992. doi:10.1109\/TSP.2009.2036477. ISSN\u00a01053-587X. S2CID\u00a07193037.\n14. ^ Mairal, Julien; Bach, Francis; Ponce, Jean; Sapiro, Guillermo (2010-03-01). \"Online Learning for Matrix Factorization and Sparse Coding\". J. Mach. Learn. Res. 11: 19\u201360. arXiv:0908.0050. Bibcode:2009arXiv0908.0050M. ISSN\u00a01532-4435.\n15. ^ Aharon, M, M Elad, and A Bruckstein. 2006. \"K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation.\" Signal Processing, IEEE Transactions on 54 (11): 4311-4322\n16. ^ Peyr\u00e9, Gabriel (2008-11-06). \"Sparse Modeling of Textures\" (PDF). Journal of Mathematical Imaging and Vision. 34 (1): 17\u201331. doi:10.1007\/s10851-008-0120-3. ISSN\u00a00924-9907. S2CID\u00a015994546.\n17. ^ Ramirez, Ignacio; Sprechmann, Pablo; Sapiro, Guillermo (2010-01-01). Classification and clustering via dictionary learning with structured incoherence and shared features. 2014 IEEE Conference on Computer Vision and Pattern Recognition. Los Alamitos, CA, USA: IEEE Computer Society. pp.\u00a03501\u20133508. doi:10.1109\/CVPR.2010.5539964. ISBN\u00a0978-1-4244-6984-0. S2CID\u00a0206591234.\n18. ^ Koniusz, Piotr; Yan, Fei; Mikolajczyk, Krystian (2013-05-01). \"Comparison of mid-level feature coding approaches and pooling strategies in visual concept detection\". Computer Vision and Image Understanding. 117 (5): 479\u2013492. CiteSeerX\u00a010.1.1.377.3979. doi:10.1016\/j.cviu.2012.10.010. ISSN\u00a01077-3142.\n19. ^ Koniusz, Piotr; Yan, Fei; Gosselin, Philippe Henri; Mikolajczyk, Krystian (2017-02-24). \"Higher-order occurrence pooling for bags-of-words: Visual concept detection\" (PDF). IEEE Transactions on Pattern Analysis and Machine Intelligence. 39 (2): 313\u2013326. doi:10.1109\/TPAMI.2016.2545667. hdl:10044\/1\/39814. ISSN\u00a00162-8828. PMID\u00a027019477.\n20. ^ AlMatouq, Ali; LalegKirati, TaousMeriem; Novara, Carlo; Ivana, Rabbone; Vincent, Tyrone (2019-03-15). \"Sparse Reconstruction of Glucose Fluxes Using Continuous Glucose Monitors\". IEEE\/ACM Transactions on Computational Biology and Bioinformatics. 17 (5): 1797\u20131809. doi:10.1109\/TCBB.2019.2905198. hdl:10754\/655914. ISSN\u00a01545-5963. PMID\u00a030892232. 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\section{Introduction}
Nut graphs constitute an important subclass of graphs.
They were introduced and first studied in a series of papers by Sciriha and co-workers
\cite{GutmanSciriha-MaxSing,Sc1998,ScOnRnkGr99,Sc2007,Sc2008,Prop23}.
A \emph{nut graph} has a singular adjacency matrix, the spectrum of which contains exactly one zero eigenvalue,
with a corresponding non-trivial kernel eigenvector
that has no zero entries.
In chemistry, this property has significant consequences for the modelling of carbon frameworks
at molecular and nano scales.
The graphs that are possible models for unsaturated carbon frameworks are the chemical graphs.
A \emph{chemical graph} is simple (without multiple edges or loops), connected, and of maximum
degree $\le 3$ (to allow for
each four-valent carbon atom retaining one valence for participation in a $\pi$ system).
Eigenvectors and eigenvalues of chemical graphs correspond to the distributions of $\pi$ molecular orbitals and
their energies within the
qualitative H{\"u}ckel model \cite{Streitwieser_1961}.
To qualify as a model of a carbon $\pi$-system, a nut graph must also be a chemical graph.
In this model, the kernel eigenvector of a nut graph
corresponds to a non-bonding orbital, occupation of which by a single electron
leads to spin density, and hence radical reactivity distributed over all carbon centres
\cite{Prop23}.
Nut graphs also have unique status as \emph{strong omniconductors} \cite{PWF_CPL_2013,PWF_JCP_2014} of nullity one,
in the {H{\"u}ckel-based} version of the SSP (source-and-sink potential)
model of ballistic molecular conduction \cite{Ern_JCP_2007a,Ern_JCP_2011b}.
For obvious reasons, we call a nut graph that satisfies the conditions
for a chemical graph a \emph{chemical nut graph}.
Given their chemical interest, a natural question is: for which combinations of
order ($n$, number of vertices) and size ($m$, number of edges) do chemical nut graphs
exist? It turns out to be possible to supply a complete answer to this question.
\section{Initial considerations}
A nut graph has various useful properties.
These include the following: a nut graph is connected;
it is non-bipartite; it has no vertices of degree $1$ \cite{ScirihaGutman-NutExt}.
We will refer to graphs that have no vertices of degree $1$ as \emph{leafless}.
Since the number of vertices of odd degree of a simple graph is even,
a chemical nut graph is leafless, with
$v_2$ vertices of degree $2$,
and $v_3$ vertices of degree $3$, where $v_3$ is even.
Order and size of a chemical nut graph are related to these numbers by
$n = v_2 + v_3$ and $m = v_2 + (3v_3)/2$.
This suggests a mapping of chemical nut graphs onto a grid parameterised by
even integers $v_3 \ge 0$ and integers $v_2 \ge 0$.
Hence, our initial question can be rephrased as: for what combinations $(v_3, v_2)$ do chemical nut graphs exist?
In other words: which pairs $(v_3,v_2)$ are \emph{realisable}? In this paper we give a complete answer to this question.
We also show that only one realisable pair, namely $(20,0)$ cannot be realised by a planar chemical nut graph.
(It is realised by chemical nut graphs of genus $1$.)
Useful background information exists, in the form of a database of nut graphs of low order
produced by the computations described in \cite{CoolFowlGoed-2017} and listed on the accompanying website \cite{nutgen-site}.
Information
available
on the website includes a dataset restricted to chemical nut graphs with $n \le 22$,
which decides the existence question for small values of $v_3$ and $v_2$.
Table~\ref{tbl:census} gives the results of our
interrogation of the dataset, listing cases of parameter pairs in the range
$n \le 22$
where no chemical nut graph exists, and graph counts for the parameter pairs that are realisable by chemical nut
graphs in the range.
\begin{table}[!ht]
\centering
\begin{small}
\begin{tabular}{|c|rrrrrrrrrrrr|}
\hline
\backslashbox{$v_2$}{$v_3$} & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & 22 \\ \hline
0 & $\emptyset$ & $\emptyset$ & \bf0 & \bf 0 & {\bf 0} & {\bf 0} & \bf9 & {\bf 0} & {\bf 0} & 5541 & \bf 5 & \bf 71 \\
1 & $\emptyset$ & $\emptyset$ & \bf0 & \bf0 & {\bf 0} & {\bf 0} & {\bf 0} & \bf10 & \bf22 & \bf235 & 13602 & $-$ \\
2 & $\emptyset$ & \bf 0 & \bf0 & \bf0 & {\bf 0} & {\bf 0} & \bf2 & {\bf 0} & \bf37 & 3600 & \bf 30760 & $-$ \\
3 & 0 & \bf0 & \bf0 & \bf0 & \bf 7 & \bf9 & \bf71 & 5042 & 13474 & 168178 & $-$ & $-$ \\
4 & 0 & \bf0 & \bf0 & \bf0 & {\bf 0} & \bf10 & 225 & \bf388 & 14022 & 480051 & $-$ & $-$ \\
5 & 0 & \bf 0 & \bf0 &\bf 0 & 7 & \bf82 & 596 & 16497 & 280798 & $-$ & $-$ & $-$ \\
6 & 0 & 0 & \bf0 & \bf0 & \bf4 & 127 & 1186 & 15801 & 545237 & $-$ & $-$ & $-$ \\
7 & 0 &\bf 1 & \bf0 & \bf 8 & 212 & 1368 & 23127 & 575614 & $-$ & $-$ & $-$ & $-$ \\
8 & 0 & 0 & \bf0 &\bf 5 & 22 & 620 & 12035 & 181009 & $-$ & $-$ & $-$ & $-$ \\
9 & 0 & 1 & \bf0 & 36 & 718 & 9603 & 211501 & $-$ & $-$ & $-$ & $-$ & $-$ \\
10 & 0 & 0 & \bf 2 & 13 & 176 & 5457 & 106013 & $-$ & $-$ & $-$ & $-$ & $-$ \\
11 & 0 & 3 & \bf0 & 189 & 4427 & 60792 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\
12 & 0 & 0 & 2 & 50 & 786 & 25535 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\
13 & 0 & 3 & \bf0 & 601 & 14153 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\
14 & 0 & 0 & 11 & 118 & 3415 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\
15 & 0 & 6 & 0 & 1881 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\
16 & 0 & 0 & 13 & 309 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\
17 & 0 & 6 & 0 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\
18 & 0 & 0 & 38 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\
19 & 0 & 10 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\
20 & 0 & 0 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\
21 & 0 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\
22 & 0 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ \\ \hline
\end{tabular}
\end{small}
\caption{Census of chemical nut graphs.
The table shows the counts of chemical nut graphs for each of the allowed degree signatures
$(v_3, v_2)$
accessible to graphs with $n \le 22$. Symbol - means that the case is outside the range of the dataset. Symbol $\emptyset$
in the table means that no graph with these parameters exists.
Entries in the table were obtained by filtering the graphs from the nut graph database \cite{hog,nutgen-site,CoolFowlGoed-2017}.
Only the boldface entries are used in the proof of our main result; realisability in all other cases follows from the theory developed in Sections \ref{subsec:ext} to
\ref{subsec:nonrel}.}
\label{tbl:census}
\end{table}
As a chemical nut graph has $v_3$ even, we may consider it to be derived from a
cubic graph on $v_3$ vertices with some edges arbitrarily subdivided to reach the total count of $v_2 + v_3$. We will use this fact later
(e.g., in the proof of Theorem \ref{cor:realisx}).
Note that fast generation of chemical graphs was made possible by methods first introduced
for cubic graphs \cite{hog,cubicpaper}.
\section{Main result}
The main existence result can be stated as follows.
\begin{theorem}
\label{thm:main}
A chemical nut graph with parameters $(v_3, v_2)$, $v_2 \geq 0, v_3 \geq 0$ and $v_3$ even, exists if and only if one of the following statements holds:
\begin{enumerate}[label=(\alph*)]
\item $v_3 = 2$ and $v_2 = 7 + 2k $, $k \geq 0$\textup{;}
\item $v_3 = 4$ and $v_2 = 10 + 2k$, $k \geq 0$\textup{;}
\item $v_3 = 6$ and $v_2 \geq 7$\textup{;}
\item $v_3 \geq 8$ and
$$(v_3,v_2) \notin \{(8, 0), (8, 1), (8, 2), (8, 4), (10, 0), (10, 1), (10, 2), (12, 1), (14, 0), (14, 2), (16, 0)\}.$$
\end{enumerate}
Moreover, in each case where chemical nut graphs exist, a planar chemical nut graph may be found, except when
$v_3 = 20$ and $v_2 = 0$, where one of the $5$ chemical nut graphs has genus $1$, and the other $4$ have genus $2$.
\end{theorem}
The key to proving this theorem,
and determining all realisable parameter pairs, is
to use Table~\ref{tbl:census} as a source of seed graphs, and then to apply systematic constructions of larger nut graphs\cite{GPS,Jan}.
The proof of the main result has several
ideas. We identify each of them by a separate claim.
\subsection{Constructions for extending nut graphs}
\label{subsec:ext}
\subsubsection{The bridge construction}
The first construction that we introduce is \emph{the bridge construction} and is applicable only to
graphs with bridges, i.e.\ graphs with edges, whose removal disconnects the graph.
\begin{figure}[!h]
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\begin{tikzpicture}
\definecolor{mygreen}{RGB}{205, 238, 231}
\tikzstyle{vertex}=[draw,circle,font=\scriptsize,minimum size=13pt,inner sep=1pt,fill=mygreen]
\tikzstyle{edge}=[draw,thick]
\draw[thick,fill=gray!20!white] (-1.2,0) ellipse (1.2cm and 1cm);
\draw[thick,fill=gray!20!white] (3.2,0) ellipse (1.2cm and 1cm);
\node at (-1.2, 0) {\huge $G_1$};
\node at (3.2, 0) {\huge $G_2$};
\node[vertex,fill=yellow,label={[yshift=0pt,xshift=5pt]90:$a$}] (u) at (0, 0) {$u$};
\node[vertex,fill=yellow,label={[xshift=-5pt,yshift=0pt]90:$b$}] (v) at (2, 0) {$v$};
\path[edge] (u) -- (v);
\end{tikzpicture}
\caption{$G$}
\label{fig1a}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\begin{tikzpicture}
\definecolor{mygreen}{RGB}{205, 238, 231}
\tikzstyle{vertex}=[draw,circle,font=\scriptsize,minimum size=13pt,inner sep=1pt,fill=mygreen]
\tikzstyle{edge}=[draw,thick]
\draw[thick,fill=gray!20!white] (-1.2,0) ellipse (1.2cm and 1cm);
\draw[thick,fill=gray!20!white] (4.2,0) ellipse (1.2cm and 1cm);
\node at (-1.2, 0) {\huge $-G_1$};
\node at (4.2, 0) {\huge $G_2$};
\node[vertex,fill=yellow,label={[yshift=0pt,xshift=5pt]90:$-a$}] (u) at (0, 0) {$u$};
\node[vertex,fill=yellow,label={[xshift=-5pt,yshift=0pt]90:$b$}] (v) at (3, 0) {$v$};
\node[vertex,fill=mygreen,label={[xshift=0pt,yshift=0pt]90:$-b$}] (x) at (1, 0) {$x$};
\node[vertex,fill=mygreen,label={[xshift=0pt,yshift=0pt]90:$a$}] (y) at (2, 0) {$y$};
\path[edge] (u) -- (x) -- (y) -- (v);
\end{tikzpicture}
\caption{$B(G, uv)$}
\label{fig1b}
\end{subfigure}%
\caption{The bridge construction. The graph $G$ consists of subgraphs $G_1$ and $G_2$
that are joined by an edge $uv$, i.e.\ $G_1$ and $G_2$ are connected components of $G - uv$.
$B(G, uv)$ is the enlargement of a nut graph by insertion of two vertices on a bridge and is also a nut graph.
Values $a$, $b$ and $-a$, $-b$, $a$ and $b$
are entries in the unique kernel eigenvectors of the graph
$G$ and its expansion, $B(G, uv)$, respectively.}
\label{fig:bridge}
\end{figure}
Let $uv$ be a bridge in $G$. By $B(G,uv)$ we denote the graph in which we insert two vertices on the bridge $uv$.
\begin{proposition}
\label{prop:4}
If we insert two vertices on a bridge $uv$ of $G$, the resulting graph $B(G,uv)$ is a nut graph if and only if $G$ is a nut graph.
\end{proposition}
\par\noindent
The forward direction of this result, together
with an alternative proof using linear algebra,
can be found in Section 4.3 of \cite{ScirihaGutman-NutExt}.
\begin{proof}\
\noindent
$(\Rightarrow)$: The process of assigning entries of a new kernel eigenvector on the bridging path $u\,x\,y\,v$ in the graph $B(G, uv)$
from the graph $G$
is unique if we retain all kernel eigenvector entries
in $G_2$.
(See Figure \ref{fig:bridge}.)
\noindent
$(\Leftarrow)$: If the graph $B(G, uv)$ is a nut graph and the vertices $u$ and $v$
have kernel eigenvector entries
assigned as shown, then the entries for vertices $x$ and $y$ follow.
The reverse operation can be carried out
by switching signs of all kernel eigenvector entries in either $G_1$ or $G_2$.
Hence, the graph $G$ is a nut graph.
\end{proof}
\subsubsection{The subdivision construction}
\begin{proposition}
\label{prop:subdivision}
If we insert four vertices on an edge $uv$ of $G$, the resulting graph $S(G,uv)$ is a nut graph if and only if $G$ is a nut graph.
\end{proposition}
\par\noindent
The forward direction of this result, together
with an alternative proof using linear algebra,
can be found as Lemma 4.1 in \cite{ScirihaGutman-NutExt}.
\begin{proof}\
\noindent $(\Rightarrow)$: The process of assigning entries of a new kernel eigenvector for the four inserted vertices $w\,x\,y\,z$ is
unique if we retain all eigenvector entries in $G$. (See Figure~\ref{fig:subdiv4}.)
\noindent $(\Leftarrow)$: If the enlarged graph $S(G, uv)$ is a nut graph,
then after removing the four additional vertices and joining $u$ directly to $v$,
retaining all other eigenvector entries, the reduced graph $G$ is also a nut graph.
\end{proof}
\begin{figure}[!htbp]
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\begin{tikzpicture}
\definecolor{mygreen}{RGB}{205, 238, 231}
\tikzstyle{vertex}=[draw,circle,font=\scriptsize,minimum size=13pt,inner sep=1pt,fill=mygreen]
\tikzstyle{edge}=[draw,thick]
\draw[thick,fill=gray!20!white] (1,0) ellipse (2.2cm and 1.4cm);
\node[vertex,fill=yellow,label={[yshift=0pt,xshift=0pt]90:$a$}] (u) at (0, 0) {$u$};
\node[vertex,fill=yellow,label={[xshift=0pt,yshift=0pt]90:$b$}] (v) at (2, 0) {$v$};
\path[edge] (u) -- (v);
\end{tikzpicture}
\caption{$G$}
\label{fig2a}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\begin{tikzpicture}
\definecolor{mygreen}{RGB}{205, 238, 231}
\tikzstyle{vertex}=[draw,circle,font=\scriptsize,minimum size=13pt,inner sep=1pt,fill=mygreen]
\tikzstyle{edge}=[draw,thick]
\draw[thick,fill=gray!20!white] (2.5,0) ellipse (3.8cm and 1.4cm);
\node[vertex,fill=yellow,label={[yshift=0pt,xshift=0pt]90:$a$}] (u) at (0, 0) {$u$};
\node[vertex,fill=yellow,label={[xshift=0pt,yshift=0pt]90:$b$}] (v) at (5, 0) {$v$};
\node[vertex,fill=mygreen,label={[xshift=0pt,yshift=0pt]90:$b$}] (w) at (1, 0) {$w$};
\node[vertex,fill=mygreen,label={[xshift=0pt,yshift=0pt]90:$-a$}] (x) at (2, 0) {$x$};
\node[vertex,fill=mygreen,label={[xshift=0pt,yshift=0pt]90:$-b$}] (y) at (3, 0) {$y$};
\node[vertex,fill=mygreen,label={[xshift=0pt,yshift=0pt]90:$a$}] (z) at (4, 0) {$z$};
\path[edge] (u) -- (w) -- (x) -- (y) -- (z) -- (v);
\end{tikzpicture}
\caption{$S(G, uv)$}
\label{fig2b}
\end{subfigure}%
\caption{The subdivision construction.
The graph
$S(G, uv)$ is obtained by
inserting vertices $w, x, y, z$ into the edge $uv$ of graph $G$.
Values $a$, $b$, $-a$, $-b$
are entries in the unique kernel eigenvectors of the graph
$G$ and its subdivision expansion, $S(G, uv)$.
}
\label{fig:subdiv4}
\end{figure}
\subsubsection{The \lq Fowler construction\rq }
The third construction is applicable to any graph $G$ and
any of its vertices $v$ with degree $d$, $d > 1$.
Let $G$ be a graph and let $v$ be a vertex of degree $d$
in $G$. Let the neighbourhood of $v$ be
$N_G(v) = \{u_1,u_2,\ldots, u_d\}$.
We remove the $d$ edges incident on $v$,
add $2d$ vertices, and connect them to the rest of the graph as shown in Figure~\ref{fig:3}.
Let $F(G,v)$ denote the resulting graph, which has been
called the \emph{Fowler construction} on $G$ \cite{GPS}.
The case where $d = 3$ was already described in \cite{Sc2008}.
Let the kernel eigenvector $\mathbf{x}$ be assigned to the vertices of $G$ in such a way that $\mathbf{x}(v) = x, \mathbf{x}(u_1) = x_1, \mathbf{x}(u_2) = x_2, \ldots , \mathbf{x}(u_d) = x_d$.
Then there is a unique way to carry over this kernel vector to $F(G,v)$. Moreover, we have the following theorem,
proved by Gauci et al.\ in \cite{GPS}.
\begin{theorem}[\cite{GPS}]
\label{thm:6}
$G$ is a nut graph if and only if $F(G,v)$ is a nut graph.
\end{theorem}
\begin{figure}[!h]
\centering
\subcaptionbox{$G$}
[.5\linewidth]{
\begin{tikzpicture}[scale=1.4]
\definecolor{mygreen}{RGB}{205, 238, 231}
\tikzstyle{vertex}=[draw,circle,font=\scriptsize,minimum size=13pt,inner sep=1pt,fill=mygreen]
\tikzstyle{edge}=[draw,thick]
\node[vertex,fill=mygreen,label={[yshift=0pt,xshift=0pt]90:$x$}] (v) at (0, 0) {$v$};
\node[vertex,fill=mygreen,label={[xshift=2pt,yshift=0pt]180:$x_1$}] (u1) at (-2, -1) {$u_1$};
\node[vertex,fill=mygreen,label={[xshift=2pt,yshift=0pt]180:$x_2$}] (u2) at (-0.7, -1) {$u_2$};
\node[vertex,fill=mygreen,label={[xshift=-2pt,yshift=0pt]0:$x_d$}] (ud) at (2, -1) {$u_d$};
\node at (0.5, -1) {$\ldots$};
\path[edge] (v) -- (u1);
\path[edge] (v) -- (u2);
\path[edge] (v) -- (ud);
\end{tikzpicture}
}%
\subcaptionbox{$F(G, v)$}
[.5\linewidth]{\begin{tikzpicture}[scale=1.4]
\definecolor{mygreen}{RGB}{205, 238, 231}
\tikzstyle{vertex}=[draw,circle,font=\scriptsize,minimum size=13pt,inner sep=1pt,fill=mygreen]
\tikzstyle{edge}=[draw,thick]
\node[vertex,fill=mygreen,label={[yshift=-4pt,xshift=0pt]90:$(1-d)x$}] (v) at (0, 0) {$v$};
\node[vertex,fill=mygreen,label={[xshift=2pt,yshift=0pt]180:$x_1$}] (q1) at (-2, -1) {$q_1$};
\node[vertex,fill=mygreen,label={[xshift=2pt,yshift=0pt]180:$x_2$}] (q2) at (-0.7, -1) {$q_2$};
\node[vertex,fill=mygreen,label={[xshift=-2pt,yshift=0pt]0:$x_d$}] (qd) at (2, -1) {$q_d$};
\node at (0.5, -1) {$\ldots$};
\node[vertex,fill=mygreen,label={[xshift=2pt,yshift=0pt]180:$x$}] (p1) at (-2, -2.5) {$p_1$};
\node[vertex,fill=mygreen,label={[xshift=2pt,yshift=0pt]180:$x$}] (p2) at (-0.7, -2.5) {$p_2$};
\node[vertex,fill=mygreen,label={[xshift=0pt,yshift=0pt]0:$x$}] (pd) at (2, -2.5) {$p_d$};
\node at (0.5, -2.5) {$\ldots$};
\node[vertex,fill=mygreen,label={[xshift=2pt,yshift=0pt]180:$x_1$}] (u1) at (-2, -4) {$u_1$};
\node[vertex,fill=mygreen,label={[xshift=2pt,yshift=0pt]180:$x_2$}] (u2) at (-0.7, -4) {$u_2$};
\node[vertex,fill=mygreen,label={[xshift=-2pt,yshift=0pt]0:$x_d$}] (ud) at (2, -4) {$u_d$};
\node at (0.5, -4) {$\ldots$};
\path[edge] (v) -- (q1);
\path[edge] (v) -- (q2);
\path[edge] (v) -- (qd);
\path[edge] (p1) -- (q2); \path[edge] (p1) -- (qd);
\path[edge] (p2) -- (q1); \path[edge] (p2) -- (qd);
\path[edge] (pd) -- (q1); \path[edge] (pd) -- (q2);
\path[edge] (p1) -- (u1);
\path[edge] (p2) -- (u2);
\path[edge] (pd) -- (ud);
\end{tikzpicture}}
\caption{A construction for expansion of a nut graph $G$ about vertex $v$ of degree $d$, to give $F(G, v)$.
The labelling of vertices in $G$ and $F(G, v)$ is shown within the circles that represent vertices.
Shown beside each vertex is the corresponding entry of the unique kernel eigenvector of the respective graph.}
\label{fig:3}
\end{figure}
The construction
$F(G,v)$
converts a nut graph that contains a vertex of given
degree $d$ to a nut graph with $2d$ more vertices of that degree. It has been described in the literature
for the case of general $d$, but here
the interesting cases are for degrees $2$ and $3$.
If $v$ is a vertex of degree $2$ then it belongs to a path of length at least $2$.
If we apply $F$ to this vertex, the length of the path increases by $4$.
If instead $v$ is a vertex of degree $3$, then the construction replaces it, and its neighbourhood, by the subgraph depicted in Figure~\ref{fig:4}.
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\begin{tikzpicture}
\definecolor{mygreen}{RGB}{205, 238, 231}
\tikzstyle{vertex}=[draw,circle,font=\scriptsize,minimum size=13pt,inner sep=1pt,fill=mygreen]
\tikzstyle{edge}=[draw,thick]
\node[vertex,fill=mygreen] (v) at (0, 0) {$v$};
\node[vertex,fill=mygreen] (u3) at (60:2.5) {$u_3$};
\node[vertex,fill=mygreen] (u2) at (-60:2.5) {$u_2$};
\node[vertex,fill=mygreen] (u1) at (180:2.5) {$u_1$};
\path[edge] (v) -- (u1);
\path[edge] (v) -- (u2);
\path[edge] (v) -- (u3);
\end{tikzpicture}
\caption{$G$}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\begin{tikzpicture}
\definecolor{mygreen}{RGB}{205, 238, 231}
\tikzstyle{vertex}=[draw,circle,font=\scriptsize,minimum size=13pt,inner sep=1pt,fill=mygreen]
\tikzstyle{edge}=[draw,thick]
\node[vertex,fill=mygreen] (v) at (0, 0) {$v$};
\node[vertex,fill=mygreen] (u3) at (60:2.5) {$u_3$};
\node[vertex,fill=mygreen] (u2) at (-60:2.5) {$u_2$};
\node[vertex,fill=mygreen] (u1) at (180:2.5) {$u_1$};
\node[vertex,fill=yellow] (y3) at (60:1.25) {};
\node[vertex,fill=yellow] (y2) at (-60:1.25) {};
\node[vertex,fill=yellow] (y1) at (180:1.25) {};
\node[vertex,fill=yellow] (t3) at (120:1.25) {};
\node[vertex,fill=yellow] (t2) at (0:1.25) {};
\node[vertex,fill=yellow] (t1) at (-120:1.25) {};
\path[edge] (y1) -- (u1);
\path[edge] (y2) -- (u2);
\path[edge] (y3) -- (u3);
\path[edge] (t1) -- (v);
\path[edge] (t2) -- (v);
\path[edge] (t3) -- (v);
\path[edge] (y1) -- (t1) -- (y2) -- (t2) -- (y3) -- (t3) -- (y1);
\end{tikzpicture}
\caption{$F(G, v)$}
\end{subfigure}
\caption{Construction $F(G,v)$ replaces cubic vertex $v$ and its neighbourhood $N_G(v)$ with the $10$-vertex subgraph shown on the right.}
\label{fig:4}
\end{figure}
The process of extending the nut graph can be described as follows: take a nut graph $G$ that has a vertex of degree $d$.
The non-trivial kernel eigenvector of the adjacency matrix of $G$ has entries $x$ on the vertex of interest
and entries $u_i$ ($\sum_i u_i = 0$) on its neighbours. It is possible to construct a larger nut graph
on $n+2d$ vertices with a full kernel eigenvector and the same nullity as $G$:
the construction involves interleaving two layers of $d$ vertices internally connected as a cocktail-party graph.
Untangling the cocktail-party portion of the graph shows that the larger graph inherits the genus of $G$,
when the vertex at which expansion takes place is either
of degree $2$ or degree $3$. The following proposition explains this more precisely.
\begin{proposition}
If a chemical graph $G$ is embedded in some closed surface $\Sigma$, then any bridge construction $B(G,uv)$,
subdivision construction $S(G, uv)$,
or Fowler construction $F(G,v)$ can be embedded
in the same surface $\Sigma$. In particular, $G$ is planar if and only if each of $B(G,uv)$, $S(G,uv)$ and $F(G,v)$ is planar.
Moreover,
if any one of the graphs $G$, $B(G,uv)$, $S(G,uv)$ or $F(G,v)$ is planar then all of them are planar.
\end{proposition}
\par\noindent
It is not hard to see, for instance, by Menger's Theorem, that $G$ has the same connectivity as $F(G,v)$. This implies the following proposition.
\begin{proposition}
Let $G$ be a chemical graph.
$G$ is polyhedral, i.e.\ planar and $3$-connected, if and only if for any vertex $v$ of $G$ the graph $F(G,v)$ is polyhedral.
\end{proposition}
In passing we note that if we generalise the Fowler construction in a natural way to multigraphs, then, for instance, the cube graph can be obtained as $F(G,v)$ where $G$ is the Theta graph.
A further observation relates to the logical connection between the constructed graph $F(G,v)$ and the subdivision
construction $S(G, uv)$.
\subsection{Realisability results}
The preceding section established tools that allow us to
draw the following three conclusions.
\begin{lemma}
\label{lem:9}
If $(v_3,v_2)$ is realisable by a graph with a bridge, then $(v_3,v_2+2)$ is also realisable (in the same surface) by such graph.
\end{lemma}
Lemma \ref{lem:9} follows from Proposition~\ref{prop:4}.
\begin{lemma}
\label{lem:10}
If $(v_3,v_2)$ is realisable, then $(v_3,v_2+4)$ is also realisable (in the same surface).
\end{lemma}
Lemma \ref{lem:10} can be established from Proposition~\ref{prop:subdivision}.
\begin{lemma}
\label{lem:11}
If $(v_3,v_2)$ is realisable, then $(v_3+6,v_2)$ is also realisable (in the same surface). Moreover if the former graph has a bridge, the latter also has a bridge.
\end{lemma}
Lemma \ref{lem:11} follows from Theorem~\ref{thm:6}.
\subsection{Non-realisability results}
\label{subsec:nonrel}
\begin{lemma}
\label{lem:12}
No chemical graph with $v_3 = 0$ is a nut graph.
\end{lemma}
\begin{proof}
The lemma claims that no cycle is a nut graph. By inspection of spectra, this is indeed the case. Namely, a cycle $C_n$ is singular if and only if
$n$ is divisible by four, in which case it has nullity $2$. Compare column 1 in Table~\ref{tbl:census}.
\end{proof}
\begin{lemma}
No chemical graph with
$$(v_3,v_2) \in \{((8, 0), (8, 1), (8, 2), (8, 4), (10, 0), (10, 1), (10, 2), (12, 1), (14, 0), (14, 2), (16, 0))\}$$
is a nut graph.
\label{lemma:cases}
\end{lemma}
\begin{proof}
The proof follows from computer determination of all chemical nut graphs on $n$ vertices,
$n \leq 22$, as presented in Table~\ref{tbl:census}, stratified by $(v_3,v_2)$ parameters. The entries covered by this lemma are shown as boldface zeros
in the table.
\end{proof}
\par\noindent
Note that Table~\ref{tbl:census} also shows non-existence of some other pairs $(v_3,v_2)$ with $v_3+v_2 \leq 22$,
but these follow from Lamma \ref{lem:12} and Corollary \ref{cor:13}
It turns out that only half of the pairs $(v_3, v_2)$ for $v_3 \in \{2, 4\}$ are only realisable (depending on the parity of $v_2$). To prove this, we need the following result.
\begin{theorem}
\label{cor:realisx}
Let $G$ be a leafless chemical graph with $v_2 \geq \frac{9}{2}v_3 + 1$. Then the pair $(v_3,v_2)$ is realisable if and only if the pair $(v_3,v_2-4)$ is realisable. If the latter pair is not realisable
then none of the pairs $(v_3,v_2 + 4k)$, $k \in \{0,1,2,\ldots\}$, is realisable.
\end{theorem}
\begin{proof}
Since $G$ is a leafless chemical graph it can be viewed as a general subdivision of a cubic graph $H$ on $v_3$ vertices.
Hence, $H$ has $m = 3 v_3 / 2$ edges. And $G$ has its $v_2$ vertices of degree $2$ placed on these edges, forming corresponding paths.
As $v_2 \geq \frac{9}{2}v_3 + 1 = 3m+1$, by the Pigeonhole Principle, when forming $G$ from $H$, at least one of the edges of $H$ must be
replaced by a path containing at least $4$ degree-$2$ vertices.
The last claim of this theorem follows by repeated application of Lemma~\ref{lem:10}.
\end{proof}
In the case $v_3 = 2$, Theorem~\ref{cor:realisx} applies to $v_2 \geq 10$; in the case $v_3 = 4$, it applies to $v_2 \geq 19$.
\begin{corollary}
\label{cor:13}
Since pairs $(v_3,v_2) \in \{(2,6),(2,8),(4,15),(4,17)\}$ are not realisable, this means that no pair $(2,2k)$ and $(4,2k+1)$ is realisable, for $k \geq 0$.
\end{corollary}
By using Corollary 13, we have generated two infinite series of zeros (non-realisable pairs), namely, for $v_3 = 2$ and $v_3 = 4$.
For the case $v_3 = 0$ we use Lemma 2 to obtain a third infinite series of zeros. The remaining zeros are finite in number
and are covered by Lemma 3 based on the database of chemical graphs in House of Graphs.
The final step in the proof of Theorem 1 is to establish that outside the triangular region of
parameter space covered by the database, every pair that does
not belong to one of the infinite series of zeros (unrealisable cases), is realisable.
To do this we use the notion of seed graphs.
\subsection{Seed graphs}
A \emph{seed graph} is a chemical nut graph that cannot be generated from any smaller chemical nut graph by
some combination of the three constructions.
We will show that all realisable pairs can be generated from a small set of seed graphs by repeated application of the constructions.
\begin{lemma}
Chemical nut graphs with the following parameters exist:
\begin{enumerate}[label=(\alph*)]
\item
Planar, with a bridge (B):
$$(v_3,v_2) \in \{(2,7),(4,10),(6,7),(6,8),(8,5),(8,6),(10,4),(10,5),(21,3),(14,4),(20,2)\} $$
\item
Planar $2$-connected (P):
$$(v_3,v_2) \in \{ (10,3),(12,2),(14,1),(16,1),(16,2),(18,1),(22,0) \}$$
\item
Polyhedral (i.e.\ planar $3$-connected) ($\mathit{\Pi}$):
$$(v_3,v_2) \in \{ (12,0),(26,0), (28, 0) \}$$
\item
Non-planar, genus one (N): $$(v_3,v_2) \in \{ (20,0) \}$$
\end{enumerate}
\end{lemma}
\begin{proof}
By inspection of adjacency data presented in the appendices
for each seed chemical nut graph.
\end{proof}
Note that other chemical nut graphs with these parameters
may exist and that other choices of seed graphs are possible.
\subsection{Proof of Theorem~{\ref{thm:main}}}
\begin{proof}[Proof]
To prove the realisability result we construct Table 2 according to the following rules.
We start by indicating the parameters of the $22$
seed graphs listed in Lemma 14 using boldface B, P, $\Pi$, N as appropriate.
A bold X is used to indicate a non-realisable entry derived from Table 1.
Other entries are filled in according to three rules based on
lemmas:
\begin{enumerate}[label=(\roman*)]
\item
(Lemma~{\ref{lem:9}}) if we have B in the table (meaning a planar chemical nut graph with a bridge) then we can insert
B two entries below;
\item (Lemma~{\ref{lem:10}}) if we have P or $\Pi$ in the table (meaning a planar chemical nut graph without a bridge) then we
can insert P four entries
below;
\item (Lemma~\ref{lem:11}) if we have B, P or $\Pi$ in a row of the table then three steps to the right we have B, P or $\Pi$, respectively.
\end{enumerate}
We proceed initially by columns.
Column $v_3 = 0$ is entirely non-realisable (see Lemma \ref{lem:12}).
The first boldface entry in each of columns $v_3 = 2$
and $v_3 = 4$ gives a series of realisable cases.
All other entries in these columns
are non-realisable: this follows from Corollary \ref{cor:13}.
In column $v_3 = 6$,
consecutive boldface B entries prove realisability for all $v_2 \geq 7$. Likewise, in column $v_3 = 8$, entry
$(3, 8)$ in combination with $(6, 8)$ proves realisability for all $v_2 \geq 5$ (by a planar graph containing a bridge).
Similarly, in column $v_3 = 10$
the
entries $(4, 10)$ and $(5, 10)$ prove realisability for all $v_2 \geq 4$. Having established the rectangle
of six
cases outlined in the table (with corners
$(7, 6)$, $(7, 10)$, $(8, 6)$, and $(8, 10)$), we fill the whole quadrant to the right and below.
Rows $v_2 = 7$ and $v_2 = 8$ are generated by the Fowler construction and all entries below by repetition of the bridge construction.
We are left with rows for $v_2 < 7$. For each of them we start with a seed graph and then repeat the Fowler construction.
Non-realisability of remaining entries follows from Table 1.
By using $(0, 26)$ as a seed graph we were able to show the existence of planar graphs for all entries to the right.
\end{proof}
\vspace{\baselineskip}
\newcommand{\mbg}{}
\newcommand{\ybg}{}
\newcommand{\bbg}{}
\begin{table}[!h]
\centering
\begin{small}
\begin{tabular}{|c|rrr:rrrrrrrrrrrrr|}
\hline
\backslashbox{$v_2$}{$v_3$} & 0 & 2 & \multicolumn{1}{r}{4 } & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & 22 & 24 & 26 & 28 & $\cdots$ \\ \hline
$0$ & $\emptyset$ & $\emptyset$ & \multicolumn{1}{r}{${\bf X}$} & ${\bf X}$ & ${\bf X}$ & ${\bf X}$ & ${\bf \Pi}$ & ${\bf X}$ & ${\bf X}$ & $F\mathit{\Pi}$ & ${\bf N}$ & $\ybg {\bf P}$ & $\ybg F\mathit{\Pi}$ & $\ybg {\bf \Pi}$ & ${\bf \Pi}$ & $\Rightarrow$ \\
$1$ & $\emptyset$ & $\emptyset$ & \multicolumn{1}{r}{${\bf X}$} & ${\bf X}$ & ${\bf X}$ & ${\bf X}$ & ${\bf X}$ & $\ybg {\bf P}$ & $\ybg {\bf P}$ & $\ybg {\bf P}$ & $FP$ & $FP$ & $FP$ & $FP$ & $FP$ & $\Rightarrow$ \\
$2$ & $\emptyset$ & ${\bf X}$ & \multicolumn{1}{r}{${\bf X}$} & ${\bf X}$ & ${\bf X}$ & ${\bf X}$ & ${\bf P}$ & ${\bf X}$ & $\ybg {\bf P}$ & $\ybg FP$ & $\ybg {\bf B}$ & $FP$ & $FP$ & $FB$ & $FP$ & $\Rightarrow$ \\
$3$ & $X$ & ${\bf X}$ & \multicolumn{1}{r}{${\bf X}$} & ${\bf X}$ & $\ybg {\bf B}$ & $\ybg {\bf P}$ & $\ybg {\bf B}$ & $FB$ & $FP$ & $FB$ & $FB$ & $FP$ & $FB$ & $FB$ & $FP$ & $\Rightarrow$ \\
$4$ & $X$ & ${\bf X}$ & \multicolumn{1}{r}{${\bf X}$} & ${\bf X}$ & ${\bf X}$ & $\ybg {\bf B}$ & $\ybg P$ & $\ybg {\bf B}$ & $FB$ & $P$ & $FB$ & $FB$ & $FP$ & $FB$ & $FB$ & $\Rightarrow$ \\
$5$ & $X$ & ${\bf X}$ & \multicolumn{1}{r}{${\bf X}$} & ${\bf X}$ & $\ybg B$ & $\ybg {\bf B}$ & $\ybg B$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $\Rightarrow$ \\
$6$ & $X$ & $\bbg X$ & \multicolumn{1}{r}{${\bf X}$} & ${\bf X}$ & $\ybg {\bf B}$ & $\ybg B$ & $\ybg P$ & $FB$ & $FB$ & $FP$ & $FB$ & $FB$ & $FP$ & $FB$ & $FB$ & $\Rightarrow$ \\
\cline{5-7} \cdashline{8-16}
$7$ & $X$ & $\bbg {\bf B}$ & ${\bf X}$ & \multicolumn{1}{|r}{$\mbg {\bf B}$ } & $\mbg B$ & \multicolumn{1}{r|}{$\mbg B$ } & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $\Rightarrow$ \\
$8$ & $X$ & $\bbg X$ & ${\bf X}$ & \multicolumn{1}{|r}{$\mbg {\bf B} $} & $\mbg B$ &\multicolumn{1}{r|}{$\mbg B$ } & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $FB$ & $\Rightarrow$ \\
\cline{5-7}
$9$ & $X$ & $\bbg B$ & ${\bf X}$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $\Rightarrow$ \\
$10$ & $X$ & $X$ & ${\bf B}$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $\Rightarrow$ \\
$11$ & $X$ & $B$ & ${\bf X}$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $\Rightarrow$ \\
$12$ & $X$ & $X$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $\Rightarrow$ \\
$13$ & $X$ & $B$ & ${\bf X}$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $\Rightarrow$ \\
$14$ & $X$ & $X$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $\Rightarrow$ \\
$15$ & $X$ & $B$ & $\bbg X$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $\Rightarrow$ \\
$16$ & $X$ & $X$ & $\bbg B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $\Rightarrow$ \\
$17$ & $X$ & $B$ & $\bbg X$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $\Rightarrow$ \\
$18$ & $X$ & $X$ & $\bbg B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $B$ & $\Rightarrow$ \\[-3pt]
$\vdots$ & $\Downarrow$ & $\Downarrow$ & \multicolumn{1}{r}{$\Downarrow$} & $\Downarrow$ & $\Downarrow$ & $\Downarrow$ & $\Downarrow$ & $\Downarrow$ & $\Downarrow$ & $\Downarrow$ & $\Downarrow$ & $\Downarrow$ & $\Downarrow$ & $\Downarrow$ & $\Downarrow$ & \\
\hline
\end{tabular}
\end{small}
\caption{Parameter pairs $(v_3,v_2)$ for which chemical nut graphs exist. Notation: $\emptyset$ - no graph exists; $X$ - no nut graph exists; $B$ - a planar graph with a bridge exists ($FB$ - obtained by Fowler construction); $P$ - a $2$-connected planar graph exists ($FP$ - obtained by Fowler construction); $\mathit{\Pi}$ - a polyhedral graph exists ($F\mathit{\Pi}$ - obtained by Fowler construction); $N$ - only non-planar graphs exist; $\Rightarrow$ - the pattern extends indefinitely to the right; $\Downarrow$ - the pattern extends indefinitely below. Boldface symbols were determined from the graph database: {\bf B, P, $\bf \Pi$, N} represent seed graphs, while {\bf X} represents non-existence that follows from Table 1.
Dashed lines indicate the quadrant that can be completely filled with bridged cases, once the boxed rectangle is established.
}
\label{tbl:map}
\end{table}
\section{Polyhedral and toroidal chemical nut graphs}
Before we conclude this paper we want to address a special class of chemical nut graphs, namely the polyhedral chemical nut graphs, i.e.\
$3$-connected planar chemical nut graphs. In practice this means that only cubic graphs, $v_2 = 0$, are eligible. The smallest chemical nut graphs of this type were found
in \cite{Prop23} and the list was extended in
\cite{hog,nutgen-site,CoolFowlGoed-2017,GPS}.
There are two cubic polyhedral nut graphs on $12$ vertices,
illustrated in \cite{Prop23}. One of these (Figure 17(b) \cite{Prop23}) is the Frucht graph \cite{GPS}; the other
(Figure 17(a) \cite{Prop23})
can be realised with $C_2$ point-group symmetry.
Since the result of applying the Fowler construction to a polyhedral nut graph is a polyhedral nut graph, these seed graphs give rise to an infinite series of parameters
$v_3 = 12,18,24, \ldots $ that admit polyhedral nuts.
In \cite{GPS}, a cubic nut graph with $v_3 = 26$
is identified. In fact, this is one of four.
By the same argument
we obtain another infinite series $v_3 = 26,32,38, \ldots $. From Table 2 one can see that no polyhedral nut graph exists for $v_3 = 14,16,,20$.
There is no cubic polyhedral nut graph with $v_3 =22$ (\cite{Prop23},
confirmed in \cite{hog,nutgen-site,CoolFowlGoed-2017,GPS}).
There are 316 chemical polyhedral nut graphs on $v_3 = 28$ vertices \cite{hog,nutgen-site,CoolFowlGoed-2017}. This gives rise to the final infinite series: $v_3 = 28,34,40, \ldots$. The adjacency
data of the three polyhedral
seeds are given in the Appendix $\Pi$.
These observations completely characterise admissible parameters and hence prove the following proposition.
\begin{proposition}
A polyhedral chemical nut graph with parameters $(v_3,v_2)$ exists if and only if $v_2 = 0$ and $v_3$ is even with $v_3 = 12$, $v_3 = 18$ or $v_3 \geq 24$.
\end{proposition}
A subset of the cubic polyhedra of special interest
for the chemistry of carbon cages
is that of the fullerenes. A \emph{fullerene} has a cubic polyhedral molecular graph in which exactly $12$ faces are of size $5$
and any other faces are of size $6$. Nut fullerenes seem
relatively rare \cite{Prop23}, and have been enumerated up to order $240$
\cite{hog,nutgen-site,CoolFowlGoed-2017}.
The smallest nut fullerene has $36$ vertices and a construction given in \cite{Prop23} (Figure 14) shows that the unique
C$_{36}$ nut fullerene can be extended by adding belts of six hexagons at a time to make larger cylindrical fullerenes that are also nut graphs. This rationalises the presence of the integers $n = 24+12k$ ($k > 0$) in the published list of orders of fullerene nut graphs \cite{hog,nutgen-site,CoolFowlGoed-2017}. The members of this sub-series all have the same six-pentagon cluster in each cylinder cap, and all are \emph{uniform} \cite{Prop23} in the sense that every vertex is surrounded in the kernel eigenvector by the same triple of entries $\{2, -1, -1\}$.
All nut fullerenes known so far have multiple pentagon adjacencies, which militates against
their stability as neutral all-carbon molecules.
Incidentally, as there is, in addition to the example on $20$ vertices, a chemical nut graph of genus $1$ on $22$ and $24$ vertices (see Appendix N),
we have an infinite series of toroidal $3$-regular chemical nut graphs. More precisely, there is a toroidal $3$-regular nut graph on $n$ vertices
if and only if $n$ is
even and $n \geq 20$.
\section{Conclusion}
The parameter space occupied by chemical nut graphs
has been characterised in terms of
allowed vertex signatures $(v_3, v_2)$,
according to Theorem {\ref{thm:main}}.
This result
has implications for the
theory of molecular conduction, in that each
chemical nut graph
corresponds to the carbon skeleton of a $\pi$-conjugated
molecule with a single non-bonding molecular orbital
and strong omni-conducting
behaviour \cite{PWF_CPL_2013,PWF_JCP_2014}.
The characterisation of chemical nut graphs by vertex degrees
also implies restrictions on their orders and sizes,
$n$ and $m$.
As a chemical nut graph has no vertices of degree $1$ and maximum degree $\Delta \le 3$, the Theorem \ref{thm:main} implies
a restricted range of Betti numbers, $m-n+1$, for these graphs.
For a chemical nut graph, $v_1 = 0$, $v_2 \ge 0$
and $v_3 > 0$ is even. Hence, $n = v_2 + v_3$,
$m = v_2 + (3/2) v_3$, and the Betti numbers span the range
\begin{equation}
1 < \frac{v_3}{2} +1 = m-n+1 \leq \left\lceil{\frac{n + 1}{2}}\right\rceil
\label{eq:range}
\end{equation}
Transforming Table 2 from $(v_3, v_2)$ to $(n, m)$
space, shows that for $n = 15$ and $n \ge 17$, there is
just one missing value in the range \eqref{eq:range}:
for odd $n \ge 15$ no chemical nut graph has $m-n+1 =3$, and for even $n \ge 18$ no chemical nut graph has $m-n+1 =2$.
For all other realisable values of $n$ (more precisely, $9 \leq n \leq 14$ and $n = 16$), there are at least two missing values in the range \eqref{eq:range}.
Although we have settled the question of the existence of chemical nut graphs, there several intriguing related topics that we can
address in future work. One of them is the question of the existence of signed chemical nut graphs. For signed graphs, see for instance recent work \cite{BeCiKoWa2019} and~\cite{GhHaMaMa2020}.
\section*{Acknowledgements}
The work of Toma\v{z} Pisanski is supported in part by the Slovenian Research Agency (research program P1-0294
and research projects J1-2481, N1-0032, J1-9187 and J1-1690),
and in part by H2020 Teaming InnoRenew CoE.
The work of Nino Bašić is supported in part by the Slovenian Research Agency (research program P1-0294
and research projects J1-2481, J1-9187, J1-1691 and N1-0140).
\bibliographystyle{amcjoucc}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,101 |
\section{Introduction and result}
A dynamic programming (DP) algorithm is \emph{pure} if it only uses
min or max and addition operations in its recursion equations, and the
equations do not depend on the actual values of the input weights.
Notable examples of such DP algorithms include the
Bellman--Ford--Moore algorithm for the shortest $s$-$t$ path
problem~\cite{ford,Moore1957,bellman}, the Floyd--Warshall algorithm
for the all-pairs shortest paths problem~\cite{floyd,warshall}, the
Held--Karp algorithm for the Travelling Salesman
Problem~\cite{held62}, and the Dreyfus--Levin--Wagner algorithm for
the weighted Steiner tree problem~\cite{dreyfus,levin}.
It is well known and easy to show that, for some optimization
problems, already pure DP algorithms can be much better than greedy
algorithms. Namely, there are a lot of optimization problems which
are easily solvable by pure DP algorithms (exactly), but the greedy
algorithm cannot even achieve any finite approximation factor: maximum
weight independent set in a path, or in a tree, the maximum weight
simple $s$-$t$ path in a transitive tournament problem, etc.
In this paper, we show that the converse direction also holds: for
some optimization problems, greedy algorithms can also be much better
than pure dynamic programming. So, the computational powers of greedy
and pure DP algorithms are \emph{incomparable}. We will show that the
gap occurs on the (undirected) minimum weight spanning tree problem,
by first deriving an exponential lower bound on the monotone
arithmetic circuit complexity of the corresponding polynomial.
Let $K_n$ be the complete undirected graph on
$[n]=\{1,\ldots,n\}$. Assume that edges $e$ have their associated
nonnegative real weights $x_{e}$, considered as formal variables. Let
${\cal T}_n$ be the family of all $|{\cal T}_n|=n^{n-2}$ spanning trees
$T$ in $K_n$, each viewed as its \emph{set} of edges.
It is well known that ${\cal T}_n$ is the family of bases of a matroid,
known as \emph{graphic matroid}; so, on this family of feasible
solutions, both optimization problems (minimization and maximization)
can be solved by standard greedy algorithms. On the other hand, the
theorem below states that the polynomial corresponding to ${\cal T}_n$
has exponential monotone arithmetic circuit complexity; due to special
properties of this polynomial, the same lower bound also holds on the
number of operations used by pure DP algorithms solving minimization
and maximization problems on ${\cal T}_n$ (see~\cref{lem:reduct}).
The \emph{spanning tree polynomial} (known also as the \emph{Kirchhoff
polynomial} of $K_n$) is the following homogeneous, multilinear
polynomial of degree $n-1$:
\[
\ptree_n(x) = \sum_{T\in{\cal T}_n} \prod_{e\in T}x_e\,.
\]
For a multivariate polynomial $f$ with positive coefficients, let
$\arithm{f}$ denote the minimum size of a monotone arithmetic
$(+,\times)$ circuit computing~$f$. Our goal is to prove that
$\arithm{\ptree_n}$ is exponential in~$n$.
\begin{thm}\label{thm:main}
\[
\arithm{\ptree_n}=2^{\Omega(\sqrt{n})}\,.
\]
\end{thm}
A ``directed version'' of $\ptree_n$ is the \emph{arborescence
polynomial}~$\dtree_n$. An \emph{arborescence} (known also as a
\emph{branching} or a \emph{directed spanning tree}) on the vertex-set
$[n]$ is a directed tree with edges oriented away from vertex $1$ such
that every other vertex is reachable from vertex $1$. Let $\vec{\cal T}_n$
be the family of all arborescences on $[n]$. Jerrum and
Snir~\cite{jerrum} have shown that $\arithm{\dtree_n}=2^{\Omega(n)}$
holds for the arborescence polynomial
\[
\dtree_n(x) = \sum_{T\in\vec{\cal T}_n}\ \prod_{ \vec{e}\in
T}x_{\vec{e}}\,.
\]
Note that here variables $x_{i,j}$ and $x_{j,i}$ are treated as
\emph{distinct}, and cannot both appear in the same monomial. This
dependence on orientation was crucially utilized in the argument
of~\cite[p.~892]{jerrum} to reduce a trivial upper bound $(n-1)^{n-1}$
on the number of monomials in a polynomial computed at a particular
gate till a non-trivial upper bound $(3n/4)^{n-1}$. So, this argument
does not apply to the \emph{undirected} version~$\ptree_n$ (where
$x_{i,j}$ and $x_{j,i}$ stand for the \emph{same} variable). To handle
the undirected case, we will use an entirely different argument.
\paragraph{Relation to pure DP algorithms}
Every pure DP algorithm is just a special (recursively constructed)
\emph{tropical} \mbox{$(\min,+)$}~ or \mbox{$(\max,+)$}~ circuit, that is, a circuit using
only min (or max) and addition operations as gates; each input gate of
such a circuit holds either one of the variables $x_i$ or a
nonnegative real number. So, lower bounds on the size of tropical
circuits yield the same lower bounds on the number of operations used
by pure DP algorithms. For optimization problems, whose feasible
solutions all have the same cardinality, the task of proving lower
bounds on their tropical circuit complexity can be solved by proving
lower bounds on the size of monotone arithmetic circuits.
Recall that a multivariate polynomial is \emph{monic} if all its
nonzero coefficients are equal to $1$, \emph{multilinear} if no
variable occurs with degree larger than $1$, and \emph{homogeneous} if
all monomials have the same degree. Every monic and multilinear
polynomial $f(x)=\sum_{S\in{\cal F}}\prod_{i\in S} x_i$ defines two
optimization problems: compute the minimum or the maximum of
$\sum_{i\in S}x_i$ over all $S\in{\cal F}$.
\begin{lem}[\cite{jerrum,juk14}]\label{lem:reduct}
If a polynomial $f$ is monic, multilinear and homogeneous, then
every tropical circuit solving the corresponding optimization
problem defined by $f$ must have at least $\arithm{f}$ gates.
\end{lem}
This fact was proved by Jerrum and Snir~\cite[Corollary~2.10]{jerrum};
see also \cite[Theorem~9]{juk14} for a simpler proof. The proof idea
is fairly simple: having a tropical circuit, turn it into a monotone
arithmetic $(+,\times)$ circuit, and use the homogeneity of $f$ to
show that, after removing some of the edges entering $+$-gates, the
resulting circuit will compute our polynomial~$f$.
\paragraph{Greedy can beat pure DP}
The (weighted) \emph{minimum spanning tree} problem $MST_n(x)$ is,
given an assignment of nonnegative real weights to the edges of $K_n$,
compute the minimum weight of a spanning tree of $K_n$, where the
weight of a graph is the sum of weights of its edges. So, this is
exactly the minimization problem defined by the spanning tree
polynomial~$\ptree_n$:
\[
MST_n(x) = \min_{T\in{\cal T}_n} \ \sum_{e\in T}x_e\,.
\]
Since the family ${\cal T}_n$ of feasible solutions of this problem is
the family of bases of the (graphic) matroid, the problem \emph{can}
be solved by the standard greedy algorithm. In particular, the
well-known greedy algorithm of Kruskal~\cite{kruskal} solves $MST_n$
using only $O(n^2\log n)$ operations.
On the other hand, since the spanning tree polynomial $\ptree_n$ is
monic, multilinear and homogeneous, \cref{thm:main} together with
\cref{lem:reduct} implies that any \mbox{$(\min,+)$}~ circuit solving the
problem $MST_n$ must have at least
$\arithm{\ptree_n}=2^{\Omega(\sqrt{n})}$ gates and, hence, at least so
many operations must be performed by any pure DP algorithm solving
$MST_n$. This gap between pure DP and greedy algorithms is our main
result.
\paragraph{Directed versus undirected spanning trees}
The arborescence polynomial~$\dtree_n$ is also monic, multilinear
and homogeneous, so that \cref{lem:reduct}, together with the above
mentioned lower bound on $\arithm{\dtree_n}$ due to Jerrum and
Snir~\cite{jerrum}, also yields the same lower bound on the size of
\mbox{$(\min,+)$}~ circuits solving the minimization problem on the family
$\vec{\cal T}_n$ of arborescences.
But this does not separate DP from greedy, because the downward
closure of $\vec{\cal T}_n$ is \emph{not} a matroid: it is only an
intersection of two matroids (see Edmonds~\cite{edmonds67}). So,
greedy algorithms are only able to \emph{approximate} the
minimization problem on $\vec{\cal T}_n$ within the factor $2$.
Polynomial time algorithms solving this problem \emph{exactly} were
found by several authors, starting from Edmonds~\cite{edmonds73}.
The fastest algorithm for the problem is due to
Tarjan~\cite{tarjan}, and solves it in time $O(n^2\log n)$,
that is, with the same time complexity as Kruskal's greedy algorithm
for undirected graphs~\cite{kruskal}. But these are not greedy
algorithms. So, $\vec{\cal T}_n$ does not separate standard, matroid based greedy and pure DP
algorithms.
\section{Proof of Theorem~\ref{thm:main}}
A \emph{rectangle} is specified by giving two families ${\cal A}$ and ${\cal B}$
of forests in the complete graph $K_n$ on $[n]=\{1,\ldots,n\}$ such
that for all forests $A\in{\cal A}$ and $B\in{\cal B}$ (viewed as sets of their
edges), we have $A\cap B=\emptyset$ (the forests are edge-disjoint),
and $A\cup B$ is a spanning tree of $K_n$. The rectangle itself is
the family
\[
{\cal R}={\cal A}\lor{\cal B}:=\{A\cup B\colon \mbox{$A\in{\cal A}$ and $B\in{\cal B}$}\}
\]
of all resulting spanning trees. A rectangle ${\cal R}={\cal A}\lor{\cal B}$ is
\emph{balanced} if $(n-1)/3\leq |A|,|B|\leq 2(n-1)/3$ holds for all
forests $A\in{\cal A}$ and $B\in{\cal B}$; recall that every spanning tree of a
graph on $n$ vertices has $n-1$ edges. Let $\bound{n}$ be the minimum
number of balanced rectangles whose union gives the family of all
spanning trees of~$K_n$.
\begin{lem}\label{lem:tau}
For the spanning tree polynomial $\ptree_n$, we have
$\arithm{\ptree_n}\geq \bound{n}$.
\end{lem}
\begin{proof}
Let $t=\arithm{\ptree_n}$. The spanning tree polynomial $\ptree_n$
is multilinear and homogeneous of degree $n-1$: every spanning tree
$T$ of $K_n$ has $|T|=n-1$ edges. Since the polynomial $\ptree_n$ is
homogeneous of degree $n-1$, and since $\ptree_n$ can be computed by
a monotone arithmetic circuit of size $t$, the well-known
decomposition result, proved by Hyafil~\cite[Theorem~1]{hyafil} and
Valiant~\cite[Lemma~3]{valiant80}, implies that $\ptree_n$ can be
written as a sum $\ptree_n=g_1\cdot h_1+\cdots+g_t\cdot h_t$ of
products of nonnegative homogeneous polynomials, each of degree at
most $2(n-1)/3$; a polynomial is \emph{nonnegative} if it has no
negative coefficients.
Every monomial of $\ptree_n$ is of the form $\prod_{e\in T}x_e$ for
some spanning tree~$T$. Since the polynomials $g_i$ and $h_i$ in the
decomposition of $\ptree_n$ are nonnegative, there can be no
cancellations. This implies that all the monomials of $g_i\cdot h_i$
must be also monomials of $\ptree_n$, that is, must correspond to
spanning trees. Moreover, since the polynomial $\ptree_n$ is
multilinear, the forests of $g_i$ must be edge-disjoint from the
forests of $h_i$. So, if we let ${\cal A}_i$ be the family of forests
corresponding to monomials of the polynomial $g_i$, and ${\cal B}_i$ be
the family of forests corresponding to monomials of the
polynomial~$h_i$, then ${\cal A}_1\lor{\cal B}_1,\ldots,{\cal A}_t\lor{\cal B}_t$ are
balanced rectangles, and their union gives the family of all spanning trees
of $K_n$. This shows $\bound{n}\leq t=\arithm{\ptree_n}$, as
desired.
\end{proof}
So, it is enough to prove an exponential lower bound on $\bound{n}$.
When doing this, we will concentrate on spanning trees of $K_n$ of a
special form. Let $m$ and $d$ be positive integer parameters
satisfying $(d+1)m=n$, $m=\Theta(\sqrt{n})$ and $m\leq d/32$; we will
specify these parameters later.
A \emph{star} is a tree with one vertex, the \emph{center}, adjacent
to all the others, which are \emph{leaves}. A $d$-\emph{star} is a
star with $d$ leaves. A \emph{spanning star-tree} consists of $m$
vertex-disjoint $d$-stars whose centers are joined by a path. A
\emph{star factor} is a spanning forest of $K_n$ consisting of $m$
vertex-disjoint $d$-stars. Note that each spanning star-tree contains
a unique star factor (obtained by removing edges between star
centers).
Let ${\cal F}$ be the family of all star factors of $K_n$. For a rectangle
${\cal R}$, let ${\cal F}_{{\cal R}}$ denote the family of all star factors $F$ of $K_n$
contained in at least one spanning tree of ${\cal R}$; in this case, we also
say that the factor $F$ is \emph{covered} by the rectangle~${\cal R}$.
\begin{lem}\label{lem:main}
There is an absolute constant $c>0$ such that for every balanced
rectangle ${\cal R}$, we have $|{\cal F}_{{\cal R}}|\leq |{\cal F}|\cdot 2^{-c\sqrt{n}}$.
\end{lem}
Note that this lemma gives a lower bound $\bound{n}\geq 2^{c\sqrt{n}}$
on the minimum number of balanced rectangles containing all spanning
trees of $K_n$. Indeed, let ${\cal R}_1,\ldots,{\cal R}_t$ be $t=\bound{n}$
balanced rectangles whose union is the family of all spanning trees of
$K_n$. Every star factor $F\in{\cal F}$ is contained in at least one
spanning tree (in fact, in many of them). So, every star factor
$F\in{\cal F}$ must be covered by at least one of these $t$ rectangles. But
\cref{lem:main} implies that none of these rectangles can cover more
than $h:=|{\cal F}|\cdot 2^{-c\sqrt{n}}$ star factors $F\in{\cal F}$. So, we need
$\bound{n}=t\geq |{\cal F}|/h\geq 2^{c\sqrt{n}}$ rectangles. Together with
\cref{lem:tau}, this yields the desired lower bound
$\arithm{\ptree_n}\geq 2^{c\sqrt{n}}$ on the monotone arithmetic
circuit complexity of the spanning tree polynomial~$\ptree_n$.
The rest of the paper is devoted to the proof of~\cref{lem:main}.
\begin{proof}[Proof of~\cref{lem:main}]
We can construct every star factor $F\in{\cal F}$ using the following
procedure.
\begin{enumerate}
\item Choose a subset of $m$ centers in $[n]$; $\binom{n}{m}$
possibilities.
\item Divide the remaining $n-m$ vertices into $m$ blocks of size
$d$, and connect all vertices of the $i$th block to the $i$th
largest of the chosen centers; there are
$\binom{n-m}{d,\ldots,d}=\frac{(n-m)!}{d!^m}$ possibilities to do
this.
\end{enumerate}
Since different realizations of this procedure lead to different
star factors, we have
\begin{equation}\label{eq:all}
|{\cal F}|=\binom{n}{m}\frac{(n-m)!}{d!^m}\,.
\end{equation}
Fix a balanced rectangle ${\cal R}={\cal A}\lor{\cal B}$ containing at least one
spanning star-tree $T_0=A_0\cup B_0$ with $A_0\in{\cal A}$ and $B_0\in{\cal B}$, and let $c_1,\ldots,c_m$ be
the centers of stars of $T_0$. Every vertex $v\in
[n]\setminus\{c_1,\ldots,c_m\}$ is connected in $T_0$ by a
\emph{unique} edge $e_v$ to one of the centers
$c_1,\ldots,c_m$. This gives us a partition $U\cup V$ of the
vertices in $[n]\setminus\{c_1,\ldots,c_m\}$ into two sets
determined by the forests $A_0$ and $B_0$:
\[
U=\{v\colon e_v\in A_0\}\ \ \mbox{ and }\ \ V=\{v\colon e_v\in
B_0\}\,.
\]
We will concentrate on the bipartite complete subgraph $U\timesV$
of $K_n$, and call the edges of $K_n$ lying in this subgraph
\emph{crossing edges}. Since our rectangle ${\cal R}$ is balanced, we
know that both $|A_0|$ and $|B_0|$ lie between $(n-1)/3$ and
$2(n-1)/3$. So, since $m=o(n)$, for $n$ large enough, we have
\begin{equation}\label{eq:balance}
|U|,|V|\geq \tfrac{1}{3}(n-1)-m \geq \tfrac{1}{4}n\,.
\end{equation}
The property that every graph $A\cup B$ with $A\in{\cal A}$ and $B\in{\cal B}$
must be cycle-free (must be a spanning tree of $K_n$) gives the
following restriction on the rectangle ${\cal R}={\cal A}\lor{\cal B}$.
\begin{clm}\label{clm:comb}
For all forests $A\in{\cal A}$ and $B\in{\cal B}$, and vertices $u\inU$ and
$v\inV$, we have $|A\cap(\{u\}\timesV)|\leq m$ and
$|B\cap(U\times \{v\})|\leq m$.
\end{clm}
That is, no forest $A\in{\cal A}$ can contain more than $m$ crossing edges
incident to one vertex in $U$, and no forest $B\in{\cal B}$ can contain more
than $m$ crossing edges incident to one vertex in~$V$.
\begin{proof}
Assume contrariwise that some vertex $u\inU$ has $l\geq m+1$
crossing edges $\{u,v_1\},\ldots,\{u,v_l\}$ in the forest $A$.
Since these edges are crossing and $u\inU$, all vertices
$v_1,\ldots,v_l$ belong to $V$. In the (fixed) spanning
star-tree $T_0=A_0\cup B_0$ (determining the partition $U\cupV$ of
vertices in $[n]\setminus\{c_1,\ldots,c_m\}$) each of these $l$ vertices is joined by an edge of
the forest $B_0$ to one of the centers $c_1,\ldots,c_m$ of stars of
$T_0$.
Since $l > m$, some two of these vertices $v_i$ and $v_j$ must be
joined in $B_0$ to the same center $c\in\{c_1,\ldots,c_m\}$. Since
${\cal R}$ is a rectangle, the graph $A\cup B_0$ must be a (spanning)
tree. But the edges $\{u,v_i\}, \{u,v_j\}$ of $A$ together with
edges $\{v_i,c\}, \{v_j,c\}$ of $B_0$ form a cycle $u\to v_i\to c\to
v_j\to u$ in $A\cup B_0$, a contradiction.
The proof of the inequality $|B\cap(U\times \{v\})|\leq m$ is the
same by using the forest $A_0$ instead of~$B_0$.
\end{proof}
So far, we only used one fixed spanning tree $T_0$ in the rectangle
${\cal R}$ to define the subgraph $U\timesV$ of $K_n$. We now use the
entire rectangle ${\cal R}={\cal A}\lor{\cal B}$ to color the \emph{edges} of $K_n$ in
red and blue. When doing this, we use the fact that the sets
$E_{{\cal A}}:=\bigcup_{A\in{\cal A}} A$ and $E_{{\cal B}}:=\bigcup_{B\in{\cal B}} B$ of edges
of $K_n$ must be disjoint:
\begin{itemize}
\item Color an edge $e\in K_n$ \emph{red} if $e\in E_{{\cal A}}$, and color $e$
\emph{blue} if $e\in E_{{\cal B}}$.
\end{itemize}
This way, the edges of every spanning tree $T\in{\cal R}$ will receive their
colors. The remaining edges of $K_n$ (if there are any) can be colored
arbitrarily.
Recall that an edge $e$ of $K_n$ is \emph{crossing} if
$e\inU\timesV$. Assume that at least half of the crossing edges is
colored in \emph{red}; otherwise, we can consider blue edges. This
assumption implies that the set $E_{\mathrm{red}}\subseteq U\timesV$ of red
crossing edges has $|E_{\mathrm{red}}|\geq \tfrac{1}{2}|U\timesV|$ edges. For
a vertex $u\in U$, the set of its \emph{good neighbors} is the set
\[
V_u=\left\{v\inV\colon \{u,v\}\in E_{\mathrm{red}}\right\}
\]
of vertices that are connected to $u$ by red crossing edges.
\Cref{clm:comb} gives the following structural restriction on star
factors covered by the rectangle~${\cal R}$.
\begin{clm}\label{clm:comb1}
For any star factor $F\in{\cal F}_{{\cal R}}$, and for any center $z$ of $F$, if $z\inU$, then $|F\cap(\{z\}\timesV_z)|\leq m$.
\end{clm}
That is, if a star factor $F$ is covered by the rectangle ${\cal R}$, then
every star of $F$ centered in some vertex $z\in U$ can only have $m$
or fewer (out of all $|V_z|$ possible) red crossing edges.
\begin{proof}
Take a star factor $F\in{\cal F}_{{\cal R}}$ having some star whose center $z$
belongs to~$U$. Since $F$ is covered by the rectangle ${\cal R}$, $F\subseteq
A\cup B$ holds for some forests $A\in{\cal A}$ and $B\in{\cal B}$. By the
definition of the edge-coloring, we have
$B\cap(\{z\}\timesV_z)=\emptyset$: all edges in $\{z\}\timesV_z$
are red, while those in $B$ are blue. So, all edges of
$F\cap(\{z\}\timesV_z)$ belong to the forest $A$, and
\cref{clm:comb} yields $|F\cap(\{z\}\timesV_z)|\leq
|A\cap(\{z\}\timesV_z)|\leq m$.
\end{proof}
We call a vertex $u$ of $K_n$ \emph{rich} if $u\inU$ and at least one
quarter of the vertices in $V$ are good neighbors of $u$, that is, if
$|V_u| \geq \tfrac{1}{4}|V|$ holds. By \cref{eq:balance}, every rich
vertex $u$ has $|V_u|\geq n/16$ good neighbors. Split the family
${\cal F}_{{\cal R}}$ of star factors covered by the rectangle ${\cal R}$ into the
family ${\cal F}^1_{{\cal R}}$ of star factors $F\in{\cal F}_{{\cal R}}$ with \emph{no} rich
center, and the family ${\cal F}^2_{{\cal R}}$ of all star factors $F\in{\cal F}_{{\cal R}}$
with \emph{at least} one rich center. We will upper-bound the number
of star factors in ${\cal F}^1_{{\cal R}}$ and in ${\cal F}^2_{{\cal R}}$ separately.
The intuition behind this splitting is that star factors
$F\in{\cal F}^1_{{\cal R}}$ have the restriction (given by \cref{lem:rich} below)
that only relatively ``few'' potential vertices of $K_n$ can be used
as centers of stars, while the restriction for the star factors
$F\in{\cal F}^2_{{\cal R}}$ (given by \cref{clm:comb1}) is that at least one of
its stars $S_z\subset F$ (centered in a rich center $z$) has
relatively ``few'' potential vertices of $K_n$ which can be taken as
leaves.
To upper-bound $|{\cal F}^1_{{\cal R}}|$, let us first show that the set
$\U^{\ast}=\left\{u\inU\colon |V_u| \geq \tfrac{1}{4}|V| \right\}$ of
all rich vertices is large enough.
\begin{clm}\label{lem:rich}
There are $|\U^{\ast}|\geq \tfrac{1}{4}|U|\geq n/16$ rich vertices.
\end{clm}
\begin{proof}
The second inequality follows from~\cref{eq:balance}. To prove the
first inequality, assume contrariwise that there are only $|\U^{\ast}| <
\tfrac{1}{4}|U|$ rich vertices in $U$. Since $|V_u| <
\tfrac{1}{4}|V|$ holds for every vertex $u \in U \setminus \U^{\ast}$,
we obtain
\begin{align*}
\tfrac{1}{2}|U\timesV|&\leq |E_{\mathrm{red}}| = \sum_{u\in\U^{\ast}}|V_u|
+ \sum_{u\inU\setminus\U^{\ast}}|V_u| < \tfrac{1}{4}|U| \cdot
|V|+|U| \cdot \tfrac{1}{4}|V| = \tfrac{1}{2}|U \times V|\,,
\end{align*}
a contradiction.
\end{proof}
Each star factor in ${\cal F}^1_{{\cal R}}$ can be constructed in the same way as
we constructed any star factor $F \in {\cal F}$ above (before
\cref{eq:all}), with the difference that centers can only be chosen
from $[n] \setminus \U^{\ast}$, not from the entire set $[n]$. Thus,
\begin{equation}\label{eq:1}
\frac{|{\cal F}^1_{{\cal R}}|}{|{\cal F}|} \leq \binom{n-|\U^{\ast}|}{m}\cdot \binom{n}{m}^{-1} \leq \mathrm{e}^{-|\U^{\ast}| \cdot m/n }=2^{-\Omega(m)}\,.
\end{equation}
Here we used \cref{lem:rich} together with the second of the two
simple inequalities holding for all $b\leq b+x < a$:
\begin{equation}\label{eq:bollobas}
\left(\frac{a-b-x}{a-x}\right)^x\leq \binom{a-x}{b}\binom{a}{b}^{-1}
\leq \left(\frac{a-b}{a}\right)^x\,.
\end{equation}
To upper bound $|{\cal F}^2_{{\cal R}}|$, we will use the restriction given by
\cref{clm:comb}. Recall that every star factor $F\in{\cal F}^2_{{\cal R}}$ has at
least one rich center. So, consider the following (nondeterministic)
procedure of constructing a star factor $F$ in ${\cal F}^2_{{\cal R}}$.
\begin{enumerate}
\item Choose a rich center $z\in\U^{\ast}$; there are at most $|\U^{\ast}|\leq
|U|\leq n$ possibilities to do this.
\item For the center $z$, do the following:
\begin{enumerate}
\item choose a subset of $i\leq m$ vertices from the set $V_z$ of
all good neighbors of $z$, and connect these vertices to $z$ by
(crossing) edges; for each $i\leq m$ there are $\binom{|V_z|}{i}$
possibilities.
\item choose a subset of $d-i$ vertices in $[n]\setminus
(V_z\cup\{z\})$ and connect them to $z$; here we have at most
$\binom{n-|V_z|-1}{d-i}\leq \binom{n-|V_z|}{d-i}$ possibilities.
\end{enumerate}
\item Choose a subset of $m-1$ distinct centers from the remaining
$n-d-1$ vertices. There are at most $\binom{n-d-1}{m-1}\leq
\binom{n-1}{m-1}=\frac{m}{n}\binom{n}{m}$ possibilities to do this.
\item Choose a partition of the remaining $n-m-d$ vertices into $m-1$
blocks of size~$d$, and connect the $i$th largest of the $m-1$
chosen centers to all vertices in the $i$th block. There are at most
$\binom{n-m-d}{d,\ldots,d}=\frac{(n-m-d)!}{d!^{m-1}}$ possibilities
to do this.
\end{enumerate}
\begin{clm}\label{clm:procedure}
Every star factor $F\in{\cal F}^2_{{\cal R}}$ can be produced by the above
procedure.
\end{clm}
\begin{proof}
Take a star factor $F\in{\cal F}_{{\cal R}}$ containing a star $S_z\subset F$
centered in a rich vertex $z\in\U^{\ast}$. The star $z$ can be picked
by Step~1 of the procedure. By \cref{clm:comb1}, the star $S_z$ can
only have $i:=|F\cap(\{z\}\times V_z)|\leq m$ good neighbors of $z$ (those
in $V_z$) as leaves, and Step~2(a) of our procedure can pick all
these $i$ leaves of $S_z$. The remaining $d-i$ leaves of the star
$S_z$ must belong to the set $[n]\setminus(V_z\cup\{z\})$. So,
Step~2(b) can pick these $d-i$ leaves of~$S_z$. Since the remaining
two steps 3 and 4 of the procedure can construct \emph{any} star factor
of $K_n\setminus S_z$, the rest of the star factor $F$ can be
constructed by these steps.
\end{proof}
The number of possibilities in Step~2 of our procedure is related to
the probability distribution
\[
h(K,n,d,i):=\prob{X=i}=\frac{\binom{K}{i}\binom{n-K}{d-i}}{\binom{n}{d}}
\]
of a hypergeometric random variable $X$: the probability of having
drawn exactly $i$ white balls, when drawing uniformly at random
without replacement $d$ times, from a vase containing $K$ white and
$n-K$ black balls. The number of possibilities in Step~2 of the
procedure (for a center $z$ picked in Step~1) is then at most $
H(|V_z|,n,d,m)\cdot \binom{n}{d}$, where
\[
H(K,n,d,m):=\prob{X\leq m}=\sum_{i=0}^m h(K,n,d,i)\,,
\]
is the probability of having drawn at most $m$ white balls. For fixed
$n,d$ and $m$, the function $H(K,n,d,m)$ is non-increasing in $K$,
implying that the maximum of $H(|V_z|,n,d,m)$ over all rich centers
$z\in\U^{\ast}$ is achieved for $K:=\min\{|V_z|\colon
z\in\U^{\ast}\}$. Hence, for every rich center $z\in\U^{\ast}$, the number of
possibilities in Step~2 is at most
\[
H(|V_z|,n,d,m)\cdot \binom{n}{d}\leq H\cdot \binom{n}{d}\,,
\]
where $H:=H(K,n,d,m)$. From the first inequality of \cref{eq:bollobas}
(applied with $x:=m$, $a:=n$ and $b:=d$) we have $\binom{n}{d} \leq
C\cdot \binom{n-m}{d}$, where $C=\left(\frac{n-m}{n-d-m}\right)^m \leq
\exp\left(\tfrac{md}{n-d-m}\right)$ is a constant since $md=O(n)$ and
$m,d=o(n)$.
Thus the total number of possibilities in all steps 1--4 and, by
\cref{clm:procedure}, also the number $|{\cal F}^2_{{\cal R}}|$ of star factors in
${\cal F}^2_{{\cal R}}$, is at most a constant times
\begin{align*}
&\underbrace{n\cdot H\cdot \binom{n-m}{d}}_{\text{Steps 1 and 2}}
\underbrace{\frac{m}{n} \binom{n}{m}}_{\text{Step 3}}
\underbrace{\frac{(n-m-d)!}{d!^{m-1}}}_{\text{Step 4}} =m\cdot
H\cdot \underbrace{\binom{n}{m}\frac{(n-m)!}{d!^m}}_{=\ |{\cal F}|} \, .
\end{align*}
Known tail inequalities for the hypergeometric distribution (see
Hoeffding~\cite{hoeffding}, or Chv\'atal~\cite{chvatal} for a direct
proof) imply that, if $m\leq (K/n-\epsilon)d$ for $\epsilon >0$, then
\begin{equation}\label{eq:tail}
H(K,n,d,m)= \prob{X\leq m}
\leq \mathrm{e}^{-2\epsilon^2 d}\,.
\end{equation}
\begin{rem}
In both papers \cite{hoeffding} and \cite{chvatal}, this upper bound
is only stated for the event $X\geq (K/n+\epsilon)d$, but using the
duality $h(K,n,d,i)=h(n-K,n,d,d-i)$ (count black balls instead of
white), the same upper bound holds also for the event $X\leq
(K/n-\epsilon)d$.
\end{rem}
In our case, $K=\min\{|V_z|\colon z\in\U^{\ast}\}\geq
\tfrac{1}{4}|V|\geq n/16$. Recall that, so far, we have only used
the conditions $(d+1)m=n$ and $m=\Theta(\sqrt{n})$ on the parameters
$m$ and $d$. We now use the last condition $m\leq d/32$. For $\epsilon
=1/32$, we then have $m\leq d/32\leq (K/n-\epsilon)d$, and
\cref{eq:tail} yields
\[
H=H(K,n,d,m)\leq \prob{X\leq d/32}\leq
\mathrm{e}^{-d/512}=2^{-\Omega(d)}\,.
\]
By taking $d:=6\sqrt{n}$ and $m:=n/(d+1)$, all three conditions on the
parameters $m$ and $d$ are fulfilled, and we obtain $|{\cal F}^2_{{\cal R}}|\leq
m|{\cal F}|\cdot 2^{-\Omega(d)}$. Together with
the upper bound \cref{eq:1}, the desired upper bound on $|{\cal F}_{{\cal R}}|$
follows:
\[
\frac{|{\cal F}_{{\cal R}}|}{|{\cal F}|}=\frac{|{\cal F}^1_{{\cal R}}|+|{\cal F}^2_{{\cal R}}|}{|{\cal F}|}\leq
2^{-\Omega(m)}+m2^{-\Omega(d)}=2^{-\Omega(\sqrt{n})}\,. \qedhere
\]
\end{proof}
\small
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,505 |
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From deserts to glaciers, tourists to Argentina have the unique experience of both extremes. Down south in the country lies Los Glaciares National Park that is full of calving glaciers, lakes and mountain views. Located in Santa Cruz, this national park is home to wildcats, ferrets, foxes, pumas and many species of bird.
The Valdes Peninsula is located at the tip of southern Argentina and sits in the Atlantic Ocean. It is a nature reserve and a World Heritage Site that has large areas of barren land, salt lakes, and a plethora of marine mammals that call it home like the elephant seal, fur seals and the Southern right whale. Orcas can be seen just off the coast and the peninsula is also the place many bird species call home.
Patagonia is a region in southern Argentina that is one of the least populated places on earth. The area is also home to the majestic mountains, glaciers and forests that can be enjoyed by those more adventurous travelers to Argentina through various tours. The area offers expedition cruises, kayak tours, hiking and trekking tours and wildlife tours for all adventure levels and abilities with duration from a few days to a few weeks.
The Cave of the Hands is located just south of the town of Perito Moreno and is a natural wonder to behold. The caves are 9,300 years old and show images of hands painted right on the rocks. Aside from the many hands are depictions of people, animals, symbols and pictures of ritualistic activities that show the history of the ancient people that left them there.
Salta is a city in the northern part of Argentina and is home to one of the world's highest railway at 14,000 feet above sea level. Here tourists to Argentina can take a 15 hour trip along the rails with amazing aerial views. Also in Salta is the Cathedral of Salta that has a statue of the Virgin Mary, circa the 16th century in its ornate interior. At night the cathedral is lit from the square in which it sits which adds to its grandeur.
Cerro San Bernando offers spectacular views of the city and travelers to Argentina can take a cable car to the top, or for the more adventurous there are stairs as well. Surrounded by mountains, this area makes for a nice change of pace from the city below.
From history to deserts to mountain ranges to forests, Argentina is a country of much variety just waiting to be explored. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,764 |
# Targeted Cyber Attacks
# Multi-staged Attacks Driven by Exploits and Malware
Aditya K Sood
Richard Enbody
Technical Editor
Peter Loshin
# Table of Contents
Cover image
Title page
Copyright
A Few Words About _Targeted Cyber Attacks_
Acknowledgments
About the Authors
Overview
Chapter 1. Introduction
References
Chapter 2. Intelligence Gathering
2.1 Intelligence Gathering Process
2.2 OSINT, CYBINT, and HUMINT
2.3 OSNs: A Case Study
References
Chapter 3. Infecting the Target
3.1 Elements Used in Incursion
3.2 Model A: Spear Phishing Attack: Malicious Attachments
3.3 Model B: Spear Phishing Attack: Embedded Malicious Links
3.4 Model C: Waterholing Attack
3.5 Model D: BYOD as Infection Carriers: USB
3.6 Model E: Direct Incursion: Network Exploitation
References
Chapter 4. System Exploitation
4.1 Modeling Exploits in Targeted Attacks
4.2 Elements Supporting System Exploitation
4.3 Defense Mechanisms and Existing Mitigations
4.4 Anatomy of Exploitation Techniques
4.5 Browser Exploitation Paradigm
4.6 Drive-By Download Attack Model
4.7 Stealth Malware Design and Tactics
References
Chapter 5. Data Exfiltration Mechanisms
5.1 Phase 1: Data Gathering Mechanisms
5.2 Phase 2: Data Transmission
References
Chapter 6. Maintaining Control and Lateral Movement
6.1 Maintaining Control
6.2 Lateral Movement and Network Reconnaissance
References
Chapter 7. Why Targeted Cyber Attacks Are Easy to Conduct?
7.1 Step 1: Building Targeted Attack Infrastructure
7.2 Step 2: Exploring or Purchasing Stolen Information About Targets
7.3 Step 3: Exploits Selection
7.4 Step 4: Malware Selection
7.5 Step 5: Initiating the Attack
7.6 Role of Freely Available Tools
References
Chapter 8. Challenges and Countermeasures
8.1 Real-Time Challenges
8.2 Countermeasures and Future Developments
References
Chapter 9. Conclusion
References
Abbreviations
# Copyright
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# A Few Words About _Targeted Cyber Attacks_
"The most complete text in targeted cyber attacks to date. Dr. Sood and Dr. Enbody are able to present the topic in an easy to read format that introduces the reader into the basics of targeted cyber attacks, how the attackers gather information about their target, what strategies are used to compromise a system, and how information is being exfiltrated out from the target systems. The book then concludes on how to build multi-layer defenses to protect against cyber attacks. In other words, the book describes the problem and presents a solution. If you are new to targeted attacks or a seasoned professional who wants to sharpen his or her skills, then this book is for you."
—Christopher Elisan, Principal Malware Scientist, RSA—The Division of EMC
"As targeted attacks become ever more prevalent, sophisticated and harmful, it's important that we understand them clearly, learn to detect them and know how to mitigate their effects. With this book, Aditya Sood and Richard Enbody have provided us with the tools to do this. Their clear, technically detailed analysis helps cut through the fear, uncertainty, doubt and hype surrounding this subject, to help us understand what's really going on and what to do about it."
—Steve Mansfield-Devine, Editor, Network Security, Computer Fraud & Security
"Dr. Aditya K Sood and Dr. Richard J Enbody have done an excellent job of taking the very complex subject of targeted attacks and breaking it down systematically so we can understand the attack techniques, tactics and procedures and build defensive mitigation strategies around them. _Targeted Cyber Attacks_ provides insights into common indicators of compromise, so your security teams can react as fast as possible and distinguish anomalous behavior from everyday normal user behavior."
—Stephan Chennette, CTO at AttackIQ, Inc.
"Sood and Enbody have taken a systematic, step by step approach to break down a pretty complex topic into bite-sized chunks that are easily digestible. They cover everything from the basics and 'need to know' of targeted attacks to the more advanced insights into the world of exploit packs, attack techniques and more."
—Dhillon Andrew Kannabhiran, Founder/Chief Executive Officer, Hack In The Box
" _Targeted Cyber Attacks_ is by far the perfect manual to dive into the dark borders of cybercrime. The book thoroughly describes the model and the mechanisms used by criminals to achieve the cyber attack to exfiltrate information or steal money. From a pen-tester's perspective, the ethical hackers will certainly find the fundamental factors to prepare a better approach to conduct high level penetration testing. Aditya and Richard deliver the secrets used by cyber-criminals to get inside the most secured companies. I learned a lot from this stunning publication authored by a BlackHat Arsenal Jedi."
—Nabil Ouchn, Founder of ToolsWatch.org and Organizer of BlackHat Arsenal
"I have always been a fan of the articles that have been published by Dr. Sood and Dr. Enbody in the past—and this book reflects that same quality of work we have come to enjoy here at CrossTalk. I found the information to be a very extensive, compelling read for anyone interested in modern cyber-attack methodologies. The information flows from chapter-to-chapter in a very logical sequence and is easily understandable by even those with limited knowledge in the cyber-security realm. I found the work to be extremely interesting and the writing style is active and enjoyable at all points. The work presented should be read by not only those in the software realm, but also the casual user who has an interest in privacy and security for themselves."
—Justin Hill, Executive Publisher of CrossTalk, the Journal of Defense Software Engineering
"Targeted attacks are one of the most virulent, dangerous cyber threats of our time. Every company, large and small, should be factoring these in as a major risk. This book brings readers up to speed, and helps them get in front of the threat, so that they can take action before they are targeted."
—Danny Bradbury, Cyber Security Journalist and Editor
# Acknowledgments
Thanks to my father, brother, and the rest of my family. I would also like to thank my mentor for providing me continuous support and inspiring me to follow this path. Thanks to my wife for all the support. This book is primarily dedicated to my new born son.
Dr. Aditya K Sood
Thanks to Aditya for the inspiration and drive to create this book. Also, thanks to my wife for yielding the time to make it happen.
Dr. Richard Enbody
Great respect to all the members of security research community for their valuable efforts to fight against cyber crime and targeted attacks!
# About the Authors
**Aditya K Sood** is a senior security researcher and consultant. Dr. Sood has research interests in malware automation and analysis, application security, secure software design, and cyber crime. He has worked on a number of projects pertaining to penetration testing, specializing in product/appliance security, networks, mobile, and web applications while serving Fortune 500 clients for IOActive, KPMG, and others. He is also a founder of SecNiche Security Labs, an independent web portal for sharing research with the security community. He has authored several papers for various magazines and journals including IEEE, Elsevier, CrossTalk, ISACA, Virus Bulletin, USENIX, and others. His work has been featured in several media outlets including Associated Press, Fox News, Guardian, Business Insider, CBC, and others. He has been an active speaker at industry conferences and presented at DEFCON, HackInTheBox, BlackHat Arsenal, RSA, Virus Bulletin, OWASP, and many others. He obtained his Ph.D. from Michigan State University in Computer Sciences.
**Dr. Richard Enbody** is an associate professor in the department of computer science and engineering. He joined the faculty in 1987, after earning his Ph.D. in computer science from the University of Minnesota. Richard received his B.A. in mathematics from Carleton College in Northfield, Minnesota, in 1976 and spent 6 years teaching high school mathematics in Vermont and New Hampshire. Richard has published research in a variety of areas but mostly in computer security and computer architecture. He holds two nanotechnology patents from his collaboration with physicists. Together with Bill Punch he published a textbook using _Python in CS1: The Practice of Computing Using Python_ (Addison-Wesley, 2010), now in its second edition. When not teaching, Richard plays hockey, squash, and canoes, as well as a host of family activities.
# Overview
This book talks about the mechanisms of targeted attacks and cyber crime. The book is written with a motive in mind to equip the users with the knowledge of targeted attacks by providing systematic and hierarchical model of different phases that happen during the process of a targeted attack. Every step in targeted attack is detailed in individual chapter explaining the insidious processes and how attackers execute successful targeted attacks. The chapter overview is presented below:
• **Chapter 1** introduces the topic of targeted attacks explaining the complete model and purpose of these attacks. This chapter lays the foundation of different phases required for successful execution of targeted attacks. Basically, the readers will get an idea of the basic model of targeted attack covering the overall idea of intelligence gathering, infecting targets, system exploitation, data exfiltration, and maintaining control over the target network. This chapter also unveils the difference between targeted attacks and advanced persistent threats.
• **Chapter 2** reveals the various types of intelligence gathering steps followed by attackers such as Open Source Intelligence (OSINT), Cyber Space Intelligence (CYBINT), and Human Intelligence (HUMINT), and how these are interconnected. This chapter discusses how attackers use Internet and openly available information about individuals and organizations from different sources such as Online Social Networks (OSNs), websites, magazines, etc., to gather intelligence about the targets before performing any kind of reconnaissance. The information collected about the targets determines the direction of targeted attacks.
• **Chapter 3** discusses the various strategies opted by attackers to infect targets to download malware and compromise the system accordingly. The chapter talks about the most widely used infection strategies used in targeted attacks such as spear phishing, waterholing, Bring-Your-Own-Device (BYOD) infection model, and direct attacks by exploiting vulnerabilities in networks and software. The sole motive behind infecting a target is to find a loophole which is exploited to plant a malware in order to take complete control over the system. Every model corresponds to a different path to trigger targeted attacks.
• **Chapter 4** unveils the complete details of system exploitation covering different types of exploits and vulnerabilities used to compromise the system. This chapter provides a hierarchical layout of different protection mechanisms designed by vendors and how these are bypassed by attackers to author successful exploits. We cover in detail about Data Execution Prevention (DEP) and Address Space Layout Randomization (ASLR) bypasses including exploit writing mechanisms such as Return-oriented Programming (ROP) and important information leakage vulnerabilities. This chapter also touches the different security solutions designed by associated companies to subvert exploit writing efforts of the attackers. In addition, details about advanced malware have been presented. We also touch base on the techniques deployed by malware for bypassing static and dynamic solutions designed by security researchers.
• **Chapter 5** covers the different data exfiltration mechanisms opted by attackers to extract data from infected systems. Exfiltration covers two sub-phases, that is, data stealing and data transmission to the attacker-controlled server. We talk about Web Injects, video and screenshot stealing, Form-grabbing, operating system information stealing, etc., and using different transmission methods such as encryption, compression over different protocol channels such as HTTP/HTTPS, Peer-to-Peer (P2P), and Internet Relay Chat (IRC). Overall, this chapter shows the sophisticated modes of data exfiltration used in targeted attacks.
• **Chapter 6** unveils the various techniques deployed by attackers to maintain control and persist in the network for a long period of time. The attackers perform network reconnaissance in the network to compromise additional systems in the network so that information can be exfiltrated at a large scale. The attacker uses custom, self-made, and public available tools such as Remote Access Toolkits (RATs) to perform different tasks such as port scanning and exploiting vulnerabilities in the target network. This step is very significant from the perspective of executing targeted attacks for a long duration of time without being detected.
• **Chapter 7** presents the importance of Crimeware-as-a-Service (CaaS) model to build components of targeted attacks through easy means. Generally, this chapter shows how easy it has become to purchase different software components and other additional elements such as compromised hosting servers by paying required amount of money. We also discuss the role of e-currency in underground market places on the Internet and processing of transactions.
• **Chapter 8** is solely dedicated to building multilayer defenses to protect against targeted attacks. The protection layers include user-centric security, end-system security, vulnerability assessment and patch management, network monitoring, and strong response plan. This chapter also presents the need and importance for building next-generation defenses to fight against ever evolving sophisticated malware.
• **Chapter 9** concludes the book by busting several myths about targeted attacks and defining the real nature of these attacks.
We refer "targeted cyber attacks" as "targeted attacks" in the context of this book.
Chapter 1
# Introduction
This chapter introduces the topic of targeted attacks explaining the complete model and purpose of these attacks. This chapter lays the foundation of different phases required for successful execution of targeted attacks. Basically, the readers will get an idea of the basic model of targeted attack covering the overall idea of intelligence gathering, infecting targets, system exploitation, data exfiltration and maintaining control over the target network. This chapter also unveils the difference between targeted attacks and advanced persistent threats.
### Keywords
Targeted Cyber Attacks; Advanced Persistent Threats; Cyber Crime; Cyber Espionage; Zero-day Exploits and Vulnerabilities
The exploitation of networks and technologies for gathering information is now commonplace on the Internet, and targeted cyber attacks are a common weapon for subverting the integrity of Internet operations. These attacks steal intellectual property, conduct cyber espionage, damage critical infrastructure and create uncertainty among users. They can achieve both tactical and strategic goals across the Internet without requiring any physical encroachment. It is clear that targeted attacks provide a tactical advantage that can play a significant role in future cyber wars.
Today, the majority of nation states are developing cyber war capabilities. Zero-day exploits ( _exploits that are designed for unknown vulnerabilities for which vendors have no awareness and no patches are available_ ) in critical software are now considered to be attack weapons that can be used to either disrupt or gain control of an opponent's network infrastructure. Government security agencies are spending millions of dollars for unknown zero-day exploits. The US government is one of the biggest buyers of these cyber weapons [1]. In fact, legitimate security companies find vulnerabilities, write zero-day exploits and sell them to governments for large amounts of money. The result is that the nation states are well equipped to launch targeted cyber attacks. In addition, even with a zero-day exploit in hand, launching a well-crafted targeted cyber attack is not cheap as substantial effort is expended in building multilayer model of attack vectors and adapting them to the target network's environment. However, targeted attacks are nation state independent and can be initiated by independent attackers around the globe.
It is easy to underestimate the impact and capabilities of targeted cyber attacks. They are capable enough to produce a kinetic effect in which command execution from a remote attacker can disrupt the physical infrastructure of a target. Examples already exist such as the Stuxnet worm [2] that targeted Industrial Control Systems (ICSs). Basically, ICS is a control system that manages and commands the behavior of a machine (equipment) used in production industries (critical infrastructure) comprising of oil, gas, water, and electricity. A well-designed cyber attack can act as a parasite that leeches critical information from the target. The value of a targeted cyber attack is directly proportional to its ability to persist and remain undetected in the target network. To succeed in the hostile environment of network resilience and counter strategies, targeted attacks require multistage attack vectors to build a cumulative attack model. On the other side, automatic breach prevention technologies are required to have the capability to assess and map the probability and effect of targeted cyber attacks.
There exists several definitions of targeted cyber attacks. We adhere to a basic definition based on the naming convention—a targeted attack is a class of dedicated attacks that aim at a specific user, company, or organization to gain access to the critical data in a stealthy manner. Targeted attacks should not be confused with broad-based attacks that are random in nature and focus on infecting and compromising large groups of users. Targeted attacks have a characteristic of discrimination and are not random in nature. It means attackers involved in targeted attacks differentiate the targets (systems/users/organizations) and wait for the appropriate opportunity to execute the attack plan. However, the term "targeted attack" is overused. We believe that the best model of a targeted attack is composed of different elements to perform insidious operations in five different phases: intelligence gathering, infecting targets, system exploitation, data exfiltration, and maintaining control. The intelligence-gathering phase consists of different information-gathering tactics used by attackers to extract data about targets. The infecting-the-target phase reveals how the targets are infected with malware through infection carriers. The system-exploitation phase shows how the target systems are fully compromised using exploits. The data-exfiltration phase is all about extracting information from the compromised systems. The maintaining-control phase shows how the attackers become persistent and remain stealthy in the network while at the same time gain access to additional number of systems in the target environment.
Some important characteristics of targeted attacks are as follows:
• Zero-day exploits against unknown vulnerabilities are used to compromise target systems so that the attacks are not easily detectable.
• Sophisticated malware families (custom coded) are used, which go unnoticed despite the presence of security solutions installed on the network and end-user systems.
• Real identity behind the attack is hidden to keep a low profile to avoid any legal problems.
• Systems having no value in the attack campaign are not infected and compromised. This in turn lowers the exposure of the attack and makes it stealthier.
• Attack is made persistent for a long period of time and operations are executed in a hidden manner.
Next is the need to understand the purpose of targeted attacks. The attackers' intentions behind launching targeted attacks are important, but targeted attacks are primarily used for earning financial gain, conducting industrial espionage, stealing intellectual property, disrupting business processes, making political statements, and subverting the operations of a nation's critical infrastructure.
Overall, targeted attacks are complex in nature because attackers have to invest substantial amount of time in selecting targets, preparing attack models and discovering zero-day vulnerabilities (known vulnerabilities can also be used). Attackers behind the targeted attacks are experts in technology, and are highly motivated to pursue intrusion campaigns. All these factors collectively provide an environment to launch targeted attacks.
A recent study on the elements of targeted attacks has shown that a sophisticated targeted attack can result in millions of dollars in losses for large organizations [3]. For Small and Medium Enterprises (SMEs), a single-targeted attack could be worth many thousands of dollars. A study conducted by Symantec [4] observed that there has been a significant increase in targeted attacks showing a jump of 42% in 2012. This number indicates that hundreds of targeted cyber attacks are happening routinely. On a similar note, Anderson et al. [5] conducted a study on measuring the cost of cyber crime, in which a framework for calculating the cyber crime cost was designed. The framework segregated the cost of cyber crime into four elements: direct costs, indirect costs, defense costs, and cost to the society. The study estimated that global law-enforcement expenditures are close to 400 million dollars worldwide to defend against cyber crime. These figures provide a glimpse of the ever increasing insecurity on the Internet.
There are other variants of targeted attacks known as Advanced Persistent Threats (APTs) [6] that exist on the Internet. Targeted attacks can be considered to be a superset of APTs [7]. Generally, APTs are highly advanced, targeted threats that use multiple attack vectors, persist (easily adaptable) in the wild and can exist undetected (stealth execution) for a long period of time. A number of researchers believe that APTs are primarily state sponsored [8], but we believe that definition to be too restrictive—an APT need not be sponsored by the state. Other researchers have explicitly mentioned that APTs are not a collection of attack vectors, rather these are well-crafted campaigns [9]. APTs are somewhat similar to targeted attacks except that APTs usually have no predefined formula or path. Adversaries behind APTs take the time to make their attack succeed. When something doesn't work, they adapt to try another approach which means they are persistent. It is not necessary for targeted attacks to be well funded, but most APT campaigns have been. The terminologies "targeted attacks" and "APTs" are often used interchangeably in the security industry to discuss the advanced sophisticated attacks that are targeted in nature. For example, several companies present APTs as state-sponsored hacker groups like Elderwood [10] that launch targeted cyber attacks against a predetermined target. The standard tactics for deception and exploitation remain the same, but attacks are differentiated based on the overall model of execution and backup support to the involved actors. Table 1.1 shows a comparison of different characteristics of targeted attacks and APTs. Generally, it's a thin line that differentiates between targeted attacks and APTs.
Table 1.1
A Comparison of Characteristics—Targeted Attacks and APTs
Tactic | Targeted Cyber Attacks | APTs
---|---|---
Deceptive | Yes | Yes
Stealthy | Yes | Yes
Exploits | Both known/unknown exploits | Primarily, zero-day exploits are used
Persistent | Depends on the design | Yes
Data exfiltration | Yes | Yes
Maintaining access | Yes: Remote Access Toolkits (RATs) | Yes: RATs
Intelligence reuse | Yes | Yes
Lateral movement | Depends on the design | Yes
Campaigns | Depends on the design | APTs are started as campaigns
State sponsored | Possible | Possible
Actors | Individual or group | Group
A number of targeted attacks have been launched recently. GhostNet was a cyber-spying operation rumored to have been launched by China and discovered by the researchers in 2009. This operation was launched against over 100 countries to gain access to critical information. GhostNet was primarily targeted against government agencies and used a RAT named "Ghost Rat" also known as "Gh0st Rat" to maintain control of the compromised systems. Operation Aurora [11], another targeted attack discovered in 2010 was launched against a wide range of large tech companies and defense contractors. The name Aurora was coined by McAfee after analyzing malicious binaries which had this name embedded in the file path configuration. The primary motive behind Operation Aurora was to gain access to the intellectual property of several organizations such as software source code and engineering documents. Organizations that were impacted by Operation Aurora include Google, Yahoo, Juniper, Morgan Stanley, Dow Chemical, and several defense contracting organizations.
Another targeted attack, Stuxnet was discovered in 2010 and targeted Iran's nuclear facilities to control their Siemens' Supervisory Control and Data Acquisition infrastructure deployed as a part of their ICSs. Stuxnet deployed a rootkit for compromising the Programmable Logic Controller (PLC) to reprogram the systems. Several security research companies and media outlets have claimed that Stuxnet was a nation state-sponsored attack under the United States campaign named "Operation Olympic Games" [12]. Duqu was a variant of Stuxnet discovered in 2011 that was designed to target global ICSs.
Symantec revealed that both the Operation Aurora and Stuxnet attacks were executed by the same attacker group which the company named as Elderwood and the overall activities have been framed in the campaign known as Elderwood Campaign. The Elderwood is considered as a group of highly skilled hackers that develop rigorous zero-day exploits to be used as weapons in targeted attacks.
Table 1.2 shows a list of targeted attacks that have occurred in the last few years (some have been labeled as APTs).
Table 1.2
Overview of History of Targeted Attacks (or APTs)
Targeted Attacks (or APTs) | Month/Year Started (Approx.) | Known/Zero-Day Vulnerability/Exploit | Targets
---|---|---|---
GhostNet (Gh0st Net) | March 2009 | Known/zero-day | Dalai Lama's Tibetan exile centers in India, London, and New York City, ministries of foreign affairs
Aurora (Hydraq) | January 2010 | Yes. Internet Explorer and Perforce software were exploited | Approx. 34 US companies such as Google, Juniper, Rackspace, etc.
Stuxnet | June 2010 | Four zero-day exploits including Shortlink (CPLINK) flaw were used including known vulnerability | Iran nuclear facilities using Siemens infrastructure
RSA breach | August 2011 | Yes. Adobe flash player | RSA and defense contractors using RSA security solutions
Duqu | September 2011 | Yes. MS Word True Type (TT) font vulnerability was used | Worldwide ICSs
Nitro attack | July 2011 | Malicious files are extracted from the attachment. Known and unknown exploits were also used | Approx. 29 chemical companies
Taidoor attack | October 2011 | Approx. nine known vulnerabilities were used | US/Taiwanese policy influencers
Flame (SkyWiper) | May 2012 | Yes. Terminal Service (TS) licensing component was exploited to generate rogue certificates | Cyber espionage in Middle East companies
There are multiple stages for launching a targeted cyber attack. Figure 1.1 shows the simple model to depict the life cycle of a targeted attack.
Figure 1.1 Generic life cycle of a targeted attack. Copyright © 2014 by Aditya K Sood and Richard Enbody.
Every stage is critical for ensuring the success of targeted attack. The stages are briefly discussed as follows:
• _Intelligence gathering and threat modeling_ : The targeted attack starts with intelligence gathering in which attackers collect information about targets from different resources, both public and private. Once the information is collected, attackers build an attack plan after modeling weaknesses associated with the target to execute attacks in stealth mode.
• _Infecting the target_ : In this phase, attacker's motive is to infect targets so that additional set of attacks can be initiated. The attackers follow different approaches such as spear phishing and waterholing attacks to coerce users to interact with malicious e-mails and web sites. The basic idea is to trick users by deploying social engineering so that malicious programs can be installed on the end-user systems.
• _System exploitation_ : In this phase, once the users are tricked to open malicious e-mails or visit infected web sites on the Internet, the malicious code exploits vulnerabilities (known and unknown) in the application software to install malicious programs on the end-user systems. The installed malware controls the various functionalities of the operating system. Once the system is compromised, the attacker can easily interact with the system and send commands remotely to perform unauthorized operations.
• _Data exfiltration_ : Data exfiltration is a process of transmitting data in a stealthy manner from the compromised system under the attacker's control. In this phase, the attackers steal sensitive data from the end-user systems. Once the system is infected with malware, the attacker has the capability to steal any data including operating system configuration details, credentials of different application software, etc. Present-day malware is well equipped to perform Man-in-the-Browser (MitB) [13] attacks to monitor, steal and exfiltrate all the critical data communicated between the end-user system and the destination server through browsers. For example, MitB attacks are heavily used in conducting banking frauds and stealing information from users that is otherwise not easily available.
• _Maintaining control and network access_ : In this phase, the primary motive of the attackers is to gain access to other systems in the network by constantly controlling the end-user system without detection. The attackers use spreading mechanisms such as USB infections, Instant Messenger (IM) infections, etc., to spread malware to additional systems. Other techniques involve spreading infections through networks by using protocols such as Remote Procedure Call (RPC), Server Message Block (SMB), Remote Desktop Protocol (RDP), and Hyper Text Transfer Protocol (HTTP). The attackers use exploits against known vulnerabilities in RPC, SMB, and RDP to compromise additional systems in the network. A number of reliable exploits are freely available for the services using the above-mentioned protocols. Successful execution of RPC/SMB/RDP exploits allows the attackers to execute arbitrary payloads and operational commands in the context of the target system. Internal web sites can also be infected with malicious HTML code hosted on enterprise servers that distribute the infections through HTTP. The attackers also harness the power of stolen data (information) from the previous phase to fingerprint network and plan accordingly to attack other systems on the network.
These five phases constitute the basic model of targeted attacks. For successful execution of targeted attacks, all phases need to succeed. As a whole, targeted attacks use standard attack vectors, but they differ significantly from traditional attacks. We will continue our discussion by concentrating on the advanced targeted cyber attacks. Each phase presented above is detailed as an individual chapter in the rest of the book.
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Chapter 2
# Intelligence Gathering
This chapter reveals the various types of intelligence gathering steps followed by attackers such as Open Source Intelligence (OSINT), Cyber Space Intelligence (CYBINT), and Human Intelligence (HUMINT), and how these are interconnected. This chapter discusses how attackers use Internet and openly available information about individuals and organizations from different sources such as Online Social Networks (OSNs), websites, magazines, etc., to gather intelligence about the targets before performing any kind of reconnaissance. The information collected about the targets determines the direction of targeted attacks.
### Keywords
Open Source Intelligence (OSINT); Cyber Intelligence (CYBINT); Human Intelligence (HUMINT); Online Social Networks (OSNs) Infection
In this chapter, we cover attack preparations, especially reconnaissance. A substantial amount of information is needed to target an individual or organization. More information creates more potential attack paths for launching an attack which increases the odds of success. This chapter delves into the information gathering tactics used by attackers to extract information to craft targeted attacks. Intelligence gathering is quite broad, so we concentrate on techniques having a direct impact on targeted attacks.
## 2.1 Intelligence Gathering Process
From an attack modeling perspective, we present intelligence gathering as the transformation of raw data into useful information.
Figure 2.1 presents a very basic intelligence gathering model covering different phases that should be completed before starting a targeted attack. The various phases are discussed below:
• _Selection and discovery_ : In this phase, attackers use publicly available resources to collect details about the target. Online Social Networks (OSNs), web sites providing individuals' identity data, government resources including reports and documents, and historical data about organizations and their employees are widely used to gather intelligence. With a data source in hand, the next step is to dig deeper to collect raw data about the target.
• _Resource extraction and mining_ : Once appropriate public resources have been located, the process of searching and collecting the data about the target starts. The data consists of details of targets including personal data, geographical location, historical data, employer data, relationships, contacts, achievements, community contributions related to research, and supportive operations, etc. A target could be an individual, group of people (such as employees in an organization and group of individuals on the social network) or a large organization, so the type of data of interest can vary. For example, individual data can be gleaned from various online portals such as social networks or web sites providing individual information, whereas information about exposed networks (targets) requires completely different operations to perform reconnaissance to detect exposed or vulnerable systems in the target network. With raw data in hand, it now must be converted into a form that is useful for the next stage of the attack. Data must be mined and correlated as appropriate for the attack.
• _Resource correlation and information processing_ : Once the raw data is collected from different resources based on the previous phases, the attackers spend time to correlate raw data before processing it. The motive is to unearth the associations of the targets and to draw a connecting line between them so that relationships can be exploited in the targeted attack. It is a critical phase because a successful attack depends on the information derived in this phase. Data correlation before final processing provides a deep insight into the target environment and behavior. The derived information is then used to develop the targeted attacks.
• _Attack modeling_ : Attack modeling refers to the process of sketching an outline of the attack by using processed information from the previous phase. In this phase, the attacker defines how the information is to be used in the targeted attack. For example, in this phase, the attacker may decide whether to use spear phishing or waterholing (refer to Chapter 3 for complete details). If the attacker is not successful in collecting sufficient behavioral patterns of an individual, but is able to get the individual's e-mail and other details, the attacker may choose to launch spear phishing instead of waterholing attack. This phase also determines how data will be exfiltrated and how to maintain control of the target systems.
Figure 2.1 Generic intelligence gathering model. Copyright © 2014 by Aditya K Sood and Richard Enbody.
## 2.2 OSINT, CYBINT, and HUMINT
Intelligence gathering can be dissected into different modes of which Open Source Intelligence (OSINT), Cyber Intelligence (CYBINT), and Human Intelligence (HUMINT) are the most viable for targeted attacks. OSINT is the process of gathering intelligence from publicly available resources (including Internet and others). OSINT should not be confused with Open Source Software (OSS) as these are both different elements. The presence of "Open Source" is used distinctively in OSINT and OSS, but refers to the same standards which are publicly available resources and software. CYBINT is the process of explicitly gaining intelligence from available resources on the Internet. CYBINT can be considered as a subset of OSINT. HUMINT is the process of gaining intelligence from humans or individuals by analyzing behavioral responses through direct interaction. OSINT, CYBINT, and HUMINT are used for both legitimate and nefarious purposes. In this book, we concentrate on intelligence gathering modes within cyber space. However, HUMINT definition can be extended with respect to Internet in which intelligence is gathered through e-communication by interacting with individuals through video sharing, messaging, etc.
The attacker's goal is to perform reconnaissance by harnessing the power of freely available information extracted using different intelligence gathering modes before executing a targeted attack. The most widely used intelligence gathering resources are the Internet, traditional mass media including magazines and newspapers, publications such as journals and conference proceedings, corporate documents, and exposed networks. Most resources post their information online so search engines such as Google and Bing can find the desired information. The attackers search and collect information about the targets by digging deep into the Internet. The information gathering process harnesses the power of search engines and custom developed tools. The extracted information helps the attackers outline a target's preferences, habits, and social culture. Intelligence gathering simply requires an Internet connection, with little additional cost incurred by the attacker.
OSINT, CYBINT, and HUMINT intelligence gathering modes can be analyzed considering following target modes:
• _Individuals_ : Here, intelligence gathering refers to querying online public resources that provide information specific to the targeted individuals. The following information is of interest to the attackers:
• Physical locations of the individuals
• OSN profiles for checking on relationships, contacts, content sharing, preferred web sites, etc.
• E-mail addresses, users' handles and aliases available on the Internet including infrastructure owned by the individual such as domain names and servers.
• Associations and historical perspective of the work performed including background details, criminal records, owned licenses, registrations, etc. This data is categorized into public data provided by official databases and private data provided by professional organizations.
• Released intelligence such as content on blogs, journal papers, news articles, and conference proceedings.
• Mobile information including phone numbers, device type, applications in use, etc.
• _Corporates and organizations_ : Here, intelligence gathering refers to the process of collecting information about target organizations. The attackers collect the required information by performing different operations as discussed:
• Determining the nature of business and work performed by target corporates and organizations to understand the market vertical.
• Fingerprinting infrastructure including IP address ranges, network peripheral devices for security and protection, deployed technologies and servers, web applications, informational web sites, etc.
• Extracting information from exposed devices on the network such as CCTV cameras, routers, and servers belonging to specific organizations.
• Mapping hierarchical information about the target organizations to understand the complete layout of employees at different layers including ranks, e-mail addresses nature of work, service lines, products, public releases, meeting, etc.
• Collecting information about the different associations including business clients and business partners.
• Extracting information out of released documents about business, marketing, financial, and technology aspects.
• Gathering information about the financial stand of the organization from financial reports, trade reports, market caps, value history, etc.
Table 2.1 shows some of the resources available online that are widely used for intelligence gathering about the targets.
Table 2.1
Intelligence Gathering Online Resources
Entity | Type of Information | Web Site
---|---|---
Electronic Data Gathering, Analysis, and Retrieval system (EDGAR) | System providing companies information pertaining to registration details, periodic reports, and other activities specific to legal aspects | <http://www.sec.gov/edgar.shtml>
Glass Door/Simply Hired | Online repositories providing information about companies work culture, jobs including salaries, employees reviews, etc. | <http://www.glassdoor.com/> | <http://www.simplyhired.com/>
Name Check/Background Check | Information about usernames and background verification of targets | <http://namechk.com/> | <http://www.advancedbackg-roundchecks.com/>
Central Operations/Robtex | Information about domain names, IP address allocation, and registrars | <http://centralops.net> | <http://www.robtex.com>
Intelius | Public records of individuals | <http://www.intelius.com/>
Jigsaw/LinkedIn | Employees information | <http://www.jigsaw.com/> | <http://www.linkedin.com/>
Spokeo | Personal information such as phone numbers | <http://www.spokeo.com/>
Hoovers | Corporate information including industry analysis | <http://www.hoovers.com/>
E-mail Sherlock | Specific e-mail patterns search | <http://www.emailsherlock.com/>
Pastebin | Underground disclosures, wiki leaks, and sensitive information disclosure from various online attacks | <http://pastebin.com/>
Github | Source codes and other software centric information | <http://www.github.com>
Google Dorks Database | Database for finding exposed network devices and servers on the Internet | <http://www.hackersforcharity.org/ghdb/> | <http://www.exploit-db.com/google-dorks/>
Google Blogosphere | Content (blog posts) released by the target | <http://www.blogspot.com>
Pentest Tools | Network information gathering tools repository | <http://pentest-tools.com>
iSeek | Target information by querying various resources and presenting in graph format | <http://iseek.com/>
Wigle | Information about WiFi networks | <https://wigle.net/>
Whois | Details about the registered domains and associated organizations | <http://www.internic.net/whois.html>
Institute of Electrical and Electronics Engineers (IEEE) | Information about research papers, journals, conferences proceedings, and associated people | <http://www.ieee.org/index.html>
Internet Assigned Numbers Authority (IANA) | Information about DNS root servers, IP address allocations, and Internet protocol resources | <https://www.iana.org/>
One can also check the OSINT toolkit [1] and repository [2] for a large collection of resources. Overall, the information gathering serves as a base for constructing a launchpad for instantiating a targeted attack. To dig deeper, we choose OSNs such as Facebook as a case study to show the level of details that can be gathered about targets.
A List of Widely Used Publicly Available Tools
_Shodan search engine_ : Shodan [3] is a search engine that has the capability to detect exposed network systems on the Internet across the globe. A simple search query on Shodan could reveal a large number of critical systems, including, but not limited to SCADA systems, control centers of power and nuclear points, routers, navigation systems, etc. Researchers use the Shodan for their research purposes, but as this search engine is publicly available, even bad guys can harness the power of this engine. The results provided by the Shodan search engine show how vulnerable the existing Internet is to exploitation by attackers. In addition, this engine has an Application Programming Interface (API) which can be used to create automated tools for data extraction. The Shodan search engine has the potential to provide interesting information about the target networks.
_Maltego_ : Maltego [4] is OSINT software, a commercial software that performs aggressive data mining and correlation on the data extracted from freely available public resources on the Internet and derives a graph of interconnected elements creating a link platform for link analysis. Maltego is broadly incorporated into reconnaissance operations as it makes the process simple and easy for the attackers. For example, Maltego can easily query OSNs, web portals, online groups, and Internet infrastructure systems to collect and mine data for analyzing relationships to targets.
_FOCA_ : Fingerprinting Organizations with Collected Archives (FOCA) [5] is a Windows-based tool used for automated fingerprinting and information gathering. This tool analyzes metadata from different file formats such as Word and PDF collected from different search engines to enumerate users, shares, e-mails addresses and other devices information. It also has the capability to search domains and target servers for specific information in addition to metadata extraction. The tool also helps to highlight data leakage vulnerabilities which expose sensitive information about the target environment.
_Search engines and hacking databases_ : Search engines such as Google and Bing are considered to be the attacker's best friend for their ability to search specific patterns resulting in a listing of exposed targets. The Google Hacking Database can find many patterns of vulnerabilities (termed Google Dorks). The Bing search engine is capable of performing Facebook graph search queries. There are additional custom-designed search engines available that perform only specific search queries.
## 2.3 OSNs: A Case Study
OSNs are quite useful for collecting information about targets. If the target's profile is not public, one effective technique is to build a fake profile with a phony identity and request the target to add the identity to his or her own circles. Let's look at some of the widely used OSNs as presented in Table 2.2 by defining the type of information revealed.
Table 2.2
OSNs as Sources of Information
OSNs | Overview | Web Sites
---|---|---
Facebook/My Space | A source to collect information about individuals and their personal resources including their relationships with other individuals | <https://www.facebook.com> | <https://myspace.com>
Twitter | A source to gather knowledge about day-to-day updates of an individual or group including what trends are going on | <https://www.twitter.com>
LinkedIn | A platform for finding information about employees and organizations | <https://www.linkedin.com>
Google+ | A source to map information about real-time sharing of content and thoughts that is useful for deriving conclusions about the target | <https://plus.google.com>
Pinterest | A platform for looking into individuals (or groups) choice of preferred content by simply pinning it on shared wall among users | <https://www.pinterest.com/>
Tumblr | Another content sharing platform | <https://www.tumblr.com/>
Instagram | A photo and video sharing platform used extensively by individuals. A good resource for extracting data | <http://instagram.com/>
Friendster | A source to collect information about college friends and old school friends. A dating web site with online gaming feature to build a fun environment | <http://www.friendster.com/>
YouTube | A widely used video sharing platform. Good for analyzing the types of videos shared by the individuals | <https://www.youtube.com>
Ning | A platform used to create small social networks encompassing smaller groups having similar choices and thoughts | <http://www.ning.com/>
Meetup | A platform to collect information about business meetings and different public events happening in a specific area | <http://www.meetup.com/>
Flickr | A centralized repository of images shared by large sections of individuals, groups, and organizations | <https://www.flickr.com/>
Let's take an example of OSINT targeting the most popular OSN, Facebook. The attacker finds that the target is an avid user of Facebook. If the attacker succeeds in adding the target to his circles (or social network) on the Facebook, the attacker can now obtain the following information about the target using features provided by Facebook. Let's have a look at the features and respective information:
• Photo sharing allows the attackers to watch for the associated persons with the target.
• Likes and comments to various posts help the attackers to understand how the target thinks.
• Contact lists and network of friends allow the attackers to determine other potential targets that can be included in the target campaign. Tagging is another way to find additional targets related to the primary target.
• Sharing of resources and videos on Facebook reveals some web sites followed by the target.
• In addition to personal information present in the target profile, the employer of the target including location may be revealed.
When combined, those details provide a lot of information. However, some users expose or share little information among peers. In those cases, there are a number of other ways attackers can exploit the integrity of Facebook by exploiting the privacy functionality to access private information of targets. Some of the supporting factors are:
• Social networks such as Facebook are complex networks which are based on interdependency among profiles. The presence of security vulnerabilities in the social networks can allow attackers to retrieve information through automated means including hijacking of accounts and accessing restricted information.
• Complex functionality such as Facebook graph search can also be exploited by attackers to retrieve information about profiles and their preferred content through data mining. Facebook graph can search for specific information about users. It is restricted to Facebook so that more people spend time on Facebook, resulting in higher advertisement revenue. FBStalker [6] is an example of a Facebook search tool that not only searches but also helps attackers to fingerprint the interests of the targets by searching using specific queries.
A number of additional techniques are used by the attackers to extract information about targets which is not otherwise easily accessible. These techniques involve social engineering attacks and less sophisticated malware to extract specific information from the targets. Social engineering attacks involve phishing and other attacks whereas malware is used to extract sensitive information. As a result, the target is tricked into providing sensitive information which is otherwise not available through OSINT.
• _Information extraction—phishing attacks_ : Phishing attacks are based on the concept of social engineering in which users are tricked to visit malicious web sites that steal information from users. Often attackers initiate a standard phishing attack by sending phishing e-mails embedded with links to malicious web sites. However, OSNs can also be used as launchpads for phishing attacks. The attackers use social engineering tricks to coerce victims to visit a malicious web site that asks for personal or specific information about the target. Organizations such as banks, educational institutions, and e-commerce web sites are often targeted by these attacks. Figure 2.2 shows a phishing e-mail attempt launched against Michigan State University (MSU) students. This phishing e-mail (refer to Figure 2.2) is a good example of a well-crafted and sophisticated phishing attempt. You can check that e-mail is sent by MSU Helpdesk but the sender's e-mail address belongs to "macicchino@email.wm.edu" which does not belong to the domain "msu.edu". The social engineering trick is the fear posed by the message in which the attacker talks about the illegal use of students' passwords by some unknown identity in accessing the MSU accounts. In addition, the attacker tricked the users to visit the embedded link "<http://cse-msuaccountreviewauthenticationforum.yolasite.com>" which redirects the users to malicious web sites asking for sensitive information. There is no doubt that some students would have fallen prey to this attack. In this case, the attacker is targeting MSU students and if successful, is gathering a specific set of information from the students which might not be available through OSINT.
• _Information extraction using malware_ : The attackers also use a certain family of malware such as Trojans (malicious programs or backdoors that reside on the end-user systems without their knowledge, steal data and upload it on the remote servers managed by the attackers) to steal specific information from the target users by infecting their systems. The malware is capable of executing a wide variety of operations including data exfiltration, launching attacks for stealing information, attacking other computers on the network, etc. The stolen information collected through malware can also be used in additional attacks such as targeted attacks. For example, fake injections (HTML/JavaScript content that is not sent by the legitimate server but injected by malware residing in the system) in a web browser allow malware to ask for specific information such as Social Security Numbers (SSNs) and e-mail addresses. Since the malware resides inside the system, it is hard to differentiate the malicious injections from normal content. Users are tricked, and as a result, sensitive information is exfiltrated to remote attackers. Such information can then be used in drafting a targeted attack.
Figure 2.2 Social engineered—phishing attack. Copyright © 2014 by Aditya K Sood and Richard Enbody.
Overall, the intelligence gathering using OSINT, CYBINT, HUMINT, and other methods is crucial for successful execution of targeted attacks. The attackers armed with extensive information about the targets have enhanced their ability to circumvent a wide variety of obstacles on the path to a successful attack.
## References
1. Benavides B. OSINT 2oolkit on the go, <<http://www.phibetaiota.net/wp-content/uploads/2013/07/2013-07-11-OSINT-2ool-Kit-On-The-Go-Bag-O-Tradecraft.pdf>> [accessed 26.09.13].
2. Hock R. Links for OSINT (Open Source Intelligence) Internet training, <<http://www.onstrat.com/osint>> [accessed 26.09.13].
3. Shodan search engine, <<http://shodanhq.com>> [accessed 25.09.13].
4. Maltego, <<https://www.paterva.com/web6/products/maltego.php>> [accessed 25.09.13].
5. FOCA, <<http://www.elevenpaths.com/lab_foca.html>> [accessed 25.09.13].
6. Mimoso M. FBStalker automates the facebook graph search mining, <<https://threatpost.com/fbstalker-automates-facebook-graph-search-data-mining/102648>> [accessed 25.09.13].
Chapter 3
# Infecting the Target
This chapter discusses the various strategies opted by attackers to infect targets to download malware and compromise the system accordingly. The chapter talks about the most widely used infection strategies used in targeted attacks such as spear phishing, waterholing, Bring Your Own Device (BYOD) infection model, and direct attacks by exploiting vulnerabilities in networks and software. The sole motive behind infecting a target is to find a loophole which is exploited to plant a malware in order to take complete control over the system. Every model corresponds to a different path to trigger targeted attacks.
### Keywords
Spear Phishing; Waterholing; Bring Your Own Device (BYOD); Targeted Campaigns
In this chapter, we discuss about the most widely used mechanisms to initiate targeted attacks. This chapter not only discusses the attack model, but also details the different vectors used to attack the targets. In the last chapter, we covered the reconnaissance and information gathering tactics used by attackers to gain insight into the target environment and behavior. We continue from there and discuss how the attackers infect the targets directly or indirectly for compromise.
We classify the attacks used for infecting the target into two ways:
1. Direct attacks, in which target network is exploited using vulnerabilities to gain access to potential critical systems or to gain critical information that can be used to launch indirect attacks, for example, exploitation of web vulnerabilities.
2. Indirect attacks, in which attackers use a number of layered attacks to accomplish the process of intrusion, for example, spear phishing and waterholing attacks.
## 3.1 Elements Used in Incursion
It is important to understand the nature of the components that are used to conduct successful targeted attacks. The most widely used and effective components in targeted attacks are discussed below:
• _Social engineering_ : Social engineering deals with the techniques of manipulating the user's psychology by exploiting trust. Social engineering often exploits a user's poor understanding of technology as users are unable to determine and fail to understand the attack patterns used in targeted attacks. Social engineering is one of the predominant components of targeted attacks because it helps to initiate the attack vector.
• _Phishing e-mails_ : The term phishing was first used in the Internet literature in 1996 by the hacker group who stole America Online (AOL) accounts' credentials. Phishing is originated from Phreaking which is considered as the science of breaking into phone networks using social engineering. A phishing attack is also based on the concept of social engineering in which users are tricked to open malicious attachments or embedded links in the e-mails. These e-mails are designed and generated to look legitimate and potentially treated as baits or hooks to trap the targets (analogous to catch fishes in sea of Internet users). Phishing attacks are the most widely used attack vehicles in targeted attacks.
• _Vulnerabilities and exploits_ : Vulnerabilities in web sites and software components, both known and unknown, can be exploited as part of an attack. The most virulent exploits are based on zero-day vulnerabilities for which details are not publicly available. They are often a component of effective targeted attacks.
• _Automated frameworks_ : Automated frameworks are used in targeted attacks to ease the burden of exploitation from the attacker's side. The emergence of automated exploit kits has resulted in sophisticated and reliable exploitation of browsers. This is because a number of exploits are bundled together in one framework that fingerprints the browser for vulnerable component before serving the exploit. As a result, only vulnerable browsers are exploited and framework does not react to browsers that are patched. In targeted attacks, Remote Access Toolkits (RATs) are deployed on infected machines to ease data theft and command execution.
• _Advanced malware_ : Based on the nature of targeted attacks, advanced malware plays a crucial role in successful campaigns. The idea behind designing advanced malware is to perform operations in a stealthy manner and to go undetected for a long period so that the attack persists. Stealthy rootkits are designed for these purposes as rootkits hide themselves under the radar where antivirus engines fail to detect them. However, less sophisticated malware has also been used in targeted attacks.
• _Persistent campaigns_ : Attackers prefer to launch small campaigns in targeted attacks for a long duration of time. The motive is to persist and to monitor the target over a period of time to collect high quality and high volumes of data at the same time.
After discussing the elements of targeted attacks, the following section talks about the different attack models used to conduct targeted attacks.
## 3.2 Model A: Spear Phishing Attack: Malicious Attachments
Spear phishing attacks have been used for a long time. It is different from a generic phishing attack because spear phishing attack is targeted against a particular individual or organization. Traditional phishing attacks have been used to capture sensitive information from the end users by duping them with social engineering tactics or simply exploiting their naïve understanding of technology. Malware authors have used phishing attacks to spread malware broadly across the Internet. In targeted attacks, spear phishing plays a very effective role. Figure 3.1 shows a very generic model of spear phishing attack that is used in the wild.
Figure 3.1 Spear phishing attack model use to launch targeted attacks. Copyright © 2014 by Aditya K Sood and Richard Enbody.
The model can be explained as follows:
• The attacker conducts spear phishing attack in which devious e-mails carrying exploit codes in the form of attachments are sent to the targets.
• Target audiences believe those e-mails to be legitimate and open the attachment.
• The exploit code executes the hidden payload and exploits vulnerability in an application component to execute specific commands in the context of end-user system.
• Once the exploit is successfully executed, malware is downloaded on the end-user system to compromise and infect it.
• The malware further downloads a RAT to take complete control of the end-user system and to attack other systems on the internal network to steal potential data.
• Once the data is stolen, different channels or tunnels are used by malware to transmit the data to offshore servers managed by the attacker.
A spear phishing attack was used against the RSA Corporation which is named as "RSA Secure ID Breach." The overall damage of this attack is not determined, but it is assumed that attackers stole Secure ID product information and number of token seeds used by several companies (organizations) such as Bank of America, Lockheed, JPMorgan Chase, Wells Fargo, and Citigroup. This indicates that RSA breach resulted in the compromise of Secure IDs (authentication tokens) of a large set of users. As a result of this, the majority of the companies had to restate the authentication tokens and RSA agreed to pay the managing cost related to customer service which was approximately 95 million dollars [1] as a whole. In RSA Attack, the attacker targeted two different batches of employees over a period of 2 days with a well-crafted phishing e-mail. The e-mail carried an XLS file containing exploit code of a then unknown vulnerability. Figure 3.2 shows how the phishing e-mail targeting RSA looked like. There could be other variants, but this one was widely distributed. The attachment carried a "2011 Recruitment Plan.xls" file embedded with an exploit code. The attachment carried an exploit code of a zero-day for Adobe Flash Player vulnerability which was later identified as CVE-2011-0609. Once the exploit was successfully executed, the malware took control of internal servers. The attacker then used a RAT named as Poison Ivy [3] to take persistent control over the target servers. The stolen information was compressed and exfiltrated from the infected system using the FTP. The complete technical analysis of the exploit used in RSA breach shows how strongly the vulnerability was exploited in the embedded SWF (Adobe file format) component in the XLS file [4].
Figure 3.2 Targeted e-mail used in RSA spear phishing e-mail. Source: Wired.com [2].
## 3.3 Model B: Spear Phishing Attack: Embedded Malicious Links
In the model discussed above, the attacker can alter the attack vector. Instead of sending malicious attachments, the attacker embeds malicious links in the spear phishing e-mails for distribution to the target audience. On clicking the link, user's browser is directed to the malicious domain running a Browser Exploit Pack (BEP) [5]. Next, the BEP fingerprints the browser details including different components such as plugins to detect any vulnerability, which can be exploited to download malware. This attack is known as a drive-by download attack in which target users are coerced to visit malicious domains through social engineering [6]. The attacker can create custom malicious domains, thus avoiding the exploitation of legitimate web sites to host malware. The custom malicious domains refer to the domains registered by attackers which are not well known and remain active for a short period of time to avoid detection. This design is mostly used for broadly distributed infections rather than targeted ones. However, modifications in the attack patterns used in drive-by download make the attack targeted in nature. The context of malware infection stays the same but the _modus operandi_ varies.
Table 3.1 shows the different types of spear phishing e-mails with attachments that have been used in the last few years to conduct targeted cyber attacks. The "Targeted E-mail Theme" shows the type of content used by attackers in the body of e-mail. The themes consist of various spheres of development including politics, social, economic, nuclear, etc.
Table 3.1
An Overview of Structure of E-mails Used in Targeted Attacks in Last Years
Targeted E-Mail Theme | Date | Subject | Filename | CVE
---|---|---|---|---
Job | Socio – Political ground | 07/25/2012 |
• Application
• Japanese manufacturing
• A Japanese document
• Human rights activists in China | • New Microsoft excel table.xls (password: 8861)
• q }(24.7.1).xls
• 240727.xls
• 8D823C0A3DADE8334B6C1974E2D6604F.xls
• Seminiar.xls
| 2012-0158
Socio - Political ground | 03/12/2012–06/12/2012 |
• TWA's speech in the meeting of United States Commission for human rights
• German chancellor again comments on Lhasa protects
• Tibetan environmental situations for the past 10 years
• Public Talk by the Dalai Lama_Conference du Dalai Lama Ottawa, Saturday, 28th April 2012
• An Urgent Appeal Co-signed by Three Tibetans
• Open Letter To President Hu | • The Speech.doc
• German Chancellor Again Comments on Lhasa Protects.doc
• Tibetan environmental statistics.xls
• Public Talk by the Dalai Lama.doc
• Appeal to Tibetans To Cease Self-Immolation.doc
• Letter.doc
| 2010-0333
Socio - Political ground | 01/06/2011 | Three big risks to China's economy in 2011 | Three big risks to China's economy in 2011.doc | 2010-3333
Socio - Political ground | 01/24/2011 | Variety Liao taking – taking political atlas Liao | AT363777.7z | 44.doc | 2010-3970
Economic situation | 03/02/2012 | Iran's oil and nuclear situation | Iran's oil and nuclear situation.xls | 2012-0754
Nuclear operations | 03/17/2011 | Japan nuclear radiation leakage and vulnerability analysis | Nuclear Radiation Exposure and Vulnerability Matrix.xls | 2011-0609
Nuclear weapon program | 04/12/2011 | Japan's nuclear reactor secret: not for energy but nuclear weapons | Japan Nuclear Weapons Program.doc | 2011-0611
Anti-trust policy | 04/08/2011 | Disentangling Industrial Policy and Competition Policy in China | Disentangling Industrial Policy and Competition Policy in China.doc | 2011-0611
Organization meeting details | 06/20/2010 | Meeting agenda | Agenda.pdf | 2010-1297
Nuclear security summit and research posture | 04/01/2010 | Research paper on nuclear posture review 2010 and upcoming Nuclear security summit | Research paper on nuclear posture review 2010.pdf | 2010-0188
Military balance in Asia | 05/04/2010 | Asian-pacific security stuff if you are interested | Assessing the Asian balance.pdf | 2010-0188
Disaster relief | 05/09/2010 | ASEM cooperation relief on Capacity Building of disaster relief | Concept paper.pdf | 2010-0188
US-Taiwan relationship | 02/24/2009 | US-Taiwan exchange program enhancement | A_Chronology_of_Milestone_events.xls US_Taiwan_Exchange_in-depth_Rev.pdf | 2009-0328
National defense law mobilization | 03/30/2010 | China and foreign military modernization | WebMemo.pdf | 2009-4324
Water contamination in Gulf | 07/06/2010 | EPA's water sampling report | Water_update_part1.pdf Water_update_part2.pdf | 2010-1297
Rumours about currency reforms | 03/24/2010 | Rumours in N Korea March 2010 | Rumours in N Korea March 2010.pdf | 2010-0188
Chinese currency | 03/23/2010 | Talking points on Chinese currency | EAIBB No. 512.pdf | 2009-4324
Trade policy | 03/23/2010 | 2010 Trade Policy Agenda | The_full_Text_of_Trade_Policy_Agenda.pdf | 2010-0188
Chinese annual plenary session | 03/18/2010 | Report on NPC 2010 | NPC Report.pdf | 2009-4324
Unmanned aircraft systems | 01/03/2010 | 2009 DOD UAS ATC Procedures | DOD_UAS_Class_D_Procedures[signed].pdf | 2008-0655
Human rights | 02/26/2009 | FW: Wolf letter to secretary Clinton regarding China human rights | 2.23.09 Sec. of State Letter.pdf | 2009-0658
NBC interview | 09/08/2009 | Asking for an interview from NBC journalist | Interview Topics.doc | Unknown
Chines defense | 01/28/2010 | Peer-Review: Assessing Chinese military transparency | Peer-Review - Assessing Chinese military transparency.pdf | 2009-4324
Asian Terrorism report | 10/13/2009 | Terrorism in Asia | RL34149.pdf | Unknown
Country threats | 01/07/2010 | Top risks of 2010 | Unknown | Unknown
Counter terrorism | 05/06/2008 | RSIS commentary 54/2009 ending the LTTE | RSIS.zip | Unknown
Anti-piracy mission | 01/13/2010 | The China's navy budding overseas presence | Wm_2752.pdf | Unknown
National security | 01/20/2010 | Road Map for Asian-Pacific Security | Road-map for Asian-Pacific Security.pdf | 2009-4324
US president secrets | 11/23/2009 | The three undisclosed secret of president Obama Tour | ObamaandAsia.pdf | 2009-1862
The model of waterholing attack discussed in the following section is a variant of drive-by download attack.
## 3.4 Model C: Waterholing Attack
A waterholing attack [7] is a term coined by RSA researchers. In general terminology, waterholes are created to attract animals to hang out around a desired area so that hunting becomes easier. Waterholes are treated as traps for hunting animals. The same concept applies to Internet users (targeted users) in which specific web sites are infected to create waterholes.
Waterholing is not a new attack vector, but a variant of a drive-by download attack in which browsers are exploited against a specific vulnerability to download malware on the end-user systems. The primary difference between the traditional drive-by download and waterholing attack is in the manner the attack is initiated. In waterholing, the attacker guesses or uses the stolen data (profiling users of the target organization) to determine the known set of web sites which are visited by the employees of target organization. In case of waterholing, spear phishing is not used as a mode of engaging users, instead the knowledge of their surfing habits is used to plant the attack. Users are not coerced through e-mails or attachments to perform a specific action rather the attacker waits for the user to visit legitimate web sites that are infected. Figure 3.3 presents a model of waterholing attack.
Figure 3.3 Waterholing attack model. Copyright © 2014 by Aditya K Sood and Richard Enbody.
The model is explained as follows:
• The attacker profiles the target users based on the Open Source Intelligence (OSINT) methods or stolen information to determine the Internet surfing habits of the users to find a set of web sites that are frequently visited by them.
• Once the attacker profiles the users, the next step is to detect vulnerabilities in those web sites (likely a subset) and exploit them to inject malicious code. As a result, users visiting those web sites will get infected with malware.
• The attacker waits for the users to visit the infected web sites so that malware is installed onto their systems using the drive-by download technique.
• Once the browser is exploited and system is infected with malware, a RAT is downloaded onto the compromised system. The RAT allows the attacker to administer the system and to attack other systems on the internal network.
• Once compromised, data is stolen and exfiltrated to some attacker-controlled system on the Internet.
The waterholing attack has been broadly deployed and a number of cases have been noticed in the last few years. The Tibetan Alliance of Chicago [8] was hacked using waterholing to attack users visiting their web site. Malicious code was placed inside an iframe (an inline frame used to load HTML/JS content from third-party server) that redirected a user's browser to a malicious domain serving a backdoor. The US Department of Labor was compromised by a waterholing attack and was shut down for a long time [9]. VOHO [10] is yet another targeted attack based on the concept of waterholing. VOHO name is coined by RSA and considered as an attack campaign in which stolen FTP account credentials are used to implant malicious code on target web sites specifically present in Washington DC and Massachusetts. The infections were triggered across multiple organizations including defense, technology, educational, and government. The attackers installed Ghost RAT Trojan on the compromised machines for further maneuvering the operations happening on the system. This attack shows how stolen information is used in the targeted attacks to initiate infections which ultimately results in compromising the target systems.
## 3.5 Model D: BYOD as Infection Carriers: USB
Universal Serial Bus (USB) devices such as thumb drives or portable hard disks are an excellent medium for carrying infections from one place to another when critical systems are not connected to the Internet. Targeted attacks against critical infrastructure such as Industrial Control Systems (ICSs) are on rise and those installations are sometimes not directly connected to the Internet. Targeted attack known as Stuxnet had the capability to spread through an infected USB device which could be plugged into critical systems for performing certain operations. ICS Computer Emergency Response Team (CERT) released a report detailing a number of cases that have happened as a result of USB infection [11].
USB devices are infected to execute code in two different modes. First, an autorun.inf file calls the hidden malware present in the USB itself. Second, rogue link files (.lnk) are generated which are linked to the malicious code. When a user clicks the shortcut, malicious code is executed.
In an ICS environment, USB devices are used to backup configurations and provide updates to the computers running in a control network environment. Generally, to manage these control systems, an individual (third-party vendor or technician) is required, who manually performs operations on the critical systems. For that, a USB is used as a storage and backup device, but at the same time it acts as a carrier if infected with malware. This is a big problem with Bring Your Own Device (BYOD) arrangements which could result in compromise of the complete network when the device is plugged in and connected to the Internet. ICS-CERT reported an issue of the same kind where a third-party vendor used an infected USB to perform updates on the turbine control systems which got infected and failed to start for 3 weeks resulting in a considerable business loss. Similarly, one of a New Jersey company's critical systems [12] were infected to take control of heating vaults and air-conditioning systems. Carelessness in handling USB devices can result in serious security compromises.
## 3.6 Model E: Direct Incursion: Network Exploitation
Exploitation of vulnerabilities in the target network is a preferred mode of direct incursion. The information gained from this process can be used in conjunction with other indirect attacks. Attackers always look forward or keep an eye on target's network infrastructure and try to detect exploitable vulnerabilities. As a result of successful exploitation, advanced malware is planted on the server side to gain complete control of the critical servers. This automatically infects all the associated systems in the network.
In recent years, several firms have been hacked as a result of the targeted attacks which resulted in a substantial loss to the business of different organizations. One notorious targeted attack was launched against Bit9 [13,14]. Attackers exploited the Internet facing web server of Bit9 and conducted a successful SQL injection that provided access to the critical systems of Bit9. SQL injection is an attack technique in which unauthorized SQL statements are injected as input values to different parameters in the web applications to manipulate the backend database. In addition to data stealing, SQL injections are used to inject malicious iframes in the Internet facing vulnerable web applications. Due to insecure deployment of web applications (web server) in Bit9, SQL injection resulted in the exposure of Bit9 certificates which were stolen and used to sign malware, specifically the kernel mode drivers. Such certificates are particularly useful because newer version of Windows requires signing of the kernel mode drivers. The attackers planted advanced malware known as HiKit [15,16], a rootkit which is advanced and persistent in nature. The motive behind installation of HiKit was to infect other Bit9 systems in the network or Bit9 customers (organizations). Once the systems were infected with HiKit, attackers deployed their own self-signed certificates and installed them into local trust stores pretending to be a Root CA. In addition, attackers also turned off the kernel driver signing process by altering the registry entries. This case shows that the exploitation of Internet facing web infrastructure could result in launching targeted attacks.
A number of infection models used in targeted attacks have been discussed. Attacker can also tune some broad-based malware spreading mechanisms such as malvertisements and social network infections and use them in collaboration with targeted attacks. Malvertisements are heavily used to fool users in believing that the content presented by the server is legitimate and they execute malicious code from the third-party domain. The attackers can also host malicious software such as fake Adobe Flash software on the infected domains to lure the victims to install malware. Social network infections result in chain infections which means, if one user in the network is infected, it can result in spreading subsequent infections to the complete network easily. Since user base is so large in social networks such as Facebook, attackers are exploiting this fact at a large scale. However, these infection mechanisms are noisy in nature which means these tactics can be easily detectable by existing defenses. In order to use these tactics in the context of targeted attacks, the attackers have to take additional efforts to build stealthy malware which can be spread under the radar without detection.
In this chapter, we have discussed about different strategies opted by attackers to engage target and initiate infections. Spear phishing and waterholing models are heavily used in targeted attacks, thereby resulting in successful infections. In majority of these models, social engineering plays a vital role in initiating the infection process. Overall, the infection models presented in this chapter provide a launchpad for the attackers to compromise the target systems.
## References
1. Schwartz N, Drew C. RSA faces angry users after breach, <<http://www.nytimes.com/2011/06/08/business/08security.html>> [accessed 28.09.13].
2. Researchers uncover RSA phishing attack, hiding in plain sight, <<http://www.wired.com/threatlevel/2011/08/how-rsa-got-hacked>> [accessed 29.09.13].
3. Poison Ivy, <<http://www.poisonivy-rat.com/index.php>> [accessed 29.09.13].
4. Branco R. Into the darkness: dissecting targeted attacks, Qualys Blog, <<https://community.qualys.com/blogs/securitylabs/2011/11/30/dissecting-targeted-attacks>> [accessed 28.09.13].
5. Kotov V, Massacci F. Anatomy of exploit kits: preliminary analysis of exploit kits as software artefacts. In: Jürjens J, Livshits B, Scandariato R, eds. _Proceedings of the 5th international conference on engineering secure software and systems (ESSoS '13)_. Berlin, Heidelberg: Springer-Verlag; 2013;181–196. In: _<http://dx.doi.org/10.1007/978-3-642-36563-8_13>_ ; 2013.
6. Cova M, Kruegel C, Vigna G. Detection and analysis of drive-by-download attacks and malicious javascript code. In: _Proceedings of the 19th international conference on world wide web (WWW '10)_. New York, NY, USA: ACM; 2010;281–290. In: _<http://dx.doi.org/10.1145/1772690.1772720>_ ; 2010.
7. RSA Blog. Lions at the watering hole—the "VOHO" affair, <<http://blogs.rsa.com/lions-at-the-watering-hole-the-voho-affair>> [accessed 29.09.13].
8. Websense Security Labs. The Tibetan alliance of Chicago hit by cyber waterholing attack, <<http://community.websense.com/blogs/securitylabs/archive/2013/08/16/tibetan-compromise.aspx>> [accessed 29.09.13].
9. Goldstein S. Department of Labor web site shut down after it's found to be hacked, NY Daily News, <<http://www.nydailynews.com/news/national/department-labor-website-hacked-shut-article-1.1332679>> [accessed 30.09.13].
10. RSA Firstwatch Blog. The VOHO campaign: an in depth analysis, RSA intelligence report, <<http://blogs.rsa.com/wp-content/uploads/VOHO_WP_FINAL_READY-FOR-Publication-09242012_AC.pdf>> [accessed 30.09.13].
11. ICS monthly monitor, <<http://ics-cert.us-cert.gov/sites/default/files/ICS-CERT_Monthly_Monitor_Oct-Dec2012_2.pdf>> [accessed 30.09.13].
12. Goodin D. Two US power plants infected with malware spread via USB drive, Ars Technica, <<http://arstechnica.com/security/2013/01/two-us-power-plants-infected-with-malware-spread-via-usb-drive>> [accessed 30.09.13].
13. Morley P. Bit9 and our customers' security, Bit9 Blog, <<https://blog.bit9.com/2013/02/08/bit9-and-our-customers-security>> [accessed 28.09.13].
14. Sverdlove H. Sharing intelligence, Bit9 Blog, <<https://blog.bit9.com/2013/02/09/sharing-intelligence>> [accessed 28.09.13].
15. Kazanciyan R. The "Hikit" rootkit: advanced and persistent attack techniques (Part 1) <<https://www.mandiant.com/blog/hikit-rootkit-advanced-persistent-attack-techniques-part-1-2>> [accessed 28.09.13].
16. Glyer C. The "Hikit" rootkit: advanced and persistent attack techniques (Part 2) <<https://www.mandiant.com/blog/hikit-rootkit-advanced-persistent-attack-techniques-part-2>> [accessed 28.09.13].
17. Caballero J, Grier C, Kreibich C, Paxson V. Measuring pay-per-install: the commoditization of malware distribution. In: _Proceedings of the 20th USENIX conference on security (SEC'11)_. Berkeley, CA, USA: USENIX Association; 2011; 13–13.
Chapter 4
# System Exploitation
This chapter unveils the complete details of system exploitation covering different types of exploits and vulnerabilities used to compromise the system. The chapter provides a hierarchical layout of different protection mechanisms designed by vendors and how these are bypassed by attackers to author successful exploits. We cover in detail about Data Execution Prevention (DEP) and Address Space Layout Randomization (ASLR) bypasses including exploit writing mechanisms such as Return-oriented Programming (ROP) and important information leakage vulnerabilities. The chapter also touches the different security solutions designed by associated companies to subvert exploit writing efforts of the attackers. In addition, details about advanced malware have been presented. We also touch base on the techniques deployed by malware for bypassing static and dynamic solutions designed by security researchers.
### Keywords
Date Execution Prevention (DEP); Address Space Layout Randomization (ASLR); Rootkits; Anti-debugging; Anti-disassembly; Code Obfuscation; Anti-virtualization; Return-oriented Programming (ROP); Info-leak Vulnerabilities
In this chapter, we present the tactics of system exploitation used by attackers in targeted attacks. In the last chapter, we presented a variety of models deployed by attackers to infect end-user systems on the fly. This chapter details the different techniques that are used by attackers to successfully exploit end-user systems to compromise and maintain access. At first, we describe the different elements that support the execution of targeted attacks.
## 4.1 Modeling Exploits in Targeted Attacks
It is crucial to understand how exploits are modeled in the context of targeted attacks. Based on the analysis of targeted attacks, we categorize exploits into two different modes:
1. _Browser-based exploits_ : This class of exploit uses browsers as a launchpad and harnesses the functionalities and features of browsers to make the exploit work. This class of exploit is hosted on remote web servers and executes when a browser opens a malicious page. Waterholing and spear phishing with embedded links are examples that use this class of exploit. As discussed earlier, Browser Exploit Packs (BEPs) are composed of browser-based exploits targeting components of browsers and third-party plug-in software such as Java and Adobe PDF/Flash.
2. _Document-based exploits_ : This class of exploit is embedded in stand-alone documents such as Word, Excel, and PDF. This class of exploit is used primarily in phishing by simply attaching the exploit file in the e-mail. The file formats support inclusion of JavaScript ActiveX Controls for executing scripts, Visual Basic for Applications (VBA) macros for executing additional code and third-party software such as Flash for interoperability and enhanced functionality. The attackers can embed the exploit code inside documents using JavaScript ActiveX Controls, VBA macros, and Flash objects. In addition, the Office document extensively relies on Dynamic Link Libraries (DLLs) for linking code at runtime. Any vulnerability found in the DLL component can directly circumvent the security model of documents and can be used to write document-based exploits.
In order to write exploits, vulnerabilities are required. The security vulnerabilities can be the result of insecure programming practices, complex codes, insecure implementation of Software Development Life Cycle (SDLC), etc. Successful exploitation of security vulnerabilities could allow attackers to gain complete access to the system. Table 4.1 shows different vulnerability classes that are used to create exploits used in targeted attacks. Vendors have developed several protection mechanisms to subvert the exploits which are discussed later in this chapter. We also cover the methodologies opted by attackers to circumvent those protections by developing advanced exploits.
Table 4.1
Vulnerability Classes Brief Description
Vulnerability Classes and Subcomponents | Brief Description
---|---
Privilege escalation/sandbox issue (unsafe reflection) | Unsafe reflection is a process of bypassing security implementation by creating arbitrary control flow paths through the target applications. The attackers' control or instantiate critical classes by supplying values to the components managing external inputs. If accepted, the attackers can easily control the flow path to bypass sandbox or escalate privileges.
Privilege escalation/sandbox issue (least privilege violation) | Attackers control the access to highly privileged resources. This occurs either due to inappropriate configuration or as a result of vulnerability such as buffer overflows. Primarily, the privilege escalation state is reached, when application fails to drop system privileges when it is essential.
Stack-based buffer overflow | Attackers have the ability to write additional data to the buffer so that stack fails to handle it, which ultimately results in overwriting of adjacent data on the stack. In general, the stack fails to perform a boundary check on the supplied buffer. Once the data is overwritten, attacker controls the return address and lands the pointer to shellcode placed in the memory.
Untrusted pointer dereference | Due to application flaw, the attacker has the capability to supply arbitrary pointer pointing to self-managed memory addresses. Application fails to detect the source of supplied value and transforms the value to a pointer and later dereferences it. Pointer dereference means that application accepts pointer for those memory locations which the application is not entitled to access. Pointer dereference with write operation could allow attackers to perform critical operations including execution of code.
Integer overflows | Integer overflow occurs when attackers store a value greater than that permitted for an integer. It results in unintended behavior which can allow premature termination or successful code execution. Primarily, integer overflows are not directly exploitable, but these bugs create a possibility of the occurrence of other vulnerabilities such as buffer overflows.
Heap-based buffer overflow | Heap overflow occurs when attacker supplied buffer is used to overwrite the data present on the heap. Basically, the data on the heap is required to be corrupted as a result of which function pointers are overwritten present in the dynamically allocated memory and attacker manages to redirect those pointers to the executable code.
Out-of-the-bounds write | A condition in the application when increment/decrement or arithmetic operations are performed on the pointer (index) that cross the given boundary of memory and pointer positions itself outside the valid memory region. As a result, the attacker gains access to other memory region through the application which could be exploited to execute code.
Out-of-the-bounds read | A similar to condition to out-of-the-bounds write, except the program reads the data outside the allocated memory to the program. It helps the attacker to read sensitive information and can be combined with other critical vulnerabilities to achieve successful exploitation.
Privilege escalation/sandbox issue (type confusion) | Type confusion vulnerabilities are specific to object oriented Java architecture. This vulnerability is caused by inappropriate access control. Type confusion vulnerabilities exploit Java static type system in which Java Virtual Machine (JVM) and byte verifier component ensures that stored object should be of given type. The confusion occurs when the application fails to determine which object type to allocate for given object.
Use-after free | Use-after free vulnerabilities occur due to the inability of the applications to release pointers once the memory is deallocated (or freed) after operations. Attackers redirect the legitimate pointers from the freed memory to the new allocated memory regions. Double free errors and memory leaks are the primary conditions for the use-after free vulnerabilities.
Process control/command injection | Process control vulnerabilities allow the attackers to either change the command or change the environment to execute commands. The vulnerable applications fail to interpret the source of the supplied data (commands) and execute the arbitrary commands from untrusted sources with high privileges. Once the command is executed, the attacker now has a flow path to run privileged commands which is not possible otherwise.
## 4.2 Elements Supporting System Exploitation
To develop efficient system exploits, attackers use sophisticated toolsets and automated exploit frameworks. The following toolset is widely used in developing targeted attacks.
### 4.2.1 Browser Exploit Packs (BEPs)
As the name suggests, a BEP is a software framework that contains exploits against vulnerabilities present in browser components and third-party software that are used in browsers. A BEP's role in both broad-based and targeted attacks is to initiate the actual infection. BEPs not only have exploits for known vulnerabilities but can also contain exploits for zero-day vulnerabilities (ones that are not publicly known). A BEP is a centralized repository of exploits that can be served, once a browser has been fingerprinted for security vulnerabilities. BEPs are completely automated so no manual intervention is required to upload and execute the exploit in vulnerable browsers. BEPs are used in conjunction with drive-by download attacks (refer to Section 4.6) in which users are coerced to visit malicious domains hosting a BEP. BEPs are well equipped with JavaScripts and fingerprinting code that can map a browser's environment as well as third-party software that enable the BEP to determine whether the target browser is running any vulnerable components that are exploitable. BEPs have reduced the workload on attackers by automating the initial steps in the targeted attacks. BEPs also have GeoIP-based fingerprinting modules that produce statistics of successful or unsuccessful infections across the Internet. This information helps the attackers deduce how the infections are progressing. Apart from targeted attacks, BEPs are also used for distribution of bots to build large-scale botnets. BEPs have turned out to be a very fruitful exploit distribution framework for attackers. Table 4.2 shows a number of BEPs that have been analyzed and released in the underground community in the last few years. The BlackHole BEP has been in existence since 2011 and is widely used.
Table 4.2
Most Widely Used BEPs List from Last 5 Years
Cool Exploit Kit | BlackHole Exploit kit | Crime Boss Exploit Pack | Crime Pack | Bleeding Life
---|---|---|---|---
CritXPack | EL Fiesta | Dragon | Styx Exploit Pack | Zombie Infection kit
JustExploit | iPack | Incognito | Impassioned Framework | Icepack
Hierarchy Exploit Pack | Grandsoft | Gong Da | Fragus Black | Eleonore Exploit Kit
Lupit Exploit Pack | LinuQ | Neosploit | Liberty | Katrin Exploit Kit
Nucsoft Exploit Pack | Nuclear | Mpack | Mushroom/Unknown | Merry Christmas
Sakura Exploit Pack | Phoenix | Papka | Open Source/MetaPack | Neutrino
Salo Exploit Kit | Safe Pack | Robopak Exploit Kit | Red Dot | Redkit
T-Iframer | Sweet Orange | Siberia Private | SofosFO aka Stamp EK | Sava/Pay0C
Zopack | Tornado | Techno | Siberia | SEO Sploit pack
Yang Pack | XPack | Whitehole | Web-attack | Unique Pack Sploit 2.1
Yes Exploit | Zero Exploit Kit | Zhi Zhu | Sibhost Exploit Pack | KaiXin
We performed a study on the different aspects of BEPs. Our research covered different exploitation tactics [1] chosen by attackers in executing exploits through BEP frameworks. The research presented the design of the BlackHole BEP and a general description of BEP behavior. In addition, we presented the mechanics of exploit distribution [2] through BEPs. The study covered the tactics used by BEPs to serve malware to end-user systems.
### 4.2.2 Zero-Day Vulnerabilities and Exploits
Zero-day vulnerability is defined as a security flaw that has not yet been disclosed to the vendor or developers. When attackers develop a successful exploit for zero-day vulnerability, it is called a zero-day exploit. It is very hard for developers and security experts to find all security flaws so attackers expect that they exist and expend substantial effort to discover security vulnerabilities. The result is an "arms race" between the attackers and the security industry.
Zero-day exploits are sold in the worldwide market [3]. A reliable zero-day exploit that allows remote code execution can be worth $100,000 or more. Because of their value in cyber warfare, even governments are purchasing zero-day exploits from legitimate security companies [4].
Zero-day exploits provide a huge benefit to attackers because security defenses are built around known exploits, so targeted attacks based on zero-day exploits can go unnoticed for a long period of time. The success of a zero-day exploit attack depends on the vulnerability window—the time between an exploit's discovery and its patch. Even a known vulnerability can have a lengthy vulnerability window, if its patch is difficult to develop. The larger the vulnerability window, the greater the chance of the attack going unnoticed—increasing its effectiveness.
Even if a patch is developed to fix vulnerability, many systems remain vulnerable, often for years. Often, a patch can be disruptive to the existing systems causing side effects and instability with damaging consequences. Large institutions can have difficulty finding all dependencies while small institutions and home users may be reluctant to install a patch because of fear of side effects. Therefore, while the value may be diminished, but still known vulnerabilities can be fruitful. As Java is ubiquitous, Java vulnerabilities are popular among attackers. Java is widely deployed in browser plug-ins and there are many Java-based applications. In addition, many users do not update the Java Runtime Environment (JRE) for several reasons. Waterholing and spear phishing attacks use embedded links to coerce browsers to visit malicious web sites embedded with malicious code that trigger Java exploits in browsers. As a result, the majority of Java-based exploits are executed through browsers.
For these reasons, both zero-day and known vulnerabilities are used in conducting targeted attacks. The software most exploited in targeted attacks are presented in Table 4.3.
Table 4.3
Most Exploited Software in Targeted Attacks
Infection Model | Major Exploited Software
---|---
Spear phishing (embedded links) |
• Browsers: Internet Explorer, Mozilla Firefox, etc.
• Oracle: JRE
• Adobe: PDF Reader/Flash Player
• Apple: QuickTime
Spear phishing (attachments) |
• Microsoft Office: MS Word, Power Point, Excel, etc.
• Adobe: PDF Reader/Flash Player
Waterholing model |
• Browsers: Internet Explorer, Mozilla Firefox, etc.
• Oracle: JRE
• Adobe: PDF Reader/Flash Player
• Apple: QuickTime
Targeted attacks that use spear phishing with attachments often use exploits against Microsoft Office components and Adobe PDF Reader or Flash. This is because files containing these exploits are easily sent as attachments in phishing e-mails. Attacks using spear phishing with embedded links prefer plugins (Java, Adobe, etc.) and browser (components) exploits that are primarily served in drive-by download attacks. An attacker's preference and the extracted target's environment information determine which attack to use and the type of exploits that will be successful in compromising the end-user systems.
Attackers have used a wide variety of exploits to compromise end-user systems. Table 4.4 shows the software-specific vulnerabilities that were exploited in targeted attack campaigns.
Table 4.4
Exploited Software in Real Targeted Attacks
Targeted Attack Campaigns | CVE Identifier | Exploited Software
---|---|---
RSA Breach | CVE-2011-0609 |
• Adobe Flash Player embedded in Microsoft XLS document
Sun Shop Campaign | CVE-2013-2423 |
• JRE component in Oracle Java SE 7
• Oracle Java SE 7
CVE-2013-1493
Nitro | CVE-2012-4681 |
• JRE component in Oracle Java SE 7 Update 6 and earlier
NetTraveler | CVE-2013-2465 |
• JRE component in Oracle Java SE 7 Update 21 and earlier
MiniDuke | CVE-2013-0422 |
• Oracle Java 7 before Update 11
• Adobe Reader and Acrobat 9.x before 9.5.4, 10.x before 10.1.6, and 11.x before 11.0.02
• Microsoft Internet Explorer 6 through 8
CVE-2013-0640
CVE-2012-4792
Central Tibetan Administration/Dalai Lama | CVE-2013-2423 |
• JRE component in Oracle Java SE 7 Update 17 and earlier
Red October Spy Campaign | CVE-2011-3544 |
• JRE component in Oracle Java SE JDK and JRE 7 and 6 Update 27 and earlier
DarkLeech Campaign | CVE-2013-0422 |
• Oracle Java 7 before Update 11
Chinese Dissidents—Council of Foreign Ministers (CFR) | CVE-2013-0422 |
• Oracle Java 7 before Update 11
• JRE component in Oracle Java SE JDK and JRE 7 and 6 Update 27 and earlier
• Microsoft Internet Explorer 8
CVE-2011-3544
CVE-2013-1288
Operation Beebus | CVE-2011-0611 |
• Adobe Flash Player before 10.2.154.27 and earlier
• Adobe Reader and Adobe Acrobat 9 before 9.1, 8 before 8.1.3, and 7 before 7.1.1
• Adobe Flash Player before 10.3.183.15 and 11.x before 11.1.102.62
CVE-2009-0927
CVE-2012-0754
Deputy Dog Operation | CVE-2013-3893 |
• Microsoft Internet Explorer 6 through 11
Sun Shop Campaign | CVE-2013-1347 |
• Microsoft Internet Explorer 8
Duqu Targeted Attack | CVE-2011-3402 |
• Microsoft Windows XP SP2 and SP3, Windows Server 2003 SP2, Windows Vista SP2, Windows Server 2008 SP2, R2, and R2 SP1, and Windows 7 Gold and SP1
Operation Beebus | CVE-2010-3333 |
• Microsoft Office XP SP3, Office 2003 SP3, Office 2007 SP2, Office 2010, Office 2004 and 2008
• Microsoft Office 2003 SP3, 2007 SP2 and SP3, and 2010 Gold and SP1
CVE-2012-0158
Stuxnet | CVE-2008-4250 |
• Microsoft Windows 2000 SP4, XP SP2 and SP3, Server 2003 SP1 and SP2, Vista Gold and SP1, Server 2008, and 7 Pre-Beta
• Windows Shell in Microsoft Windows XP SP3, Server 2003 SP2, Vista SP1 and SP2, Server 2008 SP2 and R2, and Windows 7
• Microsoft Windows XP SP2 and SP3, Windows Server 2003 SP2, Windows Vista SP1 and SP2, Windows Server 2008 Gold, SP2, and R2, and Windows 7
• Microsoft Windows XP SP3
CVE-2010-2568
CVE-2010-2729
CVE-2010-2743
Taidoor | CVE-2012-0158 |
• Microsoft Office 2003 SP3, 2007 SP2 and SP3, and 2010 Gold and SP1
A number of targeted attack campaigns as shown in Table 4.4 utilized different exploits against software provided by Microsoft, Adobe, and Oracle. One of the major reasons for large-scale exploitability of Oracle's Java, Adobe's PDF Reader/Flash and Microsoft's Internet Explorer/Office is that, these software are used in almost every organization. Recent trends have shown that Java exploits are widely deployed because of its platform independent nature, that is, its ability to run on every operating system. Java is deployed on 3 billion devices [5], which projects the kind of attack surface it provides to the attackers. The study also revealed that patches released by Oracle against known vulnerabilities are not applied immediately to 90% of devices, that is there exists a window of vulnerability exposure for at least a month after which the patch is released. That means most devices are running an outdated version of Java which put them at a high risk against exploitation.
BEPs are primarily built around exploits against Java, PDF Reader/Flash, QuickTime, etc., because these components run under browser as a part of the plug-in architecture to provide extensibility in browser's design. Plug-ins are executed in a separate process (sandbox) to prevent exploitation, but attackers are sophisticated enough to detect and work around the sandbox. Generally, sandbox is developed for restricted execution of code by providing low privilege rights and running code as low integrity processes. Sandbox is designed to deploy process-level granularity, that is, _n_ new processes are created for a code that is allowed to execute in the sandbox. On the contrary, privilege escalation vulnerabilities allow the attackers to gain high privilege rights by running code as high (or medium) integrity processes. Exploits for MS Office components are not embedded in BEPs because the majority of MS Office installations are stand-alone components.
## 4.3 Defense Mechanisms and Existing Mitigations
The attackers have designed robust exploitation tactics to create reliable exploits even if defense mechanisms are deployed. However, Microsoft has made creating exploits an increasingly difficult task for the attackers. The details presented in Table 4.5 come from discussions of Microsoft's recent Enhanced Mitigation Experience Toolkit (EMET) [6] about the latest exploit mitigation techniques. EMET is also provided as a stand-alone package that can be installed on different Windows versions to dynamically deploy the protection measures.
Table 4.5
Exploit Mitigation Tactics Provided by Microsoft
Mitigations | Descriptions
---|---
Structure Exception Handling Overwrite Protection (SEHOP) | Subverts the stack buffer overflows that use exception handlers. SEHOP validates the exception record chain to detect the corrupted entries. Also termed as SafeSEH in which exception handler is registered during compile time.
DEP | Marks the stack and heap memory locations as nonexecutable from where payload (shellcode) is executed. DEP is available in both software and hardware forms.
Heap spray allocations | Prevents heap spray attacks by preallocating some commonly and widely used pages which result in failing of EIP on the memory pages
Null page allocations | Prevents null pointer dereferences by allocating memory page (virtual page) at address 0 using NtAllocateVirtualMemory function
Canaries/GS | Protects stack metadata by placing canaries (random unguessable values) on stack to implement boundary checks for local variables.
RtlHeap Safe Unlinking | Protects heap metadata by adding 16 bit cookie with arbitrary value to heap header which is verified when a heap block is unlinked
ASLR | Prevents generic ROP attacks (more details about ROP is discussed later in this chapter) by simply randomizing the addresses of different modules loaded in the process address space
Export Address Table (EAT) Filtering | Blocks shellcode execution while calling exported functions from the loaded modules in the target process address space. Filtering scans the calling code and provides read/write access based on the calling functions
Bottom-Up Randomization | Randomizes the entropy of 8 bits base address allocated for stack and heap memory regions
ROP mitigations |
• Monitors incoming calls to LoadLibrary API
• Verifies the integrity of stack against executable area used for ROP gadgets
• Validates critical functions called using CALL instruction rather RET
• Detects if stack has been pivoted or not
• Simulates call execution flow checks of called functions in ROP gadgets
Deep hooks | Protects critical APIs provided by the Microsoft OS
Anti-detours | Subverts the detours used by attackers during inline hooking
Banned functions | Blocks certain set of critical functions provided as a part of APIs
In the following section, we present a hierarchical layout of the development of exploit writing tactics and how attackers have found ways to bypass existing anti-exploit defenses.
## 4.4 Anatomy of Exploitation Techniques
The attacker can use different exploitation mechanisms to compromise present-day operating systems and browsers by executing arbitrary code against different vulnerabilities. In our discussion, we focus on exploitation tactics that are widely used to develop exploits used in targeted attacks.
### 4.4.1 Return-to-Libc Attacks
Return-to-Libc (R2L) [7] is an exploitation mechanism used by attackers to successfully exploit buffer overflow vulnerabilities in a system that has either enabled a nonexecutable stack or used mitigation techniques such as Data Execution Prevention (DEP) [8,9]. DEP can be enforced in both hardware and software depending on the design. The applications compiled with DEP protection make the target stack nonexecutable. The R2L exploit technique differs from the traditional buffer overflow exploitation strategy. The basic buffer exploitation tactic changes a routine's return address to a new memory location controlled by the attacker, traditionally on the stack. Shellcode is placed on the stack, so the redirected return address causes a shell (privileged) to be executed. The traditional exploitation tactics fail because the stack does not allow the execution of arbitrary code. The R2L attack allows the attackers to rewrite the return address with a function name provided by the library. Instead of using shellcode on the stack the attackers use existing functions (or other code) in the library. The function executables provided by libc do not reside on any stack and are independent of nonexecutable memory address constraints. As a result, stack protections such as DEP are easily bypassed. R2L has some constraints. First, only functions provided in the libc library can be called; no additional functions can be executed. Second, functions are invoked one after another, thereby making the code to be executed as a straight line. If developers remove desired functions from the libc, it becomes hard to execute a successful R2L attack.
To defend against R2L exploits, canaries [10] have been introduced to prevent return address smashing through buffer overflows. Canary is an arbitrary value that is unguessable by the attacker and generated by the compiler to detect buffer overflow attacks. Canary values can be generated using null terminators, entropy (randomly), and random XOR operations. Canaries are embedded during compilation of an application between the return address and the buffer (local variables) to be overflowed. The R2L exploits require the overwriting of the return address which is possible only when canaries are overwritten. Canaries are checked as a part of the return protocol. Canaries can protect against buffer overflow attacks that require overwriting of return addresses. Canaries fail to provide any protection against similar vulnerabilities such as format string, heap overflows and indirect pointer overwrites. Stack Guard [11] and ProPolice [12] are the two software solutions that implement the concept of canary to prevent buffer overflow attacks.
### 4.4.2 Return-oriented Programming
The exploitation mechanism that allows the arbitrary execution of code to exploit buffer overflow vulnerabilities in the heap and stack without overwriting the return address is called Return-oriented Programming (ROP) [13–16]. ROP is an injection-less exploitation technique in which attackers control the execution flow by triggering arbitrary behavior in the vulnerable program. ROP provides Turing completeness [17] without any active code injection. ROP helps attackers to generate an arbitrary program behavior by creating a Turing-complete ROP gadget (explained later) set. Turing-complete in the context of ROP attack means that different instructions (ROP gadgets) can be used to simulate the same computation behavior as legitimate instructions. ROP is based on the concept of R2L attacks, but it is modified significantly to fight against deployed mitigation tactics. ROP attacks are executed reliably even if the DEP protection is enabled. ROP constructs the attack code by harnessing the power of existing instructions and chaining them together to build a code execution flow path. The attackers have to construct ROP gadgets which are defined as carefully selected instruction sequences ending with RET instructions to achieve arbitrary code execution in the context of a vulnerable application. In other words, any useful instruction followed by a RET instruction is good for building ROP gadgets. Researchers have indexed the most widely used ROP gadgets in different software programs to reduce the manual labor of creating and finding ROP gadgets [18] every time a new vulnerability is discovered. This approach adds flexibility that eases the development of new exploits.
When ROP gadgets are used to derive chains to build a code execution flow path, it is called ROP chaining. The utilized instructions can be present inside the application code or libraries depending on the design. ROP attacks are extensively used in attack scenarios where code injection (additional instructions) is not possible; attackers build sequences containing gadget addresses with required data arguments and link them to achieve code execution. As discussed earlier, ROP attacks overcome the constraints posed by the R2L or traditional buffer overflow exploitation model. First, ROP attacks do not require any explicit code to be injected in writable memory. Second, ROP attacks are not dependent on the type of functions available in the libc or any other library including the code segment that is mapped to the address space of a vulnerable program.
For successful writing of ROP exploits, a number of conditions are necessary. ROP attacks first place the attack payload (shellcode) in the nonexecutable region in memory and then make that memory region executable. Generally, ROP exploits require control of both the program counter and the stack pointer. Controlling program counter allows the attackers to execute the first gadget in the ROP chain whereas the stack pointer supports the subsequent execution of instructions through RET in order to transfer control to the next gadgets. The attacker has to wisely choose based on the vulnerability where the ROP payload is injected either in memory area or in stack. If the ROP payload is injected in memory, it becomes essential to adjust the stack pointer at the beginning of ROP payload. How is this possible? It is possible through stack pivoting [13,19] which can be classified as a fake stack created by the attacker in the vulnerable program's address space to place his own specific set of instructions, thereby forcing the target system to use the fake stack. This approach lets the attackers control the execution flow because return behavior can be mapped from the fake stack. Stack pivoting is used in memory corruption vulnerabilities that occur due to heap overflows. Once the heap is controlled, stack pivoting helps immensely in controlling the program execution flow. Listing 4.1 shows an example of stack pivoting.
Listing 4.1 A simple stack pivoting code.
In the listing, the contents of EAX register are moved to the ESP. The XCHG instruction exchanges the content of the ESP and EAX registers. Finally, the ADD instruction is used to add an extra set of bytes to the content present in the ESP. The stack pivot allows attackers to control stack pointer to execute return-oriented code without using RET instructions.
Based on the above discussion ROP attacks can be classified into two types:
1. _ROP with RET_ : This class of ROP attacks uses ROP gadgets that are accompanied with RET instructions at the end.
2. _ROP with indirect control transfer instruction_ : This class of ROP attacks uses ROP gadgets that use replicas of RET instructions that provide the same functionality as RET [20] or a set of instructions that provide behavior similar to a RET instruction. For example, instead of a RET instruction, an Update-Load-Branch instruction set is used to simulate the same behavior on x86. Basically, the instruction sequences end with JMP *y where y points to a POP x; JMP x sequence. Researchers call this tactic as Bring Your Own Pop Jump (BYOPJ) method.
The above categories of ROP show that ROP exploits can be written with or without RET instructions. To generalize the behavior of an ROP attack, a simple buffer overflow attack work model is presented below:
• The authorized user executes a target program and the process address space is mapped in the memory region by loading requisite libraries to export desired functions for program execution. At the same time, user-supplied input is saved in a local buffer on the stack.
• The attacker supplies arbitrary data as input to the program to overflow the buffer to overwrite the return address in order to corrupt the memory. The arbitrary data comprises of an input buffer and ROP gadgets containing libc instructions.
• The original return address is overwritten with the address of the first ROP gadget. The program execution continues until return instruction is encountered which is already present at the end of the ROP gadget.
• The processor executes the return instruction and transfers control to the next ROP gadget containing a set of instructions. This process is continued to hijack the execution flow in order to bypass DEP.
• Finally, the ROP attack uses a _VirtualAlloc_ function to allocate a memory region with write and executable permissions. The payload (shellcode) is placed in this region and program flow is redirected to this memory region for reliable execution of the malicious payload. Several other Application Programming Interfaces (APIs) help bypass the DEP. The _HeapCreate_ function allows creation of heap objects that are private and used by the calling process. The _SetProcessDEPPolicy_ functions allow disabling the DEP policy of the current process when set to OPTIN or OPTOUT modes. The _VirtualProtect_ function helps to change the access protection level of a specific memory page that is marked as PAGE_READ_EXECUTE. Finally, the _WriteProcessMemory_ function helps to place shellcode at a memory location with execute permissions. Of course, using any of these depends on the availability of these APIs in the given operating system.
Overall, ROP attacks are used to exploit systems that are configured with exploit mitigations such as GS cookies, DEP, and SEHOP.
### 4.4.3 Attacking DEP and ASLR
To protect against ROP attacks, the Address Space Layout Randomization (ASLR) [21,22] technique has been implemented in which memory addresses used by target process are randomized and allocated in a dynamic manner, thereby removing the ability to find memory addresses statically. It means the ASLR randomly allocates the base address of the stack, heap and shared memory regions every time a new process is executed in the system. Robust ASLR means that addresses of both library code and application instructions are randomized. Traditional exploits are easy to craft because majority of the applications have base addresses defined during linking time, that is, the base address is fixed. To protect against attackers capitalizing on static addresses, Position Independent Executable (PIE) support is provided by OS vendors to compile the binaries without fixed base addresses. Both ASLR and PIE are considered to be a strong defense mechanism against ROP exploits and other traditional exploitation tactics in addition to DEP.
Given time, attackers have developed responses. Exploit writing and finding mitigation bypasses are like an arms race. The attackers are very intelligent and can find mitigation bypasses to reliably exploit target systems. At the same time, researchers develop similar tactics, but their motive is to enhance the defenses by responsibly disclosing the security flaws and mitigation bypasses to the concerned vendors. It is possible to write effective exploits that can reliably bypass the DEP and ASLR. For that, attackers require two different sets of vulnerabilities. First, memory corruption vulnerability (buffer overflows heap or stack, use-after free and others) is required in the target software that allows an attacker to bypass DEP reliably. As mentioned earlier, R2L and ROP attacks are quite successful at this. Second, to bypass ASLR, attackers require additional vulnerability to leak memory address that can be used directly to execute the code. Both vulnerabilities are chained together to trigger a successful exploit on fully defended Windows system. A few examples of exploits of this category are discussed as follows:
• Browser-based Just-in-Time (JiT) exploits are authored to bypass ASLR and DEP by targeting third-party plug-ins such as Adobe Flash. JiT exploitation is based on manipulating the behavior of JiT compilation because JiT compiler programs cannot be executed in nonexecutable memory and DEP cannot be enforced. For example, an exploit targeting memory corruption vulnerability in the Flash JiT compiler and memory leakage due to Dictionary objects [23] is an example of this type exploitation. JiT exploits implement heap spraying (discussed later on) of JavaScript or ActionScript. Exploits against Adobe PDF frequently use this technique.
• JiT spraying is also used to design hybrid attacks that involve embedding of third-party software in other frameworks to exploit the weaknesses in design and deployed policies. Document-based attacks [24] are based on this paradigm. A number of successful Microsoft Office document exploits have been created by embedding Flash player objects (malicious SWF file) to exploit vulnerability in Flash. Due to interdependency and complex software design, even if the MS Office software (Excel, Word, etc.) is fully patched, the vulnerability in third-party embedded components still allows reliable exploitation. This approach enables attackers to utilize the design flaws in a Flash sandbox to collect environment information. The targeted attack against RSA utilized vulnerability in Flash and exploited it by embedding the Flash player object in an Excel file. Windows Management Instrumentation (WMI) and Component Object Model (COM) objects can also be used to design document-based exploits.
Researchers have also looked into the feasibility of attacking the random number generator [25] used to calculate the addresses applied by ASLR before the target process is actually started in the system. For reliable calculation of the randomization values, it is required that the process should be initiated by the attacker in the system to increase the probability of bypassing the ASLR.
Is there a possibility of bypassing ASLR and DEP without ROP and JiT? Yes, there is. Researchers have also designed an attack technique known as Got-it-from-Table (GIFT) [26] that bypasses ASLR and DEP without the use of ROP and JiT and this technique reliably works against use-after free or virtual table overflow vulnerabilities even without heap sprays. This technique uses the virtual function pointer table of WOW64sharedinformation, a member of the __KUSER_SHARED_DATA_ structure, to harness the power of the _LdrHotPatchRoutine_ to make the exploit work by creating fake pointers. The _KUSER_SHARED_DATA_ structure is also known as SharedUserData. It is a shared memory area which contains critical data structure for Windows that is used for issuing system calls, getting operating system information, processor features, time zones, etc. The SharedUserData is mapped into the address space of every process having predictable memory regions. The data structure also holds the SystemCall stub which has SYSENTER instruction that is used to switch the control mode from userland to kernelland. _LdrHotPatchRoutine_ is a Windows built-in function provided as a part of hotpatching support (process of applying patches in the memory on the fly). The _LdrHotPatchRoutine_ function can load any DLL from a Universal Naming Convention (UNC) path provided as value to the first parameter. The overall idea is that the vulnerability allows the attacker to provide a UNC path of the malicious DLL by calling _LdrHotPatchRoutine_ and arbitrary code can be executed in the context of the operating system.
Windows-on-Windows (WOW) is basically an emulated environment used in Windows OS for backward compatibility. This allows the Windows 64-bit (x64) versions to run 32-bit (x86) code. Although certain conditions are required for GIFT to work, this type of exploitation technique shows research is advancing.
### 4.4.4 Digging Inside Info Leak Vulnerabilities
Successful exploitation of vulnerabilities to attack DEP also requires presence of information leak vulnerabilities in order to bypass the ASLR. However, information leak vulnerabilities are also desired in other exploitation scenarios in addition to ASLR. The idea is to use the leaked address of base modules or kernel memory to map the memory contents (addresses) to be used by the exploits. In other words, info leak vulnerabilities are frequently used with ROP programming to exploit systems that use mitigations such as GS cookie, SEHOP, DEP, and ASLR. On the whole, Table 4.6 shows the different type of vulnerabilities that can be exploited to leak memory addresses [27].
Table 4.6
Info Leaking Vulnerabilities Description
Info Leaking Vulnerabilities | Description
---|---
Stack overflow—partial overwrite | Overwriting target partially and returning an info leaking gadget to perform write operations on the heap
Heap overflows—overwriting string.length field and final NULL [w]char |
• Reading the entire address space by overwriting the first few bytes of the string on the allocated heap
• Reading string boundaries by overwriting the last character of [w]char on the allocated heap
Heap massaging—overflowing the JS string and object placed after heap buffer
Type confusion | Replacing the freed memory block with attacker controlled object of same size
User after free conversion (read and write operations, controlling pointers, on demand function pointers and vtables) | Forcing pointer to reference the attacker generated fake objects and further controlling uninitialized variables.
Use-after free conversion/application-specific vulnerabilities | Utilizing use-after free scenarios to combine with application layer attacks such as Universal Cross-site Scripting (UXSS)
## 4.5 Browser Exploitation Paradigm
Browser exploitation has become the de-facto standard for spreading malware across large swaths of the Internet. One reason is that browser exploitation allows stealthy execution of arbitrary code without the user's knowledge. Spear phishing and waterholing attacks that coerce the user to visit infected web sites are based on the reliable exploitation of browsers. In addition, from a user's perspective, the browser is the window to the Internet, so it is a perfect choice for attackers wishing to distribute malicious code. Attackers are well acquainted with this fact and exploit browsers (or their components) to successfully download malware onto users' systems. Drive-by download attacks get their name from infection when the user merely visits a page: simply "driving by" and malware is downloaded. The visited sites host automated exploits frameworks that fingerprint the browsers and then serve an appropriate browser exploit. The exploit executes a hidden payload that in turn downloads malware onto the end-user system.
An earlier study on Internet infections revealed that millions of URLs [28] on the Internet including search engine queries serve drive-by download attacks. The study further concluded that drive-by download attacks are also dependent on user surfing habits and their inability to understand how malware infects them. It has also been determined that drive-by downloads are triggered using Malware Distribution Networks (MDNs) [29] which are a large set of compromised web sites serving exploits in an automated manner. BEPs significantly ease the process of browser exploitation as discussed earlier. BEPs are sold as crimeware services [30] in the underground market which is an effective business approach for earning money. In addition, drive-by download attacks have given birth to an Exploit-as-a-Service (EaaS) [31] model in which browser exploits including zero-days are sold in the underground market. The motive behind building CaaS including EaaS is to provide easy access to crimeware. So in order to understand the insidious details of browser exploitation, it is imperative to dissect the drive-by download attack model.
## 4.6 Drive-By Download Attack Model
The drive-by download [32] attack revolves around the BEP that executes the exploit to initiate the malware download. Attackers must install a BEP somewhere and then drive users to that site. In fact, attackers can even install a single exploit on the compromised domain as it depends on the design and how many exploits attackers want to deploy. We will look into all three parts: installation of the BEP, techniques to drive users to the BEP, and how the BEP works. Along the way, attackers need to compromise sites to install hidden iframes (inline frames used to embed another child HTML document in the parent web page) that drive users to the BEP. Also, we look at ways to use social engineering to drive users to those hidden iframes. Figure 4.1 shows the high level overview of the drive-by download attack.
Figure 4.1 Hierarchical steps in drive-by download attack. Copyright © 2014 by Aditya K Sood and Richard Enbody.
A drive-by download attack requires a web site to host the BEP that will infect a user's computer. A good candidate is a high-traffic site or a site that users are directed to from a high-traffic site. In order to install the BEP, the site must first be compromised. When a user visits the site after BEP installation, the BEP will exploit vulnerability in the user's browser to infect the user's computer. We will discuss more details about BEPs in the following sections.
### 4.6.1 Compromising a Web Site/Domain
The first step in drive-by download attack is to compromise a web site or to control a domain so that the attacker can install exploitation framework on it. As described in last chapter, the users have to visit the compromised web site in order to get infected. Since the World Wide Web (WWW) is interconnected and resources and content are shared across different web sites, attackers can follow different methods to compromise domains/web sites.
• _Exploiting web vulnerabilities_ : An attacker can exploit vulnerabilities in a web site to gain the ability to perform remote command execution so a BEP can be installed or redirection code can be injected. Useful vulnerabilities include Cross-site Scripting (XSS), SQL injections, file uploading, Cross-site File Uploading (CSFU), and others. The attack scenarios are explained below:
• XSS allows the attacker to inject scripts from third-party domains. This attack is broadly classified as reflective and persistent. There also exists another XSS variant which is Document Object Model (DOM) based. In this, the XSS payload is executed by manipulating the DOM environment. DOM-based XSS is out of scope in the context of ongoing discussion. We continue with the other two variants. Reflective XSS injections execute the scripts from third-party domains when a user opens an embedded link in the browser. Reflective XSS payloads can be distributed via e-mail or any communication mechanism where messages are exchanged. These attacks are considered to be nonpersistent in nature. Persistent XSS is considered as more destructive because XSS payloads are stored in the application (or databases) and execute every time the user opens the application in the browser. Persistent XSS attacks can also be tied with SQL injections to launch hybrid attacks.
• SQL injections enable the attackers to modify databases on the fly. It means SQL injections facilitate attackers to inject malicious code in the databases that is persistent. Once the malicious code is stored in the databases, the application retrieves the code every time it dynamically queries the compromised database. For example, encoded iframe payloads can easily be uploaded in the databases by simply executing an "INSERT" query. Compromised databases treat the malicious code as raw code, but when an application retrieves it in the browser, it gets executed and downloads malware on the end-user system.
• CSFU allows the attacker to upload files on behalf of active users without their knowledge. Exploitation of these vulnerabilities allows attackers to inject illegitimate content (HTML/JS) that is used for initiating the drive-by download attacks. Even the existence of simple file uploading vulnerabities has severe impacts. If the applications or web sites are vulnerable to these attacks, the attackers can easily upload remote management shells such as c99 (PHP) [33] to take control of the compromised account on the server which eventually results in managing the virtual hosts. Basically, c99 shell is also treated as backdoor which is uploaded on web servers to gain complete access.
• _Compromising hosting servers:_ Attackers can directly control the hosting servers by exploiting vulnerabilities in the hosting software. Shared hosting is also called "Virtual Hosting" [34] in which multiple hosts are present on the same server sharing the same IP addresses that map to different domain names by simply creating the virtual entries in the configuration file of web servers. Virtual hosting is different than dedicated hosting because the latter has only a single domain name configured for a dedicated IP address. Shared hosting is a popular target because exploitation of vulnerability in one host on the server could impact the state of an entire cluster. For example, there are toolsets available called "automated iframe injectors" in the underground market that allow the attackers to inject all the potential virtual hosts with arbitrary code such as malicious iframes (inline frames that load malicious HTML document that loads malware). Think about the fact that vulnerability present in one host (web site) can seriously impact the security posture of other hosts present on the server. There are many ways to compromise hosting servers:
• The attacker can upload a remote management shell onto a hosting server to control the server which can be used to infect the hosts with BEPs.
• The attacker can compromise a help-support application which has a wealth of information about the tickets raised by the users. This information can be mined for clues about potential vulnerabilities.
• The attacker can use credentials stolen from infected machines across the Internet to gain access to servers and web sites. For example, if a user logins into his/her FTP/SSH account on the hosting server, the malware can steal that information and transmit it to the Command and Control (C&C) server managed by the attacker. In this way, the attacker can take control of the hosting server from anywhere on the Internet. Considering the mass infection process, the attackers need to inject a large set of target hosts for which the complete process is required to be automated. For example, the attackers automate the process of injecting hosts (virtual directories) through FTP access (stolen earlier) by iterating over the directories present in the users' accounts. In this way, a large number of hosts can be infected as attackers perform less manual labor.
• _Infecting Content Delivery Networks_ : Co-opting a Content Delivery Networks (CDN) is particularly useful because these networks deliver content to a large number of web sites across the Internet. One use of CDNs is the delivery of ads so a malicious advertisement (malvertisement) can be distributed via CDN. Alternatively, an attacker can modify the JavaScript that a site is using to interact with a CDN opening a pathway into the CDN. Since a number of legitimate companies such as security companies' explicitly harness the functionality of CDN they may be vulnerable to infections. A number of cases have been observed in the wild in which webpages utilizing the functionality of CDNs have been infected [35].
### 4.6.2 Infecting a Web Site
An infected web site contains malicious code in the form of HTML that manipulates the browser to perform illegitimate actions. This malicious code is usually placed in the interactive frames known as iframes. An iframe is an _inline_ frame that is used by browsers to embed an HTML document within another HTML document. For example, the ads you see on a web page are often embedded in iframes: a web page provides an iframe to an advertiser who fetches content from elsewhere to display. From an attacker's viewpoint, an iframe is particularly attractive because it can execute JavaScript, that is, it is a powerful and flexible HTML element. In addition, an iframe can be sized to be 0×0 so that it effectively isn't displayed while doing nefarious things. In the context of drive-by downloads, its primary use is to stealthily direct a user from the current page to a malicious page hosting a BEP. A basic representation of iframe is shown in Listing 4.2.
Listing 4.2 Example of a normal and obfuscated iframe.
The "I-1" and "I-2" representations of iframe codes are basic. The "I-3" represents the obfuscated iframe code which means the iframe code is scrambled so that it is not easy to interpret it. Basically, attackers use the obfuscated iframes to deploy malicious content. The "I-3" representation is an outcome of running Dean Edward's packer on "I-2". The packer applied additional JavaScript codes with eval functions to scramble the source of iframe by following simple compression rules. However, when both "I-2" and "I-3" are placed in HTML web page execute the same behavior. The packer uses additional JavaScript functions and performs string manipulation accordingly by retaining the execution path intact.
Once a web site is infected, an iframe has the ability to perform following operations:
• _Redirect_ : The attacker injects code into the target web site to redirect users to a malicious domain. A hidden iframe is popular because it can execute code. One approach is for the iframe to simply load malware from a malicious domain and execute it in the user's browser. If that isn't feasible or is blocked, an iframe can be used to redirect the browser to a malicious domain hosting a BEP. The iframe may be obfuscated to hide its intent.
• _Exploit:_ The attacker deploys an automated exploit framework such as BEP on the malicious domain. A malicious iframe can load specific exploit directly from the BEP.
The attacker can also perform server side or client side redirects [36,37] to coerce a browser to connect to a malicious domain. Generally, iframes used in targeted attacks are obfuscated, so that code interpretation becomes hard and web site scanning services fail to detect the malicious activity.
### 4.6.3 Hosting BEPs and Distributing Links
BEP frameworks are written primarily in PHP and can be easily deployed on the web servers controlled by the attackers. To purchase a BEP, the underground market is place to look. Since it is a part of CaaS model, BEPs are sold as web applications. The attacker does not have to spend additional time in configuring and deploying the BEP on the server. All the fingerprinting and exploit service modules are automated. BEPs have a built-in functionality of fingerprinting end-user environment and serving appropriate exploits on the fly without user's knowledge. In addition, BEPs have a well-constructed system for traffic analysis and producing stats of successful infections among targets.
### 4.6.4 Fingerprinting the User Environment
Let's now look at what happens when a user visits a targeted web site. On visiting the web site, a malicious iframe in the web page loads the web page hosted on a malicious domain running a BEP. In this way, the exploitation process starts. It begins with the BEP gathering information about the browser's environment—a process called fingerprinting. The two most widely used fingerprinting techniques are discussed next.
_User-agent strings:_ A user-agent is defined as a client that acts on the behalf of a user to interact with server-side software to communicate using a specific protocol. Primarily, user agent is defined in the context of browsers. In the context of the Internet, browsers act as user agents that communicate with web servers to provide content to the users. Every browser is configured to send a user-agent string that is received by a server during the negotiation process and based on that, the server determines the content. The user-agent string reveals interesting information about the end user's browser environment, including but not limited to: browser type, version number, installed plug-ins, etc. One of the legitimate uses of sending the user-agent string is that the server provides different types of content after extracting the environment information from the user-agent strings. This process helps to handle the complexities of software mismatch and optimization issues. Although, it is also easy to spoof the user-agent string as many browsers provide a configuration option to update the user-agent string which is used to communicate with end-point servers. Spoofing of user-agent strings helps the security researchers and analyst to fool end-point servers. For example, a user can easily configure a Google Chrome browser user-agent string and force Mozilla Firefox to send an updated user-agent string to the end-point servers (web servers). This is possible in Mozilla Firefox by adding _general.useragent.override_ code in the browser's configuration (about: config) page. At the same time, user still surfs the Internet using the Mozilla Firefox browser. On similar front, researchers can manipulate the user-agent of the testing system to receive live malware from attacker's server for analysis by spoofing the identity of the client. User agents can be manipulated in every browser [38] with an ease.
The BEP can use the information sent in a user-agent string to fingerprint several interesting details of the browser's environment. One example of extracting information from a user-agent string is shown below: first the string followed by what it reveals. Figure 4.2 shows how the information in user-agent string is interpreted on the server side.
Figure 4.2 User-agent information.
Note that this protocol is useful for proper and robust communication on the Internet, but attackers exploit this functionality to determine the environment of end-user systems and thereby fingerprinting the information to serve an appropriate exploit.
_JavaScript/DOM objects_ : JavaScript/DOM objects are also used by BEPs to fingerprint browsers' environments. Basically, the navigator object [10] is used to extract information about browsers, operating system, plug-ins, etc., as described here [39]. The majority of the BEPs use open source code for detecting plug-ins in the browsers known as PluginDetect [40]. Figure 4.3 below provides a glimpse of the type of information revealed through a navigator object.
Figure 4.3 Plugin Checker against security vulnerabilities.
Mozilla and Qualys provide online services named as Plugin Checker [41] and BrowserCheck [42] to test the security of plug-ins based on fingerprinting their installed versions in the browser. Figure 4.4 shows how the plug-in detection code used in a number of targeted campaigns that extracts information about plug-ins installed in the victims' browsers running on the end-user machines.
Figure 4.4 Interpreting plug-ins information—PluginDetect in action. Copyright © 2014 by Aditya K Sood and Richard Enbody.
The purpose of fingerprinting is to determine if there are any vulnerable components in the user's browser. The BEP checks the list of components against its collection of exploits. If there is a match, the appropriate exploit is supplied.
### 4.6.5 Attacking Heap—Model of Exploitation
Browser-based exploits frequently manipulate the behavior of the browser's heap using predefined sequences of JavaScript objects in order to reliably execute code. The idea is to control the heap prior to the execution of heap corruption vulnerabilities. Since the heap is controlled by the attacker, it becomes easy to launch the exploit without any complications. Two different techniques are used to efficiently exploit heap corruption vulnerabilities that are discussed as below:
### 4.6.6 Heap Spraying
Heap spraying is a stage of browser exploitation where a payload is placed in a browser's heap. This technique exploits the fact that it is possible to predict heap locations (addresses). The idea is to fill chunks of heap memory with payload before taking control of the Extended Instruction Pointer (EIP). The heap is allocated in the form of blocks and the JavaScript engine stores the allocated strings to new blocks. A specific size of memory is allocated to JavaScript strings containing NOP sled (also known as NOP ramp) and shellcode (payload) and in most cases the specific address range points to a NOP sled. NOP stands for No operation. It is an assembly instruction (x86 programming) which does not perform any operation when placed in the code. NOP sled is a collection of NOP instructions placed in the memory to delay the execution in the scenarios where the target address is unknown. The instruction pointer moves forward instruction-by-instruction until it reaches the target code. When the return address pointer is overwritten with an address controlled by the attacker, the pointer lands on the NOP sled leading to the execution of the attacker supplied payload. Basically, the heap exploitation takes the following steps:
• First, create what is known as a nop_sled (NOP sled), a block of NOP instructions with a Unicode encoding which is an industry standard of representing the strings that is understood by the software application (browser, etc.). The "\0×90" represents the NOP instruction and the Unicode encoding of NOP instruction is "%u90". The nop_sled is appended to the payload and written to the heap in the form of JavaScript strings mapping to a new block of memory. Spraying the heap by filling chunks of memory with payload results in payload at predictable addresses.
• Next, a browser's vulnerability in a component (such as a plug-in) is exploited to alter the execution flow to jump into the heap. A standard buffer overflow is used to overwrite the EIP. It is usually possible to predict an appropriate EIP value that will land execution within the NOPs which will "execute" until the payload (usually shellcode) is encountered.
• The shellcode then spawns a process to download and execute malware. By downloading within a spawned process, the malware can be hidden from the user (and the browser).
A simple structure of heap spray exploit is shown in Listing 4.3 that covers the details discussed above.
Listing 4.3 Heap spraying example in action.
### 4.6.7 Heap Feng Shui/Heap Massage
Heap Feng Shui [43] is an advanced version of heap spraying in which heap blocks are controlled and rearranged to redirect the execution flow to the attacker's supplied payload or shellcode. This technique is based on the fact that the heap allocator is deterministic. It means that the attacker can easily control or hijack the heap layout by executing operations to manage the memory allocations on the heap. The overall idea is to determine and set the heap state before exploiting vulnerability in the target component. Heap Feng Shui allows the attacker to allocate and free the heap memory (blocks/chunks) as needed. Heap Feng Shui helps attackers in scenarios where exploitation of vulnerabilities requires overwriting of locations to determine the path to shellcode. Researchers have discussed a number of techniques [44] to write exploits for heap corruption vulnerabilities. Some well-known techniques are: patching all calls to virtual functions when modules are loaded into the memory, verifying the state of Structure Exception Handler (SEH) when hooking is performed and hooking universal function pointers. Researchers further advanced [45] the Heap Feng Shui technique to attack JavaScript interpreters by smashing the stack by positioning the function pointers reliably. This technique involves five basic steps. (1) defragment the heap, (2) create holes in the heap, (3) arrange blocks around the holes, (4) allocate and overflow the heap, and (5) execute a jump to the shellcode.
After successful drive-by download attack, the target systems are compromised and additional operations are performed to exfiltrate data in a stealthy manner.
## 4.7 Stealth Malware Design and Tactics
We have discussed about the different exploit writing techniques used by the attackers to exploit target systems. As we know, once the loophole is generated after exploiting the target, malware is downloaded onto the target systems. It is essential to understand the basic details of how the advanced malware is designed because these are the agents that are required to remain active in the target system and perform operations in a hidden manner. To understand the stealth malware, Joanna [46] has provided taxonomy detailing a simple but effective classification based on the modifications (system compromise point of view) performed by malware in the userland and kernelland space of the operating system. The taxonomy classifies the malware in following types:
• Type 0 Malware does not perform any modification to the userland and kernelland. However, this type of malware can perform malicious operations of its own.
• Type 1 Malware modifies the constant resources in the operating system such as code sections in the memory present in both userland and kernelland. Code obfuscation techniques are used to design this type of malware. Refer to the Section 4.7.2 for understanding different code obfuscation techniques.
• Type 2 Malware modifies the dynamic resources in the operating system such as data sections in both the userland and kernelland. These resources are also modified by the operating system itself during program execution, but the malware executes malicious code in a timely fashion thereby going unnoticed.
• Type 3 Malware has the capability to control the complete operating system. Basically, this type of malware uses virtualization technology to achieve the purpose.
Understanding the malware design helps to dissect the low level details of system compromise.
The attackers use advanced techniques such as hooking, anti-debugging, anti-virtualization, and code obfuscation to act stealthy and at the same time subvert the static and dynamic analysis conducted by the researchers. In the following few sections, we take a look into how the malware is equipped with robust design to fight against detection mechanisms used by the researchers. The majority of rootkits, bots, or other malware families use the hooking to manipulate the internal structures of operating system to hijack and steal information on the fly without being detected. Let's discuss briefly what hooking is all about.
### 4.7.1 Hooking
It is a process of intercepting the legitimate function calls in the operating system and replacing them with arbitrary function calls through an injected hook code to augment the behavior of built-in structures and modules. Basically, hooking is extensively used by the different operating systems to support the operational functionalities such as patching. With hooking, it becomes easy to update the different functions of operating systems on the fly. The attacker started harnessing the power of hooking for nefarious purposes and that's how advanced malware came to exist. Hooking is designed to work in both userland and kernelland space of operating system, provided sufficient conditions are met. For example, kernel level hooking allows the hook code to be placed in the ring 0 so that kernel functions can be altered. Table 4.7 presents brief details of the different hooking techniques used for circumventing the userland applications and designing malware.
Table 4.7
Userland Hooking Techniques
Userland Hooking Technique | Details
---|---
Import Address Table (IAT) hooking | IAT is generated when a program requires functions from another library by importing them into its own virtual address space. The attacker injects hook code in address space of the specific program and replaces the target function with the malicious one in the memory by manipulating the pointer to the IAT. As a result, the program executes the malicious function as opposed to the legitimate one.
Inline hooking | Inline hooking allows the attackers to overwrite the first few bytes of target function and place a jump instruction that redirects the execution flow to the attacker controlled function. As a result, malicious function is executed whenever the target function is loaded in the memory. After the hook code is executed, the control is transferred back to the target function to retain the normal execution flow.
DLL injections | It is a process of injecting malicious DLLs in the virtual address space of the target program to execute arbitrary code in the system. It allows the attackers to alter the behavior of the process and associated components on the fly. DLL injection is implemented in following ways:
• By specifying the DLL in the registry entry through AppInit_DLLs in HKLM hive.
• Using SetWindowsHookEx API
• Using CreateRemoteThread and WriteProcessMemory with VirtualAlloc APIs
Table 4.8 talks about the brief details about the different kernelland hooking techniques.
Table 4.8
Kernelland Hooking Techniques
Kernelland Hooking Technique | Details
---|---
System Service Descriptor Table (SSDT) Hooking | SSDT is used for dispatching system function calls from userland to kernelland. SSDT contains information about the additional service tables such as Service Dispatch Table (SDT) and System Service Parameter Table (SSPT). SSDT contains a pointer to SDT and SSPT. SDT is indexed by predefined system call number to map the function address in the memory. SSPT shows the number of bytes required to load that system call. The basic idea is to alter the function pointer dedicated to a specific system call in SDT so that different system call (to be hooked) can be referenced in the memory.
Interrupt Descriptor Table (IDT) Hooking | IDT is processor-specific and primarily used for handling and dispatching interrupts to transfer control from software to hardware and vice versa during handling of events. IDT contains descriptors (task, trap, and interrupt) that directly map to the interrupt vectors. IDT hooking is an art of manipulating the interrupt descriptor by redirecting entry point of the descriptor to the attacker controlled location in the memory to execute arbitrary code.
Direct Kernel Object Manipulation (DKOM) | In this technique, a device driver program is installed in the system that directly modifies the kernel objects specified in the memory through Object Manager. DKOM allows the system tokens manipulation and hiding of network ports, processes, device drivers, etc., in the operating system.
I/O Request Packets (IRPs) Function Table Hooking | IRP packets are used by userland application to communicate with kernelland drivers. The basic idea is to hook the IRP handler function tables belonging to other device drivers running in the kernelland and then executing malicious device driver when IRP event is triggered. The IRP entries in the IRP handler function table is hooked and redirected to nefarious functions by altering the associated pointers.
Some of these techniques are specific to certain operating systems. There have been continuous changes in the new versions of the operating systems that have rendered some of these techniques useless. Hooking techniques that worked in Windows XP might not work in Windows 8. For example, Stuxnet [47] implemented IRP function table hooking to infect latest version of Windows as opposed to SSDT and IDT hooking. However, with few modifications, these age-old techniques still provide a workaround to implement hooks. In addition, techniques like inline hooking in the userland space are universal and work on the majority of operating systems. A complete catalog of hooking techniques with a perspective of backdooring the system has been released in the Uninformed journal [48] that provides insightful information to understand how backdoors are placed in Windows operating system. Hooking can impact any component of the operating system and that's why it is widely used in the majority of malware families.
### 4.7.2 Bypassing Static and Dynamic Detection Mechanisms
The following methods and procedures are used by malware authors to subvert static and dynamic analysis techniques used by security solutions to detect and prevent malware. The attackers often use these methods to design robust malware to withstand the solutions built by the security vendors.
• _Code obfuscation/anti-disassembly_ : It is an art of protecting the malware code from reverse engineering efforts such as disassembly, which means the malware binary is transformed from machine language to assembly instructions for understanding the design of malware. However, to circumvent reverse engineering efforts, the attackers perform code obfuscation or use anti-disassembly techniques. Code obfuscation is a process of altering instructions by keeping the execution intact whereas anti-disassembly is a process to trick disassemblers to result in wrong disassembly. As a result, disassembly of malware code with obfuscation produces an assembly code which is hard to decipher. There are a number of code obfuscation models [49,50] that are used by the attackers to transform the layout of the code. An overview is presented in Table 4.9.
Table 4.9
Different Code Obfuscation Models
Obfuscation Models | Details | Detection Artifacts
---|---|---
Encryption | The malware binary is encrypted with the same algorithm and different key for every new infection. The encrypted malware binary is decrypted in the memory during runtime and executes itself. | The decryption engine does not change so signature-based detection model works.
Polymorphic encryption | It is a model of obfuscation in which attacker is capable of generating a new variant of decryption engine through code mutation with every new infection. | Emulated environment allows fetching unencrypted code in memory and signature-based detection works.
Metamorphic encryption | It is a model of obfuscation in which both the encryption and decryption engines change through self-mutation. | Memory snapshots or swap space analysis. Possible if morphing engine is mapped.
The attacker uses techniques listed in Table 4.10 to implement obfuscation and anti-disassembly models.
• _Anti-debugging_ : It is an art of embedding certain specific code snippets in the malware binary to detect and prevent debugging (manually or automated) performed on the malware binary by the researchers. Peter Ferrie [51] has already detailed on a number of anti-debugging techniques used by malware and it is one of the best references to dig deeper into the world of anti-debugging. For the purpose of this book, we cover the basic details of anti-debugging.
Table 4.10
Techniques Supporting Anti-disassembly and Code Obfuscation
Code Obfuscation/Anti-disassembly Techniques | Details
---|---
Code substitution | Replacing the original instructions with the other set of instructions that are equivalent in nature and trigger the same behavior as primary instructions while execution.
Code reassignment/register swapping | A specific set of instructions are reassigned by simply switching registers.
Embedding garbage code | Simply placing the garbage code in the malware binary which embeds additional instructions on disassembly. For example, No Operation (NOP instruction) is heavily used as garbage code.
Code randomization | A specific set of instructions in subroutines are reordered to generate random code with same functionality.
Code integration | The malware integrate itself in the target program through decompilation and generate a new version of target program.
Code transposition | Reordering the set of instructions in the original code using conditional branching and independent instructions.
Return address change | The default return address of the function is changed in conjunction with garbage code.
Program flow transformation using conditional jumps | Additional conditional jumps are placed in code to manipulate the programs execution flow.
The scope of this book does not permit us to provide complete details of above-mentioned technique, but it provides a substantial overview of the anti-debugging methods. For understanding the details of APIs referred in Table 4.11, we suggest the readers to refer Microsoft Developer Network (MSDN) documents.
• _Anti-virtualization_ : It is a process [52,53] of detecting the virtualization environment through various resources available in the system. The basic idea of embedding anti-emulation code is to equip the malware to detect whether it is executed in the virtualized environment or vice versa. As a result, the malware completely behaves differently to avoid tracing its operational functionalities. Since running malware inside virtualized environment has become a prerequisite to analyze malware behavior, a number of malware families deploy this technique to trick automated solutions. Table 4.12 lists a number of methods by which virtualized environment is detected. We refer to virtualized environment that is built using VMware and VirtualPC solutions.
Table 4.11
Widely Used Anti-debugging Techniques
Anti-debugging Technique | Details
---|---
Debugger—Win32 API calls | Presence of following calls in the IAT suggests the existence of debugger
• IsDebuggerPresent
• CheckRemoteDebuggerPresent
• NtSetDebugFilterState
• DbgSetDebugFilterState
• NtSetInformationThread
• OutputDebugStringA
• OutputDebugStringW
• RtlQueryProcessDebugInformation
• WaitForDebugEvent
• ContinueDebugEvent
System querying information with specific arguments and parameters—API calls | Presence and usage of following functions with NtQuerySystemInformation in the IAT table suggest the existence of debugger:
• SystemKernelDebuggerInformation
• ProcessDebugFlags Class
• ProcessDebugObjectHandle Class
• ProcessDebugPort
Hardware/software breakpoint based | The attackers can deploy code to check if breakpoints have been placed in the malware code during runtime.
Device names/Window handles | Presence of debugger device names can also be checked for the presence of debugger in the system. In addition, FindWindow can be used to get a handle of opened debugger window.
Table 4.12
Widely Used Anti-Emulation Techniques
Anti-Emulation Technique | Details
---|---
Hardware identifiers | Hardware-based information can be used to detect the presence of virtualized system. For example, virtual machines have only specific set of identifiers in Media Access Control (MAC) address. VMWare also uses IN instruction as a part of I/O mechanism which can also be used to detect it.
Registry based | Virtual machines have unique set of information in the registry. For example, registry can be queried for Virtual Machine Communication Interface (VMCI) and disk identifiers for detecting virtual machines.
CPU instructions | Location of Local Descriptor Table (LDT), Global Descriptor Table (GDT), Interrupt Descriptor Table (IDT) and Task Register (TR) is queried in the memory using SLDT, SGDT, SIDT and STR instructions respectively. The "S" refers to store as these instructions store the contents of respective tables.
Exception driven | A simple check to verify whether exceptions are generated inside virtual systems when invalid instructions are executed. For example, VirtualPC does not trigger any exceptions when invalid instruction is executed in it.
Other techniques | Additional set of procedures can also be used for detecting virtualized environments as follows:
• Process identifiers
• Keyboard detection and mouse activity
• Hypervisor detection
• BIOS identifiers
• Scanning for Window handles
• Pipe Names
• DLLs
• System manufacturer name scanning
• Shared folders
Other researchers [54,55] have also talked about building static and dynamic analysis system to detect anti-debugging, anti-emulation, etc., in the malware. Our earlier study released [56,57] in Virus Bulletin magazine also talks about the malware strategies to defend against static and dynamic solutions. On the real note, there are a number of identifiers (not limited to Table 4.12) that can be used to detect the VMware and VirtualPC environments and malware authors have an edge to deploy any identifier.
In this chapter, we have discussed potential attack techniques and insidious vulnerabilities that are used by attackers to perform system exploitation. Attackers utilize the techniques discussed above to craft reliable browser-based and software-centric exploits to subvert the deployed protections for compromising the target systems. Browser-based heap overflow exploitation is frequently used by attackers to infect target systems. Browser-based exploits are served through BEPs that ease out the initial infection and exploitation processes. Exploits transmitted as attachments are created to bypass generic security protections. It is very crucial to understand malware design and various types of anti-debugging, anti-virtualization, and code obfuscation techniques. Overall, targeted attacks effectiveness depend on the use of robust exploits and stealthy downloading of advanced malicious code on the target systems.
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Chapter 5
# Data Exfiltration Mechanisms
This chapter covers the different data exfiltration mechanisms opted by attackers to extract data from infected systems. Exfiltration covers two sub-phases, that is, data stealing and data transmission to the attacker-controlled server. We talk about Web Injects, video and screenshot stealing, Form-grabbing, operating system information stealing etc., and using different transmission methods such as encryption, compression over different protocol channels such as HTTP/HTTPS, Peer-to-Peer (P2P), and Internet Relay Chat (IRC). Overall, this chapter shows the sophisticated modes of data exfiltration used in targeted attacks.
### Keywords
Data Exfiltration; Information Stealing; Data Gathering
In this chapter, we talk about the different data exfiltration methods used by attackers to steal sensitive information from the infected end-user machines running in home and corporate networks. Data exfiltration is a process in which attackers extract critical information from compromised computers by opting stealthy tactics. We live in a data-driven world where the value of data is substantial so exfiltration of data has a significant impact. As a result, most malware is designed for stealing data—sometimes different malware specializes in targeting different data. Several cases are discussed below:
• Malware families such as Zeus, SpyEye, ICE IX, and Citadel are designed specifically for launching financial attacks, so data such as account information, and credit card data are the high-value targets. These malware families are also treated as toolkits because anyone can purchase these toolkits and deploy them accordingly for nefarious purposes. These toolkits are well designed and consist of different modular components which can be configured as per the requirement. The components include malicious agent (bot), management software, and additional information stealing components. Basically, these toolkits are designed to build botnets on the Internet.
• Flame malware [1] was used in targeted attack that launched against Middle Eastern countries for the purpose of cyber espionage. Flame author exploited the MD5 collision (chosen-prefix) flaw in Microsoft Terminal Services Licensing certificate to sign the components of the Flame which looked legitimate to have been originated from Microsoft Licensing authority. Flame searched and attacked the Windows update mechanism in local networks over HTTP by registering itself as a fake proxy server which sent the malicious updates signed with a fake certificate. Flame searched specifically for intellectual property such as AutoCAD designs or sensitive PDF files.
• Malware such as Stuxnet and Duqu are designed to extract information from Supervisory Control and Data Acquisition (SCADA) Systems to attack Industrial Control Services (ICS) explicitly. ICS systems have become prone to targeted attacks because these systems are not air-gapped completely. Air-gapped is a security principle which requires that critical controls systems should never interact with other systems on the Internet. In recent times, it has been noticed that ICS systems have not been secured properly and a number of critical systems can be accessed in an unauthorized manner on the Internet which makes the ICS systems vulnerable to compromises.
• Remote Access Toolkits (RATs) such as Ghost Rat [2] are designed to extract data from internal infected hosts in the enterprise network and used to control the complete network.
As you can see, most malware families hunt for data. Sometimes the data is the final target; other times data is gathered for use in a future attack.
Data exfiltration basically has two phases: (1) gathering data followed by (2) transmission of information from the end-user system. We discuss both phases next.
## 5.1 Phase 1: Data Gathering Mechanisms
In this phase, we discuss the most widely used data gathering techniques by different malware families.
_Custom keylogging_ : Keylogging (software-centric) is an old school technique in which all the keystrokes entered by a user on a keyboard are captured and stored in a temporary file on the system. Once collection is complete, the file is transferred to the Command and Control (C&C) panel for later use. C&C is a management software that is used to control and instruct the malicious agents (bots) deployed on the infected systems after exploiting vulnerability. The stolen data is transmitted to C&C panels which is stored in the database for later use. The attacker sends all types of commands to the malicious agent through C&C panel. It is considered as the epicenter of the botnets.
Keyloggers fall into two categories. First is a system-level keylogger in which all keystrokes are recorded without any filtering. Second is an application-specific keylogger in which keystrokes are recorded only for specific applications such as browsers. Application-specific keylogging makes keystroke analysis easier. One of the drawbacks of keylogging is that it produces a plethora of garbage data which must be sifted through to extract useful data.
Keylogging typically has three steps. The first step monitors whether the target process is active or not. If the target process is not active and remains dormant then unload the component. Second, if the target process is active, inject a hook [3] that sniffs keyboard traffic. Third, trigger a callback function upon successful hook and write the sniffed data to a temporary log file.
Keylogging works by hooking the keyboard message-handling mechanism. For example, in Windows, the WH_KEYBOARD procedure is hooked to capture the traffic exchanged during WM_KEYDOWN and WM_KEYUP messages. In addition, the KeyboardProc callback function is used to process the keyboard messages such as WM_KEYDOWN and WM_KEYUP. The keyboard hooks can be used in conjunction with mouse procedures such as WH_MOUSE and the MouseProc callback function to deploy more filtering in the application.
Keylogging is an old technique, but it is still used in the wild for data theft.
_Form-grabbing_ : Form-grabbing is a technique that allows the attackers to instruct the installed malware to extract the HTTP POST data from the browsers. Form-grabbing steals data in HTTP POST requests issued by a browser when an HTML form is submitted to a server. As the name suggests, the basic idea is to grab the content of the form. How is that possible? The malware implements the concept of hooking which allows the malware to intercept data flowing between components of the browser and the destination server. Since the malware is residing in the infected machine, it is easy for the malware to monitor browser activities and operations performed within the browser. As a result, malware can hook into browser components to grab the desired data and then release the hook to allow the components to continue their normal operation. This technique has a number of benefits. First, the data stolen through Form-grabbing is well formatted and contains detailed information with identifiers and names with associated values. Second, it provides an edge over keylogging because it reduces the amount of work that the malware author has to perform to clean and sift through the data to remove unwanted elements. This attack is stealthy so the user has no idea it is happening. Figure 5.1 shows an output taken from the ICE 1X botnet C&C panel showing how the malware author sees form-grabbed data from his server.
Figure 5.1 Form-grabbed data extracted by ICE 1X Bot from infected machine.
A simple example of Form-grabbing includes the hooking of the pr_write function exported by nspr4.dll as this library is used by the Firefox browser to handle the data that should be written to a buffer before transmitting to the server. The malware author can easily hook this pr_write function and read all the data from the buffer in memory. As a result, the malware can grab all the HTTP POST data written to the buffer when the form is submitted. A proof-of-concept of this technique can be found online [4].
_Web Injects_ : Web Injects is a technique of injecting unauthorized web content into incoming HTTP response data. The malware waits for the user to open the target web site in the browser against which the Web Injects payloads are to be injected. The Web Injects payloads are embedded in the binary configuration file created during the build time for the malware. Basically, Web Injects is an Man-in-the-Browser (MitB) attack in which malware intercepts communication on the channel between a browser and a web server. The malware manipulates the communication channel to and from the browser—most commonly on the response. This technique is used in the scenarios where critical information such as Social Security Number (SSN) or Personal Identification Number (PIN) is otherwise not easily available. Using this technique, the malware coerces the user to provide the critical information by exploiting the user's inability to determine that the added content in web page is illegitimate. Web Injects can be used to perform the following:
• Inject HTML content such as HTML input fields in forms to request a user's SSN, ATM PIN, etc.
• Inject unauthorized pop-ups to request sensitive information (SSN, PIN, etc.)
• Steal or rewrite active session cookies to manipulate the active session
• Inject JavaScript code to perform nefarious operations in web pages as they are rendered in the browser.
• Inject HTML possibly with JavaScript code into active sessions to launch Automated Transfer System (ATS) attacks. This attack enables the malware to execute fraudulent transactions on the user's behalf without revealing any details to the user. For example, the malware can manipulate the account information shown to the user by rewriting the account balances on the client side in the browser.
Web Injects first appeared in Zeus; a number of subsequent botnets have implemented the same technique. The rules that govern Web Injects must be static because of limitations of the hooking technique. As a result, these rules are embedded in the malware at build time. A generic Web Injects rule has following parameters:
• _set_url_ : specifies the target URL to be monitored by the malware for injection.
• _data_before:_ specifies the pattern of data in the page that should appear _before_ the actual injection payload (specified in the data_inject parameter).
• _data_inject:_ specifies the HTML or text or JavaScript to be injected in the specified part of the web page.
• _data_after:_ specifies the pattern of the data that should be present _after_ the actual injection payload (specified in the data_inject parameter).
Listing 5.1 shows an actual example of Web Injects used by Zeus and SpyEye to perform cookie injection in the context of an active session with a Citibank web site. Notice how set_url specifies the target web site, data_before specifies where the injection is to occur, and data_inject specifies what to inject.
Listing 5.1 Cookie injection using Web Inject—a Citibank example.
In a similar example, Figures 5.2 and 5.3 show fake pop-ups injected by malware in the browser using Web Injects asking for a user's credit card information including CVV, SSN, and verification information such as mother's maiden name and Date-of-Birth (DOB).
Figure 5.2 Fake HTML pop-up injected in the American Express session using Web Injects—card information.
Figure 5.3 Fake HTML pop-up injected in the American Express session using Web Injects—verification details.
_Screenshot stealing_ : A number of malware families capture screenshots when a user interacts with the browser or operating system. Screenshot capturing is popular in financial malware using a type of MitB attack. The technique of screenshot capturing can be used to capture any screen. The general idea is that when a user surfs critical web sites such as banks from the infected systems, the malware captures the screenshots of critical pages (account information, account balance, etc.) to gain potential insight into the target (user) assets.
Malware authors use Windows' built-in API functions to generate a screenshot capturing module which is embedded in the malware code itself. Screenshot module uses hooking to intercept mouse events to trigger a screenshot. For example, the malware waits for the user to open the browser and starts to input the credentials on a critical web site. As soon as the mouse is used to point to a particular area in the browser, the screenshot capturing module is executed and all the user actions are captured in the form of screenshots and sent back to the malware author. Windows provides a number of mouse event functions to map the execution of mouse events [5]. For example, MOUSE_DOWN, MOUSE_UP, and MOUSE_WHEEL can be used in the screenshot capturing module. A generic screenshot capturing module is discussed below:
• Phase A—MOUSE_DOWN event: When this event is triggered, the malware performs following steps.
• Cursor position is extracted using _GetCursorPos_ function.
• The handle to a specific window (browser window) is retrieved using the _WindowFromPoint_ function.
• A custom screenshot capturing module is called after the handle is retrieved.
• Phase B—MOUSE_UP event: When this event is triggered after the MOUSE_DOWN event, the following steps are performed by the malware to capture screenshots silently.
• A box is drawn around the selected area in the target window.
• A memory device is created using the _CreateDompatibleDC_ function that is compatible with the given dialog (or window).
• A compatible bitmap is generated using the _CreateCompatibleBitmap_ function.
• The _SelectObject_ function is used to map the bitmap image into memory.
• _BitBlt_ function is used to copy the content of a bitmap image from a source to a destination by transferring pixels.
• The _SaveBitmap_ function is used to store the bitmap image into memory and then transfer it to the C&C panel using custom-generated sockets. For transmission, the image (screenshot) is encoded and compressed.
There are a variety of ways in which screenshot capturing can be implemented. Additional sophistication can be included such as an ability to switch channels, if the primary channel fails to exfiltrate data.
_Video stealing_ : Malware often has the capability to record video on an infected system using the installed (or built-in) camera. For example, the malware may provide the capability to record an active session that a user has with a bank web site. A number of sophisticated botnet malware families provide the functionality to record video of a user having a session with the bank web site. This functionality is a part of secure fall backup mechanism to assist the situations in which primary data stealing techniques such as Form-grabbing, keylogging or Web Injects fail to serve the purpose. In those scenarios, the malware activates the video stealing component and starts recording the videos of those web sites that are specified in the configuration. The recording allows the attackers to view the recorded sessions and extract desired information later on. A number of malware families such as Citadel and Flame use this technique for data stealing. Malware authors use Windows' built-in APIs to accomplish this task. Microsoft provides a detailed reference on using audio/video APIs [6].
Malware authors can also access a user's camera using JavaScript in the browser. APIs provided in HTML5 facilitate this capture [7]. The Citadel botnet uses HTML5 to record videos from infected machines. The navigator.getUserMedia method is used to access a user's camera [8]. The concept is simple as discussed below:
• Create appropriate HTML video elements to be injected into a web page.
• Create video event listeners based on which video settings are generated.
• Execute video event listeners by accessing the device using the navigator.getUserMedia method.
• Call the play method to access the live video connection.
There can be many variations of this attack depending on the needs of the malware.
_Colocation services data stealing_ : Advanced malware such as Flame has the capability to steal information from colocation services running in the environment. For example, Flame steals data from Bluetooth devices running in the vicinity of the infected computers. In this case, malware first verifies that the infected system has the Bluetooth capability using Windows APIs. For example, a malware author can use Windows' built-in socket functions to extract Bluetooth information [9]. Bluetooth uses a Binary Large Object (BLOB) structure to handle transportation of data for different operations. Standard socket operations such as recv, send, recvfrom, and sendto functions are used for reading and retrieving data from the devices available in the pico network. A pico network is a network that is created during a wireless Bluetooth connection between two devices. So, it is also possible to steal data from the legitimate mobile devices that are sharing the network with a malicious device.
_Operating system information stealing_ : This technique is a part of almost every malware family. Malware authors extract configuration details of the infected system to map security before taking further actions. Once the system is infected, malware can retrieve any type of information from the operating system and the running applications. The type of information extracted by malware includes:
• Configuration details of the installed applications including antivirus engines, host firewalls, etc
• Information about the installed version of the operating system and running browser
• Opened ports and running services
• Boot configuration
• Information about the active users present on local and domain network
• Information from security policy files, entries in the file system and registry elements
The basic idea behind collecting information about the infected system is to gain an understanding of the environment to find weaknesses.
_Software credential grabbing_ : Malware is also capable of stealing the credentials of software installed on the infected system. The idea is to simply grab the credentials used by different software based on Post Office Protocol (POP3), Simple Mail Transfer Protocol (SMTP), File Transfer Protocol (FTP), etc., protocols. For example, if a user tries to login to a hosting server using FTP client such as FileZilla from the infected machine, the account credentials are easily stolen by the malware. This attack is possible because of the malware's ability to hook into standard Dynamic Link Libraries (DLLs) used by the software. For example, for stealing FTP credentials of different clients, the malware hooks the send function exported by WS2_32.dll to grab FTP logins. Almost any software with a login is susceptible to credential stealing: messaging and e-mail clients to mention just a few.
## 5.2 Phase 2: Data Transmission
Data transmission is always a challenge for the attackers to successfully transmit stolen information from the infected systems without being detected by the network security devices. For this, the attackers have to deploy stealthy modes of data transmission that slip under the radar. The stolen data is transmitted in an unauthorized manner without hinting users and security devices, so the process has to be stealthy and undetectable. Attackers prefer to transform the data or use those communication channels which look legitimate as normal traffic bypasses through existing security defenses. Before understanding the data transmission in detail, it is important to understand the design of communication channel used to contact Command and Control (C&C) servers.
The most widely used C&C topologies are discussed below showing how differently the malware (botnets) communicate with C&Cs:
• Centralized C&C means that single server instructs and gives command to all the other bots (malware) on the Internet following the master–slave relationship. Basically, it is based on the star topology. Protocols such as HTTP and Internet Relay Chat (IRC) support this design.
• Decentralized or locomotive C&C mean that there exists no single C&C server. It means every bot in the botnet acts as a server and gives commands to other bots in the network. Basically, it uses Peer-to-Peer (P2P) architecture in general and uses supporting protocols for C&C communication. Locomotive here refers to the feature of decentralized botnets to have high mobility in its C&C structure.
• Hybrid C&C utilizes the power of both centralized and decentralized C&Cs. For example, hybrid C&C helps to create secure fallback mechanisms to handle scenarios where primary mechanism fails.
To protect C&C from exposure, several techniques are deployed as discussed below:
• The malware authors have started using Domain Generation Algorithms (DGAs) [10] to generate pseudorandom domain names using a predefined algorithm and a dynamic seed value. The malware generates a number of domains which are queried by infected system. Out of these domains, one specific domain is resolved to C&C server enabling the communication. These domains are registered for a short period of time and do not remain active for long. DGAs have made the Domain Name System (DNS) blacklisting approach useless as domains are generated often thereby making the C&C detection harder. However, DGAs can be reversed to detect domains upfront but it is a very time-consuming process and hence not that beneficial in real-time scenarios. Malware families such as Conficker, Zeus, and Kelihos use this as secure fallback mechanism in case primary channel fails.
• DNS fast fluxing [11] is another approach in which IP address resolving to C&C domain is changed frequently thereby making the detection process harder. This is generally achieved by specifying the very low Time-to-Live (TTL) values in the DNS queries generated by the infected system. As a result, the DNS cache for C&C domain does not remain populated for a long period and new query is issued to contact C&C domain which resolves to a new IP address. Due to this nature, the tracking of C&C server becomes a daunting task.
• Malware authors have designed built-in components in C&C server software known as gates which restrict the direct access to C&C over HTTP channel. It is basically an authentication system in which the malware from the infected systems have to authenticate their identities using a special encrypted message before any data communication takes place with the C&C. Failed authentication disrupts the communication channel.
Data transmission happens in three steps as discussed below:
_Step 1—Selecting protocol for delivery_ : Different malware families choose different models of data transmission. It depends on the design of malware and the type of protocol used to transmit data across the network. Recent malware families have tended to use the HTTP or HTTPS protocol as their preferred medium of transmitting data to the C&C server. There are several reasons: (1) the stolen data looks like the normal HTTP data on the wire, (2) most firewalls allow HTTP data to pass through, and (3) HTTP data is proxy aware and easily routed through proxies.
Table 5.1 shows how different malware families' use encrypted HTTP POST requests to transmit data from the infected machines.
RC4 is a stream cipher that is used for symmetric encryption in which same keys are used for encryption and decryption purposes. The attackers embed the encryption key in the binary configuration file, which malware and the attacker use to encrypt/decrypt the data. For example, the bot (malware) configuration file is encrypted with a RC4 key and transmitted to the malware on the infected machine which decrypts it with the same RC4 key embedded and encrypted in the malware executable. Advanced Encryption Standard (AES) is also used for symmetric encryption which uses block cipher instead of stream. Both AES and RC4 are used for encryption/decryption of different malware components to secure the malware activities from being defaced. For example, Flame built the fraudulent certificate based on Microsoft Terminal Services Licensing certificate by using MD5 collision tactic. In addition, it also used some simple encryption tactics such as substitution and simple XOR encryption to transform the layout of components thereby reducing the size accordingly.
Other sets of protocols have been used to transmit data in a secure fashion. A recent example is the Flame malware [1], which also uses Secure Shell (SSH) as a backup mode of transmitting data securely from the end-user system. From the malware's perspective, it is challenging to have a proper SSH client (Flame uses a Putty-based library) installed on the end-user machine that connects back to SSH using an appropriate authentication mechanism to establish the secure tunnel for transmitting data. Putty [12] is an open source client application used for transferring files/data that supports a number of network level protocols including SSH. Similarly, the FTP and SMTP protocols can be used to exfiltrate data from the infected computers.
P2P protocols are also used to exfiltrate stolen information. This approach is decentralized as it involves placing stolen data in the form of files on a P2P network so they can be downloaded across a number of geographical areas making them difficult to completely remove from the network. The peer computers communicate among themselves rather than with one C&C server directly [13]. Basically, P2P nodes work like distributed C&C panels as commands are sent and received using push and pull mechanisms. Information is hidden and retrieved through distributed hash tables which are used to perform lookups of different objects on peers. Generally, a random identifier is chosen for a specific peer to build an object containing hash value to generate entries in the distributed hash table. The key:value pair present in the distributed hash table is used to store and access the data among peers. Over time the ownership of data varies across different nodes which increases the complexity of the network.
DNS protocol can also be used to exfiltrate information using DNS queries. Feederbot [14] is an example of this type that exchanges data using DNS Resource Record (RR) queries. The data is encrypted using RC4 which is further base 64 encoded and enabled with CRC32 checksum to fit the DNS TXT. Ebury [15] is another SSH rootkit which transmits stolen SSH keys in the form of hexadecimal strings sent with DNS queries. We expect that DNS protocol will be used efficiently at a large scale in the coming times for C&C communication.
Malware authors are also using The Onion Router (Tor) [16] anonymous network to perform data exfiltration [17]. Tor is a distributed overlay network that anonymizes TCP-based applications for web browsing, shell operations, and P2P communications. Basically, distributed overlay network means a network that is created on the top of another network which contains systems (nodes) that are distributed in nature. Tor is explicitly used to avoid detection strategies based on the DNS protocol. Tor provides anonymity between end points and it is hard to trace the source and destination systems based on location. Each node in Tor keeps a track of the network connection in memory until the connection is opened so nothing is stored on disk. As soon the data is received, it is forwarded to the next node. This builds a node circuit and once the connection is closed, the node does not have information of the previous end point. Generally, the exit node decrypts the traffic before transmitting it to the final destination, otherwise the data is completely encrypted while within Tor.
_Step 2—Data compression_ : Malware families use compression before encryption to reduce the size of data transmitted on the network. Reduced packet size lowers detection. In addition, reducing the size of transmitted files can improve both security and reliability. The attackers prefer to choose lossless compression. In lossless compression, the file can easily be recreated on the server side, whereas in lossy compression recreation of file results in loss of some part of original data. Basically, reconstruction of real data from the compressed data is a characteristic feature of the lossless compression libraries, whereas in lossy compression some part of data is lost while transformation and reconstruction. The most widely used compression libraries are presented in the Table 5.2.
For example, Flame used PPM in conjunction with BZIP2 whereas Duqu and Stuxnet used LZO compression schemes. Botnets like SpyEye also used LZO compression for fast data transmission.
_Step 3—Internet socket generation_ : Once a protocol is selected and compression mechanism is deployed, malware authors use the built-in socket capabilities of the operating system to exfiltrate the data to the C&C server [18]. The basic idea of using sockets (end points created by application software for communication between client and server) is to create a new channel from the infected machine for transmission of data. The sockets created by malware become active only at the time of data transmission and become dormant afterward in order to be stealthy. In addition, sockets are protocol independent. The majority of operating systems provide socket APIs that can be used directly to program sockets on the infected system. Sockets can be based on UDP or TCP or they can be raw depending on the device and its usage. The overall mechanism is to create a client–server communication model between the infected system and the C&C panel.
That's how exactly the data is exfiltrated from the end-user systems infected with malware.
Table 5.1
Encryption Modes for Data Transmission Used by Sophisticated Malware Families
Malware Family | Transmission Mode
---|---
Zeus | Uses RC4 algorithm to encrypt HTTP POST data. P2P variants exists
Taidoor | Deploys RC4 mechanism to encrypt HTTP POST data
Citadel | Implements XOR mechanism with AES to encrypt HTTP POST data
Lucky Cat | Uses compressed cab file to transmit through HTTP POST request
Duqu | Transfers JPEG files encrypted with AES through HTTP/HTTPS POST request
Stuxnet | Uses XOR mechanism to pad data in HTTP POST request
Flame (SkyWiper) | Uses public-key cryptography to encrypt data transmitted over HTTPS channel. XOR encryption with static key and monoalphabetic substitution is used
Table 5.2
Different Compression Schemes Used by Malware Authors During Data Transmission
Compression Libraries | Details
---|---
LZMA | Lempel–Ziv–Markov chain algorithm
LZO | Lempel–Ziv–Oberhumer—lossless data compression algorithm
BZIP2 | Based on Burrows–Wheeler algorithm
PPM | Prediction by Partial Matching algorithm
DEFLATE | Uses LZ77 algorithm and Huffman coding
GZIP | Based on DEFLATE
In this chapter, we have presented a data exfiltration model including data stealing and transmission capabilities of the advanced malware used in targeted attacks. The chapter provides details about the different data exfiltration mechanism opted by attacker to extract information from targets. Data exfiltration process heavily depends on attackers' ability to maintain control and accessibility in the target's environment. The next chapter discusses about the procedures used by attackers to maintain control over the compromised environment.
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Chapter 6
# Maintaining Control and Lateral Movement
This chapter unveils the various techniques deployed by attackers to maintain control and persist in the network for a long period of time. The attackers perform network reconnaissance in the network to compromise additional systems in the network so that information can be exfiltrated at a large scale. The attacker uses custom, self-made, and public available tools such as Remote Access Toolkits (RATs) to perform different tasks such as port scanning and exploiting vulnerabilities in the target network. This step is very significant from the perspective of executing targeted attacks for a long duration of time without being detected.
### Keywords
Lateral Movement; Marinating Control; Backdoors; Remote Access Toolkits (RATs)
In this chapter, we discuss in detail different strategies adopted by attackers to maintain continuous access and perform lateral movements in a compromised network to increase the sphere of targeted attacks. Lateral movement refers to the process of performing reconnaissance in the network from the compromised system to fingerprint other potential vulnerable systems. The idea is to maintain control of the compromised system and gain access to more systems in the network.
## 6.1 Maintaining Control
In this phase, we cover the different techniques used by attackers in targeted attacks to maintain control over the compromised system.
_Melting/Self-destructing_ : A number of malware families use dropper programs. Droppers basically act as wrappers around malware. To begin with, the dropper is downloaded onto the compromised system. The dropper installs the malware on the compromised system and then self-destructs itself. The idea is to remove all traces of the program that installed the malware in the system. To accomplish that, attackers embed a batch script code in the dropper program which executes after the malware is installed on the system. This process is also known as melting because the wrapper (dropper) program melts (removes its existence) after the system is infected. Figure 6.1 shows the disassembly of one component of a dropper program used by ICE 1X bot (variation of a Zeus bot). The code represents the x86 instructions that show how exactly the batch script is executed. The x86 instructions of the malware is obtained using reverse engineering which is a process of disassembling the components of a compiled program to understand the low-level details by extracting the hardware instructions used by the CPU.
Figure 6.1 Melting code (batch script) used in dropper programs.
The embedded batch script simply uses a "del" command with a "/F" parameter to force deletion of the installation files.
_Insidious backdoors or Remote Access Toolkits (RATs)_ : The first action of attackers after successful exploitation is to install backdoors on the compromised machines to retain control. In computer terminology, backdoor [1] is defined as an insidious mechanism deployed in the target computer system to gain unauthorized access. Backdoors are sophisticated programs that are hidden from the users and execute commands on the infected system without their knowledge. Backdoors have sufficient capability to retrieve all types of information from end users' machines. Backdoors are classified as symmetric and asymmetric. Any user on the system can use symmetric backdoors whereas asymmetric backdoors can only be used by the attackers who install them. Asymmetric backdoors use cryptographic algorithms to communicate with the attacker through an encrypted channel. In addition, asymmetric backdoors validate the end points (C&C servers), which are used to send stolen data and to receive operational commands. Generally, RATs can be considered as a type of backdoor. RATs come bundled with applications encompassing both client and server components that provide complete administration capabilities to the remote users (attackers). Backdoors and RATs capabilities are discussed below. They can:
• perform Man-in-the-Browser (MitB) attacks to hook a browser to steal sensitive information such as accounts, passwords, and credit card numbers
• execute commands remotely without any restriction and also instantiate remote command shells
• download and upload files from third-party domains
• create, delete, and modify the existing processes
• control all the Operating System (OS) messages such as keystrokes, mouse events, logs, etc.
• dump passwords, configuration files, registry details, etc., from compromised machines
• perform privilege escalation operations or add new users to the system
• generate additional TCP sessions from the system to exfiltrate data
• create additional services in the system without the user's knowledge
• alter configuration of the system such as disabling or removing firewalls, antivirus engines, and DNS entries
• scan stealthily for exposed services on the internal network
These are some of the functionalities provided by backdoors and RATs to maintain access to the compromised (infected) machines.
The attackers can use publicly available backdoors or custom-designed backdoors to stealthily operate and exfiltrate data. The choice is generally based on the nature of the targeted attack and the preference of the attackers. Table 6.1 shows some widely used RATs that have been used in the earlier targeted attacks and broad-based attacks [1].
Table 6.1
Backdoors (RATs) Used in Attacks
Backdoors (RATs) | Attacks
---|---
Poison Ivy | RSA Secure ID breach 2011
Ghost RAT | Ghost Net Espionage 2009 (early years)
Hupigon (Grey Pigeon) | Yes, Tibetan campaign 2012
TorRAT | Not specific. Used against Dutch users for conducting financial fraud
FAKEM (Terminator) | Yes, Taiwan Campaign 2013
Black Shades NET | Yes, Syrian campaign 2012
Xtreme RAT | Yes, Syrian campaign 2012
Dark Comet | Yes, Syrian campaign 2012
Nuclear RAT | Used in botnet attacks
Bifrost Trojan | Used in botnet attacks
Frutas RAT | Yes, high-profile organizations in Asia and Europe
### 6.1.1 Deploying BackConnect Servers
A big challenge to attackers is the presence of a Network Address Translation (NAT) environment created by firewalls and gateways [2]. Most organizations employ a NAT configuration to have a single IP address and distribute that address to create an internal network such as a Local Area Network (LAN). The NAT device is placed at the network perimeter where it performs address translation from an externally faced interface to the internal network. As a result, devices on that LAN share a common external IP address. There are two primary benefits of creating a NAT. First, it helps in extending the longevity of IPv4 addresses because a single IP address maps to an entire LAN. Second, the complete LAN is treated as one client for all the external network interfaces. This mask helps prevent intruders from initiating a direct communication channel with the internal systems. In addition, NAT impacts one-to-one connectivity of the systems by participating in communication. Attackers find it difficult to communicate directly with the systems as firewalls and NATs prevent the direct connection to the remote server if ports are filtered. In this way, a NAT disrupts the communication channel between the compromised machine and the attacker. Attackers use a variety of techniques to bypass NAT.
Attackers implement a technique known as BackConnect in which the infected system is forced to run a custom server using different protocols so that the attacker acts as a client and initiates connection back to the infected machine. When the installed malware steals information and fingerprints the network environment, the attacker determines whether the system is present inside a NAT or is directly accessible. If the system is found to be inside NAT, the malware is directed to download additional programs that are used to trigger a BackConnect server. Sometimes, lightweight servers are embedded in the malware itself in the form of plug-ins, which can be activated easily by the attacker.
In BackConnect, the compromised system acts as a server (starts the service) and the attackers use protocol-specific clients to connect to it for retrieving data and exploring the system for information. The attackers prefer to start BackConnect servers on legitimate port numbers, if the ports are not already in use by the desired protocols. This approach favors the attackers because the traffic appears to have originated from legitimate ports and can bypass network filters (if deployed). With BackConnect in place, attackers can connect to the infected systems from across the globe.
BackConnect is loosely based on the concept of reverse proxying as it helps to hide the traces of the C&C server. For example, the attacker can use a different IP address than used by the C&C server to initiate BackConnect. As a result, it becomes difficult to deploy firewall rules to block the traffic originating through BackConnect. Multiple BackConnect servers have been found in the wild:
• _SOCKS BackConnect_ : Secure Sockets (SOCKS) is a network protocol that is designed to route traffic (packets) through an intermediate proxy server. SOCKS is the preferred choice of attackers because it can be used to bypass Internet filtering deployed at the perimeter, for example, by a firewall. By burying itself within the SOCKS protocol, the attacker's traffic easily routes through firewalls. The idea is to install a SOCKS server on the infected computer and initiate connections through it. An additional advantage of SOCKS is that attackers can use open source SOCKS software.
• _FTP/HTTP BackConnect_ : The attacker use File Transfer Protocol (FTP)/ Hyper Text Transfer Protocol (HTTP) BackConnect server to explore a compromised system. A lightweight FTP/HTTP server is started on the infected computer and attacker connects to it. A server of this type helps the attacker enumerate the directory structure and transfer critical files.
• _RDP/VNC BackConnect_ : This BackConnect server is based on the Remote Desktop Protocol (RDP) or Virtual Network Connection (VNC) protocol. The attacker starts a RDP/VNC server on the compromised machine to explore the system using Graphical User Interface (GUI) from any location on the Internet. Such a GUI helps the attackers to enumerate the components of the system.
Proxifier [3] software has also been used to help the infected system pass network traffic through a proxy server by manipulating the socket library calls.
The BackConnect technique has been widely implemented as a part of botnet frameworks such as Zeus, SpyEye, and Citadel.
### 6.1.2 Local Privilege Escalation
The very first goal of an attacker after compromising a system is to obtain administrative rights on the infected system. If the compromised account doesn't already have administrative rights, privilege escalation can be achieved by exploiting vulnerabilities and flaws in configuration design of the compromised system. In the context of an OS, privilege escalation means gaining access rights to execute code directly in ring 0 from the ring 3. This ability is critical for successful lateral movement and reconnaissance because accessing different resources on the system requires administrative access rights. For example, executing system-level commands and kernel resources requires elevated privileges. If the attacker wants to dump LAN Manager (LM/NTLM) [4] hashes present in the Security Accounts Manager (SAM), System, or Active Directory (AD) databases from the compromised machine, an administrative account or access rights are required. In a targeted attack, local privilege escalation is common because of the capabilities it offers. Some of the most useful capabilities are:
• Accessing critical files on the system which are locked and cannot be copied and accessed directly. For example, password hashes and the boot key containing files such as " C:\windows\system32\config\sam" and " C:\windows\system32\config\system."
• Accessing system specific registry entries for write operations which require administrative rights.
• Installing or removing services in the compromised system to install backdoors or malicious programs.
In the case of Windows, several vulnerabilities exist in the wild that can be used by attackers to escalate privileges. Table 6.2 shows some of the escalation vulnerabilities that can help attackers to fetch escalated privileges on the compromised machine.
Table 6.2
A Compressed List of Windows Privilege Escalation Vulnerabilities
CVE Identifier | Details
---|---
MS13-075 | Vulnerability in Microsoft Office IME (Chinese) could allow elevation of privilege
MS13-076 | Vulnerabilities in kernel-mode drivers could allow elevation of privilege
MS13-077 | Vulnerability in Windows service control manager could allow elevation of privilege
MS13-053 | Vulnerabilities in Windows kernel-mode drivers could allow remote code execution and elevation of privilege
MS13-063 | Vulnerabilities in Windows kernel could allow elevation of privilege
MS13-058 | Vulnerability in Windows defender could allow elevation of privilege
Local privilege escalation vulnerabilities are not restricted to Windows; they also exist for the Linux and Mac operating systems as well as software security products. The vendors release appropriate patches which are supplied as a part of OS updates to eradicate the concerned vulnerabilities. The privilege escalation vulnerabilities are not remotely exploitable rather attackers require access to the vulnerable systems to exploit them.
## 6.2 Lateral Movement and Network Reconnaissance
In this part, we discuss the techniques and tactics used by attackers to perform lateral movement to distribute infections across the target's network. The first step is network reconnaissance which involves unauthorized and stealthy detection of running services, vulnerabilities, and existence of other systems on the network. The idea is to determine vulnerable and easily compromised systems on the network so that critical data can be stolen. The attackers utilize operating system capabilities to perform internal reconnaissance and information gathering.
We cover the most widely lateral movement and network reconnaissance tactics used in the targeted attacks.
### 6.2.1 Information Reuse Attacks
Information reuse plays a critical role in chaining different attacks together: the information gained from one attack can be used as an input to the next attack. Attacks such as credentials dumping, hash replay, etc. fall under this category. To illustrate the importance of information reuse, we describe password dumping including an associated attack that uses dumped hashes to attack other systems.
#### 6.2.1.1 Credentials Dumping
Once the end-user system is compromised, the attacker's next step is to dump password hashes and obtain clear-text passwords for the configured accounts. Attackers have complete knowledge about the storage of password hashes and how to dump them. Techniques such as Dynamic Link Library (DLL) injection and tools such as pwdump are readily available. The most primitive components in Windows that are targeted by attackers to dump hashes and extract credentials are presented in the Table 6.3.
Table 6.3
Locations Providing Accounts' Passwords and Hashes in Windows OS
Component | Description
---|---
SAM database | SAM database is compromised
LSASS | Local Security Authority Subsystem Service process memory is injected with rogue DLL to dump hashes
AD | Domain and AD database is compromised. Reading "NTDS.DIT" file
CredMan | Credential Manager store is hijacked and information is extracted
LSA—Registry | Local Security Authority secrets are read in the registry
Hashes are dumped directly from the memory by injecting a DLL in critical processes such as Local Security Authority Subsystem Service (LSASS) which control password handling, access token generation and authentication (logging) mechanism in Windows. In addition, an attacker with administrative access can easily query the SAM database, AD database, Credentials Manager (CredMan) store and LSA secrets in the registry.
Obtaining passwords for local accounts requires hash extraction for which administrative access is required. Similarly, elevated privileges are required for obtaining hashes from a remote computer on the network. As discussed earlier, privilege escalation plays a critical role in extracting credentials from different components. Once the hashes are dumped, the attacker can extract information such as username, computer details, user ID, Hash type (LM or LM+NTLM), LM hash, NT hash, password, audit time, status (disabled or locked) and description. There are several public tools available that facilitate hash dumping and password extraction process as shown in the Table 6.4.
Table 6.4
Most Widely Used Hash Dumping and Password Extraction Tools
Password (Hash) Dumping Tools | Description
---|---
Fgdump | Dumps SAM hashes. Works on Windows Vista and later versions
Pwdump7 | Extracts hashes after dumping SAM and SYSTEM file from the file system
Pwdump 6 | Performs DLL injection in lsass.exe process in Windows to dump hashes
Gsecdump | Extracts hashes from SAM, AD, and active logon sessions by performing DLL injection in lsass.exe
PwDumpX | Extracts domain password cache, LSA secrets and SAM databases through on-memory attacks
Powerdump | Dumps password hashes from SAM component through registry
Mimikatz | Dumps LSA secrets, SAM databases and password history using both registry and in-memory attacks
Incognito | Dumps active session tokens
Find_token | Dumps active session tokens
Cachedump | Extracts cached domain logon information using in-memory attacks
Lslsass | Dumps active logon password hashes using in-memory attack technique
The effectiveness of tools depends on their support and how broadly applicable they are. A number of tools mentioned above only work on a specific version of Windows and use specific techniques for dumping hashes from the compromised system. In addition, attackers can use old school tricks in which "regedit.exe" and "reg.exe" are used to save the SAM and SYSTEM hive for offline extraction of passwords.
#### 6.2.1.2 Pass-the-Hash Attack Model
Attackers deploy attacks such as Pass-the-Hash (PtH) [5] in scenarios where cracking of hashes is time consuming or not possible with available resources and constraints. This attack allows the attacker to compromise a number of other systems including domain controllers and servers on the LAN by reusing stolen credentials (hashes). Instead of cracking the password hashes, attackers pass the hash itself for an account. PtH attacks do not depend on the account type compromised during infiltration of the system. However, in order to dump hashes from the compromised system attackers require administrative rights or admin-equivalent privileges. For PtH attacks to work, the attacker requires the following:
• Password hashes of the compromised account along with authentication tokens
• Hashes can be passed only for those accounts which have Network Logon user rights
• Direct access to other computer systems on the network
• Target systems should be running services in listening mode to accept connections
PtH attacks are particularly effective in networks configured with an AD because successful execution of PtH attacks can result in gaining access to administrative accounts belonging to entire domains or enterprises. Additionally, dissecting the PtH accounts from normal authorized operations on the network is a difficult task because these attacks exploit the inherent design and working nature of authentication protocols such as LM, NTLM, NTLMv2, and Kerberos. PtH does not exploit any vulnerability in the authentication protocols, instead it manipulates the functionality of the given protocols. PtH attacks are particularly potent when different systems on the network use the same account credentials. For example, if a backup account that is available on all active systems on the network is compromised, the impact is widespread.
### 6.2.2 File Sharing Services (Shared Access)
Often several systems on a local intranet are configured to run in shared mode. In this setup, systems share their resources such as files, documents, and network devices. Windows supports protocols such as NetBIOS, Windows Internet Naming Service (WINS), and Common Internet File System/Server Message Block (CIFS/SMB) to provide shared access. NetBIOS [6] stands for Network Basic Input/Output System which provides name registration service for Windows clients in a LAN including datagram distributions service and session service for connectionless and connection-oriented communication respectively. NetBIOS provides identity to Window's clients on the LAN similar to hostnames on the Internet. In order for NetBIOS to work, WINS [7] is required to resolve NetBIOS names of Windows clients to their respective IP addresses. SMB [8] is an application layer protocol which is used to provide shared access among different clients (nodes) in a LAN. SMB can be deployed directly or used in conjunction with NetBIOS on the top of session layer. Earlier, SMB was also known as CIFS. Table 6.5 shows the system ports involved in communication during shared access of services.
Table 6.5
Port Numbers and Services Used in Shared Access on Windows
Port Number | Service
---|---
135 | WINS over TCP (WINS manager, client–server communication, exchange manager, DHCP manager, remote procedure call)
137 | WINS NetBIOS over UDP (name service)/WINS over TCP (registration)
138 | NetBIOS over UDP (distribution service—connectionless communication)
139 | NetBIOS over TCP (session service—connection-oriented communication, NetBT service sessions, DNS administration)
445 | SMB over TCP/IP (shared access)
After compromising an enterprise network, attackers look for specific ports to scan and verify the availability of shared access on the network. If these ports are open on another system on the LAN, attackers deploy batch scripts to gain shared access and transfer sensitive information. If the remote shares (Admin$, C$, F$—disk access) are not enabled by default, attackers can use dumped password (credentials gained earlier) to login into other systems to obtain the shared access and then steal critical data. Newer versions of Windows restrict the default (null share IPC$) sharing of resources on the network to enhance security.
For additional file sharing services, attackers look for open FTP and Trivial File Transfer Protocol (TFTP) services running on TCP port 21 and UDP port 69 respectively. Basically, these services are used to gain access to network servers or devices. In a LAN, end-user systems are often configured with these services. Exploiting shared access configurations on end-user systems in an enterprise network is an important step in lateral movement and network reconnaissance.
### 6.2.3 Batch Scripting: Command Execution and Scheduling
Batch scripting has been existing in Windows OS since its origin. Even Disk Operating System (DOS) uses it. Batch scripts are written to execute a number of commands through a script interpreter which is present in the Windows OS by default. The idea behind batch scripting is fairly simple and involves constructing a text file (.bat) and adding a number of commands provided by Windows OS to interact with various components of Windows. The attackers build a batch file on the compromised host to execute critical commands that help them to perform reconnaissance. Batch scripting enhances the process of executing more than one command at a time, thereby helping to automate the process of multiple command execution. Table 6.6 shows some of the standard commands that are used in the targeted attacks.
Table 6.6
Commands Deployed in Batch Scripts Used in Targeted Attacks
Commands | Functionality
---|---
sc.exe | Interacts with Windows service controller to perform operations on the installed services
shutdown.exe | Restarts or reboots the Windows
systeminfo.exe | Retrieves information about the system
tasklist.exe | Extracts information about running processes in the system
taskkill.exe | Kill tasks or running processes in the system
runas.exe | Execute applications and processes with different access rights
regsvr32.exe | Register DLL files as command components in registry
rexec.exe | Run commands on remote systems running rexec service
net user | Add or modify accounts on the system
net share | Manage share resources on the system
net localgroup | Manage (add/remove/modify) local groups on the system
net use | Interacts (connect/disconnect) with the shared resource
netstat | Retrieves information about TCP/UDP connections
net view | Retrieves information about domains, resources etc. shared by the target computer
net group | Manage (add/remove/) groups in domains
Ipconfig.exe | Retrieves the IP configuration of the system
Getmac.exe | Retrieves the MAC address of the system
attrib.exe | Set the attributes of the files
There are a number of other commands provided by Windows OS [9] in addition to commands presented in the Table 6.6 that can help attackers to attain more information about the running system. Listing 6.1 shows an example of batch script that can perform nefarious tasks in a few seconds. This example can be considered as an alias to the batch scripts used in the targeted attacks. Let's say the filename is explode.bat.
Listing 6.1 A simple information reconnaissance script.
Windows has a built-in capability to schedule commands using a scheduler service that executes the defined script or command at a given interval of time. This capability is used by the attackers to trigger time-specific command execution in the context of the compromised system. Windows OS provides a command named "at.exe" that is useful in the execution of programs at a given date and time. Using this functionality, the target program remains dormant and executes at a specified time and date. In this way, time-based logic bombs can be made. The "at.exe" command can be used to set the execution of "explode.bat" script as "at.exe 17:50 /every: 1, 3, 5, 7 explode.bat". This command schedules service on the compromised machine to execute the "explode.bat" script on the 1st, 3rd, 5th, and 7th day of each month at 5:30 PM.
In addition to the tools discussed above, attacks also deploy other custom and publicly available tools for extracting information from other computers and to execute commands remotely. Microsoft itself provides a classic package of remote administration tools known as "Ps Tools" as presented in the Table 6.7. These tools are useful for both internal network reconnaissance and lateral movement in targeted attacks.
Table 6.7
Selective Commands from Ps Tools Package Used in Targeted Attacks
Ps Tools | Description
---|---
PsExec | Executes commands (processes) remotely (or local)
PsService | Enumerates and control services remotely (or local)
PsPasswd | Modifies account passwords remotely (or local)
PsLoggedOn | Verifies the status of active user account on remote system (or local)
PsInfo | Enumerates system information remotely (or local)
PsFile | Enumerates opened files remotely (or local)
PsGetSID | Displays SID of the remote (or local) computer
PsList | Displays list of active process on remote (or local) computer
As you can see attackers have taken advantage of publicly available tools that are designed for legitimate purposes.
### 6.2.4 USB Spreading
Universal Serial Bus (USB) devices are used for both storage and migrating data across systems. USB devices were designed to be a portable and flexible device for storing data, but they have also proven to be convenient for data transfer. However, USB devices pose grave risks because they have been used to transfer malicious code from one system to another without the users' knowledge. For example, the Stuxnet worm had the capability to spread through infected USB devices inserted in control systems. Next, we present how a USB device can be infected when connected with a compromised Windows machine.
• After the successful infection of the end-user machine, the malware becomes silent and monitors operations performed by the operating system. If the malware has a built-in component to infect USB devices, it waits for the user to insert a USB device in the machine. A USB spreading component can be implemented in many ways, but there are two primary techniques. First, the malware hooks into a process such as explorer.exe running in the application layer. Next, the malware installs a Win 32 service as a stand-alone component, which starts automatically on every reboot.
• When a user inserts a USB device in the machine:
• The malware hooks the _RegisterDeviceNotification_ function to scan all the notifications received by the operating system, Windows in this case. If a USB-related notification is detected, the malware loads the Class ID (CLSID) which is a Globally Unique Identifier (GUID) used for fingerprinting the device in the operating system through Component Object Model (COM) interface.
• The OS calls WM_DEVICECHANGE which notifies the target application (malware) about the possible changes happening in the device configuration. The DBT_DEVICEARRIVAL device event is broadcast when the USB is inserted.
• The malware scans the DEV_BROADCAST_DEVICEINTERFACE structure to retrieve information about the USB device. The DEV_BROADCAST_HDR structure contains the headers which map the information related to device events sent through WM_DEVICECHANGE.
• The malware queries the DEV_BROADCAST_VOLUME structure to obtain the logical drive information and fetch the drive letter (G:, H:, K:, etc.) allocated to the USB device.
• Once the handle to the USB is obtained, the malware calls either _CopyFileW_ or _CreateFileW_ functions (or variants) to create or copy the malicious code onto the USB. Malware also uses the _SetFileAttributesW_ function to apply a specific set of file attributes such as making the file hidden on the USB device.
These steps show how the USB device can be used on Windows for spreading infections through a physical medium. The API calls mentioned above are a part of the Windows management functions [10]. Once the malicious code is copied onto the USB, the attacker enables the copied code to execute automatically when inserted in new machine. Different techniques have been used:
• _AutoRun infection_ : Windows has a built-in capability to support AutoRun and AutoPlay components that help automate code execution. Generally, an AutoRun.inf [11] file is created which has a certain set of parameters that specify the operations to be performed when a physical device such as a USB is attached to the system. There are several parameters defined in the AutoRun.inf file [12]. The "open" parameter specifies the type of file to be executed by the AutoRun. The "shellexecute" parameter defines the file that AutoRun will call to execute Win32 API _ShellExecuteEx_. Figure 6.2 shows the disassembly of UPAS malware that uses the _CreateFile_ function to generate "autorun.inf:" with the "open" parameter pointing to the "ws_a.exe" malicious executable.
• _LNK (.PIF) infections_ : LNK stands for Windows shortcut links whereas PIF stands for program information file. A Windows .LNK (shortcut) [13] vulnerability was exploited by the Stuxnet worm to execute its malicious code. The flaw existed in the handling of ".lnk" and ".pit" files by "explorer.exe" when shortcuts are displayed in control panels. The shortcut should be pointing to a nonsystem ".cpl", that is, control panel file. The explorer failed to verify the shortcut to a ".cpl" file and attackers exploited this flaw by injecting third-party code that gets validated. As a result, unauthorized and arbitrary code can be executed using this vulnerability. Both the Flame and Stuxnet malware used this .LNK vulnerability to trigger infections. A proof of concept code for this vulnerability exists in the wild and is named "USBSploit" [14]. It generates a malicious .LNK file, which upon execution opens a shell on the target system. It has been incorporated into the Metasploit framework [15].
Figure 6.2 UPAS malware—creating "autorun.inf" File using CreateFile function.
Figure 6.3 shows an output of C&C panel of UPAS (botnet name) malware that infected a number of USB devices. It shows how the notifications are stored on the attacker-controlled domain.
Figure 6.3 UPAS botnet C&C panel—successful USB infections.
Microsoft has taken additional steps to restrict the execution of AutoRun component in USB devices. For example, Windows 7 does not support "open" and "shellexecute" commands in the Autorun.inf file. Windows Vista restricts the hidden execution of Autorun files. However, with AutoPlay in Windows XP SP2/SP3, AutoRun functionality is still supported. As a result of this, attackers are finding vulnerabilities which work across all the versions of Windows. AutoRun and .LNK infections can be found in the wild. These techniques can be used in conjunction with other tactics to successfully execute malicious code through the USB devices.
In this chapter, we covered several techniques used by attackers to maintain control and perform lateral movement. These techniques are dependent on how stealthy malware executes so some tactics are more advanced and generate less noise as compared to others. Overall, gaining and maintaining access on compromised systems is an important step in the successful execution of targeted attacks.
## References
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3. Wikipedia, Comparison of proxifiers, <<http://en.wikipedia.org/wiki/Comparison_of_proxifiers>> [accessed 7.11.13].
4. Warlord, Attacking NTLM with pre-computed hash tables, uninformed research paper, 2006, http://uninformed.org/?v=3&a=2 [accessed 12.11.13].
5. Jungles P, Margosis A, Simos M, Robinson L, Grimes R. Mitigating pass-the-hash (PtH) attacks and other credential theft techniques, <<http://www.microsoft.com/en-us/download/details.aspx?id=36036>> [accessed 11.11.13].
6. MSDN, The NetBIOS interface, <<http://msdn.microsoft.com/en-us/library/bb870913(v=vs.85).aspx>> [accessed 10.11.13].
7. MSDN, WINS service, <<http://msdn.microsoft.com/en-us/library/windows/desktop/aa373133(v=vs.85).aspx>> [accessed 10.11.13].
8. MSDN, Microsoft SMB protocol and CIFS protocol overview, <<http://msdn.microsoft.com/en-us/library/windows/desktop/aa365233(v=vs.85).aspx>> [accessed 10.11.13].
9. MSDN, Windows command line reference, <<http://technet.microsoft.com/en-us/library/cc754340.aspx>> [accessed 7.11.13].
10. MSDN, Device management functions, <<http://msdn.microsoft.com/en-us/library/aa363234(v=vs.85).aspx>> [accessed 5.11.13].
11. Thomas V, Ramagopal P, Mohandas R. McAfee avert labs, the rise of auto run based malware, <<http://vxheaven.org/lib/pdf/The%20Rise%20of%20AutoRunBased%20Malware.pdf>> [accessed 5.11.13].
12. MSDN, Autorun.inf entries, <<http://msdn.microsoft.com/en-us/library/windows/desktop/cc144200(v=vs.85).aspx>> [accessed 5.11.13].
13. Microsoft Security Bulletin MS10-046. Vulnerability in Windows shell could allow remote code execution (2286198), <<http://technet.microsoft.com/en-us/security/bulletin/MS10-046>> [accessed 5.11.13].
14. Poli X. USBSploit, <<http://packetstormsecurity.com/files/100087/USBsploit-0.6.html>> [accessed 5.11.13].
15. Metasploit, <<http://www.metasploit.com>>.
16. Leec M, Ganis M, Lee Y, Kuris R, Koblas D, Jones L. SOCKS protocol version 5, <<http://www.ietf.org/rfc/rfc1928.txt>> [accessed 7.11.13].
Chapter 7
# Why Targeted Cyber Attacks Are Easy to Conduct?
This chapter presents the importance of Crimeware-as-a-Service (CaaS) model to build components of targeted attacks through easy means. Generally, the chapter shows how easy it has become to purchase different software components and other additional elements such as compromised hosting servers by paying required amount of money. We also discuss the role of e-currency in underground market places on the Internet and processing of transactions.
### Keywords
Crimeware-as-a-Service (CaaS); Underground cyber market; E-currency; Pay-per-infections (PPI); Rootkits
Targeted attacks can be launched with ease using Crimeware-as-a-Service (CaaS) [1] model. CaaS makes it easy for the buyers to purchase a variety of components from the underground market. Before digging into how the buyers (attackers) can purchase and deploy various components to conduct targeted attacks, some background is essential:
• _Underground markets_ : This market refers to a centralized place on the Internet that involves the buying and selling of various components to be used in cyber attacks. Underground market primarily consists of Internet Relay Chat (IRC) channels and underground forums where attackers meet and advertise illegal software, stolen information, etc. with associated costs. Other components include zero-day exploits, Browser Exploit Packs (BEPs), spam services, bulletproof hosting servers, botnets, Pay-per-Infection (PPI) services, stolen credentials including e-mails, etc. The identities of the participating individuals are hidden behind hacker handles, which are short unique names used in the underground world. However, the hacker handles are also used by whitehats and community researchers. The sellers and buyers in underground market are often available on IRC channels and underground forums. Contact details are posted in the advertisements. The complete money flow occurs using e-currencies such as WebMoney and Western Union. The transactions happening through e-currencies are untraceable and irreversible, as once the money is sent, it cannot be traced and returned to the sender. Overall, underground markets rely on exploiting the functionalities of the Internet to build obscure and anonymous channels for illegal activities, thereby earning money through shortcuts.
• _E-currency_ : Electronic currencies are widely used in the underground market. E-currency holds an equivalent value to fiat currency such as dollars and pounds based on the current exchange rates. When buyers buy any crimeware service, the transaction happens in the e-currency because transactions are online and buyers and sellers do not want a standard financial trail. One consequence is that all the transactions are irreversible in nature and it is hard to trace them back. E-currencies include WebMoney, Bitcoin, Perfect Money, Western Union, etc.
_Note: From a terminology perspective, we will use the buyer and seller when underground economy (market place) is discussed and attacker is used for a person who is planning (conducting) the attack. A buyer can be considered as an attacker who is purchasing the attack components from the underground market sellers._
## 7.1 Step 1: Building Targeted Attack Infrastructure
The very first step in launching a targeted attack is to build an attack infrastructure that provides the launchpad to execute attacks stealthily. A number of critical components are discussed next.
_Command and Control (C &C) infrastructure_: C&C servers are used to execute commands on target machines through installed malware. In addition, C&C servers collect large amount of information exfiltrated by the malware from infected computers. C&C servers are installed with well-constructed management software (web application, etc.) that makes it easy for attackers to manage the infected machines remotely. C&C servers are also used in broad-based attacks, but their sole purpose is to manage infected machines from any part of the globe. To set up C&C infrastructure, the following components are required:
• _Hosting servers_ : These are the servers used for hosting different types of resources including C&C management software, malware, etc. In standard terminology, hosting servers can be deployed as dedicated or shared. A dedicated hosting server is used to host a single instance of a web site (application) with no sharing of bandwidth. A shared hosting server hosts a number of virtual hosts each for a different web site. A dedicated server has one host and one IP address, whereas a shared server has one IP address for multiple hosts. Some attackers prefer to purchase bulletproof hosting services in which hosting servers allow the buyers to host any type of content as long as it falls under given guidelines. The majority of these servers run on compromised hosts which are then sold in the underground market. A buyer simply responds to the advertisements displayed in underground forums. Figure 7.1 shows an advertisement placed in an underground market forum for a compromised host (web site) account.
• _C &C management software_: Once hosting servers are purchased, it is the time to deploy C&C management software. Most C&C management software setups are web applications written in Personal Home Page (PHP) and hosted on web servers such as Apache and Nginx. These applications are also called C&C panels and provide the attacker with the capability of managing the infected systems from any place on the Internet. The application is termed as C&C panel because all the commands used to manage the remote bots are defined collectively to build a centralized command execution facility that is accessible from any place on the Internet. The infected hosts connect back to C&C panel and provide information about the system. As a result, the C&C database is updated as new hosts are added to the botnet. C&C management software can be purchased directly from the underground community. Alternatively, the buyer (client) can ask the seller (bot herder) to deploy the C&C management software on servers and configure the necessary components. However, advanced versions of C&C management software come with thorough instructions on deployment, including secure installation and configuration. There are a variety of ways to configure C&C management software.
Figure 7.1 Advertisement for selling shelled web site access. Copyright © 2014 by Aditya K Sood and Richard Enbody.
## 7.2 Step 2: Exploring or Purchasing Stolen Information About Targets
This step is always crucial because information about targets plays a critical role in launching targeted attacks. The attacker can explore publicly available repositories and social networks to learn their online surfing habits, other people in their social network, content shared on social networks, etc. This kind of information helps determine the mindset of the targets and is required for effective spear phishing. It is a time-consuming process to draw inferences about the target from the collected information. However, the time spent pays dividends by increasing the effectiveness of attacks.
When using CaaS, the buyer (client) can post a request in the underground market for information about specific entities or organizations and the amount to be paid for that service. Sometimes, the underground sellers advertise the type of data available including raw logs and banking information (contact details) of specific users in a given organization. For example, there is a big market for the banking information of customers of major banks which includes e-mail addresses, geographical locations, etc., of the users. In addition, selling stolen e-mail addresses of a given organization for phishing attacks or other malicious purposes is a major crimeware service in the underground market. More information about targets improves the success of a targeted attack.
## 7.3 Step 3: Exploits Selection
As discussed earlier (refer to Chapter 4), zero-day exploits are written for vulnerabilities that are unknown to vendors and the general public. The sale of zero-day exploits [2,3] has become a major underground business with some reported to sell for hundreds of thousands of dollars. Costs are usually a function of the vulnerability window, the time between discovery of the exploit and the installation of a patch.
Most hacker groups behind targeted attacks find their own zero-day exploits through aggressive research to detect vulnerabilities in widely used software. In addition, some hacker groups' primary interest is monetary from the sale of zero-day exploits in the underground community. Exploits (both zero-day and others) get packaged for sale in BEP. The basic model is to integrate a number of exploits in a framework and let the framework choose which exploit to use against target software based on the information collected. A common mechanism has the buyer posting a request in an underground market for a zero-day exploit to which a seller can respond. However, advertisements by sellers also exist. Once the exploit code is purchased, one can tweak the code accordingly for use in a targeted attack.
## 7.4 Step 4: Malware Selection
The choice of malware design plays a significant role in determining how the target is to be dissected and queried for extracting information. More sophisticated malware is harder to detect, so control of the infected system can be maintained for a longer period of time. There are many choices available for malware design, but the two most widely used malware classes in targeted attacks are discussed below.
One can select rootkits to be distributed as payloads. Rootkits are hidden programs that are installed in the compromised machine to hook the critical functions of the Operating System (OS) in order to subvert protections allowing theft of information without detection. Rootkits can be designed for userland, kernelland, or hybrid (works both in userland and kernelland). Userland refers to user space of the OS in which every process has its own virtual memory. The majority of the software applications such as browsers and other software are installed in the userland. Kernelland refers to kernel space (privileged space) of the OS which runs subsystems, drivers, kernel extensions, etc. Kernelland and userland are separated on the basis of virtual memory and privilege modes. Code running in kernelland is in complete control as compared to userland. Kernelland is also called as ring 0 and userland as ring 3 based on the concepts of protection rings. Not surprisingly, rootkits are also sold in the underground market. In the world of botnets, when a malware has rootkit capabilities with additional features, it is called a bot, an automated malicious program.
Remote Access Toolkits (RATs) are also widely used in targeted attacks. Basically, RATs provide a centralized environment in the infected machine through which attackers can easily gather information without making modifications to the underlying OS. For example, the Poison Ivy RAT has been used in a number of targeted attacks and is widely available. RATs provide an elegant Graphical User Interface (GUI) to operate the installed agent on the infected machine allowing commands to exfiltrate information. Figure 7.2 shows the GUIs of the ProRAT and DarkMoon toolkits that have been used in targeted attacks. The RAT agents can be rootkits or simply information extracting programs.
Figure 7.2 DarkMoon and ProRAT toolkits. Copyright © 2014 by Aditya K Sood and Richard Enbody.
Basically, getting access to these programs is trivial. Anyone with the basic knowledge and understanding of client-server architecture can easily operate these well-designed toolkits. Sometimes, if advanced functionality is desired, the attacker will extend functionality for a fee. On the other hand, sophisticated attackers can reverse engineer old toolkits to work with newly modified systems.
## 7.5 Step 5: Initiating the Attack
After the primary components have been purchased and configured, it is time to start the attack. After buying the exploits in the form of a BEP, the attacker has to deploy them on the compromised (or purchased) hosting space. For that, the attacker has to select a model (refer to Chapter 3) for triggering the attack. For example, let's say the attacker makes the following choices:
• _Waterholing attack_ : If the attacker wants to conduct a waterholing attack, he will have to penetrate or purchase access to a web site that is frequently visited by the target. Remember, the attacker has already obtained this information in the information-gathering phase of the attack. Since the attacker has the access to the compromised web site, it is easy to embed malicious code (iframe) pointing to a BEP. The attacker then waits for the target to visit the web site and gets infected. The attacker may set this up himself or purchase the service to deploy code on the web site. For example, if the targets are the employees of a big organization, the underground seller may charge the user (buyer) based on the number of successful infections. This approach assures the user that a certain number of targets have been compromised and loaded with malware. Such a complete service is called PPI in the underground market. Figure 7.3 shows the glimpse of this service and what charges are incurred by the buyers.
• _Spear phishing_ : Alternatively, spear phishing may be employed as follows:
• Based on the information-gathering stage, draft an e-mail with a specific theme to make the target sufficiently curious to lure him to open the e-mail and its attachments. This approach is based on the old paradigm: "Curiosity Kills the Cat." Targeted attacks frequently use political, economic, and social themes in emails based on the research conducted for the target. An important choice is whether the exploit is sent as an attachment or embedded as a hyperlink, which redirects the target user to a web site hosting a BEP.
• Once the e-mail is finalized, the next step is to select the Simple Mail Transfer Protocol (SMTP) servers that will be used to deliver the e-mail to the targets. There are several options available for this. First, one can easily configure their own server with an anonymous SMTP configuration. The anonymous configuration removes the sender's name and IP address from the SMTP headers to increase the difficulty of tracing the source. Alternatively, one can buy compromised SMTP servers from the underground market and use those servers to distribute e-mails to the target. In this scenario, if the source is traced, it does not matter because compromised accounts have been used to deliver the e-mails, so the real culprit cannot be identified. Finally, it is also possible to use free SMTP services available on the Internet to send e-mails to the targets. Attackers have multiple options to choose from.
Figure 7.3 PPI infections overview and involved cost. Copyright © 2014 by Aditya K Sood and Richard Enbody.
After performing all the steps discussed above, the targeted attack is under way. Now the attacker simply waits for the targets to fall victim to the attack. As soon as the targets get infected, C&C panels are updated with the appropriate entries and attackers can move on to the next stage to perform further reconnaissance or exfiltrate data.
## 7.6 Role of Freely Available Tools
The security community shares a number of freely available tools that are built to assist researchers and analysts for strengthening security on the Internet. Unfortunately, a number of these tools can also be used for nefarious purposes by attackers. In addition, these tools are easily available on the Internet. An obvious advantage of freely available tools is that attackers do not need to reinvent the wheel. Here are some tools that have already been or can be used in the targeted attacks.
• _Metasploit penetration testing framework_ : Metasploit [4] is an active penetration testing framework that is freely available and is composed of a number of exploits against known vulnerabilities. The idea for designing this framework is to assist penetration testers and researchers to perform aggressive penetration testing and research for assessing the stability of systems. For example, certain BEPs such as Phoenix have used the exploit code that can be found in Metasploit. In addition, the exploits provided by the Metasploit have been used for exploitation [5] by embedding the code in malicious web pages.
• _Social Engineering Toolkit (SET)_ : SET [6] automates the process of conducting social engineering attacks. SET is written to perform aggressive social engineering attacks to test the user awareness and robustness of deployed security in an organization. SET can be easily integrated with Metasploit to automate the exploitation process once the victim falls for the social engineering trick. In this way, SET coerces the user to fall prey to the social engineering attack leading to compromise of the underlying system. SET assists in creating phishing e-mails and can clone a target by building fake replicas of legitimate e-mails and web sites with embedded exploit payloads. SET is designed for legitimate work, but attackers can use it for nefarious purposes.
• _Remote Access Toolkits (RATs)_ : In a number of targeted attacks, attackers have used freely available RATs. The best example of the malicious usage of a publicly available RAT is Poison Ivy [7]. It was designed for supporting administrators to manage systems remotely, but is ideal for attackers who want to control remote systems in a stealthy manner. Poison Ivy agent is served during infection and installed on the compromised system allowing complete remote control of the system. As a result, the attacker can easily execute any command or steal any data without the user's knowledge. Other free RATs exist such as DarkMoon. Overall, free RATs can significantly help an attacker to perform insidious operations.
• _Reconnaissance tools_ : A number of reconnaissance tools are freely available, which also support attackers in targeted attacks. Toolsets such as Sysinternals [8] help attackers to execute commands against other remote systems from a compromised system in a local area network. With the help of these utilities, attackers can target peer systems in the network. A number of password dumping tools such as pwdump help attackers in stealing hashes of all the accounts present on the compromised system. Other tools are available for information gathering, exploiting, privilege escalation, and so on, which are not only a part of the penetration testing process but can also be used in targeted attacks. A long list of tools is available here [9].
The above examples cover only few cases where freely available tools can be used directly or further customized by attackers to serve their needs in targeted attacks.
We have seen how easy it has become to launch targeted attacks with the availability of CaaS. Basically, CaaS provides an easy launchpad for attackers to purchase and use existing attack components. It becomes a component installation process, where you glue a number of components together to build a final product. In addition, CaaS has enabled an underground market place that allows nefarious people to earn black money by operating illegal activities. The result is that if someone has a few thousand dollars, he/she can easily initiate a targeted attack that might not be sophisticated but still be effective. Thanks to the crimeware services provided by the underground community this process is readily accessible to all.
## References
1. Sood A, Enbody R. Crimeware-as-a-service (CaaS): a survey of commoditized crimeware in the underground market. _Int J Crit Infrastruct Prot_. 2013;6(1):28–38.
2. Krebs B, Java zero-day exploit on sale for five digits, <<http://krebsonsecurity.com/2012/11/java-zero-day-exploit-on-sale-for-five-digits>> [accessed 27.12.13].
3. Lee M, Alleged zero day first to bust Adobe's Reader sandbox, <<http://www.zdnet.com/alleged-zero-day-first-to-bust-adobes-reader-sandbox-7000007085>> [accessed 27.12.13].
4. Metasploit, <<http://www.metasploit.com>> [accessed 27.12.13].
5. Sood A, Malware retrospective: infected Chinese servers deploy Metasploit exploits, SecNiche Security Blog, <<http://secniche.blogspot.com/2013/03/malware-retrospective-infected-chinese.html>> [accessed 27.12.13].
6. Social engineering toolkit, <<https://github.com/trustedsec/social-engineer-toolkit/>> [accessed 27.12.13].
7. Poison Ivy, <<http://www.poisonivy-rat.com>> [accessed 27.12.13].
8. Sysinternals, <<http://technet.microsoft.com/en-us/sysinternals/bb545021.aspx>> [accessed 27.12.13].
9. List of tools in BackTrack, <<http://secpedia.net/wiki/List_of_tools_in_BackTrack>> [accessed 27.12.13].
Chapter 8
# Challenges and Countermeasures
This chapter is solely dedicated to building multilayer defenses to protect against targeted attacks. The protection layers include user-centric security, end system security, vulnerability assessment and patch management, network monitoring, and strong response plan. This chapter also presents the need and importance for building next-generation defenses to fight against ever evolving sophisticated malware.
### Keywords
Next-generation Defenses; Cyber Attacks Countermeasures; User Centric Security; Network Level Security; System Level Security
In this chapter, we discuss existing challenges faced by the industry in preventing targeted cyber attacks. In addition, we also present countermeasures and solutions that can result in mitigating the risk of targeted attacks.
## 8.1 Real-Time Challenges
In this section, we present the practical challenges faced by the industry in combating targeted attacks. A number of issues are discussed in the following sections.
### 8.1.1 Persisting False Sense of Security
There exist several misconceptions about the security technologies and standards that can pose a grave risk in certain situations. A number of misconceptions about security are discussed as follows. The list is not meant to be exhaustive.
• _Deployment of Secure Sockets Layer (SSL) prevents you from all types of attacks._ SSL provides defense against network attacks including interception and injection attacks in the network known as Man-in-the-Middle (MitM). These attacks help the attackers to hijack the encrypted communication channels and decrypt them accordingly to retrieve or inject plaintext data. The truth is that SSL does not protect you from all types of data-stealing attacks. SSL does not provide any protection against malware that resides in the end-user system. For example, SSL does not protect us from Man-in-the-Browser (MitB) attacks in which attackers steal the data way before it is encrypted by the SSL.
• _Implementation of firewall as a network perimeter defense makes the environment bulletproof_. Firewalls are designed basically for restricting the unauthorized traffic by dissecting packets at the perimeter level that are flowing to and fro from the internal network. However, firewall functionality is misunderstood in the sense that it makes the network bulletproof. This postulate is also false. A firewall definitely restricts the traffic flow by blocking ports, but the majority of Hypertext Transfer Protocol (HTTP) and HTTP Secure (HTTPS) traffic is allowed through firewalls. Advanced malware uses HTTP/HTTPS as a communication protocol and performs data exfiltration and management through the HTTP/HTTPS channel.
• _Custom encryption provides similar strength as standardized cryptographic algorithms._ Users believe that encryption is good for securing data at rest and in motion. It is certainly a good practice. However, the use of custom encryption algorithms is often equivalent to having no encryption. Publicly vetted encryption (reviewed publicly and analyzed for security flaws by academic and industry researchers) is the only encryption to trust. In particular, if a scheme relies on hiding anything other than keys, it should be suspected. Users need to understand the fact that encrypting data does not mean that it is totally secure.
• _Usage of Two-factor Authentication (TFA) protects from all types of fraudulent activities_. TFA mechanism is based on the concept of out-of-band authentication which is a form of multifactor authentication. It means authentication process is completed using two different channels involving two or more factors possessing some knowledge. TFA is misunderstood by users in determining that it protects from all types of frauds related to money (banking transactions or more). This is not true, as TFA is a function of strong authentication, but it does not protect from the data exfiltration occurring through end-user systems. The second channel in TFA validates and verifies that the authentic user is performing the transaction—nothing more.
• _Deployment of security policies eradicates the risks_. No doubt security policies act as vital components in an organization to harden the security posture. Security policies define the procedures that should be implemented to reduce the risk of business loss occurring due to security breaches or external threats. These policies provide proactive defense to combat threats and subsequent impacts on the organization. However, it does not mean that all the risks are eliminated. Sometimes the policies are not well tested in a production environment, potentially leaving unforeseen holes to be exploited. The considerable point is that how well these policies are followed by users. In addition, security policies should be audited regularly to eradicate the existing loopholes, but a majority of organizations fail to implement this process.
### 8.1.2 Myths About Malware Infections and Protection
There are a number of misconceptions and myths in the industry about malware infections and protection technologies that impact the security countermeasures in fighting malware infections. A number of issues are detailed as follows:
• _Anti-virus (AV) engines provide robust protection_. AV engines are software programs that are installed in the operating systems to prevent the execution of malware and protect legitimate installed applications against any infections. AV engines use techniques such as signature drafting, heuristics, and emulation. Some believe that AV engines protect the end-user system from all types of attacks and malware. For example, some users feel that if an AV solution is installed, they can surf anywhere on the Internet without getting infected. Unfortunately, such users get infected based on this false sense of security. AV engines fall short of providing robust security against zero-day attacks in which attackers use exploits for undisclosed vulnerabilities. Sophisticated malware such as rootkits having administrative access can easily tamper the functioning of AV engines thereby making them inefficient. In addition, AV engines are not considered as a strong security solution to defend against malware classes using polymorphic or metamorphic code which mutates itself on every execution.
• _Deployment of an Intrusion Prevention System (IPS) or Intrusion Detection System (IDS) protects malicious code from entering my network_. The majority of IPS and IDS are signature based, so detecting infection or malicious traffic requires a signature. But attackers can easily bypass IPS and IDS using techniques like unicode encoding, canonicalization, null byte injection, overlapping TCP segments, fragmentation, slicing, and padding [1,2].
• _Malware is distributed primarily through shady and rogue web sites such as torrents and warez_. While rogue sites do distribute malware, many more-trustworthy sites also deliver malware. For example, in targeted attacks based on waterholing (refer to Chapter 3), legitimate and highly ranked web sites are infected with malicious code that downloads malware onto user machines through drive-by download attacks. It is hard to flag sites as secure to ensure users that they are interacting with legitimate web sites free of malware.
• _Email filtering mechanisms only allow secure and verified attachments to be delivered with emails._ Email filtering is a process of filtering out the emails containing malicious attachments and illegitimate links that instantiate infections in the organization network. As described earlier (refer to Chapter 3), social engineered emails are used extensively in targeted attacks. In the corporate world, employees believe that their personal inboxes receive only secure emails with attachments from verified identities. This is not true because attackers can use several tactics such as social engineering with zero-day attacks to slip malware through enterprise email solutions and successfully deliver the malicious emails. The idea is to embed a zero-day exploit inside an attached file that bypasses through the filter and successfully delivers to the target. This technique has been seen in a number of recent targeted attacks.
• _Malware infections are specific to certain operating systems_. For example, Mac OS is much more secure than Windows and is less prone to exploitation. This is false. Mac OS also gets infected with malware and has been targeted by attackers, the recent Flashback [3] malware being one of many. In addition, malware families such as DNS changer [4,5] are platform independent and infect almost all operating systems.
• _Mobile devices are completely secure._ A number of users believe that mobile platforms are secure. Well, that's not true. There has been a significant growth in Android-based mobile platforms, and attackers are targeting these devices to steal information. In this way, mobile devices provide a plethora of information that can help to carry out targeted attacks. For example, contact information is stolen from the infected mobile devices.
• _Virtualization technologies are untouched by malware_. Virtualization is based on the concept of building security through isolation. Virtualization is implemented using hypervisors which are virtual machine monitors that run Virtual Machines (VMs). Hypervisors can be bare-metal (installed directly on the hardware) and hosted (installed in the operating system running on underlying hardware). In virtualized environments, guest VMs are not allowed to access the resources and hardware used by other guest VMs. Virtualization also helps in building secure networks as access controls can be restricted to target networks. Infected virtualized systems can be reverted back to previous snapshots (system state) in a small period of time as opposed to physical servers. Patching is far easier in virtualized servers, and migration of virtualized servers is easy among infrastructure which shows how virtualization provides portability. A number of users believe that hypervisors are immune from malware infections. Unfortunately, virtualized hypervisor malware does exist in the real world. Malware such as Blue Pill [6] is a VM-based rootkit that exploits the hypervisor layer, so that it can circumvent the virtualization model. Basically, when blue pill type of malware is installed in the operating system, the malware creates a new hypervisor on the fly and this hypervisor is used to control the infected system which is now treated as a virtualized system. As a result, it is very hard to detect the malware as it resides in the hypervisor and has the capability to tamper the kernel. These are sophisticated attacks that are difficult but not impossible to implement. A large set of users use VMs for critical operations such as banking which they think provide a secure mode of Internet surfing. The potential compromise of VMs (guest OS) in a network is vulnerable to the same set of attacks as the host OS. In addition, compromising VMs could result in gaining access to other hosts in the network. Several current families of malware are VM-centric which means they incorporate techniques that can easily detect whether the malware is running inside the virtualized machine or not. Based on this information, malware can alter the execution flow. Full hardware-based virtualization (host OS kernel is different from guest OS kernel) prevents malware from gaining access to the underlying host, but the malware can still control the complete guest OS. Partial virtualization (sharing same OS kernel as host) in which privilege restrictions are heavily used to manage virtual file systems can be easily circumvented by malware, if the kernel is exploited.
## 8.2 Countermeasures and Future Developments
In this section, we discuss remediation in the form of countermeasures to manage the risk of targeted attacks including the need for next-generation defenses.
### 8.2.1 Building a Strong Response Plan
Response plans play a crucial role in handling situations when security breaches happen in an organization. The actions defined and applied early minutes in a security breach could have a significant impact on the recovery. This mechanism also works for targeted attacks. As a start, an organization should have a thorough understanding of its critical data, so it knows which data is most important to protect. A good policy is to deploy granular controls at various layers in the organization to have multiple layers of detection and responses (as well as multiple layers of defense). The organization should at least cover these points:
• Understand the storage security of critical data, for example, how the data is secured while at rest and in motion.
• Understand the controls deployed in the network to segregate access to critical data (and servers). Basically, to determine which users have what permissions.
• Understand the inherent control and security mechanisms deployed in the network to detect data exfiltration.
• Understand the effectiveness of the traffic monitoring system to detect anomalies originating in and out of the network.
The overall model of designing a response plan is to react effectively to a security breach in a short period of time to prevent huge losses.
### 8.2.2 End System Security
End-user systems are frequently the target from where the information is stolen. It is highly advised to have a fully secured end-user system by implementing the following:
• The systems should be properly patched with the latest updates installed. There should be minimum delay in deploying patches and updates once the vendor releases them.
• Third-party applications, such as plug-ins and add-ons, should be updated on a regular basis. Avoid using older versions of plug-ins such as Java, Adobe PDF reader, and Flash player as these are the most exploited software because of their integration into browsers.
• Choose a browser for surfing the Internet which has a strong sandbox environment and strong privilege separation controls to avoid execution of exploits, if vulnerabilities are exploited. For example, Google Chrome provides a good sandbox environment for applications to run.
• Deploy a good end-user malware detection and prevention solution. The use of AV systems is still needed because these protect from known risks. For assurance against known malware, AV engines should be frequently updated with new rule sets to avoid infections.
### 8.2.3 User Centric Security
Users are the main targets of attackers. Users' knowledge and understanding of the security principles play a significant role in combating targeted cyber attacks.
• Users should be somewhat paranoid in their surfing habits on the Internet. It is highly advised that users should not click tempting links about which they are not sure of.
• Users should practice safe computing principles such as implementing strong and complex passwords, avoiding usage of same passwords at different web sites, ignoring unsolicited attachments in emails, changing passwords at regular time intervals, and adhering to software security policies. These practices definitely help in making the process harder for attacks, if not impossible.
• Users should use VMs for dedicated surfing or visiting untrusted web sites so that the primary host system can be secured and does not get infected. However, this security posture is based on the choice of an individual, but users eventually benefit from this practice.
• Users should not supply their personal and critical information on web sites (Phishing webpages) or to individuals (rogue phone calls) that are not obviously legitimate. Always verify before providing information and do not rely on the fact that if someone knows some information about you, then he or she is legitimate.
• Naive users that fail to understand the impact and business risks of a security breach that happens based on their actions can be a big problem for organizations. For this reason, organizations should provide regular training to users about ongoing threats and the best practices to mitigate and resist them.
• Users should not use their personal USB devices or peripheral storage devices in an enterprise network because it is a significant risk, as USB devices can carry malicious codes from one place to another.
### 8.2.4 Network Level Security
The network infrastructure and communication channels should be secured to deploy additional layers of security as discussed as follows:
• Organization should deploy robust network perimeter security defenses such as IPS and IDS, email filtering solutions, and firewalls to restrict the entry of malicious code in the internal network. Organizations should install robust Domain Name System (DNS) sinkholes to prevent resolution of illegitimate domains to restrict malicious traffic. Sinkholing is primarily based on the DNS protocol, and the servers are configured to provide falsified information (nonroutable addresses) to the compromised machines running malware. As a result, the malware fails to communicate with the control server and hence data exfiltration is stopped. This strategy should be opted to implement aggressive detection and prevention mechanism to subvert the communication with the malicious servers on the Internet. The sinkholes restrict the occurrence of infections in a silent way. Nothing is bulletproof, but perimeter defenses such as sinkholes add a lot to the security posture of the organization.
• Implementation of Honeynet is also an effective strategy to understand the nature of malware. Honeynet is a network of interconnected systems called as Honeypots that are running with vulnerable configuration, and the malware is allowed to install successfully in one of the systems in the Honeynet. This helps in understanding the malware design and behavior. The harnessed knowledge is used to build secure network defenses.
• Strong traffic monitoring solutions should be deployed on the edges of the networks to filter the egress and ingress traffic flowing through the network infrastructure. The motive is to determine and fingerprint the flow of malicious traffic in the network. At the same time, active monitoring helps to understand user surfing habits and the domains they connect to. In addition, Security Information and Event Management (SIEM) solutions help administrators detect anomalous events occurring in the network. The primary motive behind building SIEM platform is that it takes the output from various resources in the network, that is, events happening in the network, associated threats, and accompanying risks to build strong intelligence feed. This helps the analysts to reduce the impact on the business and to harden the security posture. SIEM is a centralized system that performs correlation and data aggregation over the network traffic to raise alerts.
• Sensitive data flowing to and from the network should be properly encrypted. For example, all the web traffic with sensitive data should be sent over an HTTPS channel. HTTPS means that all the data sent using HTTP is encrypted using SSL. Basically, HTTP is served over SSL and to implement this, the webserver has to first run the SSL service on a specific port and it should serve the SSL certificate to the browser (any client) before starting communication over HTTP. This results in encryption of all the HTTP data exchanged between the browser and the webserver. However, SSL implementation is also available for different protocols. This protocol prevents active MitM attacks which allow malicious intruder to inject arbitrary code to decrypt the encrypted communication channel on the fly.
• Administrators should analyze server logs on a regular basis to find traces of attacks or malicious traffic. The administrators look for the attack patterns related to several vulnerabilities such as injection attacks, file uploading attacks, and brute force attacks. Log provides a plethora of information such as source IP address, timestamps, port access, and number of specific requests handled by the server. Administrators can dissect the malicious traffic to understand the nature of attack. Regular log analysis should be a part of the security process and must be performed on a routine basis.
• Enterprise networks should be properly segregated using well-constructed virtual local area networks which segment the primary network into smaller networks, so that strong access rights can be configured. Basically, dividing network into small segments can help the administrators to deploy security at a granular level.
• Administrators should follow the best secure device configuration practices such as configuring strong and complex passwords for network devices such as routers, printers, and switches, using Simple Network Management Protocol (SNMP) strings that cannot be guessed, avoiding the use of clear text protocols, disabling unrequired services, deploying least privilege principles for restricting access to resources, configuring software installation and device change management policy, and out-of-band management features.
### 8.2.5 Security Assessment and Patch Management
All the network devices and web applications facing the Internet as well as running on the Intranet should be audited regularly against known and unknown vulnerabilities. Security assessments help solidify network security and eradicate critical security flaws. With robust penetration testing, different attacks can be emulated to show how attackers can obtain access to critical components in the network infrastructure. Security teams disclose flaws to the development teams to remove all the critical issues before the applications and devices are deployed in the production environment (Internet facing). Most organizations are using security assessment process as a part of Software Development Life Cycle (SDLC) to have built-in security right from the scratch.
A patch management strategy is necessary and cannot be avoided. Such a strategy supports the administrators to determine the type of patches to be deployed on network devices and software (proprietary and third party) in a tactical and automated manner to boost the overall security model of the organization. Patch management helps administrators to deploy security without affecting the accessibility.
### 8.2.6 Next-generation Defenses
Considering the state of ongoing targeted attacks, the industry requires development of next-generation defenses to protect against zero-day exploits, unknown vulnerabilities, and insidious attack vectors on the fly. The motive is to reduce the infection time, which in turn impacts the persisting nature of the malware. Several new companies are building next-generation defenses and choosing different types of engineering technologies as discussed as follows:
• A machine learning approach [7] can be used to build an effective behavioral-based malware detection system. The collected malware intelligence is used to craft sparse vector models that help in designing malware classification systems. Machine Learning [8] techniques such as Support Vector Machines (SVMs), Naive Bayes (NB), Decision Trees (DTs), and Rule-based (RB) classifiers can be used to build classification models by executing data mining operations on large sets of data. DTs are based on the concept of predictive modeling, and in data mining, DTs are used to predict value of a target parameter based on the conditions (features) provided as part of the input. NB is an independent feature model in which different features present in the base class are independent of each other. NB is based on Bayes theorem and it forces the individual feature in the class to compute the probability of a given condition to make the final decision. NB is also used for supervised learning model. SVM is a machine learning technique based on the concept of regression analysis, and the primary aim is to perform classification from the base class. RB classifier uses predefined rules to make decisions and to classify the data accordingly. These techniques help to generate classifiers based on which malicious patterns can be detected easily. Researchers are also looking into Objective-oriented Association (OOA) mining based classification to build classification system to perform modeling of the malicious code. OOA is based on the concept of defining object classes and defining relations among them to mine the data. The objects can be associated in a unary, binary, and ternary format to define multiple relationships. In addition, ensemble methods are used heavily in which a small set of classifiers are used to achieve better classification based on structural features extracted from a portable executable image.
• Advanced web-based solutions can be used to build effective JavaScript sandboxes which restrict the execution of attacks in browsers. The idea is to effectively utilize the JavaScript with existing web technologies to strengthen the security of client-server communication model.
• Building more robust hardware-based virtualization solutions [9,10] that use technologies such as Intel Virtualization Technology (VT) to detect and restrict the execution of malware by isolating the tasks and processes in the operating system. The idea is to build robust hypervisors that have the ability to identify user-initiated tasks and at the same time configure hardware to isolate the execution of the suspicious process to specific micro VMs. In this way, if the malware running inside micro VM tries to access or write critical sections of the host OS, mandatory access policies can easily restrict the execution of the malware.
• Application whitelisting [11] is another interesting approach followed by organizations to deploy whitelisted software that is allowed to run by restricting the execution of the rest. This approach is based on a client-server model in which a centralized database system is maintained that issues notifications to the clients present in the network to apply software whitelisting. At the same time, the client sends log messages back to the centralized server for later analysis.
This chapter presents countermeasures and proactive steps that are required to defend against targeted attacks. Overall, multilayer defenses are needed to build robust protections against targeted attacks. In addition, existing technologies are not enough to stand against advanced attacks which show that the time demands next-generation defenses.
## References
1. IDefense Team. Intrusion detection system (IDS) evasion, <<http://evader.stonesoft.com/assets/files/read_more/2006_VeriSign_IntrusionDetectionSystemEvasion_iDefenseSecurityReport.pdf>>; [accessed 21.12.13].
2. Barnett R. Evasion: bypassing IDS/IPS systems, <<http://docs.huihoo.com/modsecurity/breach_labs_HTTP_evasion_bypassing_IDS_and_IPS.pdf>>; [accessed 21.12.13].
3. OSX/Flashback. <<http://go.eset.com/us/resources/white-papers/osx_flashback.pdf>>; [accessed 21.12.13].
4. Meng W, Duan R, Lee W. DNS changer remediation study, <<http://www.maawg.org/sites/maawg/files/news/GeorgiaTech_DNSChanger_Study-2013-02-19.pdf>>; [accessed 21.12.13].
5. DCWG. Checking OSX (MAC) for infections, <<http://www.dcwg.org/detect/checking-osx-for-infections>>; [accessed 21.12.13].
6. Desnos A, Filol E. Detection of an HVM rootkit, <<http://www.eicar.org/files/eicar2009tr_hyp.pdf>>; [accessed 22.12.13].
7. Firdausi I, Lim C, Erwin A, Nugroho A.S. Analysis of machine learning techniques used in behavior-based malware detection. In: Proceedings of the Second International Conference on advances in computing, control and telecommunication technologies (ACT '10), pp. 201, 203, 2–3 Dec. 2010. Jakarta, Indonesia. Avaliable from: <http://dx.doi.org/10.1109/ACT.2010.33>.
8. Komashinskiy D, Kotenko I. Malware detection by data mining techniques based on positionally dependent. In: Proceedings of the 18th Euromicro International Conference on parallel, distributed and network-based processing (PDP '10), pp. 617, 623, 17–19 Feb. 2010. Pisa, Italy. Avaliable from: <http://dx.doi.org/10.1109/PDP.2010.30>.
9. Zhao H, Zheng N, Li J, Yao J, Hou Q. Unknown malware detection based on the full virtualization and SVM. In: Proceedings of international conference on management of e-commerce and e-government (ICMECG '09), pp. 473, 476, 16–19 Sept. 2009. Nanchang, China. Avaliable from: http://dx.doi.org/10.1109/ICMeCG.2009.114.
10. Dinaburg A, Royal P, Sharif M, Lee W. _Ether: malware analysis via hardware virtualization extensions_. _Proceedings of the 15th ACM Conference on computer and communications security (CCS '08)_ New York, NY, USA: ACM; 2008; _<http://dx.doi.org/10.1145/1455770.1455779>_ ; 2008; pp.51–62.
11. Gates C, Li N, Chen J, Proctor R. _CodeShield: towards personalized application whitelisting_. _Proceedings of the 28th annual computer security applications conference (ACSAC '12)_ New York, NY, USA: ACM; 2012; _<http://dx.doi.org/10.1145/2420950.2420992>_ ; 2012; pp. 279–288.
Chapter 9
# Conclusion
During the course of this book, we have dissected different phases of targeted attacks as discussed below:
• The first phase started with intelligence gathering about the targets. The attackers spend a considerable amount of time collecting information that is beneficial in constructing the attack surface. The attackers search for online social networks and other repositories containing background details of the target. This process is called Open Source Intelligence (OSINT) in which freely available resources on the Internet are queried to extract information to support initial steps in targeted attacks. OSINT should not be confused with Open Source Software (OSS) as these are both different elements. The presence of "Open Source" is used distinctively in OSINT and OSS but refers to the same standards which are publicly available resources and software. This phase is crucial for success.
• In the second phase, we covered strategies that attackers use to engage their targets so they can infect them with malware. These strategies include spear phishing, waterholing, USB infections, and direct network exploitation. The most widely used strategies are spear phishing and waterholing. In spear phishing, targets are tricked to open exploit-embedded attachments, whereas in waterholing, targets are tricked to visit infected domains serving malware. Both these strategies use social engineering.
• In the third phase, we presented details of how systems are exploited during targeted attacks. We focused on two modes. First, browser-based exploits that exploit vulnerabilities in browsers or third-party browser plugins. The attackers use Browser Exploit Pack (BEP) that fingerprints the browser to find vulnerable components and deliver an appropriate exploit. Second, document-based exploits that exploit vulnerabilities present in documents, for example, Word and Excel. We examined the complex nature of zero-day exploit techniques such as Return-to-libc (R2L) and Return-oriented Programming (ROP) attacks. We also looked at techniques to bypass protection mechanisms such as Data Execution Prevention (DEP) and Address Space Layout Randomization (ASLR). Circumventing those protections allows the execution of arbitrary code often in the context of the operating system.
• In the fourth phase, we detailed exfiltration methods starting with data gathering and finishing with the transmission of stolen data to Command and Control (C&C) servers. Several techniques such as Form-grabbing and Web Injects are widely used in online banking attacks. The stolen information is silently transmitted from the browser to the C&C often using the HTTP channel. In addition, attackers exfiltrate other types of data such as videos, screenshots, and information from colocation services. Attackers bypass peripheral security devices by compressing and otherwise altering the structure of data. Taken together, data exfiltration involves a collection of sophisticated techniques.
• In the final phase, attackers move on to compromise other systems in the network, so that data exfiltration can expand and continue. The attackers perform reconnaissance to search for other vulnerable and misconfigured systems in the network and widen the attack surface. Attackers use Remote Access Toolkits (RATs), batch scripts, and other freely available utilities to take over more targets. Stealthily maintaining control and expanding is considered to be an integral part of targeted attacks.
Based on the information presented in this book, there are some important facts about targeted attacks that bust a number of myths:
• _Targeted attacks must be highly sophisticated to succeed_. Some targeted attacks are indeed quite sophisticated, but simplicity can also succeed. Flame [1] used a very sophisticated MD5 collision attack to spoof Microsoft certificates to mask malicious code as legitimate updates. In contrast, the Taidoor targeted attacks [2] used a simple HTTP backdoor to control compromised machines. Other targeted attacks used publicly available malware (RATs). For example, both RSA (organization) Secure ID and Nitro targeted attacks used Poison Ivy [3], a well-known RAT available on the Internet.
• _Targeted attacks are nation–state sponsored and primarily conducted against government agencies_. Naturally, governments are high-value targets, but targeted attacks also hit high-profile organizations such as mining, chemical, and finance. For example, the Nitro attack [4] targeted chemical companies in order to steal intellectual property such as chemical formulas, production processes, and design documents. Operation Aurora was a highly sophisticated attack that used zero-day exploits and possessed an effective design of command and control architecture [5]. Operation Aurora [6] targeted high-end technology companies such as Google, Adobe Systems, and Juniper Networks. On the question of whether nation–states are behind the targeted attacks, it is always hard to prove their involvement.
• _Targeted attacks are not chained_. Targeted attacks are not necessarily independent of each other. Often, attacks use the stolen information and variants of exploits from previous attacks. Some targeted attacks have employed social engineering based on themes in recent news to design enticing e-mails. For example, attackers designed phishing e-mails based on the Syrian chemical attacks [7] to start new targeted attack campaigns. Both Duqu and Stuxnet [8] targeted Iran's nuclear facilities and used stolen digital certificates and keys.
• _Targeted attacks are uncommon and are only deployed against large organizations_. Almost 80% of attacks are based on opportunity [9], but the remaining 20% represents a significant number. Both small- and large-scale businesses are victims of targeted attacks. The Symantec study [10] revealed that 50% of the targeted attacks are launched against small and mid-sized businesses. Smaller targets are ripe for many targeted attacks using sophisticated techniques. However, small and mid-sized businesses still rely on old defenses and are not well equipped to counter these attacks.
In summary, we need to develop more robust technologies to counter targeted attacks. At the same time, users need to be better educated to understand how attackers utilize social engineering backed by sophisticated techniques to launch cyber attacks.
_Stay Secure!_
## References
1. Analyzing the MD5 collision in Flame, <<http://trailofbits.files.wordpress.com/2012/06/flame-md5.pdf>> [accessed 26.01.14].
2. Wuest C. Dissecting advanced targeted attacks: separating myths from facts, RSA conference, <<http://www.rsaconference.com/writable/presentations/file_upload/spo-301.pdf>> [accessed 25.01.14].
3. Poison Ivy: assessing damage and extracting intelligence, FireEye labs whitepaper, <<http://www.fireeye.com/resources/pdfs/fireeye-poison-ivy-report.pdf>> [accessed 25.01.14].
4. The Nitro attacks: stealing secrets from the chemical industry, <<http://www.symantec.com/content/en/us/enterprise/media/security_response/whitepapers/the_nitro_attacks.pdf>> [accessed 25.01.14].
5. Damballa labs whitepaper, the command structure of Aurora Botnet, <<https://www.damballa.com/downloads/r_pubs/Aurora_Botnet_Command_Structure.pdf>> [accessed 25.01.14].
6. Zetter K. Google hack attack was ultra sophisticated, new details show, <<http://www.wired.com/threatlevel/2010/01/operation-aurora>> [accessed 26.01.14].
7. Chemical attack in Syria used as enticement in targeted attack, <<http://www.symantec.com/connect/blogs/chemical-attack-syria-used-enticement-targeted-attack>> [accessed 26.01.14].
8. Duqu and Stuxnet: a timeline of interesting events, <<http://www.secureviewmag.com/downloads/article_pdf/4th_quarter_secureview_small_file.pdf>> [accessed 26.01.14].
9. 2012 Data Breach Investigations Report, Verizon, <<http://www.verizonenterprise.com/resources/reports/rp_data-breach-investigations-report-2012_en_xg.pdf?__ct_return=1>> [accessed 26.01.14].
10. Internet Security Threat Report 2011, Symantec, <<http://www.symantec.com/content/en/us/enterprise/other_resources/b-istr_main_report_2011_21239364.en-us.pdf>> [accessed 26.01.14].
# Abbreviations
Abbreviation | Description
---|---
AD | Active Directory
AES | Advanced Encryption Standard
API | Application Programming Interface
APT | Advanced Persistent Threat
ASLR | Address Space Layout Randomization
ATM | Automated Teller Machine
ATS | Automated Transfer System
AV | Anti Virus
BEP | Browser Exploit Pack
BLOB | Binary Large Object
BYOD | Bring Your Own Device
C&C | Command and Control
CaaS | Crimeware-as-a-Service
CDN | Content Delivery Network
CIFS | Common Internet File System
CLSID | Class ID
COM | Component Object Model
CredMan | Credential Manager
CSFU | Cross-site File Uploading
CVV | Card Verification Value
CYBINT | Cyber Intelligence
DEP | Data Execution Prevention
DGA | Domain Generation Algorithm
DKOM | Direct Kernel Object Manipulation
DLL | Dynamic Link Library
DNS | Domain Name System
DOM | Document Object Model
DOS | Denial-of-Service
DT | Decision Tree
EaaS | Exploit-as-a-Service
EIP | Extended Instruction Pointer
EMET | Enhanced Mitigation Experience Toolkit
FOCA | Fingerprinting Organizations with Collected Archives
FTP | File Transfer Protocol
GDT | Global Descriptor Table
GOT | Got-it-from-Table
GUI | Graphical User Interface
GUID | Globally Unique Identifier
HTML | Hypertext Markup Language
HTTP | Hypertext Transfer Protocol
HTTPS | HTTP Secure
HUMINT | Human Intelligence
ICS | Industrial Control System
IDS | Intrusion Detection System
IDT | Interrupt Descriptor Table
IM | Instant Messenger
IPS | Intrusion Prevention System
IRC | Internet Relay Chat
IRP | I/O Request Packet
JiT | Just-in-Time
JS | JavaScript
LAN | Local Area Network
LDT | Local Descriptor Table
LM | LAN Manager
LSASS | Local Security Authority Subsystem Service
MAC | Media Access Control
MDN | Malware Distribution Network
MitB | Man-in-the-Browser
MitM | Man-in-the-Middle
NAT | Network Address Translation
NB | Naive Bayes
NetBIOS | Network Basic Input Output System
NOP | No Operation
NTLM | NT LAN Manager
OOA | Objective-oriented Association
OS | Operating System
OSINT | Open Source Intelligence
OSS | Open Source Software
P2P | Peer-to-Peer
PHP | Personal Home Page
PIN | Personal Identification Number
PLC | Programmable Logic Controller
POP3 | Post Office Protocol
PPI | Pay-per-Infection
PPM | Prediction by Partial Matching
PtH | Pass-the-Hash
R2L | Return-to-Libc
RAT | Remote Access Toolkits
RB | Rule-based
RC4 | Arc Four
RDP | Remote Desktop Protocol
ROP | Return-oriented Programming
RPC | Remote Procedure Cal
SAM | Security Accounts Manager
SCADA | Supervisory Control and Data Acquisition
SDLC | Software Development Life Cycle
SDT | Service Dispatch Table
SET | Social Engineering Toolkit
SIEM | Security Information and Event Management
SMB | Server Message Block
SME | Small and Medium Enterprise
SMTP | Simple Mail Transfer Protocol
SNMP | Simple Network Management Protocol
SOCKS | Secure Sockets
SSH | Secure Shell
SSL | Secure Sockets Layer
SSPT | System Service Parameter Table
SVM | Support Vector Machine
TCP | Transmission Control Protocol
TFA | Two-factor Authentication
TFTP | Trivial File Transfer Protocol
Tor | The Onion Router
TR | Task Register
TTL | Time to Live
UDP | User Datagram Protocol
UNC | Universal Naming Convention
USB | Universal Serial Bus
UXSS | Universal Cross-site Scripting
VBA | Visual Basic for Applications
VM | Virtual Machine
VMCI | Virtual Machine Communication Interface
VNC | Virtual Network Connection
VT | Virtualization Technology
WINS | Windows Internet Name Service
WMI | Windows Management Instrumentation
WOW | Windows Over Windows
1. Cover image
2. Title page
3. Copyright
4. A Few Words About _Targeted Cyber Attacks_
5. Acknowledgments
6. About the Authors
7. Overview
8. Chapter 1. Introduction
1. References
9. Chapter 2. Intelligence Gathering
1. 2.1 Intelligence Gathering Process
2. 2.2 OSINT, CYBINT, and HUMINT
3. 2.3 OSNs: A Case Study
4. References
10. Chapter 3. Infecting the Target
1. 3.1 Elements Used in Incursion
2. 3.2 Model A: Spear Phishing Attack: Malicious Attachments
3. 3.3 Model B: Spear Phishing Attack: Embedded Malicious Links
4. 3.4 Model C: Waterholing Attack
5. 3.5 Model D: BYOD as Infection Carriers: USB
6. 3.6 Model E: Direct Incursion: Network Exploitation
7. References
11. Chapter 4. System Exploitation
1. 4.1 Modeling Exploits in Targeted Attacks
2. 4.2 Elements Supporting System Exploitation
3. 4.3 Defense Mechanisms and Existing Mitigations
4. 4.4 Anatomy of Exploitation Techniques
5. 4.5 Browser Exploitation Paradigm
6. 4.6 Drive-By Download Attack Model
7. 4.7 Stealth Malware Design and Tactics
8. References
12. Chapter 5. Data Exfiltration Mechanisms
1. 5.1 Phase 1: Data Gathering Mechanisms
2. 5.2 Phase 2: Data Transmission
3. References
13. Chapter 6. Maintaining Control and Lateral Movement
1. 6.1 Maintaining Control
2. 6.2 Lateral Movement and Network Reconnaissance
3. References
14. Chapter 7. Why Targeted Cyber Attacks Are Easy to Conduct?
1. 7.1 Step 1: Building Targeted Attack Infrastructure
2. 7.2 Step 2: Exploring or Purchasing Stolen Information About Targets
3. 7.3 Step 3: Exploits Selection
4. 7.4 Step 4: Malware Selection
5. 7.5 Step 5: Initiating the Attack
6. 7.6 Role of Freely Available Tools
7. References
15. Chapter 8. Challenges and Countermeasures
1. 8.1 Real-Time Challenges
2. 8.2 Countermeasures and Future Developments
3. References
16. Chapter 9. Conclusion
1. References
17. Abbreviations
## List of tables
1. Tables in Chapter 1
1. Table 1.1
2. Table 1.2
2. Tables in Chapter 2
1. Table 2.1
2. Table 2.2
3. Tables in Chapter 3
1. Table 3.1
4. Tables in Chapter 4
1. Table 4.1
2. Table 4.2
3. Table 4.3
4. Table 4.4
5. Table 4.5
6. Table 4.6
7. Table 4.7
8. Table 4.8
9. Table 4.9
10. Table 4.10
11. Table 4.12
12. Table 4.11
5. Tables in Chapter 5
1. Table 5.1
2. Table 5.2
6. Tables in Chapter 6
1. Table 6.1
2. Table 6.2
3. Table 6.3
4. Table 6.4
5. Table 6.5
6. Table 6.6
7. Table 6.7
## List of illustrations
1. Figures in Chapter 1
1. Figure 1.1
2. Figures in Chapter 2
1. Figure 2.1
2. Figure 2.2
3. Figures in Chapter 3
1. Figure 3.1
2. Figure 3.2
3. Figure 3.3
4. Figures in Chapter 4
1. Listing 4.1
2. Figure 4.1
3. Listing 4.2
4. Figure 4.2
5. Figure 4.3
6. Figure 4.4
7. Listing 4.3
5. Figures in Chapter 5
1. Figure 5.1
2. Listing 5.1
3. Figure 5.2
4. Figure 5.3
6. Figures in Chapter 6
1. Figure 6.1
2. Listing 6.1
3. Figure 6.2
4. Figure 6.3
7. Figures in Chapter 7
1. Figure 7.1
2. Figure 7.2
3. Figure 7.3
## Landmarks
1. Cover Image
2. Title Page
3. Table of Contents
1. Cover
2. III
3. IV
4. V
5. VI
6. IX
7. VII
8. XIII
9. XIV
10. XV
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| {
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} | 7,851 |
\section{Introduction.}
The Jaynes-Cummings model was originally introduced to describe the near resonant
interaction between a two-level atom and a quantized mode of the electromagnetic field~\cite{JC}.
When the field is treated classically, the populations of the two levels exhibit
periodic Rabi oscillations whose frequency is proportional to the field intensity.
The full quantum treatment shows that the possible oscillation frequencies are
quantized, and are determined by the total photon number stored in the mode. For an
initial coherent state of the field, residual quantum fluctuations in the
photon number lead to a gradual blurring and a subsequent collapse of the Rabi oscillations
after a finite time. On even longer time scales, the model predicts a revival of the
oscillations, followed later by a second collapse, and so on. These collapses and revivals
have been analyzed in detail by Narozhny et al~\cite{NSE}, building on the exact solution
of the Heisenberg equations of motion given by Ackerhalt and Rzazewski~\cite{AR}.
Such complex time evolution has been observed experimentally on the Rydberg micromaser~\cite{Rempe87}.
More recently, a direct experimental evidence that the possible oscillation frequencies
are quantized has been achieved with Rydberg atoms interacting with the small coherent fields
stored in a cavity with a large quality factor~\cite{Brune96a}. Another interesting feature
of the Jaynes-Cummings model is that it provides a way to prepare the field in a linear
superposition of coherent states~\cite{Banacloche91}. In the non resonant case, closely related ideas
were used to measure experimentally the decoherence of Schr\"odinger cat states of the
field in a cavity~\cite{Brune96b}.
In the present paper, we shall consider the generalization of the Jaynes-Cummings model where
the two-level atom is replaced by a single spin $s$. One motivation for this is the phenomenon of
superradiance, where a population of identical two-level atoms interacts coherently with the
quantized electromagnetic field. As shown by Dicke~\cite{Dicke}, this phenomenon can be viewed
as the result of a cooperative behavior, where individual atomic dipoles build up to make
a macroscopic effective spin. In the large $s$ limit, and for most initial conditions, a semi-classical
approach is quite reliable. However, the corresponding classical Hamiltonian system with two degrees
of freedom is known to exhibit an unstable equilibrium point, for a large region in its parameter space.
As shown by Bonifacio and Preparata~\cite{Bonifacio}, the subsequent evolution of the system, starting
from such a state, is dominated by quantum fluctuations as it would be for a quantum pendulum
initially prepared with the highest possible potential energy. These authors have found that, at short times,
the evolution of the system is almost periodic, with solitonic pulses of photons separating quieter
time intervals where most of the energy is stored in the macroscopic spin. Because the stationary states
in the quantum system have eigenenergies that are not strictly equidistant, this quasiperiodic behavior
gives way, at longer times, to a rather complicated pattern, that is reminiscent of the collapses and
revivals in the $S=1/2$ Jaynes-Cummings model.
An additional motivation for studying the large $s$ Jaynes-Cummings model comes from recent
developments in cold atom physics. It has been shown that the sign and the strength of the
two body interaction can be tuned at will in the vicinity of a Feshbach resonance, and this
has enabled various groups to explore the whole crossover from the Bose Einstein condensation
of tightly bound molecules~\cite{Greiner03} to the BCS condensate of weakly bound atomic
pairs~\cite{Bourdel04,Zwierlein04}. In the case of a fast sweeping of the external magnetic field
through a Feshbach resonance, some coherent macroscopic oscillations in the molecular condensate
population have been predicted theoretically~\cite{Levitov}, from a description of the low
energy dynamics in terms of a collection of $N$ spins $1/2$ coupled to a single harmonic oscillator.
This model has been shown to be integrable~\cite{YKA} by Yuzbashyan et al. who emphasized its connection
with the original integrable Gaudin model~\cite{Gaudin83}. It turns out that in the cross-over region,
the free Fermi sea is unstable towards the formation of a pair condensate, and that this instability
is manifested by the appearance of two pairs of conjugated complex frequencies in a linear analysis.
This shows that for any value of $N$ (which is also half of the total number of atoms), these coherent
oscillations are well captured by an effective model with two degrees of freedom, one spin, and one
oscillator. Therefore, there is a close connection between the quantum dynamics in the neighborhood
of the classical unstable point of the Jaynes-Cummings model, and the evolution of a cold Fermi gas after
an attractive interaction has been switched on suddenly.
The problem of quantizing a classical system in the vicinity of an unstable
equilibrium point has been a subject of recent interest, specially in the mathematical
community. The Bohr-Sommerfeld quantization principle suggests that the density of states
exhibits, in the $\hbar$ going to zero limit, some singularity at the energy of the
classical equilibrium point, in close analogy to the Van Hove singularities for the
energy spectrum of a quantum particle moving in a periodic potential. This intuitive
expectation has been confirmed by rigorous~\cite{Colin94a,Colin94b,Brummelhuis95}
and numerical~\cite{Child98} studies. In particular, for a critical level of a system
with one degree of freedom, there are typically $|\ln \hbar|$ eigenvalues in an energy interval
of width proportional to $\hbar$ around the critical value~\cite{Colin94b,Brummelhuis95}.
A phase-space analysis of the corresponding wave-functions shows that they are concentrated
along the classical unstable orbits which leave the critical point in the remote past and
return to it in the remote future~\cite{Colin94a}. To find the eigenstates requires
an extension of the usual Bohr-Sommerfeld rules, because matching the components of the
wave-function which propagate towards the critical point or away from it is a special
fully quantum problem, which can be solved by reduction to a normal form~\cite{Colin94a,Colin99}.
Finally, we will show that the spin $s$ Jaynes-Cummings model is an example
of an integrable
system for which it is impossible to define global action-angle coordinates.
By the Arnold-Liouville theorem, classical
integrability implies that phase space is foliated by $n$-dimensional
invariant tori. Angle coordinates are
introduced by constructing $n$-independent periodic Hamiltonian flows on
these tori. Hence each invariant torus is equipped with a lattice of symplectic
translations which act as the identity on this torus, the lattice of periods
of these flows, which is equivalent to the data of $n$ independent cycles on the torus.
To get the angles we still must choose an origin on these cycles.
All this can be done in a continuous way for close enough nearby tori
showing the existence of local action angle variables.
Globally, a first obstruction can come from the impossibility of choosing an
origin on each torus in a consistent way. In the case of the Jaynes-Cummings
model, this obstruction is absent (see \cite{Duistermat80}, page 702). The
second obstruction to the existence of global action-angle variables comes from the
impossibility of choosing a basis of cycles on the tori in a uniform way.
More precisely, along each curve in the manifold
of regular invariant tori, the lattice associated to each torus can be
followed by continuity.
This adiabatic process, when carried along a closed loop, induces an
automorphism on the lattice attached to initial (and final) torus,
which is called the monodromy ~\cite{Duistermat80}.
Several simple dynamical systems, including the spherical
pendulum~\cite{Duistermat80} or the the champagne bottle potential~\cite{Bates91} have been
shown to exhibit such phenomenon. After quantization, classical monodromy induces topological
defects such as dislocations in the lattice of common eigenvalues of the mutually commuting
conserved operators~\cite{San99}. Interesting applications have been found, specially
in molecular physics~\cite{Sadovskii06}. In the Jaynes-Cummings model, the monodromy is
directly associated to the unstable critical point, because it belongs to a singular
invariant manifold, namely a pinched torus. This implies that the set of regular tori is not
simply connected, which allows for a non-trivial monodromy, of the so-called
focus-focus type~\cite{Zou92,Zung97,Cushman01}. The quantization of a generic system which
such singularity has been studied in detail by V\~u Ng\d{o}c~\cite{San00}. Here, we present
an explicit semi-classical analysis of the common energy and angular momentum eigenstates
in the vicinity of their critical values, which illustrates all the concepts just mentioned.
\bigskip
In section~[\ref{classical}] we introduce the classical Jaynes-Cummings model and describe its stationnary points, stable and unstable. We then explain and compute the classical monodromy when we loop around the unstable point. We also introduce a reduced system that will be important in subsequent considerations. In section~[\ref{quantique}] we define the quantum model and explain that the appearence of a default in the joint spectrum of the two commuting quantities is directly related to the monodromy phenomenon in the classical theory.
In section~[\ref{semiclassique}] we perform the above reduction directly on the quantum system and study its Bohr-Sommerfeld quantization. The reduction procedure favors some natural coordinates which however introduce some subtleties in the semi classical quantization : there is a subprincipal symbol and moreover the integral of this subprincipal symbol on certain trajectories may have unexpected jumps.We explain this phenomenon. The result is that the Bohr-Sommerfeld quantization works very well everywhere, except for energies around the critical one. In section~[\ref{semiclassiquesing}] we therefore turn to the semiclassical analysis of the system around the unstable stationnary point. We derive the singular Bohr-Sommerfeld rules. Finally in section~[\ref{evolution}] we apply the previous results to the calculation of the time evolution of the molecule formation rate, starting from the unstable state. For this we need the decomposition of the
unstable state on the eigenstates basis. We find that only a few states contribute and there is a drastic reduction of the dimensionality of the relevant Hilbert space. To compute these coefficients we remark that it is enough to solve the time dependent Schr\"odinger equation for small time. We perform this analysis and we show that we can extend it by gluing it to the time dependant WKB wave function. This gives new insights on the old result of Bonifacio and Preparata.
\section{Classical One-spin system.}
\label{classical}
\subsection{Stationary points and their stability}
Hence, we consider the following Hamiltonian
\begin{equation}
H= 2 \epsilon s^z + \omega \bar{b} b + g \left( \bar{b} s^-+ b s^+ \right)
\label{bfconspinclas1}
\end{equation}
Here $s^z = s^3, s^\pm = s^1\pm i s^2$ are spins variables, and $b,\bar{b}$ is a harmonic oscillator.
The Poisson brackets read
\begin{equation}
\{ s^a , s^b \} = - \epsilon_{abc} s^c, \quad \{ b , \bar{b} \} = i
\label{poisson1}
\end{equation}
Changing the sign of $g$ amounts to changing $(b,\bar{b})\to (-b,-\bar{b})$ which is a symplectic transformation. Rescaling the time we can assume $g=1$.
The Poisson bracket of the spin variables is degenerate. To obtain a symplectic manifold, we fix the value of the Casimir function
$$
\vec{s} \cdot \vec{s} = (s^z)^2 + s^+ s^- = s_{cl}^2
$$
so that the corresponding phase space becomes the product of a sphere by a plane and has a total dimension $4$.
Let us write
$$
H= H_0 + \omega H_1
$$
with
\begin{equation}
H_1 = \bar{b} b + s^z,\quad H_0 = 2\kappa s^z + \bar{b} s^-+ b s^+, \quad \kappa = \epsilon - \omega/2
\label{hamclas}
\end{equation}
Clearly we have
$$
\{ H_0, H_1 \} = 0
$$
so that the system in integrable\footnote[1]{This remains true for the $N$-spin system. In the Integrable Community this model is known to be a limiting case of the Gaudin model \cite{YKA}}.
Hence we can solve simultaneously the evolution equations
$$
\partial_{t_0} f = \{ H_0, f \}, \quad \partial_{t_1} f = \{ H_1, f \}
$$
Once $f(t_0,t_1)$ is known, the solution of the equation of motion $\partial_{t} f = \{ H, f \}= (\partial_{t_0}+ \omega \partial_{t_1})f$ is given by $f(t,\omega t)$. The equations of motion read
\begin{align}
\partial_{t_1} b &= - ib &\partial_{t_0} b &= - i s^- &\partial_{t} b &= - i s^- -i\omega b
\label{motionb2} \\
\partial_{t_1} {s}^z &=0 & \partial_{t_0} {s}^z &= i ( \bar{b} s^- - b s^+ ) & \partial_{t} {s}^z &= i ( \bar{b} s^- - b s^+ )
\label{motionsz2} \\
\partial_{t_1}{s}^+ &= i s^+ &\partial_{t_0}{s}^+ &= 2i \kappa s^+ -2i \bar{b} s^z &\partial_{t}{s}^+ &= 2i \epsilon s^+ -2i \bar{b} s^z \label{motions+2}\\
\partial_{t_1}{s}^- &= -i s^- &\partial_{t_0}{s}^- &= -2i \kappa s^- +2i b s^z &\partial_{t}{s}^- &= -2i \epsilon s^- +2i b s^z \label{motions-2}
\end{align}
The $t_1$ evolution is simply a simultaneous rotation around the $z$ axis of the spin and a rotation of the same angle
of the harmonic oscillator part:
\begin{equation}
b(t_0,t_1) = e^{-it_1} b(t_0),\quad s^\pm(t_0,t_1) = e^{\pm it_1}s^\pm(t_0),\quad s^z(t_0,t_1) = s^z(t_0)
\label{t1flow}
\end{equation}
In physical terms, the conservation of $H_1$ corresponds to the
conservation of the total number of particles in the system, when this model (with $N$ quantum spins 1/2)
is used to describe the coherent dynamics between a Fermi gas and a condensate of
molecules~\cite{Levitov}.
We wrote the full equations of motion to emphasize the fact that the points
\begin{equation}
s^\pm = 0, \quad s^z = \pm s_{cl},\quad b=b^\dag = 0
\label{stationary1}
\end{equation}
are {\em all} the critical points (or stationary points i.e. all time derivatives equal zero) of {\it both} time evolutions $t_0$ and $t_1$ and hence are very special. Any Hamiltonian, function of $H_0$ and $H_1$, will have these two points among its critical points. However, it may have more.
For instance when $H=H_0+\omega H_1$ an additional family of stationnary points exist when $\omega^2 \epsilon^2 < s_{cl}^2$. They are given by
\begin{equation}
s^{\pm} = \sqrt{s_{cl}^2 -\epsilon^2 \omega^2}\; e^{\pm i\varphi}, \quad
s^z = -\epsilon \omega, \quad b= -{1\over \omega} \sqrt{s_{cl}^2 -\epsilon^2 \omega^2}\; e^{-i\varphi}
\quad \forall \varphi
\label{stationary2}
\end{equation}
their energy is given by
$$
E' = -{1\over \omega}(s_{cl}^2 + \omega^2 \epsilon^2)
$$
The energies of the configurations Eq.~(\ref{stationary1}) are
$$
E = \pm 2 \epsilon s_{cl}
$$
so that
$$
E-E' = {1\over \omega}(s_{cl} \pm \epsilon \omega)^2 >0
$$
and we see that $E'$ represents the degenerate ground states of $H$, breaking rotational invariance around the $z$ axis.
\bigskip
We are mostly interested in the configurations Eq.~(\ref{stationary1}). To fix ideas we assume $\epsilon <0$ so that among these two configurations, the one with minimal energy is the spin up. It looks like being a minimum however it becomes unstable for some values of the parameters. To see it, we perform the analysis of the small fluctuations around this configuration.
We assume that $b, \bar{b}, s^{\pm}$ are first order and $s^z = s_{cl} e + \delta s^z$ where $e=\pm 1$
according to whether the spin is up or down. Then
$\delta s^z$ is determined by saying that the spin has length $s_{cl}$
$$
\delta s^z = - {e\over 2s_{cl}} s^- s^+
$$
This is of second order and is compatible with Eq.~(\ref{motionsz2}). The linearized equations of motion (with respect to $H$) are
\begin{eqnarray}
\dot{b} &=& -i \omega b - i s^- \label{motionb1}\\
\dot{s}^- &=& -2i \epsilon s^- +2is_{cl} e b \label{motions-1}
\end{eqnarray}
and their complex conjugate.
We look for eigenmodes in the form
$$
b(t)=b(0)e^{-2iEt},\quad s^- = s^-(0) e^{-2iE t}
$$
We get from Eq.~(\ref{motions-1})
$$
s^-(0) = -{s_{cl}} {e\over E-\epsilon} b(0)
$$
Inserting into Eq.~(\ref{motionb1}),
we obtain the self-consistency equation for $E$:
\begin{equation}
E = {\omega\over 2} - {s_{cl}\over 2} {e \over E-\epsilon }
\label{eqE1}
\end{equation}
The discriminant of this second degree equation for $E$ is $ \kappa^2 - 2 s_{cl} e$.
If $e=-1$ it is positive and we have a local energy maximum. However if $e=+1$, the roots become complex when $\kappa^2 \leq 2s_{cl}$. In that case one of the modes exponentially increases with time, i.e. the point is {\em unstable}.
Since we are in a situation where $\kappa \leq 0$, the transition occurs when $\kappa = - \sqrt{2s_{cl}}$.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 10cm]{phaseportrait3Dbis.eps}
\caption{The image in phase space of the level set of the critical unstable point $H_0=2\kappa s_{cl}, H_1=s_{cl}$ is a pinched two dimensional torus.}
\label{figphase3D}
\end{center}
\nonumber
\end{figure}
The level set of the unstable point, i.e. the set of points in the four-dimensional phase space of the spin-oscillator system, which have the same values of $H_0$ and $H_1$ as the critical point $H_0=2\kappa s_{cl}, H_1=s_{cl}$ has the topology of a pinched two-dimensional torus see Fig.[\ref{figphase3D}]. This type of stationary point is known in the mathematical litterature as a
focus-focus singularity~\cite{Zou92,Zung97,Cushman01}.
The above perturbation analysis shows that in the immediate vicinity of the critical point, the pinched
torus has the shape of two cones that meet precisely at the critical point. One of these cones is associated to
the unstable small perturbations, namely those which are exponentially amplified, whereas the other cone
corresponds to perturbations which are exponentially attenuated. Of course, these two cones are connected, so that
any initial condition located on the unstable cone gives rise to a trajectory which reaches eventually
the stable cone in a finite time. Note that this longitudinal motion from one cone to the other is
superimposed to a spiraling motion around the closed cycle of the pinched torus which is generated by the action
of $H_1$.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 7cm]{momentClas10.eps}
\caption{The image in $R^2$ of the phase space by the map $(H_1\equiv m-s_{cl},H_0)$. The green dots are the critical points. The stable one is the point on the vertical axis $(m=0,-2\kappa s_{cl})$ and the unstable one is located at $(m=2s_{cl}, 2\kappa s_{cl})$. The upper and lower green boundaries are obtained as explained in the text. }
\label{figmoment}
\end{center}
\nonumber
\end{figure}
An important object is the image of phase space into $R^2$ under the map $(H_1,H_0)$. It is shown in Fig.[\ref{figmoment}]. It is a convex domain in $R^2$. The upper and lower green boundaries are obtained as follows. We set
$ x= \bar{b} b$ and
\begin{equation}
H_1 = x + s^z \equiv m-s_{cl}, \quad {\rm Max}(0,m-2s_{cl}) \leq x \leq m
\label{xvariation}
\end{equation}
where the parameter $m$ is introduced to ease the comparison with the quantum case. Then we set
$$
b = \sqrt{x}e^{i\theta}, \bar{b} = \sqrt{x}e^{-i\theta},\quad s^{\pm} = \sqrt{s_{cl}^2 - (s^z)^2} e^{\mp i\varphi}
= \sqrt{(m-x)(2s_{cl}-m+x)}e^{\mp i\varphi}
$$
so that
$$
H_0= 2\kappa(m-s_{cl}-x) + 2 \sqrt{x(m-x)(2s_{cl}-m+x)} \cos(\theta-\varphi)
$$
It follows that
$$
2\kappa(m-s_{cl}-x) - 2 \sqrt{x(m-x)(2s_{cl}-m+x)} \leq H_0 \leq 2\kappa(m-s_{cl}-x) + 2 \sqrt{x(m-x)(2s_{cl}-m+x)}
$$
To find the green curves in Fig.[\ref{figmoment}] we have to minimize the left hand side and maximize the right hand side in the above inequalities when $x$ varies in the bounds given in Eq.~(\ref{xvariation}).
\bigskip
In the unstable case, because an initial condition close to the critical point leads to a large
trajectory along the pinched torus, it is interesting to know
the complete solutions of the equations of motion. They are easily obtained as follows. Remark that
\begin{eqnarray*}
(\dot{s}^z )^2 &=& - ( \bar{b} s^- - b s^+ )^2 = - ( \bar{b} s^- + b s^+ )^2
+4 \bar{b} b s^+ s^- \\
&=& - (H_0-2\kappa s^z)^2 +4 (H_1-s^z )(s_{cl}^2-(s^z)^2)
\end{eqnarray*}
or
$$
(\dot{s}^z )^2 = 4 (s^z)^3 -4(H_1 +\kappa^2) (s^z)^2 + 4(\kappa H_0 - s_{cl}^2) s^z + (4s_{cl}^2 H_1 -H_0^2)
$$
Hence, $s^z$ satisfies an equation of the Weierstrass type and is solved by elliptic functions.
Indeed, setting
$$
s^z = {1\over 3}( H_1+\kappa^2) - X, \quad \dot{s}^z = i Y
$$
the equation becomes
\begin{equation}
Y^2 = 4 X^3 - g_2 X - g_3
\label{ellipticcurve}
\end{equation}
with
$$
g_2 = {4\over 3} ( H_1^2 - 3 H_0 \kappa + 2 H_1 \kappa ^2 + \kappa ^4 + 3 s_{cl}^2)
$$
$$
g_3 = {1 \over 27} (-27 H_0^2 - 8 H_1^3 + 36 H_0 H_1 \kappa - 24 H_1^2 \kappa ^2 +
36 H_0 \kappa ^3 - 24 H_1 \kappa ^4 - 8 \kappa ^6 + 72 H_1 s_{cl}^2 -
36 \kappa ^2 s_{cl}^2)
$$
Therefore, the general solution of the one-spin system reads
\begin{equation}
s^z = {1\over 3}( H_1+\kappa^2) - \wp(it + \alpha)
\label{fullsolutionsz}
\end{equation}
and
\begin{equation}
\bar{b} b = H_1 - s^z = {1\over 3} (2 H_1 - \kappa^2) + \wp(it+ \alpha)
\label{fullsolutionb2}
\end{equation}
where $\alpha $ is an integration constant and $\wp(\theta)$ is the Weierstrass function associated to the curve Eq.~(\ref{ellipticcurve}).
Initial conditions can be chosen such that when $t=0$, we start from a point intersecting the real axis
$Y=0$. This happens when $\alpha $ is half a period:
$$
\alpha = \omega_1~{\rm or}~ \omega_2~{\rm or}~ \omega_3 = - \omega_1 - \omega_2
$$
\bigskip
This general solution however is not very useful because the physics we are interested in
lies on Liouville tori specified by very particular values of the conserved quantities $H_0, H_1$.
For the configuration Eq.~(\ref{stationary1}) with spin up, the values of the Hamiltonians are
$$
H_0 = 2\kappa s_{cl}, \quad H_1 = s_{cl}
$$
In that case we have
$$
g_2 = {4 \over 3} \Omega^4,\quad g_3 = {8 \over 27} \Omega^6
$$
and
$$
4 X^3 - g_2 X - g_3 = 4\left(X + {1\over 3}\Omega^4 \right)^2\left(X - {2\over 3}\Omega^2 \right)
$$
where we have defined
\begin{equation}
\Omega^2 = 2 s_{cl} - \kappa^2 >0 {\rm ~~in~the~} unstable~{\rm case.}
\label{defomega}
\end{equation}
Hence, we are precisely in the case where the elliptic curve degenerates. Then the solution of the equations of motion is expressed in terms of trignometric functions
$$
X(t) = {2\over 3} \Omega^2 -\Omega^2\tanh^2 (\Omega (t-t_0) )
$$
and
\begin{equation}
x(t) \equiv \bar{b} b(t) = {\Omega^2
\over \cosh^2\Omega(t-t_0)}
\label{bbarclas}
\end{equation}
This solution represents a single solitonic pulse centered at time $t_0$. When the initial
condition is close but not identical to
the unstable configuration with spin up and $b=\bar{b}=0$, this unique soliton is replaced
by a periodic sequence of pulses that can be described in terms of the Weiertrass
function as shown above. Note that in the context of cold fermionic atoms,
$\bar{b}b$ represents the number of molecular bound-states that have been formed in the system.
\subsection{Normal form and monodromy}
\label{secclassicalmonodromy}
The dynamics in the vicinity of the unstable equilibrium can also be visualized
by an appropriate choice of canonical variables in which the quadratic parts in the
expansions of $H_0$ and $H_1$ take a simple form. Let us introduce the angle $\nu \in ]0,\pi/2[$
such that $|\kappa|=\sqrt{2s_{cl}}\cos\nu$ and $\Omega=\sqrt{2s_{cl}}\sin\nu$. As we
have shown, instability occurs when $\kappa^{2}<2s_{cl}$, which garanties that $\nu$ is real.
Let us then define two complex coordinates $A_s$ and $A_u$ by:
\begin{equation}
\left(\begin{array}{c} A_{s} \\ A_{u}
\end{array}\right)= \frac{1}{\sqrt{2\sin\nu}}
\left(\begin{array}{cc} e^{-i\nu/2} & e^{i\nu/2} \\
e^{i\nu/2} & e^{-i\nu/2}
\end{array}\right)
\left(\begin{array}{c} b \\ \frac{s^{-}}{\sqrt{2s_{cl}}}
\end{array} \right)
\end{equation}
The classical Poisson brackets for these variables are:
\begin{eqnarray}
\{A_s,\bar{A}_{s}\} & = & \{A_u,\bar{A}_{u}\} = 0 \\
\{A_s,\bar{A}_{u}\} & = & \{\bar{A}_{s},A_{u}\} = 1
\end{eqnarray}
Here we have approximated the exact relation $\{s^{+},s^{-}\}=2is^{z}$ by $\{s^{+},s^{-}\}=2is_{cl}$,
which is appropriate to capture the linearized flow near the unstable fixed point.
After subtracting their values at the critical point which do not influence the dynamics,
the corresponding quadratic Hamiltonian $H_0$ and global rotation generator $H_1$ read:
\begin{eqnarray}
H_0 & = & 2\Omega \Re(\bar{A}_{s}A_{u})+ \kappa H_1 \\
H_1 & = & 2\Im(\bar{A}_{s}A_{u})
\end{eqnarray}
With the above Poisson brackets, the linearized equations of motion take the form:
\begin{eqnarray}
\dot{A}_{s} & = & (-\Omega - i \kappa ) A_{s} \label{evolAs}\\
\dot{A}_{u} & = & (\Omega - i \kappa) A_{u} \label{evolAu}
\end{eqnarray}
With these new variables, the pinched torus appears, in the neighborhoood
of the unstable point, as the union of two planes intersecting transversally.
These are defined by $A_{u}=0$ which corresponds to the stable branch, and
by $A_{s}=0$ which gives the unstable branch. The global rotations generated
by $H_1$ multiply both $A_{s}$ and $A_{u}$ by the same phase factor $e^{-it_1}$. We may then
visualize these two planes as two cones whose common tip is the critical point, as
depicted on Fig.~\ref{figphase3D}. The stable
cone is obtained from the half line where $A_{s}$ is real and positive and $A_{u}=0$,
after the action of all possible global rotations. A similar description holds for the
unstable cone. As shown by Eq.~(\ref{bbarclas}),
any trajectory starting form the unstable branch eventually reaches the stable
one. This important property will play a crucial role below in the computation
of the monodromy.
For latter applications, specially in the quantum case, it is useful to write down
explicitely $A_{s}$ and $A_{u}$ in terms of their real and imaginary parts:
\begin{eqnarray}
A_{s} & = & (P_{1}-iP_{2})/\sqrt{2} \\
A_{u} & = & (X_{1}-iX_{2})/\sqrt{2}
\end{eqnarray}
This reproduces the above Poisson brackets, provided we set $\{X_{i},X_{j}\}=0$,
$\{P_{i},P_{j}\}=0$, $\{P_{i},X_{j}\}=\delta_{ij}$ for $i,j\in\{1,2\}$.
The quadratic Hamitonian now reads:
\begin{eqnarray}
H_0 & = & \Omega (X_{1}P_{1}+X_{2}P_{2}) + \kappa H_1 \\
H_1 & = & X_{1}P_{2}-X_{2}P_{1}
\end{eqnarray}
which is the standard normal form for the focus-focus
singularity~\cite{Zou92,Zung97,Cushman01}.
We are now in a position to define and compute the monodromy attached to a closed path
around the unstable point in the $(H_0,H_1)$ plane as in Fig.~\ref{figmoment}.
Let us consider a one parameter family of regular invariant tori
which are close to the pinched torus. This family can be described by
a curve in the $(H_0,H_{1})$ plane, or equivalently, in the complex plane
of the $\bar{A}_{s}A_{u}$ variable. Let us specialize to the case of a closed loop.
Its winding number around the critical value in the $(H_0,H_{1})$ plane is the same
as the winding number of $\bar{A}_{s}A_{u}$ around the origin in the complex plane.
Let us consider now the path $\chi\in[0,2\pi]\rightarrow\bar{A}_{s}A_{u}=\eta^{2}e^{i\chi}$
where $\eta>0$ is assumed to be small. To construct the monodromy, we need to define,
for each value of $\chi$, two vector fields on the corresponding torus, which are
generated by $\chi$-dependent linear combinations of $H_0$ and $H_1$ and whose flows
are $2\pi$-periodic. Furthermore we require that the periodic orbits
of these two flows generate a cycle basis $(e_1,e_2)$ in the two-dimensional homology of the torus.
This basis evolves continuously as $\chi$ increases from 0 to $2\pi$. The monodromy
expresses the fact that $(e_1,e_2)$ can turn into a different basis after one closed loop
in the $(H_0,H_{1})$ plane. The discussion to follow has been to a large extent inspired by
Cushman and Duistermat~\cite{Cushman01}.
One of these two flows can be chosen as the $2\pi$ global rotation generated by $H_1$.
The corresponding orbits are circles which provide the first basic cycle $e_1$.
For the other one, we first emphasize that the flow generated by
$H_0$ is not periodic in general. So, to get the other basic cycle $e_2$, we have to consider
the flow generated by an appropriate linear combination of $H_0$ and $H_1$.
To construct it, it is convenient to consider an initial
condition $(A_{s}(0),A_{u}(0))$ such that $\bar{A}_{s}(0)A_{u}(0)=\eta^{2}e^{i\chi}$ and
$|A_{s}(0)| \ll |A_{u}(0)|$, that is close to the unstable manifold of the pinched torus.
Let us pick a small enough $\zeta>0$ so that the linearized equations of motion are
still acurate when $|A_{s}(0)|^{2}+|A_{u}(0)|^{2}<\zeta^{2}$, and let us also assume that
$\eta \ll \zeta$. Using the global rotation invariance of the dynamics, we may choose
$A_{u}(0)=\zeta$ and $A_{s}(0)=\frac{\eta^{2}}{\zeta}e^{-i\chi}$. If we let the system evolve
starting from this initial condition, $|A_{s}|$ will first decrease and $|A_{u}|$
will increase, so the trajectory becomes closer to the unstable torus, along its
unstable manifold. After a finite time $t_{1}$, the trajectory reappears in the neighborhood
of the unstable fixed point, but now, near the stable manifold. This behavior can be seen
either as a consequence of Eq.~(\ref{bbarclas}) for the trajectories on the pinched torus
and extended to nearby trajectories by a continuity argument, or can be checked directly from
the explicit solution of the classical dynamics Eqs.~(\ref{fullsolutionsz},\ref{fullsolutionb2})
in terms of the Weierstrass function. We may thus choose $t_1$
such that $|A_{s}(t_1)|=\zeta$. A very important property of $A_{s}(t_1)$ is that it has
a well defined limit when $\eta$ goes to zero, because in this limit, one recovers the
trajectory on the pinched torus such that $A_{s}(0)=0$ and $A_{u}(0)=\zeta$.
As a result, when $\eta$ is small enough,
the argument of $A_{s}(t_1)$ in polar coordinates weakly depends on $\chi$ and
the winding number of $A_{s}(t_1)$ when $\chi$ goes from 0 to $2\pi$ vanishes.
We then let the system evolves until time $t_2$ using the linearized flow.
The time $t_2$ is chosen in order to recover $|A_{u}(t_2)|=\zeta$. Using Eq.~(\ref{evolAu})
and the fact that $\bar{A}_{s}A_{u}$ is conserved,
this gives $t_{2}-t_{1}=(2/\Omega)\ln(\zeta/\eta)$. Furthermore, from
$\bar{A}_{s}(t_1)A_{u}(t_1)=\eta^{2}e^{i\chi}$ and Eq.~(\ref{evolAu}), we deduce:
\begin{equation}
A_{u}(t_2)=e^{i\chi}\exp\left(-2i{\kappa \over \Omega}\ln\frac{\zeta}{\eta}\right)A_{s}(t_1)
\end{equation}
In general, we have no reason to expect that $A_{u}(t_2)$ should be equal to $A_{u}(0)$ so,
to get a periodic flow on the torus, the evolution generated by $H_0$ during the time $t_2$
has to be followed by a global rotation of angle $\beta (\chi)$ given by:
\begin{equation}
e^{i\beta (\chi)}=e^{i\chi}\exp\left(-2i{\kappa \over \Omega} \ln\frac{\zeta}{\eta}\right)
A_{s}(t_1)/\zeta
\end{equation}
The sign of $\beta$ reflects the fact that the $H_1$ flow applied during the time $\beta$
multiplies $A_{s}$ and $A_{u}$ by $e^{-i\beta}$.
Note that because the flows associated to $H_0$ and $H_1$ commute, the composition
of the $H_0$-flow during time $t_2$ and of the $H_1$ flow along an angle $\beta$ can also
be viewed as the flow generated by the linear combination $H_0+(\beta/t_2)H_1$ during time $t_2$.
We have established the periodicity of this flow for the orbit starting from
$A_{u}(0)=\zeta$ and $A_{s}(0)=\frac{\eta^{2}}{\zeta}e^{-i\chi}$. But any other orbit on the
invariant torus charaterized by $\bar{A}_{s}A_{u}=\eta^{2}e^{i\chi}$ can be deduced from this
one by a global rotation, which commutes with the flow of $H_0+(\beta/t_2)H_1$. This establishes
the periodicity of this latter flow, and therefore allows us to construct the other basic cycle
$e_2(\chi)$, which depends smoothly on $\chi$.
Because the winding number of $A_{s}(t_1)$ vanishes provided $\eta$ is small enough, we have the
crucial relation:
\begin{equation}
\beta (2\pi)=\beta (0)+2\pi
\end{equation}
In other words, when we follow by continuity the periodic flow generated by
$H_0+(\beta/t_2)H_1$ during time $t_2$, the final flow is deduced from the initial one
by a global rotation of $2\pi$.
To summarize, as $\chi$ increases smoothly from $0$ to $2\pi$, $e_1$ is left unchanged, whereas $e_2$
evolves into $e_2+e_1$. In this basis, the monodromy matrix of the focus-focus singularity
is then:
\begin{equation}
M=\left(\begin{array}{lr} 1 & 1 \\ 0 & 1 \end{array} \right)
\end{equation}
\subsection{Reduced system.}
As we have discussed before, the common level set of $H_0$ and $H_1$ which contains the unstable critical point
is a pinched torus. Since we are mostly interested in the time evolution of the oscillator energy $\bar{b}b$, and
because this quantity is invariant under the Hamiltonian action of $H_1$, it is natural to reduce the dynamics
to the orbits of $H_1$. For an initial condition on the pinched torus, this amounts to discard the spiraling
motion around the torus, and to concentrate on the longitudinal motion from the stable cone to the
unstable one. In other words, this reduction procedure reduces the pinched torus into a curve which has a cusp
at the critical point. This is illustrated by the thick curve on Fig.~\ref{phaseportrait3D1}.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 5cm]{phaseportrait3D1bis.eps}
\caption{The sphere represents the reduced phase-space obtained by fixing the conserved quantity $H_1$ to its value $s_{cl}$
at the critical point. Here, this point corresponds to the north pole on the sphere. The red curve is the image of the pinched torus
after the reduction procedure. It is obtained by setting the other integral of motion $H_0$ to its critical value $H_0=2\kappa s_{cl}$.
For this, we have used the expression of $H_0$ given in Eq.~(\ref{H0reducedclassical}). The singular nature of the pinched
torus is reflected by the cusp of the red curve at the north pole.}
\label{phaseportrait3D1}
\end{center}
\nonumber
\end{figure}
An other description of this trajectory is given by the thick curve on Fig.~\ref{figphase} that will be discussed below.
In practice, the first thing to do to implement this reduction is to fix the value of $H_1$ to its critical value
\begin{equation}
H_1 = \bar{b} b + s^z = s_{cl}
\label{H1critical}
\end{equation}
This defines a submanifold ${\cal H}_{s_{cl}}$ in phase space. Since $H_0$ and $H_1$ Poisson commute we may consider the reduced system
obtained by performing a Hamiltonian reduction by the one parameter group generated by $H_1$.
The reduced phase space is
the quotient of ${\cal H}_{s_{cl}}$ by the flow generated by $H_1$ (the trivial $t_1$ phases in Eq.~(\ref{t1flow})). It is of dimension 2.
Explicitly, we set:
$$
x = \bar{b}b >0 \Longrightarrow b = \sqrt{x} e^{i\theta},\quad \bar{b} = \sqrt{x} e^{-i\theta}
$$
From Eq.~(\ref{H1critical}) and from the condition $(s^z)^2+ s^+ s^- = s_{cl}^2$ we deduce
\begin{equation}
s^z = s_{cl}-x, \quad {\rm and} \quad s^\pm = \sqrt{ x(2s_{cl}-x)} e^{\mp i\varphi}
\label{sreduit}
\end{equation}
the reduced Hamiltonian $H_0$ reads
\begin{equation}
H_0 = 2\kappa s_{cl} + 2x \sqrt{2s_{cl} -x} \cos(\theta-\varphi) - 2\kappa x
\label{H0reducedclassical}
\end{equation}
Notice that $x$ and $\theta-\varphi$ are invariant by the $H_1$ flow, and they can be taken as coordinates on the reduced phase space. These coordinates are canonically conjugate as we easily see by writing the symplectic form:
$$
\omega = -i \delta b \wedge \delta \bar{b} + i {\delta s^+ \over s^+} \wedge \delta s^- =
\delta x \wedge \delta (\theta-\varphi)
$$
In the following, when talking about this reduced system, we will simply set $\varphi=0$. These coordinates are very convenient but one has to be aware that they are singular at $x=0$ and $x=2s_{cl}$. The whole segment $x=0, 0 \leq \theta < 2\pi$ should be identified to one point and similarly for the segment $x=2s_{cl}, \, 0 \leq \theta < 2\pi$.
The equations of motion of the reduced system read
\begin{eqnarray*}
\dot{x} &=& 2 x \sqrt{2s_{cl} -x} \sin \theta \\
\dot{\theta} &=& -2\kappa +2 \left( \sqrt{2s_{cl} -x} - {x\over 2 \sqrt{2s_{cl} -x}} \right)\cos \theta
\end{eqnarray*}
From this we see that the critical points are given by
$$
x=0,\quad \cos \theta = {\kappa \over \sqrt{2s_{cl}}},\quad {\rm or}\quad z^2 - {2\kappa \over \sqrt{2s_{cl}}} z + 1 = 0, \quad z=e^{i\theta}
$$
In our case $\kappa$ is negative, so that the solutions of the quadratic equations will be defined as
$$
z_\pm = {\kappa \pm i \Omega \over \sqrt{2s_{cl}}} = - e^{\mp i\nu} , \quad \Omega^2 = 2s_{cl} - \kappa^2, \quad 0 \leq \nu \leq \pi/2
$$
These points exist precisely in the unstable regime $\kappa^2 \leq 2s_{cl}$. They correspond to only one point in phase space whose energy is given by
\begin{equation}
H_0= 2\kappa s_{cl} \equiv E_{c}
\label{defEc}
\end{equation}
hence they correspond to the unstable point. The image of the pinched critical torus after the reduction procedure
is shown on Fig.~\ref{figphase} as the thick line connecting $z_{+}$ and $z_{-}$. Note that this line intersects
the $x=0$ segment with a finite angle. This angle reflects precisely the pinching of the torus at the
unstable critical point in the original four-dimensional phase space.
\bigskip
Another critical point is obtained by setting $\theta = \pi$. Then
\begin{equation}
x = 2s_{cl} - X^2, \quad {\rm with } \quad X={1\over 3} (-\kappa + \sqrt{\kappa^2 + 6 s_{cl}} )
\label{groundstate}
\end{equation}
Since $0\leq x \leq 2s_{cl}$, we should also have $0 \leq X^2 \leq 2s_{cl}$ which is the case
when $\kappa \geq -\sqrt{2s_{cl}} $, that is to say in the unstable region.
The energy is given by
\begin{equation}
H_0= 2\kappa s_{cl} -2(X+\kappa) (2s_{cl} -X^2)
\label{ground0clas}
\end{equation}
We see from that equation that, in the unstable region, $H_0-2\kappa s_{cl}$ is always negative and the configuration Eq.~(\ref{groundstate}) represents the ground state.
\bigskip
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 6cm]{phaseportrait.eps}
\caption{The phase portrait corresponding to the Hamiltonian $H_0$, Eq.~(\ref{H0reducedclassical}), in the $(\theta,x)$ variables, which
are natural coordinates for the reduced system.
The thick red line corresponds to the critical separatrix $H_0= 2\kappa s_{cl}$, which is the image of the
pinched critical torus after symplectic reduction. Note that the unstable critical point is mapped onto the
vertical axis at $x=0$, so that the critical separatrix is indeed a loop.
The blue dots correspond to the minimal and maximal values of $H_0$ when $H_1= s_{cl}$.}
\label{figphase}
\end{center}
\nonumber
\end{figure}
Finally, the energy maximum of this reduced system is obtained for $\theta=0$ and $x=2s_{cl}-Y^2$,
with $Y=(\kappa+\sqrt{\kappa^{2}+6})/3$. On the phase portrait shown on Fig.~\ref{figphase}, one sees
a curve which contains the point at $x=2s_{cl}$ and which looks like a separatrix. However, unlike the situation
around $x=0$, we notice that it does not correspond to any stationary point of the unreduced Hamiltonian. In fact it is simple to show that the curve is actually tangent to the vertical line $x=2s_{cl}$. On the sphere, this is the trajectory passing through the south pole.
\section{Quantum One-spin system.}
\label{quantique}
\subsection{Energy spectrum}
We now consider the quantum system. We set as in the classical case
$$
H = H_0+\omega H_1
$$
with
\begin{eqnarray*}
H_0 &=& (2\epsilon -\omega) s^z + b^\dag s^- + b s^+
= 2\kappa s^z + b^\dag s^- + b s^+, \quad\quad H_1= b^\dag b + s^z
\end{eqnarray*}
where as before
\begin{equation}
\kappa = \epsilon - \omega/2
\label{defkappa}
\end{equation}
We impose the commutation relations
$$
[b,b^\dag ] = \hbar, \quad [s^+,s^-] = 2\hbar s^z,\quad [s^z,s^\pm] = \pm \hbar s^\pm
$$
We assume that the spin acts on a spin-$s$ representation.
$$
s^z \vert m \rangle = \hbar m \vert m \rangle,\quad
s^\pm \vert m \rangle = \hbar \sqrt{s(s+1) - m(m \pm 1)} \; \vert m \pm 1\rangle,
\quad m = -s, -s+1, \cdots , s-1,s
$$
where $2s$ is integer. Of course
$$
(s^z)^2 +{1\over 2} (s^+ s^- + s^- s^+) = \hbar^2 s(s+1)
$$
We still have
$$
[ H_0, H_1] = 0
$$
and the quantum system is integrable. The Hamiltonian $H$ can be diagonalized by Bethe Ansatz, but we will not follow this path here. For related studies along this line for the Neumann model see \cite{BeTa}.
Note that recently, the non equilibrium dynamics of a similar quantum integrable model (the Richardson model) has been studied by a combination
of analytical and numerical tools.~\cite{Faribault09}
Let:
\begin{equation}
e_n = (2b^\dag)^{n} \vert 0 \rangle \otimes (s^+)^{M-n} \vert -s \rangle, \quad {\rm Sup}(0,M-2s) \leq n \leq M
\label{defen}
\end{equation}
For all these states one has:
$$
H_1 e_n = \hbar (M-s) e_n
$$
Since $H_0$ commutes with $H_1$ we can restrict $H_0$ to the subspace spanned by the $e_n$.
Hence, we write:
$$
\Psi= \sum_{n= {\rm Sup}(0,M-2s) }^{M} p_n \;\; {e_n\over ||e_n||}, \quad
$$
where the norm $||e_n|| $ of the vector $e_n$ is given by:
$$
||e_n||^2 = \hbar^{2M-n}2^{2n} {(2s)! (M-n)! n! \over (2s-M+n)!}
$$
Using
$$
H_0 e_n = 2\hbar \kappa (M-s-n) e_n +{ \hbar^2 \over 2} (M-n)(2 s +1-M+ n) e_{n+1} + 2\hbar n e_{n-1}
$$
the Schr\"odinger equation
$$
i\hbar {\partial \Psi \over \partial t} = H_0 \Psi
$$
becomes:
\begin{eqnarray*}
i\hbar {\partial p_n \over \partial t}&=& \hbar^{3\over 2} \sqrt{(n+1)(2s+1-M+n)(M-n)} p_{n+1} + \hbar^{3\over 2} \sqrt{n(2s-M+n)(M+1-n)} p_{n-1} \\
&& + 2\hbar \kappa (M-n-s) p_n
\end{eqnarray*}
and the eigenvector equation $H_0 \Psi= E \Psi$ reads:
\begin{eqnarray}
\hbar^{3\over 2} \sqrt{(n+1)(2s+1-M+n)(M-n)} p_{n+1} &+& \hbar^{3\over 2} \sqrt{n(2s-M+n)(M+1-n)} p_{n-1} \\
&+& 2\hbar \kappa (M-n-s) p_n = E p_n
\label{EigEqM}
\end{eqnarray}
In this basis, the Hamiltonian is represented by a symmetric Jacobi matrix and it is easy to diagonalize it numerically. Varying $M=0,1,\cdots$
we construct the lattice of the joint spectum of $H_0,H_1$.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 7cm]{latticeClas10.eps}
\caption{The lattice of the joint spectrum of $H_0$ and $H_1$ (or equivalently the integer $M$) in the unstable regime.
The green point represents the classical unstable point ($H_1= s_{cl}$, or $M=2s$, and $H_0=2\kappa s_{cl}$).
The parameters are $\hbar^{-1} = s=10$, and $\kappa= -0.5\sqrt{2s_{cl}}$.
This figure demonstrates the non-trivial quantum monodromy in this system. For this purpose, we follow by continuity
an elementary cell of this lattice along two closed paths. When the path encloses the image of the critical
point in the ($H_0$, $H_1$) plane, we see that an initial square cell does not preserve its shape at the
end of the cycle, but it evolves into a parallelogram with a tilt. In other words, the lattice of joint eigenvalues
exhibits a dislocation located at the critical value of $H_0$ and $H_1$.}
\label{figlattice}
\end{center}
\nonumber
\end{figure}
We see on Fig.~\ref{figlattice} that the lattice has a defect located near the unstable classical point.
This defect induces a quantum monodromy when we transport a cell of the lattice around it.
The rule to transport a cell is as follows. We start with a cell and we choose the next cell
in one of the four possible positions (east, west, north, south) in such a way that two edges
of the new cell prolongate two sides of the original cell, and the two cells have a common edge.
We apply these rules on a path which closes. If the path encloses the unstable point,
the last cell is different from the initial cell.
The precise form of this lattice defect can be related to the classical monodromy matrix $M$,
as shown by V\~u Ng\d{o}c~\cite{San99}. Choosing locally a cycle basis $e_i$ on the Arnold-Liouville
tori is equivalent to specifying local action-angle coordinates $(J_i,\phi_i)$ such that $e_i$ is
obtained from the periodic orbit generated by $J_i$. After one closed circuit in the base space
of this fibration by tori, the basic cycles are changed into $e'_{i}=\sum_{j}M_{ji}e_{j}$ and therefore,
the new local action variables $J'_{i}$ are deduced from the initial ones by:
\begin{equation}
J'_{i}=\sum_{j=1}^{n}M_{ji}J_{j}
\label{changeactionvar}
\end{equation}
Heuristically, the Bohr-Sommerfeld quantization principle may be viewed as the requirement that
the quantum wave functions should be $2\pi$ periodic in each of the $n$ phase variables $\phi_i$,
which implies that $J_i$ should be an integer multiple of $\hbar$. When $\hbar$ is small, the discrete
set of common eigenvalues of the conserved operators has locally the appearance of a regular lattice,
whose basis vectors $v_1$,...,$v_n$ are approximately given by: $dJ_{i}(v_{j})=\hbar \delta_{ij}$.
Because of the monodromy, this lattice undergoes a smooth deformation as one moves around a critical
value of the mutually commuting conserved quantities. After one complete turn, the basic lattice vectors
are changed into $v'_1$,...,$v'_n$, which are also approximately given by: $dJ'_{i}(v'_{j})=\hbar \delta_{ij}$.
Taking into account the definition of the classical monodromy matrix in Eq~(\ref{changeactionvar}), this
leads to the semi-classical monodromy on the joint spectrum:
\begin{equation}
v'_{i}=\sum_{j=1}^{n}(M^{-1})_{ij}v_{j}
\end{equation}
The discussion above was performed in the action angle variables $(J_1,J_2)$. In the $(H_0,H_1)$ variables
the basis $v, v'$ should be transformed by the Jacobian matrix of the mapping $(J_1,J_2)\to (H_0(J_1,J_2),H_1(J_1,J_2))$.
On Fig.~\ref{figlattice}, we see that after a clockwise turn around the singular value in the
$(H_1,H_0)$ plane (which corresponds to a positive winding in the $(H_0,H_1)$ plane), the lattice
vectors $v_M$ and $v_{H_{0}}$ are transformed into $v_M-v_{H_{0}}$ and $v_{H_{0}}$ respectively. This is
in perfect agreement with the classical monodromy matrix computed in section~\ref{secclassicalmonodromy}.
\bigskip
When $M=2s$, i.e. when $H_1$ takes its critical value corresponding to the unstable point,
$$
H_1 e_n = \hbar s \; e_n
$$
the Schr\"odinger equation simplifies to:
\begin{equation}
i\hbar {\partial p_n \over \partial t}= \hbar^{3\over 2} (n+1)\sqrt{(2s-n)} p_{n+1} + \hbar^{3\over 2}n \sqrt{(2s+1-n)} p_{n-1}
+ 2\hbar \kappa (s-n) p_n
\label{schroerduite}
\end{equation}
Setting
$$
x=\hbar n,\quad s_{cl}=\hbar s
$$
this equation becomes:
\begin{equation}
i\hbar {\partial p(x) \over \partial t}= (x+\hbar)\sqrt{(2s_{cl}-x)} p(x+\hbar) +x \sqrt{(2s_{cl}+\hbar-x)} p(x-\hbar) + 2 \kappa (s_{cl}-x) p(x)
\label{schroex}
\end{equation}
Introducing the shift operator
$$
e^{i\theta} p(x) = p(x+\hbar)
$$
this can be rewritten as:
\begin{equation}
i\hbar {\partial p(x) \over \partial t}= \Big[\sqrt{(2s_{cl}-x)} e^{i\theta} x +x
e^{-i\theta} \sqrt{(2s_{cl} -x)} + 2 \kappa (s_{cl}-x) \Big] p(x)
\label{defH0quant}
\end{equation}
and we recognize the quantum version of the classical reduced Hamiltonian Eq.~(\ref{H0reducedclassical}), but with a very
specific ordering of the operators.
\bigskip
The eigenvector equation $H_0 \Psi= E \Psi$ reads:
\begin{equation}
\hbar^{3\over 2} (n+1)\sqrt{(2s-n)} p_{n+1} + \hbar^{3\over 2}n \sqrt{(2s+1-n)} p_{n-1}
+( 2\hbar \kappa (s-n) -E) p_n =0
\label{EigEq}
\end{equation}
The eigenvalues are shown in Figure~\ref{figeigen1} in the stable case and
in the unstable case.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 3cm]{figeigen1.eps}
\includegraphics[height= 3cm]{figeigen2.eps}
\caption{The eigenvalues of the reduced $H_0$ in the stable regime on the left and in the unstable case on the right. The origin of the Energy axis is the classical critical energy $E_c=2\hbar s \kappa$.
In the unstable case the ground state energy lies below this line and close to the classical value given by Eq.~(\ref{ground0clas}).
( $\hbar^{-1} = s=30$, $\kappa= -2\sqrt{2s_{cl}}$ (stable case), $\kappa= -0.5\sqrt{2s_{cl}}$ (unstable case)). }
\label{figeigen1}
\end{center}
\nonumber
\end{figure}
\section{Bohr-Sommerfeld quantization}
\label{semiclassique}
In this section we examine the standard semi-classical analysis of the one spin system. There are two complications as compared to the usual formulae. One comes from the fact that the phase space of the spin degrees of freedom is a 2-sphere and hence has a non trivial topology. The second one is related to the fact that the Schr\"odinger equation is a difference equation and its symbol has a subprincipal part. Let us recall the Schr\"odinger equation:
\begin{equation}
(x+\hbar)\sqrt{ 2s_{cl} -x}\; p(x+\hbar) + x\sqrt{2s_{cl} +\hbar -x}\; p(x-\hbar) -2 \kappa x \; p(x)= \hbar \epsilon p(x)
\label{schroepsilon1}
\end{equation}
where
$$
\hbar \epsilon = E-2\hbar \kappa s, \quad x=\hbar n
$$
The first thing to do is to compute the Weyl symbol $h(x,\theta)$ of the Hamiltonian operator.
In standard notations, it is defined by:
$$
(Hp)(x) = {1\over 2\pi \hbar} \int e^{-{i\over \hbar} (x-y) \theta } h\left({x+y\over 2},\theta \right) p(y) d\theta dy
$$
It is straightforward to check that, for the Hamiltonian of eq.(\ref{schroepsilon1})
$$
h(x,\theta)= 2 \left(x+{\hbar\over 2}\right) \sqrt{2s_{cl} +{\hbar \over 2} -x} \; \cos \theta + 2 \kappa (s_{cl} -x)
$$
Expanding in $\hbar$ we find the principal and sub-principal symbols
$$
h(x,\theta) = h^{0}(x,\theta) + \hbar h^{1}(x,\theta) + \cdots
$$
where:
\begin{eqnarray*}
h^{0}(x,\theta) &=& 2 x \sqrt{2s_{cl} -x} \; \cos \theta + 2 \kappa (s_{cl} -x) \\
h^{1}(x,\theta) &=& \left( \sqrt{2s_{cl} -x} + {x\over 2 \sqrt{2s_{cl} -x} } \right) \cos \theta
\end{eqnarray*}
We recognize, of course, that $h^{0}(x,\theta)$ is the classical Hamiltonian of the system Eq.(\ref{H0reducedclassical}).
In particular the symplectic form reads:
$$
\omega = dx \wedge d\theta
$$
Note that the definition of the Weyl symbol we have used here is motivated by the simpler
case where the classical phase space can be viewed as the cotangent bundle of a smooth manifold.
In our case, the $\theta$ coordinate is $2\pi$ periodic, and we are not strictly speaking dealing with
the cotangent bundle over the line parametrized by the $x$ coordinate.
To justify this procedure, we may first imbed our system in a larger phase space, where both
$\theta$ and $x$ run from $-\infty$ to $\infty$.
Because the symbol $h(x,\theta)$ is $2\pi$ periodic in $\theta$,
the unitary operator $\mathcal{T}$ associated to the $2\pi$ translation of $\theta$ commutes with
the quantum Hamiltonian $H$ associated to the symbol $h(x,\theta)$. It is then possible to perform
the semi-classical analysis in the enlarged Hilbert space and to project afterwards the states thus obtained
on the subspace of $2\pi$ periodic wave-functions, which are eigenvectors of $\mathcal{T}$ with the
eigenvalue 1. Imposing the periodicity in $\theta$ forces $x$ to be an integer multiple of $\hbar$.
From Eq.~(\ref{schroepsilon1}), we see that the physical subspace, spanned by state vectors $|x=n\hbar\rangle$
with $0 \leq n \leq 2s$, is stable under the action of $H$.
An alternative approach would be to use a quantization scheme, such as Berezin-Toeplitz
quantization~\cite{Berezin75}, which allows to work directly with a compact classical phase-space
such as the sphere. When $H_1$ takes its critical value corresponding to the unstable point,
it is easy to show that the eigenvector equation~(\ref{EigEq})
can be cast into the Schr\"odinger equation for a pure spin $s$ Hamiltonian given by:
\begin{equation}
H_{\mathrm{eff}}=2\kappa s^{z}+s^{+}\sqrt{s_{cl}-s_{z}}+\sqrt{s_{cl}-s_{z}}\,s^{-}
\end{equation}
Starting from the known Toeplitz symbols of the basic spin operators, one may derive Bohr-Sommerfeld
quantization rules directly from this Hamiltonian $H_{\mathrm{eff}}$~\cite{Charles06}, but we shall not
explore this further in this paper.
Returning to our main discussion we now choose a 1-form $\gamma$ such that:
\begin{equation}
\gamma (X_{h^0}) = - h^1(x,\theta)
\label{subprincipalform}
\end{equation}
where $X_{h^0}$ is the Hamiltonian vector field associated to the Hamiltonian $h^0(x,\theta)$, which is tangent to the variety $h^0(x,\theta) = E$. Using $x$ as the coordinate on this manifold, we find:
$$
X_{h^0} = 2x \sqrt{2s_{cl}-x} \sin \theta \; \partial_x
$$
Hence, we may choose:
$$
\gamma = -{1\over 2} \left( {1\over x} + {1\over 2(2s_{cl}-x)}\right) \cot \theta \; dx
$$
Under these circumstances, the Bohr-Sommerfeld quantization conditions involve the form $\gamma$ and read
(see e.g. \cite{San99})
\begin{equation}
\Phi_{Reg}(\epsilon) = {1\over 2\pi \hbar }\int_{C(E)} \alpha + {1\over 2\pi}\int_{C(E)} \gamma + \mu_{C(E)} {1\over 4} = n
\label{phireg}
\end{equation}
where $C(E)$ is the classical trajectory of energy $E$, $\alpha$ is the canonical 1-form $\omega = d\alpha$,
$ \mu_{C(E)}$ is the Maslov index of this trajectory and $n$ is an integer. Note that the integral of $\gamma$
over $C(E)$ is completely specified by the constraint Eq.(\ref{subprincipalform}).
We now come to the fact that the 2-sphere has a non trivial topology.
A consequence is that the canonical 1-form $\alpha$ does not exist globally.
In the coordinates $x,\theta$ we may choose
$$
\alpha = x\: d\theta
$$
but this form is singular in the vicinity of the south pole where $x=2s_{cl}$.
Let us consider a closed path on the sphere parametrized by the segment at constant $x$,
$\theta$ running from 0 to $2\pi$. The integral of $\alpha$ around this path is $2\pi x$.
When $x$ goes to $2s_{cl}$, the integral goes to $4\pi s_{cl}$ which is the total area of the
sphere. But the limit path for $x=2s_{cl}$, corresponds on the sphere to a trivial loop, fixed
at the south pole. The consistency of Bohr-Sommerfeld quantization requires then $4\pi s_{cl}$
to be an integer multiple of $2\pi\hbar$. Hence we recover the quantization condition of the spin:
$2 s$ should be an integer.
Before checking the validity of Eq.(\ref{phireg}) in our system, it is quite instructive to
plot the integral of $\gamma$ along the curve $C(E)$ as a function of $E$. This is shown on Fig.~\ref{figsousdominant}
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 7cm]{BohrSommerfeldSing.eps}
\caption{The integral of the sub-dominant symbol as a function of energy (the red curve).
We see the logarithmic divergence for the energy corresponding to the singular critical point,
and the gap $-\pi$ at the energy corresponding to the trajectory passing through the south pole.
The green curve is the asymptotic formula Eq.(\ref{asympt}) in which we have added the gap at the south pole
to ease comparison and to show that it remains quite accurate even well beyond its expected range of validity.}
\label{figsousdominant}
\end{center}
\nonumber
\end{figure}
Here, we see two singularities at the values of $E$ where $C(E)$ goes through
either the north or the south pole. At the south pole $\int_{C(E)} \gamma$ merely jumps by $\pi$,
whereas at the north pole, a jump of $\pi$ is superimposed onto a logarithmic divergence.
We can easily find an asymptotic formula for this integral for energies close to the critical value $E_c= 2 s_{cl} \kappa$:
\begin{equation}
\oint \gamma = {\kappa \over \Omega} \log{ \sqrt{2s_{cl}} | \Delta E | \over 16 \Omega^4}
+ 2(\pi - \nu) - \pi\; \Theta(\Delta E) + O(\Delta E, \Delta E \log |\Delta E|)
\label{asympt}
\end{equation}
here $\Delta E = E-E_c$, and $\Theta(\Delta E) = 1$ if $\Delta E >0$ and $0$ if $\Delta E<0$.
We wish now to show that these jumps are merely an effect of working with polar coordinates
$(x,\theta)$. It is instructive to consider first a quantum system with one degree of freedom,
whose classical phase-space is the $(q,p)$ plane. From any smooth function $h(p,q)$, we define
the quantum operator $\mathcal{O}_{W,h}$ by:
$$
(\mathcal{O}_{W,h}(\Psi))(q) = {1\over 2\pi \hbar} \int e^{{i\over \hbar} (q-q')p} h\left(p,{q+q'\over 2}\right) \Psi(q') dp\:dq'
$$
Polar coordinates are introduced, at the classical level by:
$$
(p,q)=\sqrt{2x}(\cos \theta,\sin \theta)
$$
This definition implies that $dp \wedge dq = dx \wedge d\theta$.
At the quantum level, we start with the usual $(\hat{p}, \hat{q})$ operators.
In the Hilbert space $L^{2}(q)$, we introduce the eigenvector basis $\{|n\rangle\}$ of $\hat{p}^{2}+\hat{q}^{2}$,
such that $(\hat{p}^{2}+\hat{q}^{2}) |n\rangle=\hbar (2n+1)|n\rangle$. We may then set $\hat{x}=\sum_{n\geq 0}\hbar n|n\rangle\langle n|$.
To define $\hat{\theta}$, such that $[\hat{\theta},\hat{x}]=i\hbar$, it is natural to enlarge the Hilbert space,
allowing $n$ to take also negative integer values as well. Then we set $\exp(i\hat{\theta})|n\rangle=|n+1\rangle$.
From $a|n\rangle=\sqrt{n}|n-1\rangle$ and $a^{+}|n\rangle=\sqrt{n+1}|n+1\rangle$, we see that, in the physical
subspace $n \geq 0$, we have $a=\hbar^{-1/2}\exp(-i\hat{\theta})\hat{x}^{1/2}$ and $a^{+}=\hbar^{-1/2}\hat{x}^{1/2}\exp(i\hat{\theta})$.
The Weyl symbols, in the $(x,\theta)$ variables, of these operators are $\hbar^{-1/2}(x+\hbar/2)^{1/2}\exp(\mp i\theta)$.
Using $\hat{p}=(\hbar/2)^{1/2}(a+a^{+})$ and $\hat{q}=(\hbar/2)^{1/2}i(a-a^{+})$, we deduce that the Weyl symbols of
$\hat{p}$ and $\hat{q}$ in the $(x,\theta)$ variables are respectively $\sqrt{2x+\hbar}\:\cos\theta$
and $\sqrt{2x+\hbar}\:\sin\theta$. So the Weyl symbol is not invariant under the non-linear change of coordinates from
$(p,q)$ to $(x,\theta)$. For linear functions of $p$ and $q$ the symbol in $(x,\theta)$, denoted by $h_{\mathrm{pol}}(x,\theta)$
is obtained from $h(p,q)$ by substituting $p$ and $q$ by their classical expressions as functions of $(x,\theta)$
and then by shifting $x$ into $x+\hbar/2$. Such a simple rule does not hold for more complicated functions $h(p,q)$.
Nevertheless, one can show that the substitution of $x \rightarrow x+\hbar/2$ gives the symbol $h_{\mathrm{pol}}(x,\theta)$
up to first order in $\hbar$. Practically, this means that:
$$
h_{\mathrm{pol}}(x,\theta)=h(\sqrt{2x+\hbar}\cos\theta,\sqrt{2x+\hbar}\sin\theta) \;\;\;\mathrm{mod}\;\mathcal{O}(\hbar^{2})
$$
or, more explicitely:
$$
h_{\mathrm{pol}}(x,\theta)=h(p,q)+\frac{\hbar}{4x}\left(p\frac{\partial h}{\partial p}(p,q)+q\frac{\partial h}{\partial q}(p,q)\right)
+\mathcal{O}(\hbar^{2})
$$
where $(p,q)=\sqrt{2x}(\cos \theta,\sin \theta)$.
This has the following important consequence: even if $h(p,q)$ has no subprincipal part (i.e. it does not depend on
$\hbar$), $h_{\mathrm{pol}}(x,\theta)$ does acquire a subprincipal part $h_{\mathrm{pol}}^{1}(x,\theta)$.
Now, along any classical trajectory associated to
$h(p,q)$, we have:
$$
p\frac{\partial h}{\partial p}(p,q)+q\frac{\partial h}{\partial q}(p,q)=p\dot{q}-q\dot{p}=2x\dot{\theta},
$$
so $h_{\mathrm{pol}}^{1}(x,\theta)=\dot{\theta}/2$. This striking result implies that, when we work in polar
coordinates, the $\gamma$ form associated to $h_{\mathrm{pol}}^{1}(x,\theta)$ may be chosen as $-d\theta /2$. This
immediately explains why $\int_{C(E)} \gamma$ jumps by $-\pi$ when the $h(p,q)=E$ orbit crosses the origin of the
polar coordinates, because then $\int_{C(E)} d\theta$ jumps from 0 to $2\pi$, see Fig.~\ref{ellipses}.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 7cm]{ellipses.eps}
\caption{On this figure we see that $-{1\over 2} \oint d\theta$ equals zero if the integral is taken over the small ellipse, while it is equal to $-\pi$ if it is taken over the big ellipse enclosing the origin. This explains why the integral of the subprincipal symbol in polar
coordinates jumps by $\pm \pi$ when the orbit crosses the origin of the polar coordinates.}
\label{ellipses}
\end{center}
\nonumber
\end{figure}
Note that, in general, we expect
$h(p,q)$ to have also a subprincipal part $h^{1}(p,q)$. But this gives an additional term to $h_{\mathrm{pol}}^{1}(x,\theta)$
which is deduced from $h^{1}(p,q)$ simply by the classical change of variables from $(p,q)$ to $(x,\theta)$. This term has
no reason to display any singularity when the classical trajectory goes through the origin. Remark that around the south pole, if we set
$$
x=2s_{cl}-y, \quad p=\sqrt{2y}\cos \theta, \quad q = \sqrt{2y}\sin \theta
$$
then we can expand
$$
h_0 = -2 s_{cl}( \kappa +s_{cl}) + \kappa \left( \left(p+{\sqrt{2} s_{cl}\over \kappa} \right)^2 + q^2 \right)
$$
and we see that our system is equivalent to a shifted harmonic oscillator.
Note that on the sphere, polar coordinates have two singularities, at the north and at the south poles, which
correspond to $x=0$ and $x=2s_{cl}$ respectively. To generalize the above analysis to the sphere, we may thus use two
charts $(p,q)$ and $(p',q')$ such that:
\begin{eqnarray*}
(p,q) & = & \sqrt{2x}(\cos \theta,\sin \theta) \\
(p',q') & = & \sqrt{2x'}(\cos \theta',\sin \theta') \\
x+x' & = & 2s_{cl} \\
\theta + \theta' & = & 0
\end{eqnarray*}
We therefore get $dp \wedge dq = dx \wedge d\theta = dx' \wedge d\theta' = dp' \wedge dq'$. It is
interesting to note that the Weyl symbols $h_{\mathrm{pol}}(x,\theta)$ and $h'_{\mathrm{pol}}(x',\theta')$
are obtained from each other by the classical transformation from $(x,\theta)$ to $(x',\theta')$.
On the other hand, $h(p,q)$ and $h'(p',q')$ corresponding to the same quantum Hamiltonian in the physical
subspace $0\leq x=2s_{cl}-x' \leq 2s_{cl}$ do not coincide, because shifting $x$ into $x+\hbar/2$ amounts to
shifting $x'$ into $x'-\hbar/2$ instead of $x'+\hbar/2$. This implies that it is impossible to construct
a quantum operator in the $2s+1$ dimensional Hilbert space of a spin $s$ for which both subprincipal
parts of $h(p,q)$ and $h'(p',q')$ would vanish.
To formulate the Bohr-Sommerfeld rule, it is convenient to use the $(p,q)$ coordinates when we consider
a set of classical orbits which remain at a finite distance from the south pole. In these coordinates, the
subleading term $\int_{C(E)} \gamma$ and the Maslov index are both continuous when $C(E)$ crosses the north pole.
Going back to $(x,\theta)$ coordinates, we see that the jump in $\int_{C(E)} \gamma$ has to be compensated by a
jump in the Maslov index. So the Bohr-Sommerfeld formula~(\ref{phireg}) can be expressed in $(x,\theta)$
coordinates, provided these jumps in the Maslov index are taken into account.
To check Eq.(\ref{phireg}) we evaluate $\Phi_{Reg}(\epsilon_n)$ for the {\em exact} values $\epsilon_n$
and we define the defect $\delta_n$ by:
$$
\Phi_{Reg}(\epsilon_n) = n+ \delta_n
$$
Denoting
$$
I_{-1}= {1\over 2\pi \hbar }\int_{C(E)} \alpha, \quad I_0 = {1\over 2\pi}\int_{C(E)} \gamma
$$
\begin{eqnarray*}
\delta_n = & I_{-1}+I_0 - (n+{1\over 2}),&\quad {\rm if} \;\;\epsilon_n \le 2\hbar \kappa s \\
\delta_n = & I_{-1}+I_0 + {1\over 2} - (n+{1\over 2}),&\quad {\rm if}\;\; 2\hbar \kappa s \le \epsilon_n \le - 2\hbar \kappa s \\
\delta_n = & I_{-1}+I_0 + 1- (n+{1\over 2}) ,& \quad {\rm if} \;\; \epsilon_n \ge - 2\hbar \kappa s
\end{eqnarray*}
The excellent accuracy of this Bohr-Sommerfeld rule is clearly visible on Fig.~\ref{deltan},
which shows the defect $\delta_n$ as a function of $n$. We see that it remains small everywhere,
and we emphasize that it does not exhibit any accident when $\epsilon_n$ crosses the value $-2\kappa s_{cl}$,
corresponding to the south pole. On the other hand, something special happens around $\epsilon_n = 2\kappa s_{cl}$
(the energy of the north pole), which cannot be attributed to the use of polar coordinates,
but which reflects the crossing through the singular orbit associated to the pinched torus.
A detailed description of the energy spectrum in the vicinity of the classical unstable point is the
goal of next section.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 6cm]{energiesSC2.eps}
\caption{To test the quality of the Bohr-Sommerfeld quantization, we plot
the default $\delta_n$ as a function of $n$ which labels the energy eigenstates.
We see that the accuracy of the usual Bohr-Sommerfeld rule is very good
away from the critical energy. In particular there is nothing special at the south pole.}
\label{deltan}
\end{center}
\nonumber
\end{figure}
\section{Generalized Bohr-Sommerfeld rule}
\label{semiclassiquesing}
This region of the spectrum requires a different quantization
rule~\cite{Colin94a,Colin94b,Brummelhuis95,Child98,Colin99,San00,Keeling09} because the classical motion near the turning
point at $x_{min}$ is strongly affected by the presence of the unstable point at $x=0$. Therefore, we first need
to analyze the small $x$ behavior of the energy eigenstates, when $E$ is close to $2\kappa s_{cl}$.
In the intermediate regime, between $x_{min}(E)$ and $x_{max}(E)$, a standard WKB analysis is quite acurate.
Near $x_{max}(E)$, we have a regular turning point, which can be described, as usual, by an Airy function.
Gluing the wave function of the eigenstates between these three different regimes
will give us the generalized Bohr Sommerfeld quantization rules.
\subsection{Small $x$ analysis}
Here, we shall recast the normal form around the singular point, studied in
section~\ref{secclassicalmonodromy} within the framework of the reduced system. For this,
we assume $n<< 2s$ or $x=\hbar n << 2s_{cl}$. In that approximation the Schr\"odinger equation
Eq.(\ref{EigEq}) becomes:
\begin{equation}
E p_n= \hbar \sqrt{2s \hbar} (n+1) p_{n+1} + \hbar \sqrt{2s\hbar } \;n p_{n-1}
+ 2\hbar \kappa (s-n) p_n
\label{eigschroerduite2}
\end{equation}
This equation is linear in $n$ and can be solved by Laplace-Fourier transform. Letting:
\begin{equation}
\Psi^{\rm Small}(\theta)= \sum_{n=0}^\infty p_n e^{in \theta}
\label{Psidef}
\end{equation}
the equation reads:
$$
H_0 \Psi^{\rm Small} = E \Psi^{\rm Small}, \quad E = E_c + \hbar \epsilon
$$
with:
\begin{equation}
H_0 =-i\hbar ( \sqrt{2s_{cl}}(e^{i\theta} + e^{-i\theta} ) -2\kappa) \partial_{\theta}
+\hbar (2 \kappa s + \sqrt{2s_{cl} }e^{i\theta} )
\label{H0theta}
\end{equation}
Let us set:
$$
z = e^{i\theta},\quad {d\over d \theta} = i z {d\over d z}
$$
the equation becomes:
$$
\left[ (\sqrt{2s_{cl}} (z + z^{-1}) -2\kappa ) z{d\over dz} + \sqrt{2s_{cl}} z \right] \Psi^{\rm Small} = \epsilon \Psi^{\rm Small}
$$
We are looking for a solution which is analytic in a neighborhood of the origin in the
$z$-plane. This can be written as:
$$
\Psi^{\rm Small}(z, \epsilon) = \left(1-\frac{z}{z_+}\right)^{\Delta_+} \left(1-\frac{z}{z_-}\right)^{\Delta_-}
$$
where $z_\pm$ are the solutions of the second order equation already introduced before:
$$
z^2 -2{\kappa \over \sqrt{2s_{cl}}} z +1 = 0, \quad z_\pm = {\kappa \pm i\Omega \over \sqrt{2s_{cl}}}
= -e^{\mp i \nu}
$$
and
$$
\Delta_\pm = -{1\over 2 } \mp i{\epsilon - \kappa\over 2 \Omega} \equiv -{1\over 2 } \mp if(E)
$$
Note that, when $\hbar$ is small, the leading term in $f(E)$ is $(E-E_c)/2\hbar \Omega$.
It is quite instructive to express directly this wave-function in the $\theta$ variable. Denoting
$$
\chi(\theta) = (\cos\theta+\cos\nu)^{-1/2}\exp i\left[f(E)
\log\left(\frac{|\cos\frac{\theta-\nu}{2}|}{|\cos\frac{\theta+\nu}{2}|}\right)-\frac{\theta}{2}\right]
$$
we have
\begin{eqnarray*}
\Psi^{\rm Small}(\theta) = &\quad \chi(\theta), & \;\;\;-\pi+\nu \leq \theta \leq \pi-\nu \\
\Psi^{\rm Small}(\theta) = & i\exp(-\pi f(E))\chi(\theta),&\;\;\; \pi-\nu \leq \theta \leq \pi \\
\Psi^{\rm Small}(\theta) = & -i\exp(-\pi f(E))\chi(\theta),& \;\;\; -\pi \leq \theta \leq -\pi + \nu
\end{eqnarray*}
Notice that this has precisely the expected form in the semi-classical limit $\hbar \rightarrow 0$.
Indeed, when $E>E_c$, the wave function is mostly confined to
the interval $-\pi+\nu \leq \theta \leq \pi-\nu$, and it is exponentially small in the classically
forbidden regions $\pi-\nu \leq \theta \leq \pi$ and $-\pi \leq \theta \leq -\pi + \nu$.
Such exponentially small prefactor is reminiscent of the behavior of a tunneling amplitude. This is
consistent with our treatment of $\theta$ as an unbounded variable, subjected to a $2\pi$-periodic
Hamiltonian. Enforcing integer values of $n$ requires the wave-function $\Psi^{\rm Small}(\theta)$
to be $2\pi$-periodic in $\theta$. If $\Psi^{\rm Small}(\theta)$ were identically zero in the
classically forbidden intervals, its Fourier transform would develop a tail for negative values
of $n$. So these evanescent parts of the wave-function are required in order to ensure that the
wave function belongs to the physical Hilbert space $n \geq 0$.
Note that when $E<E_c$, the classically forbidden region becomes the intervals
$-\pi \leq \theta \leq -\pi + \nu$ and $\pi-\nu \leq \theta \leq \pi$.
In the classically allowed regions, the semi-classical wave function is expected to take the
textbook form:
\begin{equation}
\Psi^{\rm Small}(\theta)=a(\theta)\exp i \left(\frac{S_{0}(\theta)}{\hbar}+S_{1}(\theta)\right)
\end{equation}
where the three functions $a$, $S_{0}$ and $S_{1}$ take real values.
These functions satisfy the standard equations~\cite{Duistermaat74},
involving the principal and subprincipal
symbols $h^{0}(x,\theta)=E_c+2\sqrt{2s_{cl}}x(\cos\theta+\cos\nu)$ and
$h^{1}(x,\theta)= \sqrt{2s_{cl}}\cos\theta$:
\begin{eqnarray}
h^{0}(S'_{0}(\theta),\theta) & = & E \label{fonctiondephase} \\
\frac{d}{d\theta}\left(a^{2}(\theta)\partial_x h^{0}(S'_{0}(\theta),\theta)\right) & = & 0
\label{transportamplitude} \\
\partial_x h^{0}(S'_{0}(\theta),\theta)S'_{1}(\theta) & = & - h^{1}(S'_{0}(\theta),\theta)
\end{eqnarray}
From the above expressions for $\Psi^{\rm Small}(\theta)$, we can check that it has
{\em exactly} this semi-classical form. Most likely, this occurs because
$\Psi^{\rm Small}(\theta)$ is an eigenstate of a
quantum Hamiltonian which is derived by symplectic reduction
from the quadratic Hamiltonian of the normal form discussed in section~\ref{secclassicalmonodromy}.
Quadratic Hamiltonians are particularly well-behaved with respect to the $\hbar \rightarrow 0$
limit in the sense that the Bohr-Sommerfeld formula gives the exact energy spectrum, and also
that the full quantum propagator obeys similar equations as ~(\ref{fonctiondephase}) and
(\ref{transportamplitude}).
Let us now discuss the $x=n\hbar$ representation which is quite useful from the physical standpoint.
$$
p^{\rm Small}(x,\epsilon) = \oint_{C_0} {dz \over 2i\pi }z^{-{x\over \hbar} - 1} \Psi(z)= \oint_{C_0} {dz \over 2i\pi }z^{-{x\over \hbar} - 1} \left(1-\frac{z}{z_+}\right)^{\Delta_+} \left(1-\frac{z}{z_-}\right)^{\Delta_-}
$$
where $C_0$ is a small contour around the origin.
When $E$ is sufficiently far (in a sense to be precised below) from its critical value $E_c$,
we may use the saddle point approximation to evaluate $\Psi^{\rm Small}(x,\epsilon)$.
This yields again the expected semi-classical form, namely:
\begin{equation}
p^{\rm Small}(x)=b(x)e^{\pm i\pi/4}\exp -i \left(\frac{W_{0}(x)}{\hbar}+W_{1}(x)\right)
\end{equation}
where $b(x)$, $W_{0}(x)$ and $S_{1}(x)$ satisfy:
\begin{eqnarray}
h^{0}(x,W'_{0}(x)) & = & E \label{fonctiondephaseW} \\
\frac{d}{dx}\left(b^{2}(x)\partial_{\theta} h^{0}(x,W'_{0}(x))\right) & = & 0
\label{transportamplitudeW} \\
\partial_{\theta} h^{0}(x,W'_{0}(x))W'_{1}(x) & = & - h^{1}(x,W'_{0}(x))
\label{subleadingphaseW}
\end{eqnarray}
Note that $W_{0}(x)$ is the Legendre transform of $S_{0}(\theta)$, that is
$S'_{0}(\theta)=x$, $W'_{0}(x)=\theta$ and $S_{0}(\theta)+W_{0}(x)=x\theta$.
It is interesting to write down explicitely this semi classical wave-function
when $x \gg 1$. It reads:
$$
p^{\rm Small}_{\rm sc}(x)=B_{+}(E)\frac{e^{-i(\pi-\nu)x/\hbar}}{(x/\hbar)^{\frac{1}{2}-if(E)}}
+B_{-}(E)\frac{e^{i(\pi-\nu)x/\hbar}}{(x/\hbar)^{\frac{1}{2}+if(E)}}
$$
where $B_{-}(E)=\bar{B}_{+}(E)$ and:
\begin{equation}
\frac{B_{+}(E)}{B_{-}(E)}=e^{i(\nu-\pi/2)}\exp i \left(2f(E)\log\frac{4\hbar\Omega}{|E-E_c|}
+\frac{E-E_c}{\hbar\Omega}\right)
\end{equation}
We note that the phase factor diverges when $E$ reaches $E_c$. This behavior is induced
by the contribution of the subprincipal symbol to the phase of the wave-function.
The prefactor $\exp(i(\nu-\pi/2))$ in the above expression does not jump when $E$
crosses the critical value $E_c$. This is consistent with the analysis of the previous
section, because we do find a $\pi$ jump in the Maslov index that is compensated by a
$\pi$ jump in the contribution of the subprincipal symbol.
Let us now precise the validity domain of the stationary phase approximation.
When we integrate over $\theta$ to compute the Fourier transform $p^{\rm Small}(x)$,
we get an oscillating integral whose phase factor can be approximated by
$\exp(iS''_{0}(\theta(x))(\theta-\theta(x))^{2}/(2\hbar))$ where $\theta(x)$ is
one of the two saddle points defined implicitely by $S'_{0}(\theta(x))=x$.
This oscillating gaussian has a typical width $\langle\Delta\theta^{2}\rangle=\hbar/|S''_{0}(\theta(x))|$.
We find that:
$$
S''_{0}(\theta(x))=\frac{E-E_c}{2\sqrt{2s_{cl}}}\frac{\sin\theta}{(\cos\theta+\cos\nu)^{2}}
$$
When $E$ goes to $E_c$ at fixed $x$, $\theta(x)$ goes to $\pm (\pi-\nu)$, because
$\cos\theta+\cos\nu=E/(2\sqrt{2s_{cl}}x)$. So $|S''_{0}(\theta(x))|\simeq 2\Omega x^{2}/|E-E_{c}|$.
On the other hand, we have seen that the amplitude of $\Psi^{\rm Small}(\theta)$ diverges when
$\theta=\pm (\pi-\nu)$. The distance $\delta\theta$ between $\theta(x)$ and the closest
singularity goes like $|E-E_{c}|/2\Omega x$. The stationary phase approximation holds
as long as $\theta(x)$ is far enough from the singularities, that is if
$\langle\Delta\theta^{2}\rangle<<(\delta\theta)^{2}$, or equivalently,
\begin{equation}
|E-E_c| \gg 2\hbar \Omega
\end{equation}
When this condition is not fullfilled, the stationary phase approximation breaks down.
The dominant contribution to $p^{\rm Small}(x)$ comes from the vicinity of $\theta=\pm (\pi-\nu)$.
We have therefore to consider a different asymptotic regime, where $\hbar$ goes to $0$ while
$\epsilon=(E-E_{c})/\hbar$ remains fixed. In this case, the two exponents $\Delta_{+}$, $\Delta_{-}$
are fixed, and the only large parameter in the Fourier integral giving $p^{\rm Small}(x)$
is $x/\hbar$. One then obtains:
\begin{equation}
p^{\rm Small}(x)=A_{+}(E)\frac{e^{-i(\pi-\nu)x/\hbar}}{(x/\hbar)^{\frac{1}{2}-if(E)}}
+A_{-}(E)\frac{e^{i(\pi-\nu)x/\hbar}}{(x/\hbar)^{\frac{1}{2}+if(E)}}\equiv p_{+}^{\rm Small}(x)+p_{-}^{\rm Small}(x)
\label{psmallx}
\end{equation}
with $A_{-}(E)=\bar{A}_{+}(E)$ and:
\begin{equation}
\frac{A_{+}(E)}{A_{-}(E)}=e^{i(\nu-\pi/2)}\frac{\Gamma(\frac{1}{2}-if(E))}{\Gamma(\frac{1}{2}+if(E))}
\exp i \left(2f(E)\log (2\sin\nu) \right)
\end{equation}
This analysis first shows that, in spite of the breakdown of the stationary phase approximation,
$p^{\rm Small}(x)$ is still given at large $x$ by a sum of two semi-classical
wave-functions associated to the same principal and subprincipal symbols $h^{0}$ and $h^{1}$.
This implies that there will be a perfect matching between $p^{\rm Small}(x)$ and the WKB
wave functions built at finite $x$ from the full classical Hamiltonian. This will be discussed
in more detail below. The only modification to the conventional WKB analysis lies in the
phase factor $A_{+}/A_{-}$ between the ongoing and outgoing amplitudes, which differs markedly
from the semi-classical $B_{+}/B_{-}$. In particular, we see that the full quantum treatment at
fixed $\epsilon$ provides a regularization of the divergence coming from the subprincipal symbol.
Indeed, the singular factor $\log(4\hbar\Omega/|E-E_c|)$ is replaced by the constant $\log(2\sin\nu)$.
Such a phenomenon has been demonstrated before for various models~\cite{Colin99,San00}.
\subsection{WKB analysis}
We return to the Schr\"odinger equation:
\begin{equation}
(x+\hbar)\sqrt{ 2s_{cl} -x}\; p(x+\hbar) + x\sqrt{2s_{cl} +\hbar -x}\; p(x-\hbar) -2 \kappa x \; p(x)= \hbar \epsilon p(x)
\label{schroepsilon}
\end{equation}
where:
$$
\hbar \epsilon = E-E_{c}
$$
We try to solve this equation by making the WKB Ansatz:
$$
p^{\rm WKB}(x) = e^{{-i\over \hbar}(W_{0}(x)+\hbar W_{1}(x))} b(x)
$$
Expanding in $\hbar$, to order $\hbar^0$ we find
\begin{equation}
x \sqrt{2s_{cl}-x} \cos W'_{0} - \kappa x =0
\label{WKB}
\end{equation}
which is nothing but the Hamilton-Jacobi equation on the critical variety. It is
identical to Eq.~(\ref{fonctiondephaseW}) where $h^{0}(x,\theta)$ is the complete principal symbol,
and the energy is taken to be $E_c$. In this procedure, the energy difference $\hbar \epsilon$
is viewed as a perturbation, which can be treated by adding the constant term $-\epsilon$
to the complete subprincipal symbol $h^{1}(x,\theta)$.
Alternatively, we may write the Hamilton-Jacobi equation as:
$$
e^{i W'_{0}} = z(x), \quad z^2(x) -{2\kappa \over \sqrt{2s_{cl} -x}} z(x) + 1 =0
$$
The solution of the quadratic equation is:
\begin{equation}
z_\pm(x) = {1\over \sqrt{2s_{cl}-x}} (\kappa \pm i \sqrt{\Omega^2-x})=
\left( {\kappa \pm i \sqrt{\Omega^2-x} \over \kappa \mp i \sqrt{\Omega^2-x}} \right)^{1/2}
\label{zpmx}
\end{equation}
so that $z_\pm(x)\vert_{x=0} = e^{\pm i(\pi - \nu)}$. Hence:
\begin{equation}
e^{-{i\over \hbar} W_{0}(x)} = e^{\mp {i\over \hbar} \left(\pi x+\kappa \sqrt{\Omega^2-x} + {i\over 2} (2s_{cl}-x)
\log{ \kappa +i \sqrt{\Omega^2-x} \over \kappa -i \sqrt{\Omega^2-x} } \right)}
\label{soluWKB}
\end{equation}
where the two signs refer to the two branches of the classical trajectory. Notice that they also correspond to the two determinations of the square root $\sqrt{\Omega^2-x}$.
At the next order in $\hbar$ the equation for $b(x)$ is:
\begin{equation}
\frac{d}{dx}\left(b^{2}(x)x\sqrt{\Omega^{2}-x}\right)=0
\label{WKBstationary}
\end{equation}
Therefore, we may choose $b(x)=(x\sqrt{\Omega^{2}-x})^{-1/2}$.
The correction $W_{1}(x)$ to the phase function satisfies Eq.~(\ref{subleadingphaseW}) with
$h^{1}(x,\theta)$ replaced by $h^{1}(x,\theta)-\epsilon$, that is:
$$
\partial_{\theta} h^{0}(x,W'_{0}(x))W'_{1}(x) = - h^{1}(x,W'_{0}(x))+\epsilon
$$
$W'_{1}(x)$ is the sum of two terms, $W'_{1}(x)=-\gamma(x)+\epsilon \delta \beta(x)$,
where $\gamma(x)dx$ is simply the one-form $\gamma$ defined by Eq.~(\ref{subprincipalform}),
evaluated on the critical trajectory, and $\delta \beta(x)=\left(\partial_{\theta} h^{0}(x,W'_{0}(x))\right)^{-1}$.
To understand better the meaning of $\delta \beta (x)$, we start with the action integral $\int_{C(E)} \beta$,
where $\beta = \theta dx$ is closely related to the canonical 1-form $\alpha = x d\theta$,
because $\alpha+\beta=d(x\theta)$. Note that, by contrast to $\alpha$, $\beta$ is defined on the
sphere only after choosing a determination of the longitude $\theta$. Let us now study
the variation of this action integral when $E$ moves away from $E_c$ and is changed into
$E_c + \hbar \epsilon$. We have:
$$
\int_{C(E_c+ \hbar \epsilon)} \beta - \int_{C(E_{c})} \beta = \hbar \epsilon \int_{C(E_c)}
{\partial\theta\over \partial E} dx
$$
But $\theta(x,E)$ satisfies $h^{0}(x,\theta(x,E))=E$, so:
$$
\partial_{\theta}h^{0}(x,\theta(x,E))\frac{\partial \theta}{\partial E}(x,E)=1
$$
and finally:
$$
\int_{C(E_c+ \hbar \epsilon)} \beta - \int_{C(E_{c})} \beta = \hbar \epsilon \int_{C(E_c)} \delta \beta(x) dx
$$
Explicitely, the equation for $W_{1}(x)$ reads:
\begin{equation}
W'_{1}(x) = {\pm 1\over 2\sqrt{\Omega^2-x}}
\left( {\kappa \over 2(2s_{cl} -x)} + {\kappa - \epsilon \over x} \right)
\label{WKBstationary2}
\end{equation}
So we have:
\begin{equation}
e^{-iW_{1}(x)} = \exp\left(\pm {1\over 4} \log{ \kappa +i \sqrt{\Omega^2-x} \over
\kappa -i \sqrt{\Omega^2-x} } \mp i {\epsilon -\kappa\over 2 \Omega} \log{ \Omega + \sqrt{\Omega^2-x} \over \Omega - \sqrt{\Omega^2-x}}\right)
\label{expressionW1}
\end{equation}
We can now expand $p^{\rm WKB}(x)$ when $x$ is small. We find:
\begin{equation}
p_\pm^{\rm WKB}(x) \simeq A_\pm^{\rm WKB} (\epsilon)\frac{e^{\mp i(\pi-\nu){x\over \hbar}}}{(x/\hbar)^{\frac{1}{2}\mp if(E)}}
\label{wkbsmall}
\end{equation}
where:
$$
A_\pm^{\rm WKB} (\epsilon)= {1\over \sqrt{\Omega}} e^{\mp i \left[ { \kappa \Omega \over \hbar }
+(2s+\frac{1}{2})\nu+{\epsilon-\kappa \over 2\Omega}\log \left( {4 \Omega^2 \over \hbar} \right)\right]}
$$
Notice that the phase factor is the sum of two terms which have simple geometrical interpretations:
$$
S^+_{cl} = \kappa \Omega + 2s_{cl}\nu = \pi \Omega^{2}-\int_0^{\Omega^2} \theta(x)dx
= \int_{C_+} \alpha
$$
is the classical action computed on the upper half of the classical trajectory,
and
$$
\frac{\nu}{2} + {\epsilon-\kappa \over 2\Omega}\log \left( {4 \Omega^2 \over \hbar} \right) =
\int^{\Omega^2}_0\hskip-0.7cm \backprime \hskip .5cm \tilde\gamma(x) dx
$$
is the regularized integral of the subprincipal symbol $\tilde\gamma(x)=\gamma(x)-\epsilon\delta\beta (x)$
on the same trajectory:
$$
\int^{\Omega^2}_0\hskip-.7cm \backprime \hskip .55cm \tilde\gamma(x) dx = \lim_{x_A\to 0}
\left(\int_{x_A}^{\Omega^{2}} \tilde\gamma(x) dx + f(E) \log { x_{A} \over \hbar } \right)
$$
Similarly, setting $ x = \Omega - |\xi |$ and expanding in $|\xi|\over \Omega$ and $|\xi|\over \kappa$ we find to leading order:
\begin{equation}
p^{\rm WKB}_\pm(x) \simeq {1\over \Omega | \xi|^{1/4}} e^{\pm{2i \over 3 \hbar \kappa} |\xi|^{3/2}
\pm i\left( {1\over 2\kappa} - {\epsilon-\kappa \over \Omega^2}\right) |\xi|^{1/2}
}
\label{wkbairy}
\end{equation}
The relevant wave function is of course a linear combination of $p_\pm^{\rm WKB}(x)$.
\subsection{Airy function analysis}
Again, we start with the Schr\"odinger equation Eq.~(\ref{schroepsilon}). We set:
$$
p^{\rm Airy}(x) = (-1)^n \tilde{p}^{\rm Airy}(x),\quad x = n\hbar
$$
\begin{equation}
(x+\hbar)\sqrt{ 2s_{cl} -x}\; \tilde{p}^{\rm Airy}(x+\hbar) + x\sqrt{2s_{cl} +\hbar -x}\; \tilde{p}^{\rm Airy}(x-\hbar) +2 \kappa x \tilde{p}^{\rm Airy}(x)= -\hbar \epsilon \tilde{p}^{\rm Airy}(x)
\end{equation}
and we expand $x=\Omega^2 + \xi$, keeping the terms linear in $\xi$. Remembering that
$\sqrt{\kappa^2} = -\kappa$, we find:
$$
{d^2\over d\xi^2} \tilde{p}^{\rm Airy}+ {1\over \Omega^2}\left(1-{\Omega^2\over 2\kappa^2}\right)
{d\over d\xi}\tilde{p}^{\rm Airy} -{1\over \hbar^2 \kappa \Omega^2}
\left( \hbar \epsilon - {\hbar \Omega^2\over 2\kappa} - \hbar \kappa - {\Omega^2\over \kappa} \xi \right)
\tilde{p}^{\rm Airy}=0
$$
The unique solution which decreases exponentially in the classically forbiden region $\xi >0$ is proportional to the Airy function:
$$
\tilde{p}^{\rm Airy}(\xi) = e^{-c\; \xi }
{\rm Airy}( a^{1/3}(\xi- b ) )
$$
where:
$$\ a= {1\over \hbar^2 \kappa^2},\quad
b=\hbar\left({1\over 2} - { \kappa (\epsilon-\kappa)\over \Omega^2}\right), \quad
c= {1\over 2 \Omega^2} ( 1-{\Omega^2\over 2\kappa^2})
$$
When $\xi <0$ it behaves like:
$$
{\sin ( {2\over 3} |X|^{2/3} + {\pi\over 4} ) \over |X|^{1/4}}, \quad X = a^{1/3}(\xi- b )
$$
We expand in $|\xi|\over \kappa$ and $|\xi|\over \Omega$. One has:
$$
{2\over 3}|X|^{3/2} =- {2\over 3 \hbar \kappa} |\xi|^{3/2} - \left(
{1\over 2 \kappa}-{ (\epsilon-\kappa)\over \Omega^2} \right) |\xi|^{1/2}
$$
In that approximation $e^{-c\; \xi } \simeq 1$, and we get:
\begin{equation}
\tilde{p}^{\rm Airy}(\xi) = C { \sin \left({2\over 3 \hbar \kappa} |\xi|^{3/2} + \left(
{1\over 2 \kappa}-{ (\epsilon-\kappa)\over \Omega^2} \right) |\xi|^{1/2}- {\pi\over 4}\right)\over \sqrt{\pi} |\xi|^{1/4}}
\label{airy}
\end{equation}
\subsection{Gluing the parts together}
The small $x$ analysis gave:
$$
p(x) = p_+^{\rm Small}(x) + p_-^{\rm Small}(x)
$$
which is valid when $x << 2s_{cl}$. In an intermediate regime, both the small $x$
and the WKB approximation are valid at the same time, as can be seen by comparing Eq.~(\ref{psmallx})
and Eq.~(\ref{wkbsmall}). So we can glue these two wave-functions as follows:
$$
p(x) = {A_+^{\rm Small}(\epsilon) \over A_+^{\rm WKB}(\epsilon)} p_+^{\rm WKB}(x) +
{A_-^{\rm Small}(\epsilon) \over A_-^{\rm WKB}(\epsilon)} p_-^{\rm WKB}(x)
$$
We can now prolongate the $p_\pm^{\rm WKB}(x)$ functions up to the region $x \simeq \Omega^2-|\xi|$.
Recalling the asymptotic form of the WKB wave function in that region Eq.~(\ref{wkbairy}), we find
\begin{eqnarray*}
p(x) &\simeq& {A_+^{\rm Small}(\epsilon) \over A_+^{\rm WKB}(\epsilon)}
{1\over \Omega |\xi|^{1/4}} e^{{2i \over 3 \hbar \kappa} |\xi|^{3/2}
+ i\left( {1\over 2\kappa} - {\epsilon-\kappa \over \Omega^2}\right) |\xi|^{1/2}} \\
&& + {A_-^{\rm Small}(\epsilon) \over A_-^{\rm WKB}(\epsilon)}
{1\over \Omega |\xi|^{1/4}} e^{-{2i \over 3 \hbar \kappa} |\xi|^{3/2}
- i\left( {1\over 2\kappa} - {\epsilon-\kappa \over \Omega^2}\right) |\xi|^{1/2}}
\end{eqnarray*}
This has to be compatible with the Airy asymptotic formula which embodies the boundary
conditions Eq.~(\ref{airy}). Hence we find the condition:
\begin{equation}
{ A_+^{\rm Small}(\epsilon) \over A_-^{\rm Small}(\epsilon) }\;
{A_-^{\rm WKB}(\epsilon) \over A_+^{\rm WKB}(\epsilon) } \; {e^{i{\pi\over 4}} \over e^{-i{\pi\over 4}}}= -1
\label{genbohrsommerfeld}
\end{equation}
This equation determines the energy parameter $\epsilon$ and is the generalized Bohr-Sommerfeld condition.
Given the fact that:
\begin{eqnarray*}
{A^{\rm Small}_+(\epsilon)\over A^{\rm Small}_-(\epsilon)} &=& e^{i(\nu-{\pi\over 2})}
{ \Gamma \left( {1\over 2} -i {\epsilon - \kappa \over 2 \Omega} \right) \over
\Gamma \left( {1\over 2} +i {\epsilon - \kappa \over 2 \Omega} \right) }
e^{i {\epsilon -\kappa \over \Omega} \log \left({ 2 \Omega \over \sqrt{2s_{cl}}}\right)} \\
{A^{\rm WKB}_+(\epsilon)\over A^{\rm WKB}_-(\epsilon)} &=& e^{-2 i \left[ { 2\kappa \Omega \over \hbar }
+(4s+1)\nu+{\epsilon-\kappa \over \Omega} \log \left( {4 \Omega^2 \over \hbar} \right) \right]}
\end{eqnarray*}
Eq.(\ref{genbohrsommerfeld}) becomes:
\begin{equation}
{ \Gamma \left( {1\over 2} -i {\epsilon - \kappa \over 2 \Omega} \right) \over
\Gamma \left( {1\over 2} +i {\epsilon - \kappa \over 2 \Omega} \right) } \;
e^{i\left[{ 2\kappa \Omega \over \hbar}+2(2s+1)\nu+
{\epsilon-\kappa \over \Omega} \log \left({8\Omega^3 \over \hbar \sqrt{2s_{cl}}}\right)\right] } =-1
\end{equation}
Taking the logarithm, we find the quantization condition:
\begin{equation}
\Phi_{Sing}(\epsilon_n)= 2\pi \left( n +{1\over 2} \right), \quad n\in Z, \quad E_n = 2\kappa s_{cl} + \hbar \epsilon_n
\label{SBS}
\end{equation}
where:
$$
\Phi_{Sing}(\epsilon)= -i \log { \Gamma \left( {1\over 2} -i {\epsilon - \kappa \over 2 \Omega} \right) \over
\Gamma \left( {1\over 2} +i {\epsilon - \kappa \over 2 \Omega} \right) } +
{ 2\kappa \Omega \over \hbar}+2(2s+1)\nu+
{\epsilon-\kappa \over \Omega} \log \left({8\Omega^3 \over \hbar \sqrt{2s_{cl}}}\right)
$$
To test this condition, we can compute $\delta_n= \Phi(\epsilon_n^{\rm exact})- 2\pi \left( n +{1\over 2} \right)$
for the exact values of the energies. In Figure~\ref{bohrsommerfeld}, we plot $\delta_n$ as a function of $n$,
using for $\Phi(\epsilon)$ both the usual Bohr-Sommerfeld function:
$$
\Phi_{Reg}(\epsilon) = {1\over \hbar }\int_{C(E)} \alpha + \int_{C(E)} \gamma
$$
and the function $\Phi_{Sing}(\epsilon)$. Near the singularity, the singular Bohr-Sommerfeld condition is much
more accurate.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 5cm,width=15cm]{BohrSommerfeld2.eps}
\caption{To compare the usual and the singular Bohr-Sommerfeld rules, the default $\delta_n$ is plotted as a function of $n$.
The red dots are obtained by using the usual Bohr Sommerfeld rule, while the green dots are obtained by using $\Phi_{Sing}(\epsilon)$.
The dashed vertical lines represent the interval $|E-E_c| < 2 \hbar \Omega$ of validity of the singular Bohr-Sommerfeld rule.}
\label{bohrsommerfeld}
\end{center}
\nonumber
\end{figure}
In Figure~\ref{superposition} a typical example of the components of the eigenvectors is shown, comparing with the exact result obtained by direct diagonalization of the Jacobi matrix Eq.~(\ref{EigEq}) and the various results corresponding to the different
approximations: small $x$, WKB, and Airy. The agreement is very good.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 5cm,width=15cm]{superposition2.eps}
\caption{The components of an eigenvector close to the critical level. The dots are the exact result. The red curve is the result of the small $x$ analysis. The green curve is the WKB result and the blue curve is the Airy result.}
\label{superposition}
\end{center}
\nonumber
\end{figure}
From Eq.~(\ref{SBS}) we can compute the level spacing between two successive energy levels. It is given by:
\begin{equation}
\epsilon_{n+1}-\epsilon_n = {2\pi\over \Phi_{Sing}'(\epsilon_n)}= {2\pi \Omega \over
\log{\left(8\Omega^3\over \hbar\sqrt{2s_{cl}}\right)} - \Psi'({\epsilon-\kappa \over 2\Omega})}
\label{depsilonn}
\end{equation}
where we defined
$$
\Psi'(x) ={i \over 2} {d\over dx} \log {\Gamma \left( {1\over 2} - ix \right) \over \Gamma \left( {1\over 2} + ix \right)}
$$
This function has a sharp minimum located at $x=0$ where it takes the value $-(\gamma + 2 \log 2)$ where $\gamma$ is Euler's constant,
see Fig.~\ref{levelspacing}.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 5cm,width=15cm]{levelspacing.eps}
\caption{The energy level spacing $\epsilon_{k+1}-\epsilon_{k}$
as a function the the state label $k$. This plot shows clearly the accumulation of energy levels in the vicinity
of the critical point.}
\label{levelspacing}
\end{center}
\nonumber
\end{figure}
Hence to leading order, the smallest energy spacing is:
$$
\Delta \epsilon \simeq {2\pi\Omega \over |\log \hbar |}
$$
A detailed analysis of the level spacing in the vicinity of the critical level has been given recently~\cite{Keeling09},
where it has been applied to the description of the long time dynamics of the system.
\section{ Evolution of the oscillator energy}
\label{evolution}
Once the eigenvectors and eigenvalues are known, we can compute the time evolution of the oscillator energy:
$$
\bar{x}(t) = \langle s \vert e^{i{t H\over \hbar}} b^\dag b e^{-i{ tH\over \hbar}} \vert s \rangle
$$
Since the observable $b^\dag b$ commutes with $H_1$, its matrix elements between eigenstates of $H_1$
with different eigenvalues vanish. Because $\vert s \rangle$ is an eigenvector of $H_1$ with eigenvalue $\hbar s$,
we can therefore restrict ourselves to this eigenspace of $H_1$.
The time evolution can be computed numerically by first decomposing the initial state on the
eigenvector basis:
\begin{equation}
\vert s \rangle = \sum_{n } c_n \vert \Psi(E_n)\rangle
\label{defcn}
\end{equation}
so that the wave-function at time $t$ is:
$$
\vert \Psi(t) \rangle = \sum_{n } c_ne^{-iE_n t/\hbar} \vert \Psi(E_n)\rangle
$$
In the stable regime $\kappa^2 > 2s_{cl} $, we get quite regular oscillations with
a rather small amplitude as shown in Figure~\ref{nomolecules}.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 5cm,width=15cm]{30-2sqrt2Clas2Short.eps}
\caption{The excitation number of the oscillator as function of time in the stable case.
The green curve is the small time result Eq.~(\ref{xbarstable}).
($\hbar^{-1}=s=30$, $\kappa= -2\sqrt{2s_{cl}}$,
$s_{cl}=1$). Notice that the vertical amplitude is about one hundred times smaller than in the unstable case (compare with Fig.~\ref{molecules}).}
\label{nomolecules}
\end{center}
\nonumber
\end{figure}
By contrast, in the unstable regime $\kappa^2 < 2s_{cl}$, it is energetically favorable to excite the oscillator,
and we get a succession of well separated pulses shown in Figure~\ref{molecules}.
Note that the temporal succession of these pulses displays a rather well defined periodicity, but
the fluctuations from one pulse to another are relatively large.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 5cm, width=15cm]{30-05sqrt2Clas2Per.eps}
\caption{The excitation number of the oscillator as function of time in the unstable case. ($\hbar^{-1}=s=30$, $\kappa= -0.5\sqrt{2s_{cl}}$,
$s_{cl}=1$). It clearly shows an aperidodic succession of pulses. The green curve is the semiclassical result Eq.~(\ref{xbarinstable}). }
\label{molecules}
\end{center}
\nonumber
\end{figure}
The remaining part of this paper is an attempt to understand the main features of this evolution.
On Figure~\ref{figeigencoeff} we show the coefficients $|c_n|$. Only those eigenstates whose energy
is close to the critical classical energy $E_c$ contribute significantly. A good estimate of the
energy width of the initial state $|s\rangle$ is given by
$\Delta E=\left( \langle s|H^{2}|s \rangle - \langle s|H|s \rangle^{2}\right)^{1/2}$.
From Eq.~(\ref{EigEq}), we find:
\begin{equation}
\Delta E= \hbar \sqrt{2s_{cl}}=\hbar \Omega /\sin\nu
\label{deltaE}
\end{equation}
Comparing with the criterion $|\Delta E| \gg 2\hbar \Omega$ for the validity of the stationary phase
approximation, we see that most of the eigenstates which have a significant weight in the spectral
decomposition of the initial state actually belong to the singular Bohr-Sommerfeld regime.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 5cm,width=15cm]{figeigencoeff3bis.eps}
\caption{The coefficients of the expansion of the initial state $|s\rangle$ on the eigenvectors $\vert \Psi(E_n)\rangle$.
The red dots are the exact values. We see that only eigenvectors close to the critical level contribute significantly.
The curve is given by Eq.~(\ref{ce}), which is deduced from the knowledge of the
short time evolution. The agreement between this approximate result and the exact one is remarkable, even in the
immediate vicinity of the critical level.}
\label{figeigencoeff}
\end{center}
\nonumber
\end{figure}
An important consequence of this observation is that we can compute $\bar{x}(t)$
by considering only the few relevant states.
The result is shown in Figure~\ref{nt6} for a spin $s=30$.
We retained only six states and superposed the result to the exact curve obtained by keeping the 61 states.
We can hardly see any difference between the two curves.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[width=13cm]{Full_6States.eps}
\caption{The excitation number of the oscillator as function of time for $s=30$.
The blue curve is the exact result, and the red curve is obtained by keeping only 6 energy eigenstates.}
\label{nt6}
\end{center}
\nonumber
\end{figure}
We can compute the coefficients $|c_n|$ of the decomposition of the initial state $|s\rangle$. From Eq.~(\ref{defcn}), we have:
\begin{equation}
\langle s | e^{-i{Ht\over \hbar}} | s \rangle = \sum_n e^{-i{E_n t \over \hbar}} |c_n|^2 \simeq
\int_{-\infty}^{\infty} e^{-i{Et\over \hbar}} |c(E)|^2 \rho(E) dE
\label{defce}
\end{equation}
where we have approximated the sum over discrete energy levels by an integral. Since we already know that the integral is concentrated around the critical energy $E_c$, the density $\rho(E)$ can be computed from the singular Bohr-Sommerfeld phase eq.(\ref{SBS}):
$$
\rho(E) = {dn\over dE} = {1\over 2\pi } {d \over dE} \Phi_{Sing}(E)= {1\over 2\pi \hbar} {d \over d\epsilon} \Phi_{Sing}(\epsilon)
$$
On the other hand, since
$$
|s\rangle = {e_0\over ||e_0||}
$$
where $e_0$ is the state Eq.~(\ref{defen}) with the oscillator in its ground state, we get:
$$
\langle s | e^{-i{Ht\over \hbar}} | s \rangle = {1\over 2\pi} \int_0^{2\pi} \Psi(\theta,t) d\theta =p_0(t)
$$
where $\Psi(\theta,t)$ is the solution of the Schr\"odinger equation with initial condition
$\Psi(\theta,t)\vert_{t=0}=1$. Inverting the Fourier transform in Eq.~(\ref{defce}), we arrive at:
$$
|c(E)|^2 \rho(E) = {1\over 2\pi \hbar} \int_{-\infty}^\infty e^{i{E t\over \hbar}} p_0(t) dt
$$
Taking for $p_0(t)$ the small time expression Eq.~(\ref{pnt}), which is valid for times $t< t_0 \simeq {|\log \hbar |\over 2\Omega}$,
we expect to find an expression valid in the energy interval $|E-E_c| > 2 \Omega {\hbar \over |\log \hbar |}$ i.e. for energies
far enough from the critical energy. However because of the factor $1/|\log \hbar |$ this interval covers most of the range Eq.~(\ref{deltaE})
where we expect $c(E)$ to be substantially non zero. Notice also that $\hbar\over |\log \hbar | $ is the order of magnitude of level spacing
in the critical region, so that the interval where the use of the small time wave function is not legitimate contains at most a few levels.
Explicitly, computing the Fourier transform of Eq.~(\ref{pnt}), we find:
\begin{equation}
|c(E)|^2 \rho(E) ={1\over \hbar \sqrt{2 s_{cl}}} {e^{-{ \nu \alpha(E)\over \hbar \Omega}} \over 1+ e^{-{\pi \alpha(E) \over \hbar \Omega}}}
\label{ce}
\end{equation}
where:
$$
\alpha(E) = E-E_c - \hbar \kappa
$$
The coefficients $|c(E)|^2$ computed with this formula are shown in Fig.~\ref{figeigencoeff}. The agreement with the exact coefficients is excellent.
\bigskip
For longer time scales, we may also infer that the quantum dynamics exhibits the Bohr frequencies that can be
deduced from the singular Bohr-Sommerfeld quantization rule~(\ref{SBS}). As shown before, the typical
spacing between these energy levels is given by $\Delta \epsilon \simeq {2\pi\Omega \over |\log \hbar |}$.
This corresponds to a fundamental period $\Delta t \simeq |\log \hbar|/\Omega$. The fact that the energy
levels are not equally spaced, as shown on Figure~\ref{levelspacing} is responsible for the aperiodic
behavior on time scales larger than $\Delta t$. Systematic procedures to analyze the long time behavior,
starting from a nearly equidistant spectrum have been developed~\cite{Leichtle96}, but we have not yet
attempted to apply them to the present problem. Note that the quantum dynamics in the vicinity of
classically unstable equilibria has received a lot of attention in recent years~\cite{Micheli03,Boukobza09}.
\bigskip
Instead, motivated by the analysis of the previous section, we now present a discussion of the time evolution in the semi-classical regime. As for the stationary levels, we show that we have to treat separately the initial evolution, when $x$ is still
small and the system is close to the unstable point, and the motion at later times, for which the
usual WBK approach is quite reliable. As we demonstrate, this allows us to predict the appearance of pulses,
together with their main period $\Delta t$. This generalizes the early study by
Bonifacio and Preparata~\cite{Bonifacio}, who focussed on the $\kappa=0$ case.
\subsection{Small time analysis}
Another expression for the mean oscillator energy is:
$$
\bar{x}(t) = \hbar \sum_{n=0}^{2s} n |p_n(t)|^2
$$
where $p_n(t)$ is the solution of Eq.~(\ref{schroerduite}) with boundary condition:
$$
p_n(t)\vert_{t=0} = \delta_{n,0}
$$
For small time, $p_n(t)$ will be significantly different from zero only for small $n$.
So in Eq.~(\ref{schroerduite}) we may assume $n << 2s$. The equation becomes:
\begin{equation}
i\hbar {\partial p_n \over \partial t}= \hbar \sqrt{2s \hbar} (n+1) p_{n+1} + \hbar \sqrt{2s\hbar } \;n p_{n-1}
+ 2\hbar \kappa (s-n) p_n
\label{schroerduite2}
\end{equation}
As for the stationary case, this equation is linear in $n$ and can be solved by Laplace-Fourier transform.
We define:
\begin{equation}
\Psi(\theta, t) = \sum_{n=0}^\infty p_n(t) e^{in \theta}, \quad \Psi(\theta, t)\vert_{t=0} = 1
\label{Psidef2}
\end{equation}
The time evolution is given by:
\begin{equation}
\partial_t \Psi =-( \sqrt{2s_{cl}}(e^{i\theta} + e^{-i\theta} ) -2\kappa) \partial_{\theta} \Psi
-i (2 \kappa s + \sqrt{2s_{cl} }e^{i\theta} ) \Psi
\label{schroetheta2}
\end{equation}
This equation can be solved by the method of characteristics. We introduce
the function $y(\theta)$ defined by:
$$
{d y \over d\theta}= {1\over \sqrt{2s_{cl}}(e^{i\theta} + e^{-i\theta} ) -2\kappa}
$$
Explicitely:
$$
y(\theta) = {1\over 2 \Omega} \log { e^{i\theta} + e^{i \nu} \over e^{i\theta} + e^{-i \nu}}, \quad {\rm or~else} \quad e^{i\theta} = {1\over \sqrt{2s_{cl}}} \left(\kappa + i\Omega {\cosh \Omega y \over \sinh \Omega y}\right)
$$
The time dependent Schr\"odinger equation becomes:
$$
\partial_t \Psi + \partial_y \Psi = - i(2 s+1)\kappa \Psi + \Omega {\cosh \Omega y \over \sinh \Omega y} \Psi
$$
whose solution reads:
$$
\Psi(y,t) = e^{-i\kappa(2s+1)t } {\sinh \Omega y \over \sinh \Omega(y-t)} \Psi(y-t,0)
$$
Imposing the initial condition Eq.~(\ref{Psidef2}) yields $\Psi(y-t,0)=1$ and then:
$$
\Psi(y,t) = e^{-i\kappa(2s+1)t } {\sinh \Omega y \over \sinh \Omega(y-t)}
$$
Returning to the variable $\theta$, we get:
\begin{equation}
\Psi(\theta,t) = e^{-i\kappa(2s+1)t } \; {e^{i\nu}-e^{-i\nu} \over e^{i\nu-\Omega t} -e^{-i\nu+\Omega t}}\;
{1\over 1-{e^{\Omega t} - e^{-\Omega t} \over e^{i\nu - \Omega t} - e^{-i\nu + \Omega t}} \; e^{i\theta}}
\label{solution_temps_courts}
\end{equation}
As for the stationary case, we note that this wave-function has exactly the form dictated by
semi-classical analysis, where the full symbol associated to
Eqs.~(\ref{schroerduite2}),(\ref{schroetheta2}) is
$h(x,\theta)=2\sqrt{2s_{cl}}(x+\hbar/2)(\cos\theta+\cos\nu)$, after we have subtracted the energy
$E_c$ of the unstable point.
The generalization of Eq.~(\ref{fonctiondephase}) to the time dependent case is:
\begin{equation}
h^{0}(\partial_{\theta}S_{0}(\theta,t),\theta) + \partial_{t}S_{0}(\theta,t) = 0
\end{equation}
But since $\Psi$ is independent of $\theta$ at $t=0$, $\partial_{\theta}S_{0}(\theta,t=0)=0$, which
means classically that the trajectory begins at $x=0$. But because of the form of $h^{0}$, we
get $h^{0}(\partial_{\theta}S_{0}(\theta,t=0),\theta)=0$ for any $\theta$, which implies that
$\partial_{t}S_{0}(\theta,t=0) = 0$. This shows that $S_{0}(\theta,t) = 0$ identically, in agreement
with the fact that a classical trajectory starting at $x=0$ stays there for ever!
So the time evolution manifested in Eq.~(\ref{solution_temps_courts}) is of purely quantum nature,
and it is encoded in the evolution of the amplitude $a(\theta,t)$ and the subleading phase $S_{1}(\theta,t)$.
Writing $\Psi=a(\theta,t)\exp(iS_{1}(\theta,t))$, the next order in $\hbar$ gives:
\begin{equation}
\partial_{x}h^{0}(0,\theta)\partial_{\theta}\Psi+\partial_{t}\Psi+\frac{1}{2}\frac{d}{d\theta}
\left(\partial_{x}h^{0}(0,\theta)\right)_{|t}\Psi+ih^{1}(0,\theta)\Psi=0
\end{equation}
A simple check shows this is exactly the same equation as (\ref{schroetheta2}) without the
term $-i2\kappa s\Psi$, due to the subtraction of the energy $E_c$.
Expanding in $e^{i\theta}$ we find:
\begin{equation}
p_n(t) =e^{-i\kappa(2s+1)t } \; {e^{i\nu}-e^{-i\nu} \over e^{i\nu-\Omega t} -e^{-i\nu+\Omega t}}
\; \left[ {e^{\Omega t} - e^{-\Omega t} \over e^{i\nu - \Omega t} - e^{-i\nu + \Omega t}}\right]^n
\label{pnt}
\end{equation}
It is instructive to consider the large $n$ limit. Then $p_{n}(t)$
is proportional to $\exp[-\frac{i}{\hbar}(\kappa (2s_{cl}+\hbar)t+(\pi-\nu)x)]$. This is the
dominant phase factor for a WKB state of energy $E=E_c$ which is concentrated on the
outgoing branch of the critical classical trajectory. Note that there also appears a quantum
correction to the energy, corresponding to a finite $\epsilon=\kappa$. On the physical side, this
is quite remarkable, because we have just seen that the short time evolution is driven by
purely quantum fluctuations, formally described by the subprincipal symbol. Nevertheless, the subsequent
evolution is quite close to the classical critical trajectory, because the energy distribution of
the initial state is quite narrow, as we have discussed.
Further information is obtained by looking at the probability distribution of the number of emitted quantas:
$$
| p_n(t)|^2 = {\Omega^2 \over \Omega^2 + 2 s_{cl} \sinh^2 \Omega t }
\left[ {2 s_{cl} \sinh^2\Omega t \over \Omega^2 + 2 s_{cl} \sinh^2 \Omega t } \right]^n
$$
This is of the form:
$$
| p_n(t)|^2 \simeq e^{-\beta(t) n}, \quad \beta(t)= \log \left( 1 + {\Omega^2 \over 2s_{cl} \sinh^2 \Omega t} \right)
$$
Hence we have a {\em thermal} distribution with a time-dependent effective temperature.
Such behavior is due to the strong entanglement between the spin and the oscillator.
In fact we somehow expect that quantum effects, such as entanglement, are likely to be
magnified in situations where there is an important qualitative difference between the classical
and the quantum evolutions.
It is now simple to compute the mean number of molecules produced in this small time regime. We find:
\begin{equation}
\bar{x}(t) =2 \hbar s_{cl} {\sinh^2\Omega t \over \Omega^2}, \quad \Omega = \sqrt{2s_{cl}-\kappa^2}
\label{xsmalltime}
\end{equation}
The small time approximation is valid as long as the number of molecules is small and will break after a time $t_0$ such that $\bar{x}(t_0)\simeq 1$. To leading order in $\hbar$, this time scale is:
$$
t_0 \simeq -{1\over 2 \Omega} \log \hbar
$$
Note that in the {\em stable} case, we change $\Omega$ into $i\Omega$ and then:
\begin{equation}
\bar{x}(t) =2 \hbar s_{cl} {\sin^2\Omega t \over \Omega^2}, \quad \Omega = \sqrt{\kappa^2 - 2 s_{cl}}
\label{xbarstable}
\end{equation}
In that case we never leave the small $n$ regime and this approximation remains valid even for large time as can be seen on Fig.~\ref{nomolecules}.
\subsection{Periodic Soliton Pulses}
When time increases, the approximation $n << 1$ is not valid anymore and we must take into account the exact
quantum Hamiltonian. In the regime $\hbar << x = n\hbar << 2s_{cl}$, we can still perform a WKB approximation.
The previous discussion has shown that the semi classical wave-function is concentrated on the classical
orbit of $h^{0}$ with energy $E=E_c+\hbar\kappa$, that is $\epsilon = \kappa$.
We set:
$$
p(x,t) = e^{- {i\over \hbar} \kappa (2s_{cl}+\hbar) \; t} e^{-{i\over \hbar} (W_{0}(x)+\hbar W_{1}(x)} b(x,t)
$$
where $W_{0}(x)$ is given by Eq.~(\ref{soluWKB}), and $W_{1}(x)$ by Eq.~(\ref{expressionW1}) in which
$\epsilon$ is replaced by it actual value $\kappa$. The only source of time dependence, at this level
of approximation, arises from the transport equation of the amplitude. Let us define the local velocity
$v(x)$ on the classical trajectory by:
$$
v(x)=-\partial_{\theta} h^{0}(x,W'_{0}(x)) = \pm 2x \sqrt{\Omega^{2}-x}
$$
From this field, we obtain the Hamiltonian flow on the classical trajectory
by solving the differential equation $dx/dt=v(x)$. Let us denote by
$x(x_{0},t)$ the solution of this equation which starts from $x_{0}$ at $t=0$.
Likewise, we may reverse the flow and define $x_{0}(x,t)$.
It is convenient to introduce the function $u(x)$ such that $du/dx=-1/v(x)$.
The solution of the flow is then:
\begin{equation}
u(x_{0})=u(x)+t
\end{equation}
For our problem, we have:
$$
u(x) = - \int^x {1\over 2x \sqrt{\Omega^2 -x}} dx, \quad {\rm or~else} \quad x= {\Omega^2 \over \cosh^2 \Omega u}
$$
The transport equation simply states the conservation of the local density $b^{2}(x,t)$:
$$
\partial_{t}(b^{2}(x,t))+\partial_{x}(b^{2}(x,t)v(x))=0
$$
The evolution of this one dimensional conserved fluid is characterized by
the invariance of $b^{2}v$ along the flow:
\begin{equation}
b^{2}(x,t)v(x)=b^{2}(x_{0}(x,t),0)v(x_{0}(x,t)) \equiv B_{0}^{2}(u(x)+t)
\end{equation}
which simply results from the conservation law and the fact that the velocity field
does not depend on time.
Then we have:
\begin{equation}\label{semicl-evol-int}
\bar{x}(t) = \int dx {x\over [ x^2 (\Omega^2 -x)]^{1/2} } B_0(u(x)+t)^2 =
\int du {\Omega^2 \over \cosh^2 \Omega u} B_0(u+t)^2
\end{equation}
If we assume that $B_0(u)^2$ is peaked around $t_0$, say $B_0(u)^2 = \delta(u-t_0)$, we find:
\begin{equation}
\bar{x}(t) = {\Omega^2 \over \cosh^2 \Omega (t-t_0)}
\label{xbarinstable}
\end{equation}
which is just the classical expression. There is a simple way to
evaluate the time $t_0$. In fact there must be a regime where the
small time formula and this semiclassical formula should agree. i.e.
\begin{equation}
\hbar {2s_{cl} \over \Omega^2} \sinh^2\Omega t \simeq {\Omega^2 \over
\cosh^2 \Omega (t-t_0)}, \quad
{\rm or} \quad \hbar {2s_{cl} \over 4 \Omega^2} e^{2 \Omega t } \simeq 4
\Omega^2 e^{2 \Omega (t-t_0) }
\end{equation}
and this gives the time $t_0$:
\begin{equation}\label{t0}
t_0 = -{1\over 2 \Omega} \log{ 2 s_{cl} \over 16 \Omega^4} \hbar
\simeq -{1\over 2 \Omega} \log \hbar + O(\hbar^0)
\end{equation}
This reproduces well the position of the first peak. Its height is also well
reproduced if instead of the crude approximation $B_0(u)^2 =
\delta(u-t_0)$, we glue the wave functions given by the small time solution and the WKB solution at a gluing time $t_g$:
$$
B_0(u + t_g)^2 \simeq K \frac{\sinh(\Omega u)}{\cosh(\Omega u)^3}
\exp\left(-\frac{\Omega^2}{\bar x(t_g) \cosh(\Omega u)^2 }\right)
$$
where the normalization constant $K$ is given by the condition:
$$
\int_0^\infty B_0(u)^2 du =1
$$
and $\bar x(t_g)$ is the value of $\bar x$ given by Eq.~(\ref{xsmalltime}) taken at time $ t=t_g$ (see Fig.~\ref{first-peack}).
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 5cm, width=10cm]{first-peack.eps}
\caption{Time evolution of the mean oscillator energy
$\bar x(t)$ for $\kappa=1/\sqrt{2}$, $s_{cl}=1$ and
$\hbar=1/300$. The blue line corresponds to the exact solution, the
red one to the semiclassical solution and the
yellow line to the small time solution.}
\label{first-peack}
\end{center}
\end{figure}
The height of the first peak can be estimated from
Eq.~(\ref{semicl-evol-int}). The maximum is obtained when we have the
largest overlap between $1/\cosh^2{\Omega u}$ and $
B_0(u+t)^2$. This is happens with a very good approximation, for
$t=t_0$ such that the maximum of $B_0(u+t_0)^2$ is at zero:
$$
\sinh^2{\Omega (t_0-t_g)} = -\frac{1}{4}\left(1-\frac{2\Omega }{\bar x(t_g)}
\right)+ \sqrt{\frac{1}{16}\left(1-\frac{2\Omega}{\bar x(t_g)}
\right)^2+ \frac{1}{2}}
$$
The previous equation allows us also to discuss the freedom in the
choice of the gluing point $t_g$. Indeed we expect that $t_0$ does not
depend on $t_g$ as long as $t_g$ is restricted to a temporal domain where the
short time solution and the semiclassical one overlap. If we plot $t_0$ as a
function of $t_g$, we see that the curve we obtain develops a plateau,
which means that the value of $t_0$ we obtain chosing $t_g$ inside this plateau is independent of $t_g$.
\begin{center}
\begin{figure}
\includegraphics[height= 5cm, width=10cm]{tg-t0.eps}
\caption{Time $t_0$ of the first pulse as a function of the gluing time $t_g$ between
the short time solution and the WKB solution. The existence of a plateau shows that gluing
can be done in an unambiguous way. $\kappa =-1/\sqrt{2}$, $s_{cl}=1$ and
$\hbar=10^{-4}$. We can recognize a plateau between $t_g=1$ and $t_g=4$.}
\end{figure}
\end{center}
Moreover, if we plot the value of $t_0$ at the plateau, as a function of $\log(\hbar)$, we
recover with a very good accuracy the rough estimation of
Eq.~(\ref{t0}), see Fig.~\ref{t0fig}.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[height= 5cm, width=10cm]{t0-logh.eps}
\caption{Time $t_0$ of the first pulse as a function of $\hbar$.
The blue points are obtained by looking at the value of $t_0$ at the
plateau. The red line corresponds to the Eq.~(\ref{t0}). $\kappa=-1/\sqrt{2}$, $s_{cl}=1$.
}
\label{t0fig}
\end{center}
\nonumber
\end{figure}
If we return to the $delta$-function approximation, it is clear that the situation reproduces itself after a time interval $2 t_0$.
Hence we can write:
$$
\bar{x}(t) \simeq \sum_{k=-\infty}^{\infty} {\Omega^2 \over \cosh^2 \Omega(t -(2k+1)t_0)} \simeq
\Omega^2 {\rm cn}^2[\Omega(t-t_0) \vert k^2]
$$
where ${\rm cn}$ is a Jacobi elliptic function of modulus $k^2$ given approximately by:
$$
k^2 \simeq 1-{2\hbar s_{cl} \over \Omega^4}
$$
When $\kappa =0$, this is Bonifacio and Preparata's result~\cite{Bonifacio}.
Clearly, it remains a challenge to account for the lack of exact periodicity in the pulse sequence, using the time-dependent approach. In this perspective, returning to the integrable structure may be interesting in order to obtain an effective model around the critical point in terms of particle-hole excitations of Bethe pseudo-particles.
\section{Conclusion}
Let us emphasize the main results of this work. First, the Jaynes-Cummings model exhibits, in a large region of its
parameter space, an unstable fixed point which corresponds to the focus-focus singularity of an integrable system
with two degrees of freedom. As a result, there is a classical monodromy when one considers a loop which encircles the
critical value in the $(H_{0},H_{1})$ plane. At the quantum level, this phenomenon is manifested by a dislocation in the
joint spectrum of $H_{0}$ and $H_{1}$. We have then analyzed the eigen-subspace of $H_{1}$ which corresponds to the
critical point. The associated reduced phase-space is a sphere. We have shown how to perform a semi-classical analysis
using the convenient but singular coordinates $(x,\theta)$ on this sphere.
The main result here is that when the classical orbit crosses either the north or the south pole, where the longitude
$\theta$ is not well defined, the action integral associated to the subprincipal symbol jumps by $\pm \pi$, and this
is compensated by a simultaneous jump in the Maslov index. Most of the spectrum is well described by usual Bohr-Sommerfeld
quantization rules, at the exception of typically $|\log \hbar|$ eigenstates in the vicinity of the critical energy, for which
special Bohr-Sommerfeld rules have been obtained. Remarkably, the classical unstable equilibrium state, where the spin component
$s^{z}$ is maximal and the oscillator is in its ground-state, has most of its weight on the subspace spanned by these singular
semi-classical states. This fact explains rather well the three time scales observed in the evolution of the mean energy of the
oscillator. At short time, this energy grows exponentially, reflecting the classical instability of the initial condition.
At intermediate times, the energy of the oscillator exhibits a periodic sequence of pulses, which are well described by the
classical motion along the pinched torus containing the unstable point. Finally, the delicate pattern of energy levels, which are
not exactly equidistant, governs the aperiodic behavior observed for longer time scales.
This work leaves many unsolved questions. One of them is to develop a detailed analytical description of the
long time behavior. This should a priori be possible because, as we have seen, the initial state is a linear
superposition of only a small number of energy eigenstates, for which the singular WKB analysis developed here
provides an accurate modeling. Another interesting direction is the extension to several spins, which is
physically relevant to the dynamics of cold atom systems after a fast sweep through a Feshbach resonance.
It would be interesting to discuss if the notion of monodromy can be generalized with more than two degrees of freedom.
Finally, it remains to see whether the qualitative features of the time evolution starting from the unstable state
remain valid for an arbitrary number of spins, as may be conjectured from the structure of the quadratic normal form
in the vicinity of the singularity.
\section{Acknowledgements}
One of us (B.D.) would like to thank Yshai Avishai for many stimulating discussions on out of equilibrium dynamics
of cold atom systems and for an initial collaboration on this project. We have also benefited a lot from many
extended discussions with Thierry Paul on various aspects of semi-classical quantization. Finally, we wish to thank
Yves Colin de Verdi\`ere and Laurent Charles for sharing their knowledge with us.
B.D.was supported in part by the ANR Program QPPRJCCQ, ANR-06BLAN-0218-01,
O.B. was partially supported by the European Network ENIGMA, MRTN-CT-2004-5652
and L.G. was supported by a grant of the ANR Program GIMP, ANR-05-BLAN-0029-01.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,670 |
Studiò a Parigi con Carle van Loo e François Boucher, e successivamente fu attivo alla corte di Karlsruhe (Germania) per venti anni prima di stabilirsi a Strasburgo dal 1774. Dipinse ritratti e scene di argomento storico e religioso. Ottenne il secondo posto al Prix de Rome nel 1749.
Biografia
Figlio di uno scultore in legno, entrò nell'ordine degli agostiniani a Saarlouis per imparare il latino. Era lo zio di Antoine Ignace Melling. Si recò a Parigi dove lavorò per uno scultore ornamentale. Fu allievo di Charles André van Loo (1705-1765) e vinse il primo premio di disegno dall'Académie royale de peinture et de sculpture. Fu raccomandato a François Boucher (1703-1770) che lo mise in condizione di vincere il Prix de Rome nel 1750. Entrò nella scuola degli allievi protetti di van Loo e vi rimase fino al 1755. Quando partì per Roma, si arruolò al servizio di una corte tedesca e vi lavorò a lungo senza ottenere molti profitti. Andò con la sua famiglia a Strasburgo per fondarvi una scuola di disegno e realizzò in quella città molti dipinti di ogni tipo .
Opere
Presunto autoritratto dell'artista, Museo delle arti decorative di Strasburgo
Ritratto di una donna anziana, 1774
Ritratto di due sposi, 1776, Museo delle arti decorative di Strasburgo
Ritratto del dottor Isaac Ottmann, 1778 Museo delle arti decorative di Strasburgo
Ritratto di Catherine-Léonie Staedel (moglie del medico Ottmann), 1778 Museo delle arti decorative di Strasburgo
Ritratto della moglie P.-J. Staedel, 1778 Museo storico di Strasburgo ?
Ritratto di uomo, Museo di Belle Arti di Strasburgo
Le sei virtù civiche, 1796? (sei allegorie nel Salone dei vescovi, Palazzo dei Rohan
L'apoteosi del principe Massimiliano (Palazzo del governatore militare di Strasburgo)
Note
Bibliografia
Victor Beyer, « Joseph Melling », in Nouveau dictionnaire de biographie alsacienne, vol. 26, p. 2592
Pierre Brasme, La Moselle et ses artistes, Éd. Serpenoise, Metz, 2002, p. 161-162
Philippe Bronder, « Les Melling », in Histoire de Saint-Avold et de ses environs depuis la fondation de la ville jusqu'à nos jours, Nouvian, Metz, 1868, 76-77 Les Melling
Richard Melling, « Der Karlsruher Hofmaler Joseph Melling (1724–1796) und seine Familie », in Badische Heimat, n. 30, 1950, pp. 31-43
Durival, Description de la Lorraine et du Barois, chez la veuve Leclerc, 1779
Altri progetti
Paesaggisti francesi
Pittori di corte
Vincitori del Prix de Rome | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,822 |
Q: Indexed views on double linked server A server SERVER_A has some data.
A server SERVER_B defines A as a linked server, and defines some views on the data. The queries are like this:
CREATE view myview as
select * from
openquery ( SERVER_A, select .... )
A server SERVER_C defines B as a linked server.
I have full access to SERVER_C, I can ask for some changes to SERVER_B, but not to SERVER_A. All my applications will access server SERVER_C, and need the data from SERVER_A. The views on SERVER_B are quite complex, and they cause timeout before returning anything.
I think that to improve the performance on these queries I need to turn the views on SERVER_B into "indexed views". To index a view, I must create a clustered index.
Questions:
1) Where should I define the clustered index? Should it be at SERVER_B, or I can somehow define it on SERVER_C?
2) Will the use of an indexed view on SERVER_B affect in any way the performance of SERVER_A even when the view is not being used?
A: You can't create an Indexed view on linked server table.
According to BOL http://msdn.microsoft.com/en-us/library/ms191432.aspx
"The view must reference only base tables that are in the same database as the view."
A: To improve performance you may create a copy of data from server A on server C and update it on a schedule or use replication, if possible. Sometimes it works, if you don't need too 'live' data. With OPENQUERY whole dataset is transferred each time the query is executed.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,015 |
Home » Topics » Tech Giants » C&W's great escape
C&W's great escape
by Ben Rossi 10 February 2006
When Cable &Wireless (C&W), the telecommunications and data services provider, announced plans in December to put most of its US assets into Chapter 11 bankruptcy protection, few were surprised. After all, C&W was widely known to have run out of ideas – and patience – for dealing with a string of dot-com-era data services acquisitions.
Wisely, C&W's customers had been briefed well in advance. CEO Francesco Caio devised the Chapter 11 plan, and the escape route for customers, shortly after he took over in April 2003,
Bad call
The costly acquisitions that C&W is walking away from
MCI Internet – $1.5 billion – July 1998
Digital Island – $340 million – May 2001
Exodus – $750 million – November 2001
figuring it was the only way to extract the company from its North American mess.
First, he began leasing bandwidth from alternative providers in North America. Plunging bandwidth prices meant that this 'virtual' network could be put together and operated more cheaply than C&W's physical network, which was destined for Chapter 11.
Then, C&W briefed its UK and global customers about its plan and, from October 2003, quietly started transferring them to the virtual network. "They were quite relaxed about it," says Duncan Black, C&W's director of corporate solutions strategy. Few clients were lost as a result, he claims.
Customers in the US with no overseas operations were left behind with C&W America. That unit, which C&W put into administration, now seems destined to be bought by investment firm Gores Technology for 'only' $125 million.
Meta Group analyst Stan Lepeak says that even US customers left with C&W America should not be too concerned. The pre-packaged nature of C&W America's administration order, with a buyer already lined up to take over at the end of March 2004, gives them the best chance of a smooth transition, he says.
If it goes according to plan – and it remains a big 'if' for now – C&W's strategy could prove a blueprint for other data services companies that run into difficulties. Despite all the trouble at C&W and others, everyone in the industry knows there is still much more to come.
Ofcom to scrutinise UK cloud positions of AWS, Microsoft and Google
Google hit by record €4bn fine for EU Android market dominance | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 335 |
Q: Finding key in a list of dictionaries and sorting the list Having a JSON-like structure:
{
"shopping_list": [
{
"level": 0,
"position": 0,
"item_name": "Gold Badge"
},
{
"level": 1,
"position": 10,
"item_name": "Silver Badge"
},
{
"level": 2,
"position": 20,
"item_name": "Bronze Badge"
}
]
}
I'm trying to sort the list by key.
However, when trying to get them by:
k = [c.keys()[0] for c in complete_list["shopping_list"]]
I get TypeError: 'dict_keys' object does not support indexing.
*
*How to get keys?
*How to sort the list by specified key?
A: Try this to get the keys in a list of lists :
d = {
"shopping_list": [
...
}
k = [list(c.keys()) for c in d["shopping_list"]]
print(k)
Output :
[['item_name', 'level', 'position'], ['item_name', 'level', 'position'], ['item_name', 'level', 'position']]
However even if you wanted to sort this list of lists based on some specific key value say value for "level" key or "position" key, the list of lists would remain same.
A: Here it is (admitting the key you mention is 'level'):
k = sorted([c for c in complete_list["shopping_list"]], key=lambda x: x['level'])
print(k)
# >> [{'level': 0, 'position': 0, 'item_name': 'Gold Badge'}, {'level': 1, 'position': 10, 'item_name': 'Silver Badge'}, ...
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,060 |
Barry Stern (January 24, 1960April 1, 2005) was a heavy metal drummer from Chicago, Illinois.
History
Zoetrope and Trouble (1976-1995)
From 1976 until 1988, he drummed for Zoetrope, for whom he also served as vocalist and songwriter. After recording two albums with that band, Stern joined Trouble in 1989. He appeared on two of their albums, 1990's Trouble and 1992's Manic Frustration, before leaving in 1993.
In 1994, Barry replaced Joe Hasselvander in Cathedral for touring purposes.
Final musical projects and Death (2005-2007)
A guest spot on Debris Inc.'s 2005 self-titled debut album was Barry's final recording appearance.
Stern died on April 1, 2005 from complications following hip replacement surgery. He was the son of Louise Stern, the brother of Karen Stern Jarger and Arlen Stern, the brother in-law of John Jarger, as well as the Uncle of Jeremy Jaeger, Lindsay Jarger, and Alan Stern.
Prior to his death, he was the founder of the band D-Connect (his stage name). Stern was singing and drumming for the band D-Connect at the time of his death. Trouble dedicated their 2007 album Simple Mind Condition to his memory.
Bands
Former
Zoetrope - Drums (1976-1988)
Trouble - Drums (1989-1993)
D-Connect - Drums, Vocals (2003-2005)
Live
Cathedral - Drums (1994-1995)
Discography
Zoetrope
"The Right Way" b/w "Call 33" 7" (1980, self-released)
The Metal Log Vol. 1 demo tape (1983, self-released)
The Metal Log Vol. 2 demo tape (1985, self-released)
Amnesty LP (1985, Combat Records, reissued on CD with the Metal Log demos as bonus tracks by Century Media in Europe in 1999)
A Life of Crime LP (1987, Combat Records, reissued on CD with liner notes by Kevin Michael on Century Media in Europe in 1998)
Trouble
Trouble (1990)
Trouble Live Dallas Bootleg (1990, a live concert originally broadcast by radio during the band's tour for the self-titled album on Def American, available on CDR through the band's website)
Manic Frustration (1992)
Knocking on Heavy's Door (1992; Split w/ Massacria, Tanner, Meglomanic)
Debris Inc.
Debris Inc. (2005)
References
External links
Trouble comment on Barry's passing
Chicago Punk Pix's Tribute To Barry
American heavy metal drummers
American performers of Christian music
American heavy metal singers
Singers from Chicago
2005 deaths
1960 births
20th-century American singers
20th-century American drummers
American male drummers
Trouble (band) members
20th-century American male musicians
Cathedral (band) members
Septic Tank (band) members | {
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New Flight Path Information
New Flight Path Information – 27th May 2014
Public Mislead – Press Release 18th May 2014
One's Enough – Gatwick Airport Ltd (GAL)
2nd Runway Consultation Update
The 2nd Gatwick Runway Consultation has finally brought out some honesty from Gatwick Airport Ltd (GAL) and the Gatwick Diamond Group.
At the Consultation Exhibitions the GAL experts would/could not provide any proposals as to limiting and safeguarding against airborne pollution, they could only say it will be monitored and that currently most areas around Gatwick come within "legal UK limits". However, pollution will not just come from aircraft but also from increased and concentrated road traffic which would mean that at times some locations may already fail the more stringent EU limits.
The GAL experts stated that GAL will only be responsible for the infrastructure "within the airport boundary", potentially including the rerouting of the A23 and Balcombe Road and the M23 Junction 9, with everything else the responsibility of local and central government. They offered the already planned removal of the hard shoulder on the M23 and longer trains as major projects that will solve the traffic issues.
The GAL noise contour maps only show Average Noise Levels which infers that only the neighbourhoods adjacent to and at each end of the airport runways will be deemed as being moderately inconvenienced, however, the experts conceded that much of Crawley and surrounding areas will at times be subjected to more excessive noise levels.
The GAL experts would not be specific as to any real benefits for the Crawley residents. The consistent response was that "businesses and the unemployed in the South East and the wider UK would greatly benefit from new jobs and business opportunities". This response has been echoed by both GAL and Gatwick Diamond executives in the past month in either the press and/or at a meeting with Crawley Councillors.
The GAL experts, and also the Gatwick Diamond executives in the press, questioned the claim that up to 45,000 houses will be potentially be required to accommodate a future incoming migrant workforce. They had to be reminded that this 45,000 was a key point within a report commissioned jointly by W.S.C.C and Gatwick Diamond.
The "wide spaced" runway options show that additional land will be required to that currently "safeguarded for airport expansion".
This Consultation shows that whilst the South East and the wider UK will benefit from new economic and employment opportunities, Crawley and the area around Gatwick will bear the cost by having the environment, that has brought many people to the area, destroyed.
The Government's official consultation on the final proposals is due in the autumn, so again we should all ask ourselves, "although there may be employment, will our children and grand children really want to live in such a polluted, urbanised and congested environment?
Final objections can be sent by email to gatwick runway consultation before end of Friday 16th May 2014
The easiest way to say No, is to click on the following link : –
gatwickrunwayconsultation@ipsos.com simply substitute the NAME and ADDRESS in the auto-generated email and press the send button.
Note: One's Enough is a voluntary non political group of "cross party" councillors and members of the public, supporting a One Runway Two Terminal Airport with spare capacity.
Gatwick airport consultation period
Consultation Period Ending
PH(N)RA would like to remind you that if you have not done so already, you need to respond to the Gatwick Airport runway consultation before 16th May. You will find details at
www.gatwickairport.com/business-community/New-runway/Second-runway-consultation/
The online form is long-winded and some of the questions are deliberately crafted, in answering some of these questions the response could be interpreted as a yes to a further undisclosed runway option. Therefore, if you wish to object to the runway expansion outright, please send an email to this effect before the 16th.
Exhibitions – Press Release 5th May 2014 | {
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{"url":"http:\/\/cnx.org\/content\/m14314\/1.7\/","text":"# OpenStax-CNX\n\nYou are here: Home \u00bb Content \u00bb Angular quantities for general motion\n\n## Navigation\n\n### Recently Viewed\n\nThis feature requires Javascript to be enabled.\n\n# Angular quantities for general motion\n\nModule by: Sunil Kumar Singh. E-mail the author\n\nSummary: Angular quantities are not limited to rotation about fixed axis only.\n\nNote: You are viewing an old version of this document. The latest version is available here.\n\nAngular quantities like angular displacement, velocity, acceleration and torque etc. have been discussed in earlier modules. We were, however, restricted in interpreting and applying these quantities to either circular motion or pure rotational motion. Essentially, these physical quantities have been visualized in reference to the axis of rotation and a circular path.\n\nIn this module, we shall expand the meaning and application of angular quantities in very general terms, capable of representing pure as well as impure rotation and translation. We shall soon find that concepts of angular quantities in terms of general motion, as a matter of fact, provides unique analysis frame work, in which all motion types are inclusive.\n\n## General interpretation of angular quantities\n\nAngular quantities, in their basic forms, are defined in very general context. When we defined, for example, torque in the context of rotation of a rigid body or a single particle attached to a \"mass-less\" rod, we actually presented a definition in special context of a fixed axis. As a matter of fact, the definition of angular quantities does not require an axis to be defined.\n\nWe can define and interpret angular quantities very generally with respect to a \"point\" in the reference system - rather than an axis. This change in reference of measurement allows us to extend measurement of angular quantities beyond angular motion. For some, it may sound a bit inconsistent to know that we can actually associate all angular quantities even with a straight line motion or a translational motion! For example, we can calculate torque on a particle, which is moving along a straight line or we can determine angular displacement and velocity for a projectile motion! We shall work out with appropriate examples to illustrate the point.\n\nIndeed, angular quantities are found to suit rotational motion or where curvature of path is involved. For this reason, we tend to think that angular quantities are applicable only to rotation. There is nothing wrong to think so. As a matter of fact, there are many situations in real time, which suits a particular analysis technique. Consider projectile motion along a parabolic path. We employ the concept of independence of motions in mutually perpendicular directions, based on experimental facts. This paradigm of analysis suites the analysis of projectile motion in a best fit manner. But as a student of physics, it is important to know the complete picture.\n\nWe must understand here that the broadening the concept of angular quantities is not without purpose. We shall find out in the subsequent modules that the de-linking of angular concepts like torque and angular momentum from an axis, lets us derive very powerful law known as conservation of angular momentum, which is universally valid unlike Newton's law (for translational or rotational motion).\n\nThe example given below calculates average angular velocity of a projectile to highlight the generality of angular quantity.\n\n### Example 1\n\nProblem : A particle is projected with velocity \"v\" at an angle of \"\u03b8\" with the horizontal. Find the average angular speed of the particle between point of projection and point of impact.\n\nSolution : The average angular speed is given by :\n\n\u03c9 avg = \u0394 \u03b8 \u0394 t \u03c9 avg = \u0394 \u03b8 \u0394 t\n\nHere,\n\n\u0394 t = 2 v sin \u03b8 g \u0394 t = 2 v sin \u03b8 g\n\nFrom the figure, magnitude of the total angular displacement is :\n\n\u0394 \u03b8 = 2 \u03b8 \u0394 \u03b8 = 2 \u03b8\n\nPutting these values, we have :\n\n\u03c9 avg = \u0394 \u03b8 \u0394 t = 2 \u03b8 g 2 v sin \u03b8 \u03c9 avg = \u03b8 g v sin \u03b8 rad \/ s \u03c9 avg = \u0394 \u03b8 \u0394 t = 2 \u03b8 g 2 v sin \u03b8 \u03c9 avg = \u03b8 g v sin \u03b8 rad \/ s\n\nFrom this example, we see that we can indeed associate angular quantity like angular speed with motion like that of projectile, which is not strictly rotational.\n\n## Angular velocity of a particle in general motion\n\nAverage angular velocity is defined as the ratio of change of angular displacement and time interval :\n\n\u03c9 avg = \u0394 \u03b8 \u0394 t \u03c9 avg = \u0394 \u03b8 \u0394 t\n(1)\n\nInstantaneous angular velocity is obtained by taking the limit when time interval tends to zero. In other words, the instantaneous angular velocity (simply referred as angular velocity) is equal to the first differential of angular displacement with respect to time :\n\n\u03c9 = d \u03b8 d t \u03c9 = d \u03b8 d t\n(2)\n\nThe most important aspect of the definitions of angular velocity is that angle is measured with respect to a point. The measurement of angular velocity, in turn, will depend on the choice of origin. Further, the linear distance of the particle from a given point is not constant like in rotation as particle can move along any path - even straight line.\n\nIn the modules on circular motion and rotation of rigid body, we have worked with angular velocity as applicable to rotation. In the earlier example in this module, we determined average angular velocity for a projectile motion. Now, we shall further extend the concept of angular velocity to an angular motion, which is not pure translation.\n\n### Example 2\n\nProblem : A rod of length 10 m lying against a vertical wall starts moving. At an instant, the rod makes an angle of 30\u00b0 as shown in the figure. If the velocity of end \u201cA\u201d at that instant, is 10 m\/s, then find the angular velocity of the rod about \u201cA\u201d.\n\nSolution : In this case, the point about which angular velocity is to be determined is itself moving with a velocity of 10 m\/s. However, we are required to find angular velocity for a particular instant i.e. instantaneous angular velocity for which the point can be considered at rest.\n\nWe shall examine the situation with the geometric relation of the length of rod with respect to the position of its end \u201cA\u201d. This is a logical approach as the relation for the position of end \u201cA\u201d shall let us determine its velocity and equate the same with the given value.\n\nx = AB cos \u03b8 x = AB cos \u03b8\n\nAn inspection of the equation reveals that if we differentiate the equation with respect to time, then we shall be able to relate linear velocity with angular velocity.\n\nx t = AB t ( cos \u03b8 ) = - AB sin \u03b8 x \u03b8 t x t = AB t ( cos \u03b8 ) = - AB sin \u03b8 x \u03b8 t\n\nv = - AB \u03c9 sin \u03b8 \u03c9 = - v AB sin \u03b8 v = - AB \u03c9 sin \u03b8 \u03c9 = - v AB sin \u03b8\n\nPutting values,\n\n\u03c9 = - 10 10 sin 30 \u00b0 = - 2 rad \/ s \u03c9 = - 10 10 sin 30 \u00b0 = - 2 rad \/ s\n\nNegative sign here signifies that the angle \u201c\u03b8\u201d decreases with time ultimately becoming equal to 0\u00b0 as measured in anticlockwise direction from x-axis. We can, therefore, conclude that instantaneous angular velocity of point \u201cA\u201d is clockwise having a magnitude of 2 rad\/s.\n\n### Interpretation of angular velocity\n\nIn this sub-section, we shall interpret angular velocity for general motion with a reference to rotation, highlighting \"where and how they are different or same\".\n\nWe can see that the expressions of angular velocity are same as in the case of pure rotation. In the case of pure rotation, the angular velocity is perpendicular to the plane of angular displacement and is aligned either in the positive (clockwise) or negative (anticlockwise) direction of the axis. In the general case, the angular velocity is similarly perpendicular to plane of angular displacement and its sense of direction is clockwise or anticlockwise like in the case of rotation, but with respect to a point - not with respect to an axis. This makes the difference to the measurement of angular displacement and hence to the angular velocity as shown in the figure below :\n\nIn the above figure, we have considered rotation of a particle about y-axis. For visualization, we have considered the initial position of the particle in the yz - plane. The particle moves from \"A\" to \"B\" in anticlockwise direction as seen from the top. We note following important differences for general consideration for rotation of a particle :\n\n\u2022 The linear distances of the particle at two instants are not same (OA # OB) from the point \"O\" as against from the axis, which are equal (O'A # O'B).\n\u2022 The angle measured from \"O\" (general case) and \"O'\" (rotational case) are different (\u03b8 # \u03b8').\n\u2022 The moment arm from \"O\" is OO' not O'A as in the case of rotation.\n\u2022 The direction of angular velocity vector is not in the direction of y-axis as in the case of rotation.\n\nFor the sake of comparison, here, we have analyzed a pure rotation. In general, however, the plane of angular displacement, unlike pure rotation, can change and so the direction of angular velocity. We must understand that particle is free to move along any path.\n\nAs a matter of fact, angular velocity in rotation is just a special case of angular velocity defined in general. This can be visualized easily, if we make (i) the origin \"O\" to coincide with \"O'\" (ii) ensure that the magnitude of position vector (r) of the particle does not change during motion and (iii) the plane of motion is perpendicular to the axis of rotation. In that case, the particle will be constrained to rotate about an axis. Clearly, the angular velocity as defined for rotational motion is a special case of general definition with certain restrictions.\n\n## Angular and linear velocity\n\nThe relation of angular velocity of a particle with its linear velocity in general case has the same vector form as for rotation of a particle, but here again its interpretation is different as we shall evaluate angular velocity with respect to a point (not with respect to an axis) and for a motion, which need not be rotational. The angular velocity is related to linear velocity by the following vector product :\n\nv = \u03c9 x r v = \u03c9 x r\n(3)\n\nThe vector \"r\" is measured from the point about which angular velocity is being calculated. In case, the point coincides with the origin of the reference system (we normally plan so), the vector \"r\" becomes the position vector of the particle in the given reference.\n\nAngular velocity and other angular quantities are mostly defined in terms of vector product. The evaluation techniques for its magnitude and direction follow certain steps. It is always good to have a step wise plan to evaluate and associate direction in consistent manner. These techniques have already been described in the module titled \" Vector product \" in this course. However, it would be refreshing to get a grip on the process as applied to specific quantity. For this reason, we present the evaluation process for angular velocity in the sub-section here, which can be applied for other angular quantities as well in subsequent modules.\n\n### Evaluation of angular velocity as vector product\n\nEvaluation of a vector quantity involves magnitude and direction :\n\n(i) Magnitude of angular velocity\n\nWe can find the magnitude of angular velocity, using any of the following two alternatives :\n\nThe magnitude of vector product relating linear velocity with angular velocity is given as :\n\nv = \u03c9 r sin \u03b8 v = \u03c9 r sin \u03b8\n\n\u03c9 = v r sin \u03b8 \u03c9 = v r sin \u03b8\n(4)\n\nThe expression for the magnitude of vector product relating linear velocity with angular velocity can also be interpreted as :\n\nv = \u03c9 ( r sin \u03b8 ) = \u03c9 r v = \u03c9 ( r sin \u03b8 ) = \u03c9 r\n\n\u03c9 = v r \u03c9 = v r\n(5)\n\nwhere \" r r \" represents perpendicular distance of the velocity vector from the point of reference\n\n(ii) Direction of angular velocity\n\nThe direction of angular velocity is such that the plane formed by it and position vector is perpendicular to linear velocity. The angular velocity is also individually perpendicular to linear velocity vector. We must here understand that angular velocity is an operand of the vector product - not the vector product itself. As such, we can not directly apply right hand vector multiplication rule to obtain the direction of angular velocity.\n\nThe easiest directional visualization of angular velocity is gathered from the sense of angular displacement. If we curl fingers of the right hand along the motion about the point, then the direction of thumb points towards the direction of angular velocity. This estimate of direction together with fact that angular velocity is perpendicular to the plane of angular motion, completely determines the direction of angular velocity.\n\n#### Example 3\n\nProblem : The velocity of a particle confined to xy - plane is (8i \u2013 6j) at an instant when its position is (3 m, 4m). Find the angular velocity of the particle at that instant.\n\nSolution : Angular velocity is related to linear velocity by the following relation,\n\nv = \u03c9 x r v = \u03c9 x r\n\nAn inspection of the equation reveals that angular velocity is perpendicular to the direction of velocity vector. Now, velocity vector lies in xy-plane. It, then, follows that angular velocity is directed along z-axis. Let angular velocity be represented as :\n\n\u03c9 = a k \u03c9 = a k\n\nPutting the values, we have :\n\n8 i - 6 j = a k x ( 3 i + 4 j ) 8 i - 6 j = a k x ( 3 i + 4 j )\n\n8 i - 6 j = 3 a j - 4 a i ) 8 i - 6 j = 3 a j - 4 a i )\n\nComparing the coefficients of unit vectors on either side of the equation, we have :\n\n3 a = - 6 a = - 2 3 a = - 6 a = - 2\n\nand\n\n- 4 a = 8 a = - 2 - 4 a = 8 a = - 2\n\nThus,\n\n\u03c9 = - 2 k \u03c9 = - 2 k\n\nThe angular velocity, therefore, is 2 rad\/s in the negative z \u2013 direction.\n\n## Summary\n\n1: Angular quantities are general quantities, which can be defined and interpreted for any motion types \u2013 pure\/impure translation or rotation.\n\n2: Angular velocity is defined as the time rate of change of angular displacement with respect to a point :\n\n\u03c9 = d \u03b8 d t \u03c9 = d \u03b8 d t\n\n3: If the point, about which angular velocity is determined, is the origin of the reference system, then the position of the particle is represented by position vector.\n\n4: The measurement and physical interpretation of angular velocity are different than that in the case of rotational motion.\n\n5: Angular velocity in rotation is a sub-set of angular velocity in general case. In the case of rotation, moment arm is radius vector, which is perpendicular to the axis of rotation. Also, the plane of motion in rotation is perpendicular to the axis of rotation.\n\n6: The relation between linear and angular velocity has the following form,\n\nv = \u03c9 x r v = \u03c9 x r\n\nwhere \u201cr\u201d determines the position of the particle with respect to the point. The magnitude of angular velocity can be determined, using any of the following two relations :\n\n\u03c9 = v r sin \u03b8 \u03c9 = v r sin \u03b8\n\nand\n\n\u03c9 = v r \u03c9 = v r\n\nwhere \" r r \" represents perpendicular distance of the velocity vector from the point of reference\n\n7: Direction of angular velocity is perpendicular to the plane of motion. The plane of motion, however, may not be fixed like in the case of rotation. As such, direction of angular velocity may not be limited to only two directions as in the case of rotation. For this reason, we need to treat angular velocity as a general vector quantity, which can not be represented by a scalar with positive and negative signs as in the case of rotation.\n\n## Content actions\n\n### Add module to:\n\nMy Favorites (?)\n\n'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.\n\n| A lens I own (?)\n\n#### Definition of a lens\n\n##### Lenses\n\nA lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.\n\n##### What is in a lens?\n\nLens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.\n\n##### Who can create a lens?\n\nAny individual member, a community, or a respected organization.\n\n##### What are tags?\n\nTags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.\n\n| External bookmarks","date":"2014-07-12 16:49:01","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8657634258270264, \"perplexity\": 486.6872295169263}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-23\/segments\/1404776434099.83\/warc\/CC-MAIN-20140707234034-00035-ip-10-180-212-248.ec2.internal.warc.gz\"}"} | null | null |
PayPal is currently running a promotion and a chance to with $5 all the way to $10,000!
Although a bit silly, this is PayPal's way of encouraging people to sign up to their services and use their platform as a means of payment for everyday expenses.
It's called Moola Mania and they are offering Instant Prizes in the form of cash. The promotion will last from February 1, 2018 until March 15, 2018. Checkout their prizes!
You'll be sent a code that will allow you to play the game.
After that, every "qualifying" purchase you make will give you a turn to play. | {
"redpajama_set_name": "RedPajamaC4"
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The Love Song
Once again this week, we are short on time, energy, and inspiration at this blog. If the song below hadn't popped up on shuffle this morning, there probably wouldn't be anything here today, either.
In High Fidelity, Nick Hornby wrote:
What came first, the music or the misery? People worry about kids playing with guns, or watching violent videos, that some sort of culture of violence will take them over. Nobody worries about kids listening to thousands, literally thousands of songs about heartbreak, rejection, pain, misery and loss. Did I listen to pop music because I was miserable? Or was I miserable because I listened to pop music?
There's a corollary to this idea that we learn "heartbreak, rejection, pain, misery and loss" from pop songs. It's that we also learn about love through them. Yet most of what we learn is some wispy, moon/June cookie-cutter emotion that's miles wide and fractions of an inch deep. Nevertheless, songwriters and singers keep recycling the same platitudes, and listeners keep lapping 'em up as if they were new.
Do we listen to pop music because we're stupid about love, or are we stupid about love because we listen to pop music?
There's got to be more to love than the stuff most love songs are about. And there's got to be more to love than Valentine's Day, frankly. More than cards and flowers and chocolates and the other totems we're told to spend money on today. There's got to be more to it than sex. (Which singers, songwriters, and listeners frequently confuse with love—right?)
So what's love supposed to be about? What's a love lesson worth listening to, one worth learning?
This: "The Dutchman," popularized by Steve Goodman but written by a fellow Chicago folksinger, Mike Smith. This is the greatest love song ever written, because Smith understands that love, to be worth anything, has to endure through everything: not just the tribulations of thwarted infatuation, but the most difficult barriers life can put up. Only when it can do that is it a love really worth singing about.
Posted in: YouTube | {
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Табук () — многозначный термин.
Табук — провинция в Саудовской Аравии, расположенная на северо-западе страны, вдоль берегов Красного моря и границы с Египтом.
Табук — город, центр одноимённой провинции на северо-западе Саудовской Аравии.
Табук — деревня в Черемховском районе Иркутской области России.
См. также
Табук (оружие) — иракская самозарядная снайперская винтовка на базе югославской винтовки Zastava M76.
Битва при Табуке | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,447 |
Imhoff Archeological Site, also known as Site 23CP7, is a historic archaeological site located near Blackwater, Cooper County, Missouri. It is a Middle Woodland Period village site situated on a terrace in the Lamine River locality of the Missouri River Valley. The pottery and stone tools from the site belong to the technological/artistic tradition that is described as "Hopewell." The site was discovered by J. Mett Shippee in the 1930s. Marvin Kay surveyed the site and conducted very limited testing during 1971. No radiocarbon dates are available for the site. A sample of obsidian from the Imhoff site, in the George C. Nicholas collection, has been analyzed using Neutron Activation Analysis. The obsidian from the Imhoff site can be traced to the obsidian cliff in Yellowstone National Park, Wyoming.
It was listed on the National Register of Historic Places in 1972.
See also
Mellor Village and Mounds Archeological District
National Register of Historic Places listings in Cooper County, Missouri
References
Archaeological sites on the National Register of Historic Places in Missouri
National Register of Historic Places in Cooper County, Missouri
Hopewellian peoples
Native American history of Missouri
Former populated places in Missouri | {
"redpajama_set_name": "RedPajamaWikipedia"
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Q: Twig, error on compare date If I try to compare date as this:
{% if (expireDate==NULL) %}
{% if (date(expireDate) < date()) %}
DANGER!
{% elseif (date(expireDate) == date()) %}
WARNING
{% else %}
DON'T WORRY
{% endif %}
{% endif %}>
I assume that date() without params return the current date, and expireDate has a Date value.
My problem is that "WARNING" never been shown, and this mean that even if expireDate is today the (date(expireDate) < date()) is always false.
Am I doing something wrong?
UPDATE#1
This should works, but I don't know if it's correct.
{% if (expireDate==NULL) %}
{% if (date(expireDate)|date('d-m-Y') < date()|date('d-m-Y')) %}
DANGER!
{% elseif (date(expireDate)|date('d-m-Y') == date()|date('d-m-Y')) %}
WARNING
{% else %}
DON'T WORRY
{% endif %}
{% endif %}>
| {
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{"url":"https:\/\/17calculus.com\/parametrics\/volume\/","text":"## 17Calculus Parametric Equations - Volume\n\nUsing Limits\n\n### Derivatives\n\nGraphing\n\nRelated Rates\n\nOptimization\n\nOther Applications\n\n### Integrals\n\nImproper Integrals\n\nTrig Integrals\n\nLength-Area-Volume\n\nApplications - Tools\n\nApplications\n\nTools\n\n### Practice\n\nCalculus 1 Practice\n\nCalculus 2 Practice\n\nPractice Exams\n\n### Articles\n\nIf we have a parametric curve defined as $$x = X(t)$$ and $$y = Y(t)$$, we can determine the volume of the solid object defined by revolving this curve about an axis. We will limit our curve from $$t=t_0$$ to $$t=t_1$$.\n\nWhen revolved about the x-axis, the integral we use to calculate volume is $V_x = \\pi \\int_{t_0}^{t_1}{[Y(t)]^2 [dx\/dt] dt}$ This is sometimes written $V_x = \\pi \\int_{t_0}^{t_1}{y^2 ~ dx}$ where $$dx = [dx\/dt] dt$$\n\nSimilar to the x-axis integral, we use this integral to calculate volume. $V_y = \\pi \\int_{t_0}^{t_1}{[X(t)]^2 [dy\/dt] dt}$ also written as $V_y = \\pi \\int_{t_0}^{t_1}{x^2 ~ dy}$ where $$dy = [dy\/dt] dt$$\n\nNotes\n1. Notice that when revolving about x-axis, the integrand contains $$Y(t)$$ squared. And we have a similar situation for revolution about the y-axis. This may seem somewhat counter-intuitive but it makes sense when you know where the equations come from.\n2. Don't forget the $$\\pi$$ out in front of the integrals.\n\nPractice\n\nCalculate the volume of revolution when the curve $$x = t^3$$, $$y = 2t^2+1$$, $$-1 \\leq t \\leq 1$$ is rotated about the x-axis.\n\nProblem Statement\n\nCalculate the volume of revolution when the curve $$x = t^3$$, $$y = 2t^2+1$$, $$-1 \\leq t \\leq 1$$ is rotated about the x-axis.\n\nSolution\n\n### Krista King Math - 466 video solution\n\nvideo by Krista King Math\n\nLog in to rate this practice problem and to see it's current rating.","date":"2022-01-25 10:48:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9490911364555359, \"perplexity\": 794.2965964251549}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320304810.95\/warc\/CC-MAIN-20220125100035-20220125130035-00397.warc.gz\"}"} | null | null |
Beata Pawlak, née en à Garbatka dans le district mazovien de Kozienice, morte le à Kuta (Bali), est une journaliste et écrivaine polonaise.
Biographie
Après une scolarité secondaire dans un lycée de Radom, Beata Pawlak fait des études de lettres polonaises à l'Université Jagellonne de Cracovie. Au début des années 1980, elle milite au sein de l'opposition démocratique et participe à des publications clandestines. À partir de 1984, elle est en exil à Paris, où elle fréquente notamment des émigrés de pays musulmans.
Après 1989, elle revient en Pologne et travaille comme journaliste pour Gazeta Wyborcza pendant une dizaine d'années. Elle écrit sur la Pologne et le monde de l'islam. Selon Ryszard Kapuściński, Beata Pawlak écrit sur l'islam comme nul autre dans la presse polonaise. Elle a magistralement combiné connaissance avec sensibilité, passion avec responsabilité, diligence avec une remarquable détermination pour expliquer ce monde. En mars 2002, elle entreprend un voyage à travers l'Asie. De l'Inde, via le Népal, la Thaïlande et la Malaisie, jusqu'en Indonésie. Elle meurt le 12 octobre 2002 dans une attaque terroriste sur l'île indonésienne de Bali.
En 2003, un prix Beata-Pawlak, décerné depuis lors à un rythme annuel, est créé pour couronner l'auteur d'un texte en polonais sur les autres cultures, religions et civilisations.
Wojciech Tochman s'est inspiré pour son roman Córeńka (Znak, 2005) de l'histoire de sa vie.
Publications
Livres
Mamuty i petardy. Czyli co naprawdę cudzoziemcy myślą o Polsce i Polakach (« Mammouths et pétards. Que pensent vraiment les étrangers de la Pologne et des Polonais »), PWN, 2001
Aniołek (« Le Petit Ange », roman sur les parents de l'auteur avec leur propre portrait tissé, des réflexions sur la vie et la mort), Prószyński et Cie, 2003
adapté au cinéma par
Piekło jest gdzie indziej, Reportaże o świecie islamu: Algieria, Francja, Bośnia, Gaza, Izrael, Liban, Irak, Kurdystan, Egipt, Polska, Turcja, Ali Ağca i Jan Paweł II (« L'Enfer est ailleurs », Reportages sur le monde musulman : Algérie, France, Bosnie, Gaza, Israël, Liban, Irak, Kurdistan, Égypte, Pologne, Turquie, et sur Ali Ağca et Jean-Paul ), Prószyński et Cie, 2003
Ouvrages collectifs
Anna Bikont, Beata Pawlak, Mariusz Szczygieł, , , , , , , Wojciech Tochman, , Kraj raj (« Le Pays paradisiaque »), 1993
Małgorzata Szejnert, Mirosław Banasiak, Beata Pawlak, et al. Anna z gabinetu bajek (« Anna du cabinet des contes de fée »), 1999
, Jacek Hugo-Bader, Beata Pawlak, Sławomir Zagórski, , , , Dorota Karaś, , Anna Fostakowska, Anna Bikont, , Wioletta Gnacikowska, Nietykalni. Reportaże roku 1999 (« Intouchables. Reportages de l'année 1999 ») – deux reportages de Beata Pawlak sur la Pologne, Prószyński et Cie, 2000
Notes et références
Site externes
Naissance en 1957
Naissance dans la voïvodie de Mazovie
Militant polonais
Femme de lettres polonaise
Journaliste polonais
Écrivain polonais du XXe siècle
Écrivain polonais du XXIe siècle
Victime du terrorisme islamiste
Décès en octobre 2002
Décès à Bali
Décès à 45 ans | {
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Producer-handler loses pricing challenge
Judge says USDA has authority to alter exemption
By MATEUSZ PERKOWSKI
A dairy farm that distributes its own milk has lost a legal challenge against a USDA policy that subjected it to federal milk pricing regulations.
GH Dairy of El Paso, Texas, claimed the agency violated federal law by altering an exemption that previously allowed such producer-handlers to operate outside the federal milk marketing order system, regardless of size.
An administrative judge with USDA has rejected the dairy's arguments, ruling that it was within the agency's authority to change the exemption.
Traditionally, producer-handlers who packaged and shipped only their own fluid milk were generally exempt from regional federal milk marketing orders, which seek to stabilize fluctuations in the dairy industry by regulating pricing.
In April 2010, however, the USDA decided to narrow that exemption across all marketing orders so that it only applied to dairies that shipped 3 million pounds or less of milk per month.
GH Dairy and other farms that produce and distribute more than 3 million pounds of milk per month opposed the new policy. Because they'd be subjected to the milk marketing order system, such producer-handlers would have to pay money into a fund aimed at equalizing prices for farmers.
GH Dairy filed a petition asking USDA to set aside the new policy, claiming that the agency had exceeded its statutory authority by making the change.
The company also claimed the agency's decision to alter the exemption was not based on substantial evidence of its necessity.
Administrative law Judge Victor Palmer disagreed with those conclusions.
Under federal law, such federal marketing orders only apply to milk "purchased from producers or associations of producers."
Since producer-handlers use their own milk and don't purchase it, GH Dairy argued the order shouldn't apply to them.
However, the judge said "purchased" must be construed broadly, to mean "acquired for marketing," according to U.S. Supreme Court precedent.
USDA also had enough evidence that the exemption was needed, the judge said.
The agency had found large producer-handlers could cause "disorderly market conditions" since they're not subject to the same rules as their regulated competitors.
A group of producer-handlers, the American Independent Dairy Alliance, had argued the exemption shouldn't be changed because they produce less than 1 percent of the milk sold in the U.S.
Milk should be subject to the free market rather than an "arcane system of regulation" that's no longer relevant, the group said.
The National Milk Producers Federation, which represents cooperatives, said the exemption should be narrowed to level the playing field for dairy manufacturers. | {
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{"url":"https:\/\/formulas.tutorvista.com\/math\/linear-function-formula.html","text":"Top\n\n# Linear Function Formula\n\nA linear function is a polynomial function whose graph gives a straight line. The linear function formula is of the form\n\nHere x is independent variable, Y is dependent variable, m is slope and c is intercept.\n\nSuppose the equation of line passes through the points (x1,y1) and (x2,y2) in a graph then the linear function is given by\n\nThen the slope m is given by\n\n Related Calculators Linear Function Calculator Linear Calculator Calculating Linear Regression Graph Linear Equations Calculator\n\n## Linear Function Examples\n\nLets see some examples on linear function:\n\n### Solved Examples\n\nQuestion\u00a01: Solve: 3x + 5 = 0\nSolution:\n\nGiven function is 3x + 5 = 0\n3x + 5 = 0\n3x = -5\nx = $\\frac{-5}{3}$ = - 1.667\n\nQuestion\u00a02: Solve: 4x - 7 = 0\nSolution:\n\nGiven function is 4x - 7 = 0\n4x - 7 = 0\n4x = 7\nx = $\\frac{7}{4}$ = 1.75\n\n*AP and SAT are registered trademarks of the College Board.","date":"2019-05-26 20:14:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6674194931983948, \"perplexity\": 976.1696527501613}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232259452.84\/warc\/CC-MAIN-20190526185417-20190526211417-00195.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:intro}
Automated classification and parameter estimation procedures are crucial for the analysis of upcoming astronomical surveys. Planned missions such as Gaia (\citealt{2001A&A...369..339P}) and the Large Synoptic Survey Telescope (LSST; \citealt{2009arXiv0912.0201L}) will collect data for more than a billion objects, making it impossible for researchers to manually study significant subsets of the data. At the same time, these upcoming missions will probe never-before-seen regions of astrophysical parameter space and will do so with larger telescopes and more precise detectors. This makes the training of automated learners for these new surveys a difficult, non-trivial task.
Supervised machine learning methods (see \citet{br11} for review) have shown great promise for the automatic estimation of astrophysical quantities of interest---\emph{response} variables in the statistics parlance---from sets of \emph{features} extracted from the observed data. These studies include areas as diverse as photometric redshift estimation (\citealt{2004PASP..116..345C}, \citealt{2005PASP..117...79W}, \citealt{2007ApJ...663..752D}, \citealt{2010ApJ...712..511C}), stellar parameter estimation and classification (\citealt{2007A&A...470..761T}, \citealt{2010A&A...522A..88S}), galaxy morphology classification (\citealt{2004MNRAS.348.1038B}, \citealt{2008A&A...478..971H}), galaxy-star separation (\citealt{2008MNRAS.386.1417G}, \citealt{2009AJ....137.3884R}), supernova typing (\citealt{2011MNRAS.tmp..545N}, \citealt{2011arXiv1103.6034R}) and variable star classification (\citealt{2007debo}, \citealt{2011arXiv1101.2406D}, \citealt{2011rich}), among others.
These studies typically assume that the distribution of training data is representative of the set of data to be analyzed (the so-called \emph{testing} data). In reality, in astronomy the distributions of training and testing data are usually substantially different. This \emph{sample selection bias} can cause significant problems for an automated supervised method and must be addressed to ensure satisfactory performance for the testing data. For instance, standard cross-validation techniques assume that the training and testing distributions are exactly the same; when this is not the case, sub-optimal model selection can occur.
In this paper, we show the debilitating effects of sample selection bias on the problem of automated classification of variable stars from their observed light curves. Using a set of highly studied, well-classified variable star light curves from the {\it Hipparcos~} (\citealt{1997perr}) Space Astrometry Mission and the Optical Gravitational Lensing Experiment (OGLE, \citealt{1999udal}) missions, we train a classifier to automatically predict the class of each variable star in the All Sky Automated Survey (ASAS, \citealt{1997AcA....47..467P}, \citealt{2001ASPC..246...53P}). We demonstrate that this classifier results in a high error rate, a substantial number of anomalies, and low average classifier confidence. These debilitating effects are also seen in existing catalogs such as the ACVS (\citealt{2000AcA....50..177P}, \citealt{acvs}), whose use of training data from OGLE plus from an early ASAS release yields a supervised classifier that is only confident on 24\% of all sources. Upcoming surveys, whose automated prediction algorithms will be trained on data from older surveys or idealized models, will suffer from these same maladies if sample selection bias is not treated properly.
To overcome sample selection bias, we propose a few methods, including importance weighting, co-training, and active learning. On both the ASAS variable star classification problem and a simulated variable star data set, we find that active learning (AL) performs the best. AL is an iterative procedure, whereby on each iteration the testing data whose inclusion in the training set would most improve predictions over the entire testing set are queried for manual follow-up and added to the training set. AL is a semi-supervised method that leverages the known features of the testing data to make the best decision about which of these objects is most useful to the supervised learner. We argue that active learning is appropriate in many areas of astrophysics, where follow-up information can often be attained through spectroscopic observations, manual study, or citizen science projects (e.g., \citealt{2008MNRAS.389.1179L}). Furthermore, AL is a principled method for selecting objects for expensive follow-up in circumstances where it is infeasible to perform an in-depth analysis on every object. In particular, projects such as Galaxy Zoo stand to benefit from the active learning approach for candidate object selection, especially when data sizes become prohibitively large to manually analyze each source.
The structure of the paper is as follows. In \S\ref{sec:sampBias} we describe in detail the problem of sample selection bias, showing how it can arise in various astronomical settings and detailing its adverse effects in a variable star classification problem. In \S\ref{sec:methods} we propose a few methods that can be used to mitigate the effects of sample selection bias. We describe active learning in detail, focusing on its implementation with Random Forest classification. Next, we test those methods in \S\ref{sec:deb}, showing that AL attains the best results in a simulated variable star classification experiment. In \S\ref{sec:allstars} we describe our online active learning variable star classification tool, {\tt ALLSTARS}, which was developed to aid the manual study of objects in various photometric surveys. We present the result of applying active learning to classify ASAS variable stars in \S\ref{sec:results}, showing drastic improvement over the off-the-shelf classifier. Finally, we end with some concluding remarks in \S\ref{sec:conclusions}.
\section{Sample Selection Bias in Astronomical Surveys}
\label{sec:sampBias}
A fundamental assumption for supervised machine learning methods is that the training and testing sets\footnote{Throughout the paper, we call training data those objects with known response variable that are used to train the supervised model, and we call testing data the objects of interest whose unknown response is to be predicted by the model.} are drawn independently from the same underlying distribution. However, in astrophysics this is rarely the case. Populations of well-understood, well-studied training objects are inherently biased toward intrinsically brighter and nearby sources and available data are typically from older, lower signal-to-noise detectors.
Indeed, in studies of variable stars, samples of more luminous, well-understood stars are often employed to train supervised algorithms to classify fainter stars observed by newer, deeper surveys. Examples of this abound in the literature. For instance, \citet{2009A&A...506..519D} use a training set from OGLE, a ground-based survey from Las Campanas Observatory covering fields in the Magellanic Clouds and Galactic bulge, to classify higher-quality CoRoT (COnvection ROtation and planetary Transits, \citealt{2009A&A...506..411A}) satellite data. \citet{2011arXiv1101.2406D} train a classification model using a subset of the {\it Hipparcos~} periodic star catalog containing the most reliable labels from the literature and most confident period estimates. This systematic difference between the training and testing sets can cause supervised methods to perform poorly, especially for the types of object under-sampled by the training set.
In \citet{2009A&A...506..519D}, the authors recognize that a training set ``should be constructed from data measured with the same instrument as the data to be classified" and claim that some misclassifications occur in their analysis due to systematic differences between the two surveys. Because the aims and specifications of each survey are different, their observed sources usually occupy different regions of feature space. See, for example, Fig. \ref{fig:traintestoffset}, where there is an obvious absence of the combined {\it Hipparcos~} and OGLE training data in the high-frequency, high-amplitude regime where the density of the testing set of ASAS variables is high. Even if two surveys have similar specifications (e.g., cadence, filter, depth), they may be looking in different parts of the sky or with different sensitivities and thus will observe different demographics of the same sources, causing a systematic differences in the survey priors.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=6.5in]{ASAS_traintest.jpg}
\end{center}
\caption{ Sample selection bias for ASAS variable star (red $\Box$) classification using a training set of well-understood data from {\it Hipparcos~} and OGLE (black $\diamond$). Left: Large distributional mismatch exists in the period-amplitude plane. Only those ASAS data whose statistical significance of the frequency estimate is larger than the median are plotted. ASAS testing data have high density in short-period, high-amplitude and long-period, moderate-amplitude regions, where there are little training data. Right: Testing data tend to have smaller values of the QSO-like variability metric---which measures how well the observed light curve fits a damped random walk QSO model (see \citealt{2011butl})---and larger values of the statistical significance of the first frequency (compared to a null, white-noise model; see \citealt{2011rich}). \label{fig:traintestoffset} }
\end{figure}
In other areas of astrophysics and cosmology it is common practice to construct supervised models using spectroscopic samples and apply those models to predict parameters of interest for objects that fall entirely outside the support of the distribution of the spectroscopic data. For example, photometric redshift estimation methods typically train a regression model using a set of spectroscopically confirmed objects, whereby those models are extended to populations of galaxies that are fainter and (often) at higher redshift (papers that have studied this problem include \citealt{2010MNRAS.405..987B} and \citealt{2011AAS...21715004S}). Several authors have proposed novel methods to mitigate the effects of non-representative photo-z training sets using physical association of galaxies (\citealt{2010ApJ...721..456M,2010ApJ...725..794Q}) or calibration through cross-correlation (\citealt{2010ApJ...724.1305S}). Another field where these issues occur is supernova typing, where classifiers are typically trained on spectroscopically confirmed templates and then applied to classify fainter testing data (\citealt{2010kess,2011MNRAS.tmp..545N}). Recently, \citet{2011arXiv1103.6034R} studied the impact of the accuracy of a supervised supernova classification method on the particular spectroscopic strategy employed to obtain training sets, finding that deeper samples with fewer objects are preferred to surveys with shallower limits.
The situation we describe, where the training and testing samples are generated from different distributions, is referred to in the statistics and machine learning literature as \emph{covariate shift} (\citealt{shimo2000}) or \emph{sample selection bias} (\citealt{heckman1979}). This systematic difference can cause catastrophic prediction errors when the trained model is applied to new data. These problems arise for two reasons. First, under sample selection bias, standard generalization error estimation procedures, such as cross-validation, are biased, resulting in poor model selection. Off-the-shelf supervised methods are designed to choose the model that minimizes some error criterion integrated with respect to the training distribution; when the testing distribution is substantially different, this model is likely to be suboptimal for prediction on the testing data. In (\S \ref{ss:impwt}) we describe a principled weighting scheme to alleviate this complication. Second, significant regions of parameter space may be ignored by the training data---such as in the variable star classification problem shown in Fig. \ref{fig:traintestoffset}---causing catastrophically bad extrapolation of the model onto those regions. In this case, any classifier trained only on the training data will produce poor class predictions for the ignored regions of parameter space: no weighting scheme on the training data can enforce good classifier performance in these regions. This suggests that the testing data need to be used, in a semi-supervised manner, to augment the training set. In this paper, we explore two different approaches to this problem: \emph{co-training} (\S\ref{ss:cotrain} and \emph{self-training}), where testing instances with most certain class prediction are iteratively added to the training set, and \emph{active learning} (\S\ref{ss:actlearn}), where testing instances whose labels, if known, would be of maximal benefit to the supervised method, are manually studied to ascertain the value of their response (e.g. class label, redshift, etc.), and subsequently included in the training set.
\subsection{Example: Source Classification for ASAS}
\label{ss:acvs}
In this section, we demonstrate the effects of sample selection bias in classifying variable stars from the All Sky Automated Survey (ASAS). Particularly, we use an automated machine learning classifier to classify sources in the ASAS Catalogue of Variable Stars (ACVS, \citealt{2002AcA....52..397P}). ACVS verson 1.1\footnote{The ACVS catalog can be downloaded at \url{http://www.astrouw.edu.pl/asas/data/ACVS.1.1.gz}.} consists of $V$-band light curves for 50,124 stars that have passed tests of variability as described in \citet{2000AcA....50..177P}. As a training set for this classification problem, we use only the confidently labeled {\it Hipparcos~} and OGLE sources used in \citet{2007debo} and \citet{2011rich}. This data set consists of 1542 variable stars from 25 different science classes. The period-amplitude relationship of the instances in the training set of {\it Hipparcos~} and OGLE data, and in the ACVS catalog are plotted in Figure \ref{fig:traintestoffset}, where sample selection bias is obvious.
As a part of ACVS, predicted classes are provided for a fraction of the stars. As described in \citet{2002AcA....52..397P}, ACVS obtains their classifications using a neural net type algorithm trained on set of visually labeled ASAS sources, confirmed OGLE cepheids \citet{1999AcA....49..223U,1999AcA....49..437U}, and OGLE Bulge variable stars \citet{2002AcA....52..129W}. A filter is used to divide strictly periodic from less regular periodic sources. A neural net is trained on the period, amplitude, Fourier coefficients (first 4 harmonics), $J-H$ and $H-K$ colors and IR fluxes to predict the classes of the strictly periodic sources. Several ACVS objects either have multiple labels or are annotated as having low confidence classifications. For less regular periodic sources, location in the $J-H$ vs. $H-K$ plane is tested; if the object falls within an area of late-type irregular or semi-regular stars, it is assigned the label MISC, else it is inspected by eye. We find that 38,117 ACVS stars, representing 76\% of the catalog, are either labeled as MISC, assigned multiple labels, or have low class confidence. The remaining 24\% of stars have confident ACVS labels, and provide a set of classifications to compare our algorithms against. In Figure \ref{fig:traintestclass} we plot in color, in period-amplitude space, the classes of the training data and the ACVS classes of the ASAS data\footnote{Note that not all sources are actually periodic, meaning that some period estimates are nonsensical. However, we also use the statistical significance of the frequency estimate as an input feature into our classifier; thus the classifier learns to trust the only periodic features of those sources with high frequency significance, and to rely on only the non-periodic features of the low-significance data.}.
\begin{figure*}
\begin{center}
\includegraphics[angle=0,width=6.5in]{ASAS_classes.jpg}
\end{center}
\caption{ Left: Period-amplitude relationship for the 1542 training set sources from the {\it Hipparcos~} and OGLE surveys. Symbols and colors denote the true science class of each object. Right: Same for an arbitrary sample of size 10,000 from the 50,124 ASAS testing objects, where symbols and colors denote the ACVS labels. Black `U' denotes that the source is either labeled MISC, doubly-labeled, or has low confidence label by ACVS. Our goal is to use the training data set to predict the class label (and posterior class probabilities) for each ASAS object. Complicating this task is the significant distributional difference between the training and testing sets. \label{fig:traintestclass}}
\end{figure*}
As our base model, we use a Random Forest classifier (\citealt{2001brei}). Random Forest has recently been shown by \citet{2011rich} and \citet{2011arXiv1101.2406D} to attain accurate results in automated classification of variable stars. In this paper, we represent each variable star in our data set by the 59 light-curve features used by \citet{2011rich}, as well as 5 additional light-curve features from \citet{2011arXiv1101.2406D}. The Random Forest classifier is a supervised, non-parametric method that attempts to predict the science class of each star from its high-dimensional feature vector. It operates by constructing an ensemble of classification decision trees, and subsequently averaging the results. The key to the good performance of Random Forest is that its component trees are de-correlated by sub-selecting a small random number of features as splitting candidates in each non-terminal node of the tree. As a result, the average of the de-correlated trees attains highly decreased variance over each single tree, with no substantial increase in bias.\footnote{For more details about the Random Forest variable star classifier used, see \citet{2011rich}.}.
By training a Random Forest classifier on the {\it Hipparcos~} and OGLE data as in \citet{2011rich} and applying that classification model to predict the class label of each object in ACVS, we obtain a 65.5\% correspondence with the ACVS labels for the 24\% of objects that have a confident ACVS label. A table showing the correspondence of our predicted Random Forest classification labels with those of ACVS is plotted in Figure \ref{fig:rfasaspred}. The Random Forest algorithm classifies 90\% and 79\% of the Mira and RR Lyrae, FM stars identified by ACVS, but shows much lower correspondence for other classes, such as Delta Scuti, Population II Cepheid, and RR Lyrae, FO. Note that the Random Forest class taxonomy is finer than that used by ACVS, including twice as many classes; as such, the Random Forest has the ability to identify objects of rarer classes, such as T Tauri and Gamma Doradus stars.
\begin{figure*}
\begin{center}
\includegraphics[angle=0,width=6.5in]{confmat_asas_AL0.pdf}
\end{center}
\caption{Off-the-shelf Random Forest classifications of the ASAS data set, using a training set of the 1542 {\it Hipparcos~} \& OGLE sources, compared to the ACVS classifications. Rows are normalized to sum to 100\%, marginal counts are listed to the right and bottom of the table. The RF classifier finds a 65.5\% correspondence with the ACVS labels, for the 12,007 objects with ACVS label, with many major discrepancies. Particularly, the RF detects a very small number of the ACVS Cepheids, Delta Scuti, and Chemically Peculiar stars. Also, the RF finds a gross overabundance of Double Mode RR Lyrae and Wolf-Rayet stars. These artifacts result from sample selection bias. \label{fig:rfasaspred}
}
\end{figure*}
However, there are serious problems that arise by running the analysis in this manner and ignoring the significant sample selection bias between the training and testing sets. In Figure \ref{fig:traintestoffset} we saw that the distribution of the training set of {\it Hipparcos~} and OGLE sources is wildly different than the distribution of ASAS sources;
notably, regions of long-period, amplitude $<1$ sources and regions of short-period, high-amplitude sources are densely populated in ASAS but contain little or no training data. As a consequence, a large proportion of the ASAS data set has no counterpart in the training set that closely matches its feature vector, meaning that it will likely be incorrectly identified by the Random Forest classifier as belonging to a physically different class of variable star. One telling statistic is that for only 14.6\% of the ASAS objects does the Random Forest produce a posterior class probability of $\ge 0.5$, meaning that the classifier is only confident on the class predictions for 15\% of the entire ASAS ACVS catalog.
In Figure \ref{fig:rfasaspred} we find that many ASAS sources (9109 of 50,124, or 18.2\%) are identified by the Random Forest classifier as being of RR Lyrae, DM type, a relatively rare type of doubly pulsating variable star. This is far too many RR Lyrae, DM candidates; for comparison, \citet{2011AcA....61....1S} find only 91 RR Lyrae, DM candidates in the entire OGLE-III catalog, out of 16,836 total RR Lyrae candidates (0.5\%). This artifact in our classification occurs because the RR Lyrae, DM objects have multiple pulsational modes, causing their data to poorly fold around a single period. Because ASAS photometry is less precise than that of {\it Hipparcos~} or OGLE, its folded light curves are considerably more noisy. Consequently, for a large subset of ASAS sources that do not resemble any of the training data, the classifier's ``best guess" is RR Lyrae, DM because training light curves of that class most resemble ASAS data. In addition, an artificially high number of Wolf Rayet and Beta Lyrae stars are found by the RF. This deficiency of the off-the-shelf classifier illustrates the need for other approaches.
\section{Methods to Treat Sample Selection Bias}
\label{sec:methods}
Above, sample selection bias was defined, its presence in astrophysical problems motivated, and its adverse effects exemplified with an example in variable star classification. In this section, we will introduce three different principled approaches of treating sample selection bias, and argue that active learning is the most appropriate of these methods for dealing with astronomical sample biases. Later, these methods will be compared using variable star data from the OGLE and {\it Hipparcos~} missions.
\subsection{Importance Weighting}
\label{ss:impwt}
Under sample selection bias, standard generalization error estimation procedures, such as cross-validation, are biased, resulting in poor model selection for supervised methods. To remedy this, importance weighting (IW) cross-validation is often used (see \citealt{sugi2005}, \citealt{huan2007}, and \citealt{sugi2007}). Under this approach, the training examples are weighted by an empirical estimate of the ratio of test-to-training-set feature densities during the training procedure. Specifically, when evaluating the statistical risk of the statistical model over the training data, the weights
\begin{equation}
\label{eq:iw}
w_i = \frac{\mathbf{P}_{\rm Test}({\bf x}_i,y_i)}{\mathbf{P}_{\rm Train}({\bf x}_i,y_i)} = \frac{\mathbf{P}_{\rm Test}({\bf x}_i)\mathbf{P}_{\rm Test}(y_i|{\bf x}_i)}{\mathbf{P}_{\rm Train}({\bf x}_i)\mathbf{P}_{\rm Train}(y_i|{\bf x}_i)} = \frac{\mathbf{P}_{\rm Test}({\bf x}_i)}{\mathbf{P}_{\rm Train}({\bf x}_i)}
\end{equation}
are typically used, where ${\bf x}_i$ is the feature vector and $y_i$ is the response variable (i.e., class) for training object $i$. To achieve the last inequality in Equation \ref{eq:iw}, it is assumed that $\mathbf{P}_{\rm Test}(y_i|{\bf x}_i)=\mathbf{P}_{\rm Train}(y_i|{\bf x}_i)$, i.e. that the probability of a specific response given a feature vector is the same for training and testing sets. In practice, this equality will probably not hold for the types of astrophysical data sets we are interested in: though the mapping from features to response values may be the same for the training an testing sets, the prior distributions over the responses, $y$, are different, in general. Even in this situation, use of the ratio of feature densities---though imperfect---may still be useful, and is more tractable than using the joint feature-response densities\footnote{Note that we could alternatively rewrite the joint density as $\mathbf{P}(y_i)\mathbf{P}({\bf x}_i | y_i)$. It is unlikely that $\mathbf{P}_{\rm Test}({\bf x}_i | y_i) = \mathbf{P}_{\rm Train}({\bf x}_i | y_i)$ in most practical situations; however, if this were to hold then the importance weights would simply reduce to the ratio of response priors.}.
Even so, in practice the training and testing feature densities are difficult to estimate (and their ratio is even harder to estimate) because they reside in high-dimensional feature spaces. To overcome this, Eqn. \ref{eq:iw} can be estimated via distribution matching (\citealt{huan2007}) or by fitting a probabilistic classifier to the classification problem of training vs. testing set and employing the output probability estimates (\citealt{zadr2004}).
Using the weights defined in Eqn. \ref{eq:iw} when training a classifier induces an estimation procedure that gives higher importance to training set objects in regions of feature space that are relatively under-sampled by the training data, with respect to the testing density. This enforces a higher penalty for making errors in regions of feature space that are under-represented by the training set. This is sensible because, since the ultimate goal is to apply the model to predict the response of the testing data, we should attempt to do well at modeling the output in regions of feature space densely populated by testing data (and conversely ignore modeling regions devoid of testing data). For the ASAS example, importance weighting will give large weights to the training data in the region of Amplitude $< 0.5$ and Period $> 100$ and affix small weights to data in the high-amplitude clump centered around a 300-day period.
Though this approach is useful in some problems, importance weighting has been shown to be asymptotically sub-optimal when the statistical model is correctly specified\footnote{In other words, IW produces worse results than the analogous unweighted method if the parametric form of $\mathbf{P}(y|{\bf x})$ is correct.} (\citealt{shimo2000}), and with flexible non-parametric models such as Random Forest we observe very little change in performance using IW (see \S\ref{sec:deb}). An additional, more debilitating drawback is that IW requires the support of the testing distribution be a \emph{subset} of the support of the training distribution\footnote{Else the weights, defined as the ratio of test-to-training set feature densities, explode, and the theoretical properties of the method no longer hold.}, which, in the types of supervised learning problems common in astrophysics, is rarely the case.
\subsection{Co-training}
\label{ss:cotrain}
In astronomical problems, we typically have much more unlabeled than labeled data. This is due to both the pain-staking procedures by which labels must be accrued (e.g., by spectroscopic follow-up or manual assignment), and the fact that there are exponentially more dim, low signal-to-noise sources than bright, well-understood sources. Recently, supervised classification algorithms have been developed that use both labeled and unlabeled examples to make decisions. This class of models is referred to as \emph{semi-supervised} because learning is performed both on the instances with known response values and on the feature distribution of instances with no known response. Semi-supervised methods such as \emph{co-training} and \emph{self-training} slowly augment the training set by iteratively adding the most confidently-classified test cases in the previous iteration.
Co-training was formalized by \citet{blum1998} as a method of building a classifier from scarce training data. In this method, two separate classifiers, $h_1$ and $h_2$, are built on different (disjoint) sets of features, ${\bf x}_1$ and ${\bf x}_2$. In an iteration, each classifier adds its $p$ most confidently labeled test instances to the training set of the \emph{other} classifier. This process continues either for $N$ iterations or until all test data belong to the training set of both classifiers. The final class predictions are determined by multiplying the class probabilities of each classifier, i.e. $p(y|{\bf x}) = h_1(y|{\bf x}_1)h_2(y|{\bf x}_2)$. Co-training has shown impressive performance in situations where very few training examples are used to classify many test cases. \citet{blum1998} use co-training in a two-class problem, using 12 labeled web pages to classify a corpus of 1051 unlabeled pages, achieving a 5\% error rate.
In the original co-training formulation, it was assumed that each object could be described by two different `views' (i.e. feature sets) of the data that were both redundant (each view of the object gives similar information) and conditionally independent given the true class label. While this natural redundancy may be present in web page classification (e.g., the words on the web page and the words on pages linked to that page), it is not generally the case. Later papers by \citet{goldman2000} and \citet{nigam2000} argue that even when a natural feature division does not exist, arbitrary or random feature splits produce better results than self-training (\citealt{nigam2000}), where a single classifier is built on all of the features whereby the most confidently classified testing instances are iteratively moved to the training set.
In the variable star classification paper of \citet{2009A&A...506..519D}, something akin to a single iteration of self-training was performed for CoRoT classification using OGLE training data, where candidate lists obtained with the first version of the classifier were used to select very probable class members amongst the testing set data for subsequent inclusion in the training set. This augmentation procedure led to inclusion of an extra 114 sources into the training set.
Both co-training and self-training are reasonable approaches to problems that suffer from sample selection bias because they iteratively move testing data to the training set, thereby gradually decreasing the amount of bias that exists between the two sets. However, in any one step of the algorithm, only those data in a close neighborhood to existing training data will be confidently classified and made available to be moved to the training set. Thus, as the iterations proceed, the dominant classes in the training data diffuse into larger regions of feature space, potentially gaining undue influence over the testing data. In addition, co-training and self-training will never predict classes that are rare or unrepresented in the training data, even if they are prominent in the testing data. In \S\ref{sec:deb} we apply both self-training and co-training to variable star classification, finding that these methods perform poorly in terms of overall error rate, especially for classes that are under-sampled by the training data.
\subsection{Active Learning}
\label{ss:actlearn}
An important feature to supervised problems in astronomy is that we often have the ability to selectively follow up on objects to ascertain their true nature. For example, for different problems this can be achieved by targeted spectroscopic study, visualization of (folded) light curves, or querying of other databases and catalogs. Consider astronomical source classification: while it is impractical to manually label all hundred-million plus objects that will be observed by Gaia and LSST, manual labeling of a small, judiciously chosen set of objects can greatly improve the accuracy of an automated supervised classifier. This is the approach of \emph{active learning} (and in particular, pool-based active learning, \citealt{lewis1994}). Under pool-based AL for classification, an algorithm iteratively selects, out of the entire set of unlabeled data, the object (or set of objects) that would give the expected maximal performance gains of the classification model, if its true label(s) were known. The algorithm then queries the user to manually ascertain the science class of the object(s), whereby the supervised learner incorporates this information its subsequent training sets to improve upon the original classifier. For a thorough review of active learning, see \citet{sett2009}.
Active learning has enjoyed wide use in machine learning, with impressive results in many areas of application, such as text classification, speech recognition, image and video classification, and medical imaging (\citealt{lewis1994,tong2001,tong2002,yan2003,liu2004,tur2005}). Begin with a training set $\mathcal{L}$ and testing set $\mathcal{U}$. On each active learning iteration, we manually find the class of the testing set source, ${\bf x}' \in \mathcal{U}$, whose inclusion into $\mathcal{L}$ would most improve the classifier's performance on the testing data (according to some metric, see \S\ref{ss:query}). These queried active learning samples tend to be data that reside in relatively dense regions of testing set feature space, $\mathbf{P}_{\rm Test}({\bf x})$, scarcely populated regions of training set feature space, $\mathbf{P}_{\rm Train}({\bf x})$, and in regions where the class identity is uncertain.
For an appropriate selection metric, a small number of active learning samples will suffice in making the labeled set feature distribution resemble the unlabeled set distribution. This approach is similar to the importance sampling approach of \citet{zadr2004}, who show that if training set sources are resampled with respect to the appropriate (weighted) distribution, then the statistical risk of the classifier built on that data will minimize the statistical risk evaluated over all of the data. The drawback to that approach is that it needs a relatively large initial training sample and requires that for all non-zero regions of $\mathbf{P}_{\rm Test}$, $\mathbf{P}_{\rm Train}$ also be non-zero. On the other hand, the active learning approach to sample selection bias is to \emph{expand} the training set in a way that makes it most closely resemble the testing set, and thus these problems are avoided.
\subsubsection{Active Learning Query Function}
\label{ss:query}
Several strategies have been proposed to determine which testing data about which active learning will query the ``human annotator." Most of these prescriptions attempt to select data whose label, if known, would maximally help the classifier. The simplest form of querying is \emph{uncertainty sampling} (\citealt{lewis1994}), by which on each iteration, the training datum with highest label uncertainty (measured, e.g., by entropy or margin) is queried for manual identification. Though simple, this approach does not explicitly consider changes to the overall error rate of the classifier, and is prone to select outlying points that have little influence in the classification of the other testing data.
Since we have an explicit goal of minimizing the classification error rate over the entire set of testing data, it is sensible to consider this metric explicitly when queuing data for AL. This is the approach taken by the \emph{expected error reduction} strategies (\citealt{roy2001toward}), where on each iteration the algorithm queries the testing point whose inclusion into the training set would produce the smallest classification error rate (statistical risk) over the testing set. These methods operate by iteratively adding each testing point to the training set and retraining the classifier\footnote{For many machine learning algorithms, fast incremental updating algorithms exist, making this approach tractable.}. However, because the true labels of the training data are not known \emph{a priori}, one must also iterate over the possible labels of the training data, and can only compute an estimate of the expected decrease in testing error rate by approximating the error under all possible labels of all testing data. For common astronomy data sets, with $\gtrsim 10^5$ objects, expected error reduction is impractical. A viable alternative is \emph{variance reduction} (\citealt{cohn1996}), where the testing object that minimizes the classifier's variance is selected on each iteration. Since a classifier's error can be decomposed into variance plus squared-bias plus label noise\footnote{Classifier variance measures the variability in a classifier with respect to the actual training set used, classifier bias is the amount of discrepancy between the true labels and the expected prediction of a classifier (averaged over all possible training sets), and label noise is the amount of error in the training set labels.}, minimizing the variance amounts to minimizing the error rate; also, for many models, the variance can be written in closed form, circumventing any costly computations.
In this paper, we consider two different selection criteria. The first criterion is motivated by importance weighting and the second is motivated by selecting the sources whose inclusion into the training set would produce the most total change in the predicted class probabilities for the testing sources. To meet these criteria, we revisit the Random Forest classifier. For each of $B$ bootstrap samples from the training set, we build a decision tree, $\theta_b$, which predicts the class of each object from its feature vector, ${\bf x}$. The Random Forest class probability of class $y$ is simply the empirical proportion,
\begin{equation}
\label{eq:rfprob}
\widehat{P}_{{\rm RF}} (y|{\bf x}) = \frac{1}{B} \sum_{b=1}^B \theta_{b}(y|{\bf x})
\end{equation}
of the $B$ trees that predict class $y$. Additionally, the Random Forest provides a measure of the \emph{proximity} of any two feature vectors with respect to the ensemble of decision trees, defined as
\begin{equation}
\rho({\bf x}',{\bf x}) = \frac{1}{B} \sum_{b=1}^B I({\bf x} \in T_b({\bf x}'))
\end{equation}
which is the proportion of trees for which the two objects ${\bf x}$ and ${\bf x}'$ fall in the same terminal node, where $I(\cdot)$ is a boolean indicator function. Here, we use the notation $T_b({\bf x}')$ to denote the terminal node of feature vector ${\bf x}'$ in tree $b$.
Heuristically, sample selection bias causes problems in the building of a classifier principally because large density regions of testing data are not well represented by the training data. Our first AL selection procedure uses this heuristic argument to select the testing point, ${\bf x}' \in \mathcal{U}$, whose feature density is most under-sampled by the training data, as measured by the ratio of the two densities.
This amounts to choosing AL samples that maximize
\begin{equation}
\label{eq:alrf1}
S_1({\bf x}') = \frac{\mathbf{P}_{\rm Test}({\bf x}')}{\mathbf{P}_{\rm Train}({\bf x}')} \approx \frac{\sum_{{\bf x} \in \mathcal{U}} \rho({\bf x}',{\bf x})/N_{\rm Test}}{\sum_{{\bf z} \in \mathcal{L}} \rho({\bf x}',{\bf z})/N_{\rm Train}}
\end{equation}
where we estimate the training and testing set densities at ${\bf x}'$ by averaging the RF proximity measure over the set of training ($\mathcal{L}$) and testing ($\mathcal{U}$) sets, respectively. The expression $\sum_{{\bf x} \in \mathcal{U}}\rho({\bf x}',{\bf x})/N_{\rm Test}$, is the average, over the trees in the forest, of the proportion of testing data with which ${\bf x}'$ shares a terminal node. The estimate of the probability density at ${\bf x}'$ would need be normalized by the average volume of the terminal nodes of ${\bf x}'$; however, since Equation \ref{eq:alrf1} considers the ratio of two such densities at ${\bf x}'$, the average volume terms cancel, giving the above expression.
Our second AL selection criterion is to choose the testing example, ${\bf x}' \in \mathcal{U}$, that maximizes the total amount of change in the predicted probabilities for the testing data. This is a reasonable metric because it says that we will only spend time manually annotating the testing data whose labels most affect the predicted classifications. To achieve this, we create a selection metric that attempts to choose the ${\bf x}'$ that maximizes the total change, summed over the testing set, of the RF probability vectors (as measured using the $\ell_1$ norm). An approximate solution to this problem is to choose the testing data points that maximize
\begin{equation}
\label{eq:alrf2}
S_2({\bf x}') = \frac{\sum_{{\bf x} \in \mathcal{U}}\rho({\bf x}',{\bf x})(1 - \max_y \widehat{P}_{{\rm RF}}(y|{\bf x}))}{\sum_{{\bf z} \in \mathcal{L}} \rho({\bf x}',{\bf z}) +1}
\end{equation}
where the Random Forest probability, $\widehat{P}_{{\rm RF}}(y|{\bf x})$, is defined in Equation \ref{eq:rfprob}.
In Appendix \ref{app:alrf} we work out the details of deriving Eqn. (\ref{eq:alrf2}) from the stated goal of selecting testing points whose labels maximally affect the total change of the Random Forest predicted probabilities over $\mathcal{U}$.
The key elements to Eqns. \ref{eq:alrf1}--\ref{eq:alrf2} are (1) the testing set density, represented by $\sum_{{\bf x} \in \mathcal{U}} \rho({\bf x}',{\bf x})$ is in the numerator, and (2) the training set density, represented by $\sum_{{\bf z} \in \mathcal{L}} \rho({\bf x}',{\bf z})$, is in the denominator. This means that we will choose instances that are in close proximity to many testing points and are far from any training data, thereby reducing sample selection bias. In addition, $S_2$ is a weighted version of $S_1$ with the Random Forest prediction uncertainty, represented by $1 - \max_y \widehat{P}_{{\rm RF}}(y|{\bf x})$, in the numerator. This means that $S_2$ gives higher weight to those testing points that are difficult to classify thereby causing the algorithm to focus more attention along class boundaries, which should lead to better performance of the classifier.
\subsubsection{Batch-Mode Active Learning}
In typical active learning applications, queries are chosen in serial. However, in most astronomical applications, it makes more sense to query several testing set objects at once, in \emph{batch mode}; for instance, in a typical observing run multiple objects are queued for follow-up observation. In the variable star classification problem, we determine that the best use of users' time is to supply them with dozens of sources to label at one sitting.
The challenge with batch-mode AL is to determine how to best choose multiple testing instances at once. Selecting the top few candidates is typically suboptimal because those objects generally lie in the same region of feature space, as is obvious from analyzing the criteria in Eqns. \ref{eq:alrf1}-\ref{eq:alrf2}. Heuristic methods have been devised that create diversity in the batch of AL samples for a particular classifier (e.g., \citealt{brinker2003} for SVMs). In our use of AL, we sample batches of AL samples by treating the criterion function as a probability sampling density, i.e., $\mathbf{P}(\textrm{select } {\bf x}') \propto S_1({\bf x}')$. In \S\ref{sec:deb} we compare this density method, which we call AL-d, to a method that selects the top candidates on each AL iteration, which we refer to as AL-t.
\subsubsection{Crowdsourcing Labels}
\label{ss:crowdsource}
Most active learning papers assume that labels can be found, without noise, for any queried data point. In typical astronomical applications, this will not be the case. For instance, after follow-up observations of an object, its true nature might still be difficult to ascertain and will often remain unknown. Indeed, in classifying variable stars, users will sometimes have difficulty in obtaining the true class of an object, especially for noisy or aperiodic sources. This causes two complications in the AL process:
\begin{enumerate}
\item some queried sources will still have an unknown label after manual classification, and
\item a few sources will be annotated with an \emph{incorrect} label.
\end{enumerate}
The first difficulty means that we expect to receive user labels for only a fraction of the queried sources; to avoid wasting costly user time, we attempt to select AL sources that users will have a higher probability of successfully labeling (in \S \ref{ss:cost} we describe how this is achieved by using a cost function). To overcome the second complication, we use crowdsourcing, where several users are presented with the same set of AL sources. The idea behind crowdsourcing is that by using the combined set of information about each object from multiple users, we are able to suppress the noise in the manual labeling process.
A difficulty in crowdsourcing is in simultaneously predicting the best label and judging the accuracy of each annotator from a set of user responses. Users are likely to disagree on some objects, so determining a true label can often be tricky. However, because each annotator has a different skill level, we should give more credence to the labels of the more adept users in deciding on a label. In the active learning paper of \citet{donmez2009}, a novel, yet simple method called {\tt IEThresh} was introduced to filter out the less-adept users in crowdsourcing labels. Their basic approach is to start each user with the same prior skill level. Then, as the AL iterations progress, users whose responses agree with the consensus votes of the crowd are given higher `reward'. The skill level of each user is determined by the upper confidence interval (UI) of the mean reward of all their previous labels. For each subsequent iteration, only those users whose UIs are higher than $\epsilon$ times the UI of the best annotator are included in the vote for the class of that object. Even if a particular user's label is not used in a vote, their reward level can change, meaning that users are able to drift in and out of the decision-making process over time.
In \S\ref{sec:results}, we use the {\tt IEThresh} algorithm with $\epsilon=0.85$ to crowdsource the ASAS labels. In addition, for a source to be included in the training set, we require that at least 70\% of users who looked at the source return a label. This strict policy is implemented so that only the most confident AL sources are moved to the training set so as to avoid including incorrectly labeled objects.
\subsubsection{Cost of Manual Labeling}
\label{ss:cost}
Standard active learning methods assume that the cost of attaining a label is the same for every data point, and thus aim to minimize the total number of queries performed (or equivalently achieve the lowest error rate for a given number of queries). This assumption is not valid for variable star classification problem, for a variety of reasons. First, higher signal-to-noise light curves with larger number of epochs will be, on average, easier to manually label than sparser, noisier light curves. Second, a star that has been observed and cataloged by multiple surveys (for instance, it is in the SDSS footprint) will have more archival data with which to determine its true class. Third, depending on its coordinates, a star may or may not be readily available for spectral follow-up. To avoid wasting user time on impossible-to-classify objects, these factors must be taken into account when choosing AL samples.
In applying AL to variable star classification, we treat the cost as a multiplicative factor on the querying criteria. That is, the AL function is $S({\bf x}') = S_1({\bf x}')(1-C({\bf x}'))$, where the cost function, $C({\bf x}')$, is
\begin{equation}
C({\bf x}') = \mathbf{P}({\bf x}' \textrm{ cannot be manually labeled } | {\bf x}' \textrm{ is queried}),
\end{equation}
i.e., the cost function is the probability that a user (or set of users) cannot actually determine a label for that source, given that the user was given that object to manually study\footnote{Other definitions of the cost are possible, such as the time necessary for a user to manually label a source or the user disagreement rate. As formulated, our ``cost" function measures the inutility of the user on each particular source.}. High cost means that we will avoid querying that object. Inclusion of a cost function deters us from wasting valuable user time on objects that are too noisy or sparsely sampled to determine their science class. In \S\ref{sec:results} we describe how we model the cost and derive an empirical cost estimate for each object in the ASAS testing set.
\subsubsection{Stopping Criterion}
Insofar as the aim of active learning is to improve the performance of a classifier to the greatest extent possible with as little effort as possible, we must determine when to stop manually labeling sources. A reasonable rule of thumb is to stop querying data for active learning when the effort needed to acquire the new labels is larger than the benefit that those labels have on the classifier's performance. However, it is often difficult to compare these gains and losses, especially for problems where there do not exist ground truths with which to judge the classifier performance nor good metrics to measure gains and losses. Alternatively, one can track the intrinsic stability of the classifier (e.g., by measuring its average confidence over the testing set), and stop when a plateau is reached (cf. \citealt{vlachos2008,olsson2009}). In our implementation of AL, we choose to run iterations until the performance of the classifier levels off (as judged by a few intrinsic and extrinsic metrics, see \S\ref{sec:results}).
\section{Experiment: OGLE and {\it Hipparcos~} Variable Stars}
\label{sec:deb}
In this section, we test the effectiveness of the various methods proposed in \S\ref{sec:methods} in combating sample selection bias for variable star classification. Starting with the set of 1542 well-understood, confidently-labeled variable stars from \citet{2007debo}, we randomly draw a sample of 721 training sources according to a selection function, $\Gamma$, that varies across the amplitude-period plane as
\begin{equation}
\Gamma({\bf x}) \propto \log(\textrm{period }{\bf x}) \cdot \log(\textrm{amplitude }{\bf x})^{1/4}.
\end{equation}
This selection function is devised so that the training set under-samples short-period, small-amplitude variable stars. The resultant training and testing sets are plotted in the amplitude-period plane, along with the training set selection function, in Figure \ref{fig:debtrte}.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=6.0in]{debsim_traintest.jpg}
\end{center}
\caption{Training (black $\blacktriangle$) and testing (red x) data for the simulated example using OGLE \& {\it Hipparcos~} data. The 771 training data were randomly sampled from the original 1542 sources according to the sampling distribution plotted in color. Using this sampling scheme, we create sample selection bias by over-sampling long-period, high-amplitude stars and under-sampling the short-period, low-amplitude sources. \label{fig:debtrte} }
\end{figure}
Distributional mismatch between the training and testing sets causes an off-the-shelf Random Forest classifier to perform poorly for short-period small-amplitude sources. The median overall error rate for a Random Forest classifier trained on the training data and applied to classify the testing data is 29.1\%. This is 32\% larger than the 10-fold cross-validation error rate of 21.8\% on the entire set of 1542 sources (see \citealt{2011rich}; the error rate quoted here is slightly lower due to the addition of new features). The average error rate for testing set objects with period smaller than 0.5 days is 36.1\%.
To treat the sample selection bias, we use each of the following methods:
\begin{itemize}
\item Importance weighting. A single Random Forest is built on the training set, with class-wise\footnote{In importance weighting, ratios of feature densities are typically used as the weights. However, in our implementation of Random Forest, weights may only be defined by class.} importance sampling weights defined as the ratio of the testing set to training set class proportions\footnote{Since we know the true class of each object, we are able to use this information to derive the weights. In a real problem, the feature or class densities would need to be estimated.}.
\item Self-training and co-training. Each algorithm is repeated for 100 iterations, where on each iteration the most confident 3 testing set objects are added to the training set. For co-training, we use both random feature splits (CT) and periodic versus non-periodic features (CT.p).
\item Active learning. Using the metrics in Equations \ref{eq:alrf1} (AL1) and \ref{eq:alrf2} (AL2), we perform 10 rounds of active learning, with batch size of 10 objects selected on each round. The classifier is retrained on the available labeled data after each round. Testing set objects are selected for manual labeling either by treating the selection metrics as probability distributions (AL1.d, AL2.d), or by taking the top candidates (AL1.t, AL2.t). We also compare to an AL method that selects objects completely at random (AL.rand).
\end{itemize}
For each of the active learning approaches, we evaluate the error rate only over those testing set objects that are not queried by the algorithm. This way we do not artificially decrease the error rate by evaluating sources whose labels have been manually obtained. Note that for this experiment, we have assumed that the true labels can be manually obtained with no error.
Distributions of the classification error rates for each method, obtained over 20 repetitions, are plotted in Figure \ref{fig:deberr}. The largest improvement in error rate is obtained by both AL1.t and AL2.t (25.5\% error rate), followed by AL2.d (25.9\%). Quoted results for the active learning methods are after querying 100 training set objects (10 AL batches of size 10). AL1.d lags well behind the performance of these other AL querying functions. None of the other methods produces a significant decrease in the error rate of the classifier. Indeed, the ST and CT approaches cause an \emph{increase} in the overall error rate. IW produces a slight decrease in the error rate, by an average of 0.4\%, which represents 3 correct classifications. An important observation is that the AL.rand approach of randomly queuing observations for manual labeling does not perform well compared to the more principled AL approaches.
Figure \ref{fig:debal} depicts the error rate of the AL approaches as a function of the total number of objects queried. Between the AL1 and AL2 metrics, there is no clear winner, but once large numbers of samples have been observed AL2.d and AL2.t perform better than their AL1 counterparts. We also find in Figure \ref{fig:debal} that the AL.d approaches---where objects are drawn with probability proportional to the AL criterion---perform worse than the approaches that always select the top AL candidates. This is unexpected, as selecting only the top methods in batch mode produces samples of objects from the same region in feature space, causing an inefficient use of follow-up resources. However, this observed better performance by the AL.t strategies may be an artifact of using small batch sizes (10 objects); in the application of active learning to ASAS, we typically use batch sizes $>50$.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=6.5in]{deb_errorrates.pdf}
\end{center}
\caption{Error rates, evaluated over the testing set, of 10 different methods applied to the OGLE \& {\it Hipparcos~} simulated data set of 771 training and 771 testing samples. Due to sample selection bias, the default Random Forest (RF-Def.) is ineffective. Importance weighting (IW) improves upon the RF only slightly. The co-training and self-training methods produce an increased error rate. Only the active learning approaches yield any significant gains in the performance of the classifier over the testing set. Note that the AL methods were evaluated over those testing data not in the active learning sample. No large difference is found between the two AL metrics, but both outperform the random selection of AL samples. Note that each boxplot displays the 25th and 75th quantiles as the edges of the boxes, with the center line denoting the median and the whiskers extending to the minimum and maximum. \label{fig:deberr} }
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=6.5in]{deb_ALiter.pdf}
\end{center}
\caption{Performance of the active learning approaches for the OGLE \& {\it Hipparcos~} classification experiment. Both AL1 and AL2 dominate the performance of AL.rand, but there is no clear winner between these two approaches. AL1.t performs best for the first few iterations, but is overtaken by AL2.t after 100 samples are queried. AL2.d performs significantly better than AL1.d after about 50 iterations. For each method, the mean error rate---evaluated over the testing set not included in the AL sample---is plotted along with $\pm 1$ standard error bands. \label{fig:debal} }
\end{figure}
Active learning is able to significantly improve the classification error rate on the set of OGLE \& {\it Hipparcos~} testing data because it selectively probes regions of feature space where class labels, if known, would most influence the classifications of a large number of testing data. For the OGLE and {\it Hipparcos~} variable star data, sets of low-amplitude, short-period stars are selected by the AL algorithm, which in turn improve the error rates within the science classes populated by these types of stars, without increasing error rates within the classes that are highly sampled by the training set. We make this more concrete in Table \ref{tab:deb}, where the classifier error rates within a few select classes are shown. The active learning classifiers show substantial improvement, on average, over the default Random Forest for the classes which are most under-sampled by the training data and show no increase in the error rates for the classes that are most over-represented in the training set.
\begin{deluxetable}{cccrrrrrrrrrr}
\tabletypesize{\footnotesize}
\tablewidth{7.0in}
\tablecolumns{13}
\tablecaption{Error rates, in \%, over all testing data, and for those testing data within selected science classes in the OGLE \& {\it Hipparcos~} experiment. The first set of classes are those most under-represented in the training data. The second set are those most over-represented in the training data. Several methods for sample selection bias reduction are compared. \label{tab:deb}}
\tablehead{
\colhead{Science Class} &
\colhead{$N_{\rm Train}$} &
\colhead{$N_{\rm Test}$} &
\colhead{RF\tablenotemark{a}} &
\colhead{IW } &
\colhead{ST } &
\colhead{CT } &
\colhead{CT.p } &
\colhead{AL1.d\tablenotemark{b}} &
\colhead{AL1.t\tablenotemark{b}} &
\colhead{AL2.d\tablenotemark{b}} &
\colhead{AL2.t\tablenotemark{b}} &
\colhead{AL.rand\tablenotemark{b}}
}
\startdata
\hline
All &771 &771 &28.9 &28.5 &29.6 &30.0 &29.4 & 27.3 & 25.5 & 25.9 & 25.5 & 28.0\\
\hline
Delta Scuti & 25 &89 & 15.7 &15.7 &15.7 &15.7 &14.6 & 15.4 & 14.0 & 15.6 & 21.3 & 12.3\\
Beta Cephei &9 &30 & 95.0 &91.7 &96.7 &96.7 &96.7 & 90.7 & 87.5 & 88.9 & 84.0 & 90.7\\
W Ursa Maj. &16 &43 &40.7 &36.0 &51.2 &60.5 &61.6 &27.0 & 27.3 & 27.1 & 19.2 & 30.1\\
\hline
Mira & 121 &23 & 8.7 & 8.7 & 8.7 & 8.7 & 4.3 & 9.1 & 8.7 & 8.7 & 8.7 & 9.8\\
Semi-Reg. PV &33 &9 & 33.3 &33.3 &33.3 &33.3 &33.3 & 33.3 & 33.3 & 33.3 & 33.3 & 35.4\\
Class. Cepheid &122 &68 & 2.9 & 2.9 & 1.5 & 1.5 & 1.5 & 3.1 & 1.5 & 1.6 & 1.5 & 2.8
\enddata
\tablenotetext{a}{Default Random Forest.}
\tablenotetext{b}{Errors evaluated over all objects not in the active learning sample.}
\end{deluxetable}
\section{{\tt ALLSTARS}: Active Learning Light Curve Web Interface}
\label{sec:allstars}
We developed the {\tt ALLSTARS} (Active Learning Lightcurve classification Service) web based tool as the crowdsourcing user interface to our active learning software. For each active learning iteration, this website displays to a user the set of AL-queried sources. For each source, users are given access to eight external web resources in addition to several feature space visualizations to facilitate manual classification of that source. A screen shot of the {\tt ALLSTARS} web interface is in Figure \ref{fig:allstars}. Additionally, for each source a user may make a science classification, a rating of their confidence, a data quality classification, can tag the source as interesting, and also may provide comments and store a manually-determined period. This set of information is used to determine the class of each of the active learning queried sources and to decide which subset of those sources to add to the training set.
{\tt ALLSTARS} was built using a combination of {\tt javascript}, {\tt PHP}, and {\tt Python} which accesses a {\tt MySQL} database. Backend feature generators, active learning and classification algorithms were implemented using a combination of {\tt Python}, {\tt C} and {\tt R}. The interactive plots are generated using the {\tt Flot jQuery}\footnote{{\tt Flot} is a {\tt Javascript} plotting library downloadable from \url{http://code.google.com/p/flot/}.} package. External resources made available for classifying each source are:
\begin{itemize}
\item NED Extinction Calculator: \url{http://ned.ipac.caltech.edu/forms/calculator.html}
\item SDSS DR7 Explorer: \url{http://cas.sdss.org/dr7/en/tools/explore/obj.asp}
\item SDSS DR7 Navigate Tool: \url{http://cas.sdss.org/dr7/en/tools/chart/navi.asp}
\item SIMBAD Query by coordinates: \url{http://simbad.u-strasbg.fr/simbad/sim-fcoo}
\item 2MASS Interactive Image ($J$-band): \url{http://irsa.ipac.caltech.edu/applications/2MASS/IM/interactive.html}
\item SkyView Original DSS image: \url{http://skyview.gsfc.nasa.gov/cgi-bin/query.pl}
\item NVO DataScope: \url{http://heasarc.gsfc.nasa.gov/cgi-bin/vo/datascope/init.pl}
\item DotAstro LightCurve Warehouse: \url{http://dotastro.org/}
\end{itemize}
The initial page for a source includes two color-color plots: $B-J$ vs. $J-K$ and $J-H$ vs. $H-K$, using colors from the SIMBAD source which best matches the location of the given source. The source is also shown on a log-amplitude vs. log-period plot, with sources from the initial {\it Hipparcos~} and OGLE training set displayed in the background. These sources are discriminated using 21 different colors which represent most science classes to which the user may classify. An interactive magnitude vs. time light curve plot is also shown, with options to display it either unfolded, folded on any of the three most significant periods, or folded using a user entered or zoom-box generated period. The chosen period also updates a black circle on the amplitude-period plot. Also available on this initial page are the top three algorithm classifications and their confidences.
{\tt ALLSTARS} can be used to display any source available in the \url{http://dotAstro.org} Lightcurve Warehouse, allowing a registered user to make a science classification, assess data quality, note a manually found period, or add additional comments for that source. This web interface is an extremely useful tool, not only for performing active learning for variable star classification, but also for following up on outliers discovered via unsupervised learning, for finding typical examples of light curves of desired science classes, and to manually search through subsets of the {\tt dotAstro} data warehouse.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=5.2in]{allstars_page1.pdf}
\end{center}
\caption{ Screen shot of the {\tt ALLSTARS} web interface. Here, a Mira variable from the ASAS survey has been queried by the user. From top to bottom, the user is provided a (folded) ASAS light curve of the source, its location in amplitude-period space, its $J-H$ vs. $H-K$, and its $B-J$ vs. $J-K$ colors. At the top of the page are several tabs which link to external resources. On the left margin the user can make and submit a classification for the source. \label{fig:allstars}}
\end{figure}
\section{Application of Active Learning to classify ASAS Variable Stars}
\label{sec:results}
We use the active learning methodology presented in \S\ref{ss:actlearn} to classify all of ACVS (see \S\ref{ss:acvs}) starting with the combined {\it Hipparcos~} and OGLE training set. We employ the $S_2$ AL query function (Equation \ref{eq:alrf2}), treating it as a probability distribution (AL2.d in \S\ref{sec:deb}), and selecting 50 AL candidates on each of 9 iterations (except for the first iteration, where 75 AL candidates were chosen). For a cost function, we employ data from our first AL iteration to train a logistic regression model to predict cost as a function of \verb freq_signif, the statistical significance of the estimated first frequency\footnote{This will bias us away from selecting aperiodic sources, such as T Tauri. However, this is a reasonable approach because (1) there are simply too many aperiodic sources that are impossible to classify manually, and (2) in AL we draw a random sample from the $S_2({\bf x}') * (1-C({\bf x}'))$ meaning that we are still very likely to select some interesting aperiodic sources with high $S_2$ score.}.
A total of 11 users classified sources using the {\tt ALLSTARS} web interface. To help train new users, the beginning of each iteration was populated with 14-18 high-confidence sources\footnote{As to not throw away useful annotations, these classifications were used along with the AL samples.}. A total of 615 sources were observed by users (this represents 1.2\% of the ACVS catalog). The average user classified 137 sources, with a range from 21--474. User responses were combined using the crowdsourcing methodology in \S\ref{ss:crowdsource}. This led to the inclusion of 415 ASAS sources (67\% of all sources that were studied manually) into the training set. In Figure \ref{fig:asasalsel} we plot the AL queried data from one iteration in the amplitude-period plane, highlighting those which were selected for inclusion in the training set.
As described in \S\ref{ss:acvs}, the default RF only attains a 65.5\% agreement with the ACVS catalog. After 9 AL iterations, this jumps to 79.5\%, an increase of 21\% in agreement rate. The proportion of ACVS sources in which we are confident (defined as $\max_y \widehat{P}_{\rm RF}(y|{\bf x}) > 0.5$) climbs from 14.6\% to 42.9\%. This occurs because the selected ASAS data that are subsequently used as training data fill in sparse regions of training set feature space, thus increasing the chance that ASAS sources are in close proximity to training data and increasing the RF maximum probabilities. As a function of the AL iteration, both the ACVS agreement rate and the proportion of confident classifications achieved by our classifier are plotted in Figure \ref{fig:alrates}. The full evolution of the distribution of $\max_y \widehat{P}_{\rm RF}(y|{\bf x})$ is plotted in Figure \ref{fig:asasprobdist}. As the iterations proceed, power is shifted from low to high probabilities.
In Figure \ref{fig:alasaspred} we plot a table of the correspondence between our classifications after 9 AL iterations and the ACVS class. Comparing to Figure \ref{fig:rfasaspred}, we see that the AL predictions more closely match the ACVS labels across most science classes. For example, correspondence in the Classical Cepheid class raised from 24\% to 61\%, RR Lyrae, FM from 79\% to 93\%, Delta Scuti from 22\% to 60\%, and Chemically Peculiar from 1\% to 72\%. We have also identified a number of candidates for more rare classes, such as 117 RV Tauri, 177 Pulsating Be stars, and 43 T Tauri. Additionally, the number of RR Lyrae, DM candidates, which was artificially high for the original RF classifier, has diminished from 9109 to 442. A summary of our ASAS classification active learning, by class, is given in Table \ref{tab:predasas}.
As a consequence of performing active learning on the ASAS data set, we were able to detect the presence of 3 additional science classes of red giant stars. These classes were discovered by one of the AL users upon realizing that many of the queried pulsating red giant stars were low-amplitude with 10-75 day periods. A literature search revealed that these stars naturally break into small-amplitude red giant A and B subclasses (SARG A and B, see \citealt{2004MNRAS.349.1059W}). Furthermore, the presence of a red giant subclass of long secondary period (LSP, \citealt{2007ApJ...660.1486S}) stars was discovered and added. Via active learning, our classifier identified 3699 SARG A, 8823 SARG B, and 5889 LSP candidates.
Our final experiment is to compare our classification results using active learning with the classification of a Random Forest that is trained on the ACVS labels. The aim of this study is to determine whether our classifier's disagreement with ACVS is due principally to inadequacies in our classifier or mistakes and inconsistencies in the ACVS classifications. Using a 5-fold cross-validation on the ACVS labels, a RF classifier finds a 90\% agreement rate with ACVS (compared to 79.5\% using AL). A substantial proportion of our disagreement with ACVS results from the use of a finer taxonomy (where, e.g., we can correctly identify some of ACVS Mira candidates as Semi-Regular PVs). Within the classes in which the AL classifier has its poorest agreement with ACVS, the ACVS RF also does not do well: for Pop. II Cepheids, the ACVS RF finds only 37\% agreement (compared to 0\%), for Multi. Mode Cepheids it finds 45\% agreement (29\%), and in Beta Cepheid it finds 0\% agreement (0\%). This evidence points to the conclusion that the disagreement of our AL classifier to ACVS within these classes is due more to lack of self-consistency of those classes in ACVS (due either to mistakes in ACVS or absence of crucial features) than to any shortcomings in the active learning methodology.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=5.5in]{ASAS_ALsamples.jpg}
\end{center}
\caption{ Active learning samples on a single iteration of the algorithm. Yellow circles signify points that at least 65\% of users were able to classify. These points are included on subsequent iterations of the algorithm. Cyan triangles signify variable stars that were queried, but for which fewer than 65\% of users were able to classify. Black diamonds and red squares are the original training and testing data, as in Figure \ref{fig:traintestoffset}. \label{fig:asasalsel}}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=6.5in]{ASAS_ALmetrics.pdf}
\end{center}
\caption{ Left: Percent agreement of the Random Forest classifier with the ACVS labels, as a function of AL iteration. Right: Percent of ASAS data with confident RF classification (posterior probability $>0.5$), as a function of AL iteration. In the percent agreement with ACVS metric, performance increases dramatically in the first couple of iterations and then slowly levels off. In the percent of confident RF labels, the performance increases steadily. \label{fig:alrates}}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=6.5in]{RF_maxProbDist.jpg}
\end{center}
\caption{ Distribution of the Random Forest $\max_y \widehat{P}_{\rm RF}(y|{\bf x})$ values for the ASAS data, as a function of AL iteration. For the default RF classifier, most values are smaller than 0.4, meaning that the classifier is confident on very few sources. As the AL iterations proceed, much of the mass of the distribution gradually shifts toward larger values. The distribution slowly becomes multimodal: for a slim majority of sources, the algorithm has high confidence, while for a substantial subset of the data the algorithm remains unsure of the classification. \label{fig:asasprobdist}}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=6.5in]{confmat_asas_AL9.pdf}\\
\includegraphics[angle=0,width=6.5in]{confmat_asas_confident_AL9.pdf}
\end{center}
\caption{Top: Classifications of the active learning RF classifier after 9 iterations of AL. Compared to Figure \ref{fig:rfasaspred}, there is a closer correspondence to the ACVS class labels (y axis). Notably, the RRL, DM artifact has largely disappeared. Bottom: Same for only sources with classification probability $> 0.5$. Here, the agreement is even higher. The main confusion is in classifying ACVS RR Lyrae, FO and Delta Scuti as W Ursae Maj. \label{fig:alasaspred}}
\end{figure}
\begin{deluxetable}{lrrrr}
\tabletypesize{\footnotesize}
\tablewidth{4.0in}
\tablecolumns{5}
\tablecaption{Results, by class, of performing active learning to classify ASAS variable stars. \label{tab:predasas}}
\tablehead{
\colhead{Science Class} &
\colhead{$N_{\rm Train}$} &
\colhead{$N_{\rm AL Add}$\tablenotemark{a}} &
\colhead{$N_{\rm RF}$\tablenotemark{b}} &
\colhead{$N_{\rm AL}$\tablenotemark{c}}
}
\startdata
a. Mira & 144 & 20 & 3587 & 3173 \\
b1. Semireg PV & 42 & 59 & 5799 & 9223 \\
b2. SARG A & 0 & 15 & 0 & 3699 \\
b3. SARG B & 0 & 29 & 0 & 8823 \\
b4. LSP & 0 & 54 & 0 & 5889 \\
c. RV Tauri & 6 & 5 & 0 & 117 \\
d. Classical Cepheid & 191 & 16 & 324 & 494 \\
e. Pop. II Cepheid & 23 & 0 & 98 & 5 \\
f. Multi. Mode Cepheid & 94 & 4 & 162 & 263 \\
g. RR Lyrae, FM & 124 & 26 & 1714 & 1667 \\
h. RR Lyrae, FO & 25 & 14 & 51 & 317 \\
i. RR Lyrae, DM & 57 & 3 & 9109 & 442 \\
j. Delta Scuti & 114 & 19 & 822 & 1755 \\
k. Lambda Bootis & 13 & 0 & 0 & 0 \\
l. Beta Cephei & 39 & 0 & 0 & 0 \\
m. Slowly Puls. B & 29 & 0 & 0 & 0 \\
n. Gamma Doradus & 28 & 0 & 0 & 0 \\
o. Pulsating Be & 45 & 4 & 10 & 177 \\
p. Per. Var. SG & 55 & 1 & 1663 & 624 \\
q. Chem. Peculiar & 51 & 14 & 27 & 447 \\
r. Wolf-Rayet & 40 & 0 & 6683 & 1198 \\
s. T Tauri & 14 & 4 & 753 & 43 \\
t. Herbig AE/BE & 15 & 0 & 4 & 1 \\
u. S Doradus & 7 & 0 & 0 & 1 \\
v. Ellipsoidal & 13 & 0 & 0 & 0 \\
w. Beta Persei & 169 & 25 & 2110 & 3055 \\
x. Beta Lyrae & 145 & 37 & 11962 & 2603 \\
y. W Ursae Maj. & 59 & 66 & 5246 & 6108 \\
\enddata
\tablenotetext{a}{ASAS sources added to the training set after 8 AL iterations.}
\tablenotetext{b}{Number of ASAS sources classified by the default Random Forest.}
\tablenotetext{c}{Number of ASAS sources classified by the RF after 8 AL iterations.}
\end{deluxetable}
\section{Conclusions}
\label{sec:conclusions}
We have described the problem of sample selection bias (a.k.a. covariate shift) in supervised learning on astronomical data sets. Though supervised learning has shown great promise in automatically analyzing large astrophysical databases, care must be taken to account for the biases that occur due to distributional differences between the training and testing sets. Here, we have argued that sample selection bias is a common problem in astronomy, primarily because the subset of well-studied astronomical objects typically forms a biased sample of intrinsically brighter and nearby sources. In this paper, we showed the detrimental influence of sample selection bias on the problem of supervised classification of variable stars.
To alleviate the effects of sample selection bias, we proposed a few different methods. We find, on a toy problem using {\it Hipparcos~} and OGLE light curves, that active learning performs significantly better than other methods such as importance weighting, co-training, and self-training. Furthermore, we argue that AL is a suitable method for many astronomical problems, where follow-up resources are usually available (albeit with limited availability). Active learning simply gives a principled way to determine which sources, if followed up on, would help the supervised algorithm the most. We show that in classifying variable stars from the ASAS survey, AL produces hugely significant improvements in performance within only a handful of iterations. Our {\tt ALLSTARS} web interface was critical in this work, as was the participation of knowledgeable (``trained expert'') users and sophisticated crowdsourcing methods.
Though we have introduced a couple of AL querying functions, many different options are available. In particular, we argue that the $S_2$ criterion is appropriate for our classification problem because it targets objects whose inclusion in the training set would induce the largest overall change in the classification predictions over the testing set. However, for each problem, a different AL function will be appropriate. The pertinent querying function depends on the problem at hand, the type of response being modeled, and the kind of supervised algorithm employed, and typically several different choices are available.
One common cause of sample selection bias in variable star classification is that data from older surveys---whose sources have typically been observed over many epochs---are commonly used to classify data from ongoing surveys, whose sources contain many fewer epochs of observation. In addition to AL, other viable approaches to this particular problem are those of \emph{noisification}, where the training set light curves are artificially modified to resemble those of the testing set, and \emph{denoisification}, where each testing light curve is matched to a (clean) training light curve. These techniques are currently being studied by \citet{long2011}.
Our discussion of sample selection bias has revolved around the use of non-parametric tools (and in particular Random Forests). For the types of complicated classification and regression problems in astrophysics, flexible non-parametric methods are usually necessary. However, in many applications, parametric models are appropriate. In this parametric setting, there are several methods of overcoming sample selection bias, including Bayesian experimental design (\citealt{chaloner1995bayesian}).
We conclude by emphasizing the importance of treating sample selection bias for future petabyte-scale surveys such as Gaia and LSST. These upcoming surveys will collect data at such massive rates that rare, unexpected, and yet-undiscovered sources will be prevalent in their data streams. Furthermore, due to superior optics and cameras, they will probe different populations of sources than observed by any previous mission. For these reasons, any conceivable training set constructed prior to the start of these surveys will have significant sample selection bias. Through active learning, we now have a principled way to queue sources for targeted follow-up in order to augment training sets to optimize the performance of machine-learned algorithms and to maximize the science that these missions produce.
\acknowledgements
The authors acknowledge the generous support of a CDI grant (\#0941742) from the National Science Foundation. This work was performed in the CDI-sponsored Center for Time Domain Informatics (\url{http://cftd.info}). N.R.B. is supported through the Einstein Fellowship Program (NASA Cooperative Agreement: NNG06DO90A). We acknowledge the help of Christopher Klein, Adam Morgan, and Brad Cenko in helping to manually classify variable stars with {\tt ALLSTARS}. We also thank Laurent Eyer for helpful conversations.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,963 |
{"url":"https:\/\/www.khanacademy.org\/science\/health-and-medicine\/mental-health\/dementia-delirium-alzheimers\/a\/dementia-and-delirium-including-alzheimers","text":"# Dementia and Delirium, including\u00a0Alzheimer\u2019s\n\n\u201cHas anyone seen my wallet?\u201d \u201cWhat time do I have to go to the dentist?\u201d \u201cWhat\u2019s the name of that movie we went to last week?\u201d We all ask these sorts of questions from time-to-time no matter our age; and not having a perfect memory is perfectly normal. It's also normal that our longer term memory, as well as our ability to remember things that only occur occasionally, fade a little as we age. Nevertheless, as we get into our sixties, many of us worry that forgetfulness or difficulty paying attention are signs of worse things to come. One of the most widespread fears as we age is dementia due to Alzheimer\u2019s disease.\nBoth dementia (including Alzheimer\u2019s) and delirium are both common causes of memory loss, impaired thinking and understanding (cognition) and impaired behaviour. They are distinct disorders but may be difficult to tell apart:\n\u2022 Dementia is a group of symptoms that mainly affects memory, cognition and social interactions, and the ability to do everyday tasks. Symptoms start gradually often with no clear beginning, and are usually permanent.\n\u2022 Delirium, on the other hand, typically begins suddenly with a noticeable start point. It mainly affects attention, and often resolves after a few days or weeks, although it can last longer.\n\n## Structure, function and brain damage in dementia\n\nYour brain is your body\u2019s control centre. It is full of millions of interconnected brain cells, called neurons that receive, process, and send messages that control and coordinate almost everything that you do. Your brain is divided into three main parts, the brain stem, cerebellum and cerebrum. These different regions are responsible for different functions, such as movement, speech, sense perception, emotions, heart rate, cognition and memory. Information travels back and forth from the brain to the body along neurons, in the form of electrical impulses. When a nerve impulse reaches the end of a neuron a chemical neurotransmitter is released, which stimulates the electrical nerve impulse in the next neuron, allowing it to \u201cjump\u201d from one neuron to the next. Neurons in different parts of the brain use different neurotransmitters to transfer information. Some neurotransmitters stimulate brain activity, others calm the brain, and some do both. When your brain is functioning well, these neurotransmitters are all in balance.\nDiagram showing regions of the brain affected by dementia\nImage of neurotransmitters traveling across a synapse\nDementia may occur anytime neurons get damaged. The damage may be anywhere within the brain, and in more than one area at the same time. Most dementias are caused by neurodegenerative diseases, most commonly Alzheimer\u2019s disease, Lewy body dementia and frontotemporal dementia. These diseases cause clumps of abnormal proteins to build up inside neurons, damaging them, and causing them to slowly degenerate and die. Not surprisingly, this disrupts the production of neurotransmitters and interrupts brain signals. In addition to these, vascular dementia is another common cause of progressive dementia. In this case, brain damage occurs when the blood supply to the neurons is reduced or blocked, again causing them to malfunction or die. Blood vessel damage and blockage may result during a stroke, or may be caused by high blood pressure or other blood vessel disorders. There are also several other rare neurodegenerative diseases, and many other less common causes of dementia, such as infections, or dietary deficiencies.\nDelirium is an acute, transient, and usually reversible brain malfunction. What exactly happens inside the brain of someone with delirium is not well understood; although, it is thought to be brought on by multiple neurotransmitter imbalances, which alter your normal brain activity.\n\n## Symptoms of dementia and delirium\n\nDementia symptoms affect people differently depending on the area or areas of the brain that are affected, and may be progressive or reversible depending upon the cause of the damage. Sometimes the symptoms are very similar to those of delirium, making diagnosis somewhat tricky, and it is fairly common for a person with dementia to develop delirium.\nDementia: symptoms usually start gradually, are fairly constant on a day-to-day basis, and slowly and steadily become worse over the course of about a decadeDelirium: symptoms usually appear over a few hours to a few days, and may fluctuate on and off during the day, and often occurs at night.\nCognitive symptoms include:Cognitive symptoms include:\nMemory lossMemory Loss\nDifficulty speaking and communicatingDifficulty speaking and communicating\nDifficulty with complex tasksRambling or nonsense speech\nDifficulty planning and organizingDifficulty reading and writing\nDisorientationDisorientation\nLoss of coordinationWandering attention\nBecoming easily distracted\nBecoming withdrawn\nPsychological symptoms include:Psychological symptoms include:\nPersonality changesInability to focus\nInability to reasonInability to reason\nInappropriate behaviourReduced awareness of the environment\nParanoiaAgitation\nAgitationHallucinations\nHallucinationsDisturbed sleep\nFear, anxiety, anger or depression\nDelirium is usually transient rather than permanent, although how well you recover often depends on how well you were beforehand \u2014 if you were in good health before it happened, you are more likely to have a full recovery. Unfortunately, for some people, particularly those who are critically ill, delirium may lead to significant memory loss and decrease in thinking skills, as well as a general decline in health, poor recovery and increased risk of death.\nA diagram that details what each region of the brain controls\n\n## What causes dementia and delirium?\n\n### Dementia\n\nFour diseases account for most cases of dementia: Alzheimer\u2019s disease (about 50-60% of cases), vascular dementia (about 15-20% of cases), Lewy body dementia, and frontotemporal dementia.$^1$ Each of these have different characteristics:\n\u2022 Alzheimer\u2019s symptoms usually start in your mid to late 60s, although a small proportion of people get early onset Alzheimer's, with symptoms starting in their 40s or 50s. This form of Alzheimer\u2019s has a strong genetic link and typically runs in families.\n\u2022 Vascular dementia is rare if you are under 65, but usually starts more suddenly than Alzheimer\u2019s. Diagnosis may be complicated by the fact that you have Alzheimer\u2019s or another dementia at the same time. People at risk for vascular dementia often have a history of smoking, cardiac dysrhythmia, hypertension, diabetes, and coronary artery disease.\n\u2022 Lewy body dementia: Lewy bodies are abnormal clumps of protein that build up in neurons causing brain signalling malfunctions; they are often found in the brains of people with Alzheimer\u2019s and Parkinson\u2019s disease (primarily a movement disorder). Lewy body dementia is similar to Alzheimer\u2019s, but can be distinguished by unique symptoms such as rapid eye movement sleep behaviour disorder, and fluctuations between confusion and clarity.\n\u2022 Frontotemporal dementia often has an earlier onset than Alzheimer\u2019s, generally in your 50s or early 60s. The brain damage occurs at the front of your brain in regions that control personality, behaviour, and language.\nExactly why we get these dementias remains a puzzle, although there are likely genetic links in many cases. Other rare conditions have also been linked to dementia including Huntington\u2019s disease, Creutzfeldt-Jakob disease, and Parkinson\u2019s disease, as well as traumatic brain injury.\nAdditionally, there are a variety of other reasons you may get dementia when the symptoms are often reversible. These include infections (e.g., meningitis, or syphilis), nutritional deficiencies, reactions to medications, brain bleeds, substance abuse, and poisoning. It is also possible that malfunction of other vital organs including the liver or kidneys can disrupt brain function causing dementia.\nAs an older adult, any condition that ends up in a visit to the hospital increases your risk for delirium, and up to 80% of people who are critically ill will be delirious at some time during their hospital stay.$^2$ There are many different conditions that can trigger delirium with most common including dehydration, acute infections, such as urinary tract infection or pneumonia, prescribed drugs including antidepressants, sleeping pills and narcotics, and alcohol or substance abuse or withdrawal.\nA diagram showing the causes of dementia and delirium\n\n## How many people have dementia and delirium?\n\nDementia: Worldwide, there are around 50 million people living with dementia, and with 8 million new cases every year; as the average lifespan is steadily increasing, dementia is increasing fast in many countries.$^3$ Overall, almost 1 in 10 people over the age of 60 have dementia, and the prevalence increases rapidly with age.$^1$ Women are at slightly more risk for it than men, and your chances increase if you are obese, have diabetes, high blood pressure, or other cardiovascular risk factors.$^1$\nA graph illustrating the predicted number of cases of dementia in various regions of the world.\nDelirium: Delirium is very common in the hospital setting, especially when you are older and unwell. In fact, almost half of older patients are delirious when they are admitted, or develop delirium while they are there.2 Like dementia, delirium increases significantly with age.4 In contrast to dementia, delirium is a little more common among men than women.\n\n## Are there ways to prevent dementia and delirium?\n\nDementia: We don\u2019t know for sure how to prevent dementia, but a healthy active lifestyle that includes keeping your mind and body active, and that keeps you socially connected may be of benefit. Other steps you can take that may also reduce your risk are to quit smoking, lower your blood pressure, and eat a healthy diet that maintains a good balance of nutrients and vitamins, all of which may indirectly reduce your risk.\nDelirium: The best way to prevent delirium is to remove the triggers. In a hospital setting this could include systems that maintain as calm and quiet an environment as possible, while providing the individualized support and medical care that is necessary, and when possible avoiding medications known to trigger delirium.\n\n## What are the treatments for dementia and delirium?\n\nDementia: Memory loss and other symptoms of dementia may have many causes, so after reviewing your medical history, and symptoms, your doctor will likely want you to undergo other tests to evaluate your thinking and movement skills, and possibly your mental health. In addition to this, you may need a brain scan to check to see if you have had a stroke or brain bleed, or whether or not you have a brain tumour, as well as various laboratory tests to check for any metabolic and nutritional problems.\nMost types of dementia are incurable, but there are medications that regulate brain chemicals involved in brain functions including memory, judgement, and learning that may be helpful with controlling or reducing your symptoms. Two of the more commonly prescribed types of medicine are cholinesterase inhibitors and memantine, which is the first in a new class of drugs for the treatment of Alzheimer\u2019s. Other medications are also available to treat any accompanying symptoms, and occupational therapists are experts at helping with strategies that improve coping with day-to-day challenges and quality of life.\nDelirium: Your doctor will likely follow a similar process to that outlined above for dementia, to decide whether or not you are delirious. If you are, your treatment will be aimed at eliminating the underlying cause or causes, which for example could include treating an infection, or stopping or changing a particular medicine that you are taking to treat something else. Your doctor may also recommend additional supportive care, to ensure you stay hydrated, treat any pain you may have, keep you oriented with your surroundings, and help you get up and about again.\n\n## Consider the following:\n\nMemantine is the first drug in a new class of drugs (called N-methyl-D-aspartate receptor, or NMDA receptor antagonists) that has been approved to treat moderate to severe Alzheimer\u2019s disease. While cholinesterase inhibitors used to treat Alzheimer\u2019s help raise the levels of the neurotransmitter known as acetylcholine, memantine works by regulating a chemical called glutamate. Glutamate is the salt of glutamic acid. It is an essential amino acid that is involved in many chemical processes that occur in living organisms, that lead to growth, production of energy, and elimination of waste. In the brain however, glutamate is an important neurotransmitter that\u2019s involved in regulating almost everything your brain does including memory, cognition, and learning. For your brain to work well, glutamate has to be in the right place, at the right time, and in the right amount \u2014 too much or too little glutamate is harmful. In Alzheimer\u2019s disease, excess glutamate accumulates when neurons get damaged, which can be devastating for functioning neurons. Memantine works by blocking the effects of too much glutamate, preventing cell death, and delaying some of the symptoms of moderate to severe Alzheimer\u2019s disease. Because memantine has a different mechanism of action, it may be used in combination with cholinesterase inhibitors, which may improve outcomes for longer periods of time.","date":"2019-01-18 01:23:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 5, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.31233420968055725, \"perplexity\": 3201.5055310502253}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-04\/segments\/1547583659654.11\/warc\/CC-MAIN-20190118005216-20190118031216-00137.warc.gz\"}"} | null | null |
<?php
namespace Neos\Flow\Aop;
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Q: How to figure out the right size background image for iPhone I am working with media queries to create screen breaks, in order to have a website optimized on both the desktop and mobile. Currently, when I bring the site up with its current css the background pictures are stretched out of proportion. (The current background sizes are: 640px x 960px.)
Here is the current css:
@media only screen and (-webkit-min-device-pixel-ratio : 2), only screen and (min-device- pixel-ratio : 2) {
#home{width: 980px;
height: 1090px;
background-image: url(../images/landingPageRetina.jpg) 50% 0 no-repeat;
background-size: 100% 100%;}
#about{background-image: url(../images/aboutMobile.png) 50% 0 no-repeat;}
#music{background-image: url(../images/musicMobile.jpg) 50% 0 no-repeat;}
#videos{background-size: url(..images/videosMobile.jpg)50% 0 no-repeat;}
#connect{background-image: url(..images/connectMobile.jpg)50% 0 no-repeat ;}
#contact{background-image: url(../images/contactMobile.jpg)50% 0 no-repeat ;}
}
Does anyone have any suggestions on what the correct sizes are - so that the images will render proportionally?
A: iPhone5 is 640x1136. Look at this article by smashing magazine for more details: http://mobile.smashingmagazine.com/2013/03/21/responsive-web-design-with-physical-units/
The best background to use is one that won't require so many media queries in order to make it look good. Try CSS3 gradients or tiled backgrounds...
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\section{Introduction}
Many tasks in computing are prohibitively difficult to formalize and thus hard to get right. A classical example is the recognition of digits from images. Formalizing what exactly distinguishes the digit \emph{2} from a \emph{7} is in a way that captures all common handwriting styles is so difficult that this task is normally left to the computer. A classical approach for doing so is to learn a \newterm{feed-forward neural network} from pre-classified example images. Since the advent of \newterm{deep learning} (see, e.g., \cite{DBLP:journals/nn/Schmidhuber15}), the artificial intelligence research community has learned a lot about engineering these networks, such that they nowadays achieve a very good classification precision and outperform human classifiers on some tasks, such as sketch recognition \cite{DBLP:conf/bmvc/YuYSXH15}. Even safety-critical applications such as obstacle detection in self-driving cars nowadays employ neural networks.
But if we do not have formal specifications, how can we assure the safety of such a system? The classical approach to tackle this problem is to construct \emph{safety cases} \cite{Wagner2015}. In such a safety case, we characterize a set of environment conditions under which a certain output is desired and then test if the learned problem model ensures this output under all considered environment conditions. In a self-driving car scenario, we can define an abstract obstacle appearance model all of whose concretizations should be detected as obstacles. Likewise, in a character recognition application, we can define that all images that are \emph{close} to a given example image (by some given metric) should be detected as the correct digit. The verification of safety cases somewhat deviates from the classical aim of formal methods to verify correct system behavior in all cases, but the latter is unrealistic due to the absence of a complete formal specification. Yet, having the means to test neural networks on safety cases would help with certification and also provides valuable feedback to the system engineer.
Verifying formal properties of feed-forward neural networks is a challenging task. Pulina and Tacchella~\cite{DBLP:conf/cav/PulinaT10} present an approach for neurons with non-linear activation functions that only scales to small networks. In their work, they use networks with 6 nodes, which are far too few for most practical applications. They combine counterexample-triggered abstraction-refinement with \newterm{satisfiability modulo theory} (SMT) solving. Scheibler et al.~\cite{DBLP:conf/mbmv/ScheiblerWWB15} consider the bounded model checking problem for an inverse pendulum control scenario with non-linear system dynamics and a non-linear neuron activation function, and despite employing the state-of-the-art SMT solver iSAT3 \cite{DBLP:conf/fmcad/ScheiblerNMFTBF16} and even extending this solver to deal better with the resulting problem instances, their experiments show that the resulting verification problem is already challenging for neural networks with 26 nodes.
In \newterm{deep learning} \cite{DBLP:journals/nn/Schmidhuber15}, many works use networks whose nodes have piece-wise linear activation functions. This choice has the advantage that they are more amenable to formal verification, for example using SMT solvers with the theory of linear real arithmetic, without the need to perform abstract interpretation.
In such an approach, the solver chooses the \emph{phases} of (some of) the nodes, and then applies a linear-programming-like sub-solver to check if there exist concrete real-valued inputs to the network such that all nodes have the selected phases. The node phases represent which part of the piece-wise linear activation functions are used for each node. It has been observed that the SMT instances stemming from such an encoding are very difficult to solve for modern SMT solvers, as they need to iterate through many such phase combinations before a problem instance is found to be satisfiable or unsatisfiable \cite{DBLP:journals/corr/KatzBDJK17,DBLP:journals/aicom/PulinaT12}. Due to the practical importance of verifying piecewise-linear feed-forward neural networks, this observation asks for a specialized approach for doing so.
Huang et al.~\cite{DBLP:journals/corr/HuangKWW16} describe such an approach that is based on propagating constraints through the layers of a network. The constraints encode regions of the input space of each layer all of whose points lead to the same overall classification in the network. Their approach is partially based on discretization and focusses on robustness testing, i.e., determining the extent to which the input can be altered without changing the classification result. They do not support general verification properties.
Bastiani et al.~\cite{DBLP:conf/nips/BastaniILVNC16} also target robustness testing and define an abstraction-refinement
constraint solving loop to test a network's robustness against adversarial pertubations. They also employ the counter-examples that their approach finds to learning more robust networks.
Katz et al.~\cite{DBLP:journals/corr/KatzBDJK17} provide an alternative approach that allows to check the input/output behavior of a neural network with linear and so-called \newterm{ReLU} nodes against convex specifications. Many modern network architectures employ these nodes. They present a modification of the \newterm{simplex algorithm} for solving linear programs that can also deal with the constraints imposed by ReLU nodes, and they show that their approach scales orders of magnitudes better than when applying the SMT solvers \texttt{MathSAT} or \texttt{Yices} on SMT instances generated from the verification problems.
Modern neural network architectures, especially those for image recognition, however often employ another type of neural network node that the approach by Katz et al.~does not support: \newterm{MaxPool} nodes. They are used to determine the strongest signal from their input neurons, and they are crucial for \newterm{feature detection} in complex machine learning tasks. In order to support the verification of safety cases for machine learning applications that make use of this node type, it is thus important to have verification approaches that can efficiently operate on networks that have such nodes, without the need to simulate MaxPool nodes by encoding their behavior into a much larger number of ReLU nodes.
In this paper, we present an approach to verify neural networks with piece-wise linear activation functions against convex specifications. The approach supports all node types used in modern network network architectures that only employ piece-wise linear activation functions (such as MaxPool and ReLU nodes). The approach is based on combining satisfiability (SAT) solving and linear programming and employs a novel linear approximation of the overall network behavior. This approximation allows the approach to quickly rule out large search space parts for the node phases from being considered during the verification process. While the approximation can also be used as additional constraints in SMT solving and improves the computation times of the SMT solver, we apply it in a customized solver that uses the \newterm{elastic filtering} algorithm from \cite{DBLP:journals/informs/ChinneckD91} for minimal infeasible linear constraint set finding in case of conflicts, and combine it with a specialized procedure for inferring implied node phases. Together, these components lead to much shorter verification times. We apply the approach on two cases studies, namely collision avoidance and character recognition, and report on experimental results.
We also provide the resulting solver and the complete tool-chain to generate verifiable models using the Deep Learning framework \texttt{Caffe}~\cite{jia2014caffe} as open-source software.
\section{Preliminaries}
\paragraph{Feed-Forward Neural Networks:} We consider \newterm{multi-layer} (\newterm{Perceptron}) networks with \newterm{linear}, \newterm{ReLU}, and \newterm{MaxPool} nodes in this paper. Such networks are formally defined as directed acyclic weighted graphs $G = (V,E,W,B,T)$, where $V$ is a set of nodes, $E \subset V \times V$ is a set of edges, $W : E \rightarrow \mathbb{R}$ assigns a weight to each edge of the network, $B : V \rightarrow \mathbb{R}$ assigns a \newterm{node bias} to each node, and $T$ assigns a \emph{type} to each node in the network from a set of available types $\mathcal{T} \in \{\mathit{input}, \mathit{linear}, \mathit{ReLU}, \mathit{MaxPool}\}$. Nodes without incoming edges are called \newterm{input nodes}, and we assume that $T(v) = \mathit{input}$ for every such node $v$. Vertices that have no outgoing edge are also called \newterm{output nodes}.
A feed-forward neural network with $n$ input nodes and $m$ output nodes represents a function $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$. Given assignments $\mathit{in} : \{1, \ldots, n\} \rightarrow V$ and $\mathit{out} : \{1, \ldots, m\} \rightarrow V$ that define the orders of the input and output nodes (so that we can feed elements from $\mathbb{R}^n$ to the network to obtain an output from $\mathbb{R}^m$), and some input vector $(x_1, \ldots, x_n) \in \mathbb{R}^n$, we can define the network's behavior by a node value assignment function $a:V \rightarrow \mathbb{R}$ that is defined as follows:
\begin{itemize}
\item For every node $v$ with $T(v) = \mathit{input}$, we set $a(v) = x_j$ for $j={\mathit{in}^{-1}(v)}$,
\item For every node $v$ with $T(v) = \mathit{linear}$, we set $a(v) = \sum_{v' \in V, (v',v) \in E} W((v',v)) \cdot a(v') + B(v)$.
\item For every node $v$ with $T(v) = \mathit{ReLU}$, we set $a(v)= \max(B(v)+\sum_{v' \in V, (v',v) \in E} \allowbreak{} W((v',v)) \cdot a(v'),0)$.
\item For every node $v$ with $T(v) = \mathit{MaxPool}$, we set $a(v) = \max_{v' \in V, (v',v) \in E} a(v')$.
\end{itemize}
Function $f$'s output for $(x_1,\ldots,x_n)$ is defined to be $(a(\mathit{out}(1)), \allowbreak{}\ldots, \allowbreak{} a(\mathit{out}(m)))$. Note that the weights of the edges leading to $\mathit{MaxPool}$ nodes and their bias values are not used in the definition above.
Given a node value assignment function $a:V \rightarrow \mathbb{R}$, we also simply call $a(v)$ the \newterm{value} of $v$. If for a ReLU node $v$, we have $s(v)<0$ for $s(v) = B(v)+\sum_{v' \in V, (v',v) \in E} \allowbreak{} W((v,v')) \cdot a(v')$, and hence $a(v) = 0$, we say that node $n$ is in the $\leq 0$ phase, and for $s(v) \geq 0$, and hence $a(v) \geq 0$, we say that it is in the $\geq 0$ phase. If we have $s(v)=0$, then it can be in either phase. For a $\mathit{MaxPool}$ node $v$, we define it to be in phase $e \in E \cap (V \times \{v\})$ if $a(v)=a(v')$ for $e = (v',v)$. If multiple nodes with edges to $v$ have the same values, then node $v$ can have any of the respective phases.
Modern neural network architectures are \newterm{layered}, i.e., we have that every path from an input node to an output node has the same length. For the verification techniques given in this paper, it does however not matter whether the network is layered.
Networks can also be used to \newterm{classify} inputs. In such a case, the network represents a function $f' : \mathbb{R}^n \rightarrow \{1, \ldots, m\}$ (for some numbering of the classes), and we define $f'(x_1, \ldots, x_n) = \argmax_{i \in \{1, \ldots, m\}} y_i$ for $(y_1, \ldots, y_m) = f(x_1, \ldots, x_n)$.
We do not discuss here how neural networks are learned, but assume networks to be given with all their edge weights and node bias values. Frameworks such as \texttt{Caffe}~\cite{jia2014caffe} provide ready-to-use functionality for learning edge weights and bias values from databases of examples, i.e., tuples $(x_1, \ldots, x_n, y_1, \ldots, y_m)$ such that we want the network to induce a function $f$ with $(x_1, \ldots, x_n) = (y_1, \ldots, y_m)$. Likewise, for classification problems, the databases consist of tuples $(x_1, \ldots, x_n, c)$ such that we want the network to induce a function $f'$ with $f'(x_1, \ldots, x_n) = c$.
When using a neural network learning tool, the architecture of the network, i.e., everything except for the weights and the node bias values, is defined up-front, and the framework automatically derives suitable edge weights and node bias values. There are other node types (such as \newterm{Softmax} nodes) that are often used during the learning process, but removed before the deployment of the trained network, and hence do not need to be considered in this work. Also, there are network layer types such as \newterm{convolutional layers} that have special structures. From a verification point of view, these are however just sets of linear nodes whose edges share some weights, and thus do not have to be treated differently.
\paragraph{Satisfiability Solvers:}
Satisfiability (SAT) solvers check if a Boolean formula has a satisfying assignment.
The formula is normally required to be in conjunctive normal form, and thus consists of \newterm{clauses} that are connected by \newterm{conjunction}. Every clause is a disjuction of one of more \newterm{literals}, which are Boolean variables or their negation.
A SAT solver operates by successively building a valuation of the Boolean variables and \newterm{backtracking} whenever a conflict of the current \newterm{partial valuation} and a clause has been found. To achieve a better performance, SAT solvers furthermore perform \newterm{unit propagation}, where the partial assignment is extended by literals that are the only remaining ones not yet violated by the partial valuation in some clause. Additionally, modern solvers perform \newterm{clause learning}, where clauses that are implied by the conjunction of some other clauses are lazily inferred during the search process, and select variables to branch on using a \newterm{branching heuristic}. Most modern solvers also perform \newterm{random restarts}. For more details on SAT solving, the interested reader is referred to \cite{FM09HBSAT}.
\paragraph{Linear Programming:} Given a set of linear inequalities over real-valued variables and a linear optimization function (which together are called a \newterm{linear program}), the linear programming problem is to find an assignment to the variables that minimizes the objective function and fulfills all constraints. Even though linear programming was shown to have polynomial-time complexity, it has been observed that in practice \cite{Kroening2008}, it is often faster to apply the \newterm{Simplex algorithm}, which is an exponential-time algorithm.
\paragraph{Satisfiability Modulo Theory Solving:}
SAT solvers only support Boolean variables. For problems that can be naturally represented as a Boolean combination of constraints over other variable types, Satisfiability Modulo Theory (SMT) solvers are normally applied instead. An SMT solver combines a SAT solver with specialized decision procedures for other theories (such as, e.g., the theory of linear arithmetic over real numbers).
\section{Efficient Verification of Feed-forward Neural Networks}
In this paper, we deal with the following verification problem:
\begin{definition}
\label{def:mainProblem}
Given a feed-forward neural network $G$ that implements a function $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$, and a set of linear constraints $\psi$ over the real-valued variables $V = \{x_1, \ldots, x_n, y_1, \ldots, y_m \}$, the neural net (NN) verification problem is to find a node value assignment function $a$ for $V$ that fulfils $\psi$ over the input and output nodes of $G$ and for which we have $f(x_1, \ldots, x_n) = (y_1, \ldots, y_m)$, or to conclude that no such node value assignment function exists.
\end{definition}
The restriction to conjunctions of linear properties in Definition~\ref{def:mainProblem} was done for simplicity. Verifying arbitrary Boolean combinations of linear properties can be fitted into Definition~\ref{def:mainProblem} by encoding them into the structure of the network itself, so that an additional output neuron $y_{\mathit{add}}$ outputs a value $\geq 0$ if and only if the property is fulfilled. In this case, $\psi$ is then simply $y_{\mathit{add}} \geq 0$.
There are multiple ways to solve the neural network (NN) verification problem. The encoding of an NN verification problem to an SMT problem instance is straight-forward, but yields instances that are difficult to solve even for modern SMT solvers (as the experiments reported on in Section~\ref{sec:experiments} show).
As an alternative, we present a new approach that combines 1) linear approximation of the overall NN behavior, 2) irreducible infeasible subset analysis for linear constraints based on elastic filtering \cite{DBLP:journals/informs/ChinneckD91}, 3) inferring possible safe node phase choices from feasibility checking of partial node phase valuations, and 4) performing unit-propagation-like reasoning on node phases.
We describe these ideas in this section, and present experimental results on a tool implementing them in the next section.
Starting point is the combination of a linear programming solver and a satisfiability solver.
We let the satisfiability solver guide the search process. It determines the phases of the nodes and maintains a set of constraints over node phase combinations. On a technical level, we allocate the SAT variables $x_{(v,\leq 0)}$ and $x_{(v,\geq 0)}$ for every ReLU node $v$, and also reserve variables $x_{(v,e)}$ for every MaxPool node $v$ and every edge $e$ ending in $v$. The SAT solver performs unit propagation, clause learning, branching, and backtracking as usual, but whenever the solver is about to branch, we employ a linear programming solver to check a linear approximation of the network behavior (under the node phases already fixed) for feasibility. Whenever a conflict is detected, the SAT solver can then learn a conflict clause. Additionally, we infer implied node phases in the search process.
We describe the components of our approach in this section, and show how they are combined at the end of it.
\subsection{Linear Approximation of Neural Network Value Assignment Functions}
\label{subsec:linearApprox}
Let $G = (V,E,W,B,T)$ be a network representing a function $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$. We want to build a system of linear constraints using $V$ as the set of variables that closely approximates $f$, i.e., such that every node value assignment function $a$ is a correct solution to the linear constraint system, and the constraints are as tight as possible. The main difficulty in building such a constraint system is that the $\mathit{ReLU}$ and $\mathit{MaxPool}$ nodes do not have linear input-output behavior (until their phases are fixed), so we have to approximate them linearly.
Figure~\ref{fig:ReLU} shows the activation function of a $\mathit{ReLU}$ node, where we denote the weight\-ed sum of the input signals to the node (and its bias) as variable $c$. The output of the node is denoted using the variable $d$.
If we have upper and lower bounds $[l,u]$ of $c$, then we can approximate the relationship between $c$ and $d$ by the constraints $d \geq 0$, $d \geq c$, and $d \leq \frac{u \cdot (c - l)}{u-l}$, all of which are linear equations for constant $u$ and $l$. This yields the set of allowed value combinations for $c$ and $d$ drawn as the filled area in Figure~\ref{fig:ReLU}.
\begin{figure}[tb]
\centering\begin{tikzpicture}
\draw[dashed] (-1.5,0) -- (2,2);
\path[fill=black!20!white] (-1.5,0) -- (0,0) -- (2,2) -- cycle;
\draw[->] (-3,0) -- (3,0) node[right] {$c$};
\draw[->] (0,-0.5) -- (0,2) node[right] {$\,d$};
\draw[very thick] (-3,0) -- (0,0) -- (2,2);
\draw (2.0,0.2) -- (2.0,-0.2) node[below] {$u$};
\draw (-1.5,0.2) -- (-1.5,-0.2) node[below] {$l$};
\end{tikzpicture}
\caption{The activation function of a $\mathit{ReLU}$ node, with a linear over-approximation drawn as filled area.}
\label{fig:ReLU}
\end{figure}
Obviously, this approach requires that we know upper and lower bounds on $c$. However, even though neural networks are defined as functions from $\mathbb{R}^n$, bounds on the input values are typically known. For example, in image processing networks, we know that the input neurons receive values from the range $[0,1]$. In other networks, it is common to \newterm{normalize} the input values before learning the network, i.e., to scale them to the same interval or to $[-1,1]$. This allows us to use classical \emph{interval arithmetic} on the network to obtain basic lower and upper bounds $[l,u]$ on every node's values.
For the case of $\mathit{MaxPool}$ nodes, we can approximate the behavior of the nodes linearly similarly to the ReLU case, except that we do not need upper bounds for the nodes' values. Let $c_1, \ldots, c_k$ be the values of nodes with edges leading to the $\mathit{MaxPool}$ node, $l_1, \ldots, k_k$ be their lower bounds, and $d$ be the output value of the node. We instantiate the following linear constraints:
\begin{equation*}
\bigwedge_{i \in \{1, \ldots, k\}} (d \geq c_i) \ \ \wedge \ \ (c_1 + \ldots + c_k \geq d + \sum_{i \in \{1,\ldots,k\}} l_i - \max_{i \in \{1,\ldots,k\}} l_i)
\end{equation*}
Note that these are the tightest linear constraints that can be given for the relationship between the values of the predecessor nodes of a $\mathit{MaxPool}$ node and the node value of the $\mathit{MaxPool}$ node itself.
After a linear program that approximates the behavior of the overall network has been built, we can use it to make all future approximations even tighter. To achieve this, we add the problem specification $\psi$ as constraints and solve, for every variable $v \in V$, the resulting linear program while minimizing first for the objective functions $1 \cdot v$, and then doing the same for the objective function $-1 \cdot v$. This yields new tighter lower and upper bounds $[l,u]$ for each node (if the network has any ReLU nodes), which can be used to obtain a tighter linear program.
Including the specification in the process allows us to derive tighter bounds than we would have found without the specification. The whole process can be repeated several times: whenever new upper and lower bounds have been obtained, they can be used to build a tighter linear network approximation, which in turn allows to obtain new tighter upper and lower bounds.
\subsection{Search process and Infeasible Subset Finding}
\label{subsec:basicSearchProcess}
Given a phase fixture for all ReLU and MaxPool nodes in a network, checking if there exists a node value assignment function with these phases (and such that the verification constraint $\psi$ is fulfilled) can be reduced to a linear programming problem. For this, we extend the linear program built with the approach from the previous subsection (with $V$ as the variable set for the node values) by the following constraints:
\begin{itemize}
\item For every $\leq 0$ phase selected for a ReLU node $v$, we add the constraints $v = 0$ and $\sum_{(v',v) \in E} W((v', \allowbreak{} v)) \cdot v' + B(v) \leq 0$.
\item For every $\geq 0$ phase selected for a ReLU node $v$, we add the constraint $v \geq \sum_{(v',v) \in E} \allowbreak{} W((v',v)) \cdot v' + B(v)$.
\item For every phase $(v',v)$ selected for a MaxPool node $v$, we add the constraint $v = v'$.
\end{itemize}
If we only have a partial node phase selection, we add these constraints only for the fixed nodes.
If the resulting linear program is infeasible, then we can discard all extensions to the partial valuation from consideration in the search process. This is done by adding a \newterm{conflict} clause that rules out the Boolean encoding of this partial node phase selection, so that even after restarts of the solver, the reason for infeasibility is retained.
However, the reasons for conflicts often involve relatively few nodes, so shorter conflict clauses can also be learned instead (which makes the search process more efficient). To achieve this, we employ \emph{elastic filtering} \cite{DBLP:journals/informs/ChinneckD91}. In this approach, all of the constraints added due to node phase selection are weakened by \emph{slack variables}, where there is one slack variable for each node. So, for example a constraint $\sum_{(v',v) \in E} W((v',v)) \cdot v' + B(v) \leq 0$ becomes $\sum_{(v',v) \in E} W((v',v)) \cdot v' + B(v) - s_v \leq 0$. When running the linear programming solver again with the task of minimizing a weighted sum of the slack variables, we get a ranking of the nodes by how much they contributed to the conflict, where some of them did not contribute at all (since their slack variable had a $0$ value). We then fix the slack variable with the highest value to be $0$, hence making the corresponding constraints strict, and repeat the search process until the resulting LP instance becomes infeasible. We then know that the node phase fixtures that were made strict during this process are together already infeasible, and build conflict clauses that only contain them. We observed that these conflict clauses are much shorter than without applying elastic filtering.
Satisfiability modulo theory solvers typically employ cheaper procedures to compute \newterm{minimal infeasible subsets} of linear constraints, such as the one by Duterte and de Moura~\cite{DBLP:conf/cav/DutertreM06}, but the high number of constraints in the linear approximation of the network behavior that are independent of node phase selections seems to make the approach less well-suited, as our experiments with the SMT solver \texttt{Yices} that uses this approach suggest.
\subsection{Implied Node Phase Inference during Partial Phase Fixture Checking}
\label{subsec:inferredNodeDetection}
In the partial node fixture feasibility checking step from Section~\ref{subsec:basicSearchProcess}, we employ a linear programming solver. However, except for the elastic filtering step, we did not employ an optimization function yet, as it was not needed for checking the feasibility of a partial node fixture.
For the common case that the partial node fixture \emph{is} feasible (in the linear approximation), we define an optimization function that allows us to infer additional infeasible \emph{and} feasible partial node fixtures when checking some other partial node fixture for feasibility.
The feasible fixtures are cached so that if it or a partial fixture of it is later evaluated, no linear programming has to be performed.
Given a partial node fixture to the nodes $V' \subset V$, we use $-1 \cdot \sum_{v \in V \setminus V', T(v) = \mathit{ReLU}} v - \frac{1}{10} \sum_{v \in V \setminus V', T(v) = \mathit{MaxPool}} v$ as optimization function. This choice asks the linear programming solver to minimize the error for the ReLU nodes, i.e, the difference between $a(v)$ and $\max(\sum_{v' \in V, (v',v) \in E} W((v',v)) \cdot a(v')+B(v),0)$ for every assignment $a$ computed in the linear approximation of the network behavior and every ReLU-node $v$. While this choice only minimizes an approximation of the error sum of the nodes and thus does not guarantee that the resulting variable valuation denotes a valid node value assignment function, it often yields assignments in which a substantial number of nodes $v$ \emph{have} a tight value, i.e., have $a(v) = \max(\sum_{v' \in V, (v',v) \in E} W((v',v)) \cdot a(v')+B(v),0)$.
If $\mathsf{tight}$ is the set of nodes with tight values, $p$ is the partial SAT solver variable valuation that encodes the phase fixtures for the nodes $V'$, and if $p'$ is the (partial) valuation of the SAT variables that encodes the phases of the tight nodes, we can then cache that $p \cup p'$ is a partial assignment that is feasible in the linear approximation. So when the SAT solver adds literals from $p'$ to the partial valuation, there is no need to let the linear programming solver run again.
At the same time, the valuation $a$ (in the linear approximation) can be used to derive an additional clause for the SAT solver. Let $\mathsf{unfixed}$ be the ReLU nodes whose values are not fixed by $p$. If for any node $v \in \mathsf{unfixed}$, we have $a(v)>0$, then we know by the choice of optimization function and the fact that we performed the analysis in a linear approximation of the network behavior, that some node in $v$ needs to be in the $\geq 0$ phase (under the partial valuation $p$). Thus, we can learn the additional clause $\left(\bigvee_{l \in p} \neg l\right) \vee \bigvee_{v \in V, T(v) = \mathit{ReLU}, ((v,\leq 0) \mapsto \mathbf{true}) \notin p} (v,$ $\geq 0)$ for the SAT solver, provided that the values of the MaxPool nodes are valid, i.e, for all MaxPool nodes $v$ we have $a(v)=a(v')$ for some $(v,v') \in E$. This last restriction is why we also included the MaxPool nodes in the optimization function above (but with lower weight).
\subsection{Detecting Implied Phases}
\label{subsec:detectingImpliedPhases}
Whenever the SAT solver has fixed a new node phase, the selected phases together may imply other node phases. Take for example the net excerpt from Figure~\ref{fig:newExcerpt}. There are two ReLU nodes, named $r_1$ and $r_2$, and one MaxPool node. Assume that during the initial analysis of the network (Section~\ref{subsec:linearApprox}), it has been determined that the value of node $r_1$ is between $0.0$ and $1.5$, and the value of node $r_2$ is between $0.1$ and $2.0$. First of all, the SAT solver can unconditionally detect that node $r_2$ is in the $\geq 0$ phase. Then, if at some point, the SAT solver decides that node $r_1$ should be in the $\leq 0$ phase, this fixes the value of $r_1$ to $0$. Since the flow out of $r_2$ has a lower bound $>0$, we can then deduce that $m$'s phase should be set to $(r_2,m)$.
\begin{figure}[bt]
\centering\begin{tikzpicture}
\node[draw,shape=circle,fill=black!20!white] (relu1) at (0,0) {$r_1$};
\node[draw,shape=circle,fill=black!20!white] (relu2) at (4,0) {$r_2$};
\node[draw,shape=circle,fill=black!20!white] (maxpool) at (2,-1) {$m$};
\draw[->,thick] (-1,0.5) -- (relu1);
\draw[->,thick] (-0,0.6) -- (relu1);
\draw[->,thick] (1,0.5) -- (relu1);
\draw[->,thick] (3,0.5) -- (relu2);
\draw[->,thick] (4,0.6) -- (relu2);
\draw[->,thick] (5,0.5) -- (relu2);
\draw[->,thick] (relu1) -- (maxpool);
\draw[->,thick] (relu2) -- (maxpool);
\draw[->,thick] (maxpool) -- +(0,-0.6);
\end{tikzpicture}
\caption{An example neural network part, used in Subsection~\ref{subsec:detectingImpliedPhases}.}
\label{fig:newExcerpt}
\end{figure}
Similar reasoning can also be performed for flow leading into a node. If we assume that the analysis of the initial linear approximation of the network's node functions yields that the outgoing flow of $m$ needs to be between $0.5$ and $0.7$, and the phase of $r_1$ is chosen to be $\leq 0$, then this implies that the phase of $r_2$ must be $\geq 0$, as otherwise $m$ would be unable to supply a flow of $>0$.
Both cases can be detected without analyzing the linear approximation of the network. Rather, we can just propagate the lower and upper bounds on the nodes' outgoing flows through the network and detect implied phases. Doing so takes time linear in the size of the network, which is considerably faster than making an LP solver call. This allows the detection of implied phases to be applied in a way similar to classical unit propagation in SAT solving: whenever a decision has been made by the solver, we run implied phase detection to extend the partial valuation of the SAT solver by implied choices (which allows to make the linear approximation tighter for the following partial node fixture feasibility checks).
\subsection{Overview of the Integrated Solver}
To conclude this section, let us discuss how the techniques presented in it are combined. Algorithm~\ref{algo:nnVerifier} shows the overall approach. In the first step, upper and lower bounds for all nodes' values are computed. The solver then prepares an empty partial valuation to the SAT variables and an empty list $\mathit{extra}$ in which additional clauses generated by the LP instance analysis steps proposed in this section are stored. The SAT instance is initialized with clauses that enforce that every ReLU node and every MaxPool node has exactly one phase selected (using a \newterm{one-hot encoding}).
In the main loop of the algorithm, the first step is to perform most steps of SAT solving, such as unit propagation, conflict detection \& analysis, and others. We assume that the partial valuation is always labelled by \newterm{decision levels} so that backtracking can also be performed whenever needed. Furthermore, additional clauses from $\mathit{extra}$ are mixed to the SAT instance $\psi$. This is done on a step-by-step basis, as the additional clauses may trigger unit propagation and even conflicts, which need to be dealt with eagerly. After all clauses from $\mathit{extra}$ have been mixed into $\psi$, and possibly the partial valuation $p$ has been extended by implied literals, in line \ref{line:infer}, the approach presented in Sect.~\ref{subsec:detectingImpliedPhases} is applied. If it returns new implied literals (in the form of additional clauses), they are taken care of by the SAT solving steps in line~\ref{line:satSolvingSteps} next. This is because the clauses already in $\psi$ may lead to unit propagation on the newly inferred literals, which makes sense to check as every additional literal makes the linear approximation of the network behavior tighter (and can lead to additional implied literals being detected). Only when all node phases have been inferred, $p$ is checked for feasibility in the linear approximation (line~\ref{line:check}).
There are two different outcomes of this check: if the LP instance is infeasible, a new conflict clause is generated, and hence the condition in line~\ref{ref:ifClauseLabel} is not satisfied. The algorithm then continues in line~\ref{line:satSolvingSteps} in this case. Otherwise, the branching step of the SAT solver is executed. If $p$ is already a complete valuation, we know at this point that the instance is \emph{satisfiable}, as then the $\mathrm{CheckForFeasibility}$ function just executed operated on an LP problem that is not approximate, but rather captures the precise behavior of the network. Otherwise, $p$ is extended by a decision to set a variable $b$ to $\mathbf{true}$ (for some variable chosen by the SAT solver's variable selection heuristics). Whenever this happens, we employ a plain SAT solver for checking if the partial valuation can be extended to one that satisfies $\psi$. This not being the case may not be detected by unit propagation in line~\ref{line:satSolvingSteps} and hence it makes sense to do an eager SAT check. In case of conflict, the choice of $b$'s value is inverted, and in any case, the algorithm continues with the search.
\algrenewcommand\algorithmicindent{2.3em}
\begin{algorithm}[tb]
\begin{algorithmic}[1]
\Function{VerifyNN}{$V,E,T,B,W$}
\State $(\overrightarrow{\mathit{min}},\overrightarrow{\mathit{max}}) \gets \mathrm{ComputeInitialBounds}(V,E,T,B,W)$ \Comment{Section~\ref{subsec:linearApprox}}
\State $(\overrightarrow{\mathit{min}},\overrightarrow{\mathit{max}}) \gets \mathrm{RefineBounds}(V,E,T,B,W,\overrightarrow{\mathit{min}},\overrightarrow{\mathit{max}})$ \Comment{Section~\ref{subsec:linearApprox}}
\State $p \gets \emptyset$, $\mathit{extra} \gets \emptyset$
\State $\psi \gets \bigwedge_{v \in V, T(v)=\mathit{MaxPool}} \bigvee_{v' \in V, (v',v) \in E} x_{v,(v',v)}$
\State $\psi \gets \psi \wedge \bigwedge_{v \in V, T(v)=\mathit{MaxPool}, v',v'' \in V, v' \neq v'', (v',v) \in E, (v'',v) \in E} (\neg x_{v,(v',v)} \vee \neg x_{v,(v'',v)})$
\State $\psi \gets \psi \wedge \bigwedge_{v \in V, T(v)=\mathit{ReLU}} (x_{v,\leq 0} \vee x_{v,\geq 0}) \wedge (\neg x_{v,\leq 0} \vee \neg x_{v,\geq 0})$
\While{$\psi$ has a satisfying assignment}
\While{$\mathit{extra}$ is non-empty}
\State Perform unit propagation, conflict detection, backtracking, and clause
\State learning for $p$ on $\psi$, while moving the clauses from $\mathit{extra}$ to $\psi$ one-by-one.\label{line:satSolvingSteps}
\EndWhile
\State $\mathit{extra} \gets \mathrm{InferNodePhases}(V,E,T,B,W,p,\overrightarrow{\mathit{min}},\overrightarrow{\mathit{max}})$ \Comment{Section~\ref{subsec:detectingImpliedPhases}} \label{line:infer}
\If{$\mathit{extra}=\emptyset$}
\State $\mathit{extra} \gets \mathrm{CheckForFeasibility}(V,E,T,B,W,p,\overrightarrow{\mathit{min}},\overrightarrow{\mathit{max}})$ \Comment{Section~\ref{subsec:basicSearchProcess}-\ref{subsec:inferredNodeDetection}} \label{line:check}
\If{$p \models c$ for all clauses $c \in \mathit{extra}$} \label{ref:ifClauseLabel}
\If{$p$ is a complete assignment to all variables}
\State \Return \textbf{Satisfiable}
\EndIf
\State Add a new variable assignment $b \!\mapsto \mathbf{true}$ to $p$ for some variable $b$ in $\psi$.
\If{$p$ cannot be extended to a satisfying valuation to $\psi$}
\State $p = p \setminus \{b \mapsto \mathbf{true} \} \cup \{b \mapsto \mathbf{false}\}$
\EndIf
\EndIf
\EndIf
\EndWhile
\State \Return \textbf{Unsatisfiable}
\EndFunction
\end{algorithmic}
\caption{Top-level view onto the neural network verification algorithm.}
\label{algo:nnVerifier}
\end{algorithm}
\section{Experiments}
\label{sec:experiments}
We implemented the approach presented in the preceding section in a tool called \texttt{Planet}. It is written in C++ and bases on the linear programming toolkit \texttt{GLPK} 4.61\footnote{GNU Linear Programming Kit, \url{http://www.gnu.org/software/glpk/glpk.html}} and the SAT solver \texttt{Minisat 2.2.0} \cite{DBLP:conf/sat/EenS03}. While we use \texttt{GLPK} as it is, we modified the main search procedure of \texttt{Minisat} to implement Algorithm~\ref{algo:nnVerifier}.
We repeat the initial approximation tightening process from Section~\ref{subsec:linearApprox} until the cumulative changes in $\overrightarrow{\mathit{min}}$ and $\overrightarrow{\mathit{max}}$ fall below $1.0$. We also abort the process if 5000 node approximation updates have been performed (to not spend too much time in the process for very large nets), provided that for every node, its bounds have been updated at least three times.
All numerical computations are performed with \texttt{double} precision, and we did not use any compensation for numerical imprecision in the code apart from using a fixed safety margin $\epsilon = 0.0001$ for detecting node assignment values $a(v)$ to be greater or smaller than other node assignment values $a(v')$, i.e., we actually check if $a(v) \leq a(v')-\epsilon$ to conclude $a(v) \leq a(v')$, whenever such a comparison is made in the verification algorithm steps described in Sect.~\ref{subsec:basicSearchProcess} and Sect.~\ref{subsec:inferredNodeDetection}. Since the neural networks learned using the \texttt{Caffe}~\cite{jia2014caffe} deep learning framework (which we employ for our experiments in this paper) tend not to degenerate in the node weights, this is sufficient for the experimental evaluation in this paper. Also, we did not observe any differences in the verification results between the SMT solver \texttt{Yices} \cite{DBLP:conf/cav/Dutertre14} on the SMT instances that we computed from the verification problems and the results computed by our tool.
The tool is available under the GPLv3 license and can be obtained from \texttt{https://github.com/progirep/planet} along with all scripts \& configuration files needed to learn the neural networks used in our experiments with the \texttt{Caffe} framework and to translate them to input files for our tool.
All computation times given in the following were obtained on a computer with an Intel Core i5-4200U 1.60\,GHz CPU and 8 GB of RAM running an x64 version of GNU/Linux. We do not report memory usage, as it was always $< 1$\,GB. All tools run with a single computation thread.
\subsection{Collision Avoidance}
As a first example, we consider the problem of predicting collisions between two vehicles that follow curved paths at different speeds. We learned a neural network that processes tuples $(x,y,s,d,c_1,c_2)$ and classifies them into whether they represent a colliding or non-colliding case. In such a tuple,
\begin{itemize}
\item the $x$ and $y$ components represent the relative distances of the vehicles in their workspace in the X- and Y-dimensions,
\item the speed of the second vehicle is $s$,
\item the starting direction of the second vehicle is $d$, and
\item the rotation speed values of the two vehicles are $c_1$ and $c_2$.
\end{itemize}
The data is given in normalized (scaled) form to the neural network learner, so that all tuple components are between $0$ and $1$ (or between $-1$ and $1$ for $c_1$ and $c_2$). We wrote a tool that generates a few random tuples (within some intervals of possible values) along with whether they represent a colliding or non-colliding case, as determined by simulation. The vehicles are circle-shaped, and we defined a safety margin and only consider tuples for which either the safety margins around the vehicles never overlap, or the vehicles themselves collide. So when only the safety margins overlap, this represents a ``don't care'' case for the learner. The tool also visualizes the cases, and we show two example traces in Figure~\ref{fig:collisions}. The tool ensures that the number of colliding cases and non-colliding ones are the same in the case list given to the neural network learner (by discarding tuples whenever needed). We generated 3000 tuples in total as input for \texttt{Planet}.
We defined a neural network architecture that consists of 40 linear nodes in the first layer, followed by a layer of MapPool nodes, each having 4 input edges, followed by a layer of 19 ReLU nodes, and 2 ReLU nodes for the output layer. Since \texttt{Caffe} employs randomization to initialize the node weights, the accuracy of the computed network is not constant. In 86 out of 100 tries, we were able to learn a network with an accuracy of 100\%, i.e., that classifies all example tuples correctly.
\mathchardef\mhyphen="2D
We want to find out the \emph{safety margin} around the tuples, i.e., the highest value of $\epsilon > 0$ such that for every tuple $(x,y,s,d,c_1,c_2)$ that is classified to $b \in \{\mathit{colliding},\mathit{notColliding}\}$, we have that all other tuples $(x \pm \epsilon,y \pm \epsilon,s \pm \epsilon, d\pm \epsilon, c_1 \pm \epsilon, c_2 \pm \epsilon)$ are classified to $b$ by the network as well. We perform this check for the first 100 tuples in the list, use \newterm{bisection search} to test this for $\epsilon \in [0,0.05]$, and abort the search process if $\epsilon$ has been determined with a precision of $0.002$.
We obtained 500 NN verification problem instances from this safety margin exploration process. Figure~\ref{fig:cactusPlotCollision} shows the distribution of the computation times of our tool on the problem instances, with a timeout of 1 hour. For comparison, we show the computation times of the SMT solver \texttt{Yices 2.5.2} and the (I)LP solver \texttt{Gurobi 7.02} on the problem instances. The SMT solver \texttt{z3} was observed to perform much worse than \texttt{Yices} on the verification problems, and is thus not shown. The choice of these comparison solvers was rooted in the fact that they performed best for verifying networks without MaxPool nodes in \cite{DBLP:journals/corr/KatzBDJK17}. We also give computation times for \texttt{Gurobi} and \texttt{Yices} after adding additional linear approximation constraints obtained with the approach in Section~\ref{subsec:linearApprox}. The computation times include the time to obtain them with our tool.
It can be observed that the computation times of \texttt{Gurobi} and \texttt{Yices} are too long for practical verification, except if the linear approximation constraints from our approach in this paper are added to the SMT and ILP instances to help the solvers. While \texttt{Yices} is then still slower than our approach, \texttt{Gurobi} actually becomes a bit faster in most cases, which is not surprising, given that it is a highly optimized commercial product that employs many sophisticated heuristics under-the-hood, whereas we use the less optimized \texttt{GLPK} linear programming framework. \texttt{Planet} spends most time on LP solving. It should be noted that the solver comparison is slightly skewed, as \texttt{Yices} employs arbitrary precision arithmetic whereas the other tools do not.
\begin{figure}[tb]
{\centering\includegraphics[width=0.46\columnwidth]{nocollision}
$\quad$
\includegraphics[width=0.46\columnwidth]{collision}
}
\caption{Two pairs of vehicle trajectories, where the first one is non-colliding, and the second one is colliding. The lower vehicle starts roughly in north direction, whereas the other one starts roughly in east direction. The first trajectory is non-colliding as the two vehicles pass through the trajectory intersection point at different times.}
\label{fig:collisions}
\end{figure}
\begin{figure}
\input{cactus}
\caption{Cactus plot of the solver time comparison for the 500 vehicle collision benchmarks. Time is given in seconds (on a $\log$-scale), and the lines, from bottom right to top left, represent \texttt{Gurobi} without linear approximation (dashed), \texttt{Yices} without linear approximation (solid), \texttt{Yices} with linear approximation (dotted), \texttt{Planet} (solid), and \texttt{Gurobi} with linear approximation (solid).}
\label{fig:cactusPlotCollision}
\end{figure}
\subsection{MNIST Digit Recognition}
\label{subsec:digitRecognition}
As a second case study, we consider handwritten digit recognition. This is a classical problem in machine learning, and the MNIST dataset \cite{mnistlecun} is the most commonly used benchmark for comparing different machine learning approaches. The \texttt{Caffe} framework comes with some example architectures, and we use a simplified version of Caffe's version of the \texttt{lenet} network \cite{726791} for our experiments. The \texttt{Caffe} version differs from the original network in that is has piecewise linear node activation functions.
Figure~\ref{fig:images} (a)-(b) shows some example digits from the MNIST dataset. All images are in gray-scale and have $28 \times 28$ pixels. Our simplified network uses the following layers:
\begin{itemize}
\item One input layer with $28 \times 28$ nodes,
\item one convolutional network layer with $3 \times 13 \times 13$ nodes, where every node has 16 incoming edges,
\item one pooling layer with $3 \times 4 \times 4$ nodes, where each node has 16 incoming edges,
\item one ReLU layer with 8 nodes, and
\item one ReLU output layer with 10 nodes
\end{itemize}
The ReLU layers are fully connected. Overall, the network has 1341 nodes, the search space for the node phases is of size $16^{3 \cdot 4 \cdot 4} \cdot 2^8 \cdot 2^{10} = 2^{162}$, and the network has
9344 edges.
We used this architecture to learn a network from the 100000 training images of the dataset, and the resulting network has an accuracy of 95.05\% on a separate testing dataset. Note that an accuracy of 100\% cannot be expected from any machine learning technique, as the dataset also contains digits that are even hardly identifiable for humans (as shown in Figure~\ref{fig:hardlyIdentifiable2}).
We performed a few tests with the resulting network. First we wanted to see an input image that is classified strongly as a $2$. More formally, we wanted to obtain an input image $(x_{1,1}, \ldots, x_{28,28})$ for which the network outputs a vector $(y_0, \ldots, y_9)$ for which $y_2 \geq y_i + \delta$ for all $i \in \{0,1,3,4,5,6,7,8,9\}$ for a large value of $\delta$. We found that for values of $\delta=20$ and $\delta=30$, such images can be found in 4 minutes 25 seconds and 32 minutes 35 seconds, respectively. The two images are shown in Figure~\ref{fig:20er} and Figure~\ref{fig:30er}. For $\delta = 50$, no such image can be found (4 minutes 41 seconds of computation time), but for $\delta = 35$, \texttt{Planet} times out after 4 hours. \texttt{Gurobi} (with the added linear approximation constraints) could not find a solution in this time frame, either.
Then, we are interested in how much noise can be added to images before they are not categorized correctly anymore. We start with the digit given in Figure~\ref{fig:modelThree}, which is correctly categorized by the learned network as digit 3. We ask whether there is another image that is categorized as a 4, but for which each pixel has values that are within an absolute range of $\pm 8 \%$ of color intensity of the original image's pixels, where we keep the pixels the same that are at most three pixels away from the boundaries. To determine that this is not the case, \texttt{planet} requires 1 minutes 46.8 seconds. For a range of $\pm 0.12$, \texttt{planet} times out after four hours. The output of \texttt{planet} shows that long conflict clauses are learned in the process, which suggests that we applied it to a difficult verification problem.
We then considered an error model that captures noise that is likely to occur in practice (e.g., due to stains on scanned paper). It excludes sharp noise edges such as the ones in Figure~\ref{fig:30er}. Instead of restricting the amplitude of noise, we restrict the noise value differences in adjacent pixels to be $\leq 0.05$ (i.e., 5\% of color density). This constraint essentially states that the noise must pass through a linearized low-pass filter unmodified. We still exclude the pixels from the image boundaries from being modified.
Our tool concludes in 9 minutes 2.4 seconds that the network never misclassifies the image from Figure~\ref{fig:modelThree} as a $4$ under this noise model. Since the model allows many pixels to have large deviations, we can see that including a linear noise model can improve the computation time of \texttt{planet}.
\begin{figure}[tb]
\begin{center}
\subfigure[`3' digit from the MNIST dataset]{\input{digit1} \label{fig:modelThree}}
$\ \ $
\subfigure[`2' digit from the MNIST dataset]{\input{digit2} \label{fig:hardlyIdentifiable2}} %
$\ \ $
\subfigure[Image classified as digit 2 with $\delta=20$]{\input{20er}\label{fig:20er}} %
$\ \ $
\subfigure[Image classified as digit 2 with $\delta=30$]{\input{30er}\label{fig:30er}}
\end{center}
\caption{Example digit images from Section~\ref{subsec:digitRecognition}}
\label{fig:images}
\end{figure}
\section{Conclusion}
In this paper, we presented a new approach for the verification of feed-forward neural networks with piece-wise linear activation functions. Our main idea was to generate a linear approximation of the overall network behavior that can be added to SMT or ILP instances which encode neural network verification problems, and to use the approximation in a specialized approach that features multiple additional techniques geared towards neural network verification, which are grouped around a SAT solver for choosing the node phases in the network. We considered two case studies from different application domains. The approach allows arbitrary convex verification conditions, and we used them to define a noise model for testing the robustness of a network for recognizing handwritten digits.
We made the approach presented in this paper available as open-source software in the hope that it fosters the co-development of neural network verification tools and neural network architectures that are easier to verify. While our approach is limited to network types in which all components have piece-wise linear activation functions, they are often the only ones used in modern network architectures anyway. But even if more advanced activation functions such as \newterm{exponential linear units} \cite{DBLP:journals/corr/ClevertUH15} shall be used in learning, they can still be applied to learn an initial model, which is then linearly approximated with ReLU nodes and fine-tuned by an additional learning process. The final model is then easier to verify. Such a modification of the network architecture during the learning process is not commonly applied in the artificial intelligence community yet, but while verification becomes more practical, this may change in the future.
Despite the improvement in neural network verification performance reported in this paper, there is still a lot to be done on the verification side: we currently do not employ specialized heuristics for node phase branching selection, and while our approach increases the scalablity of neural network verification substantially, we observed it to still be quite fragile and prone to timeouts for difficult verification properties (as we saw in the MNIST example).
Also, we had to simplify the LeNet architecture for digit recognition in our experiments, as the original net is so large that even obtaining a lower bound for a single variable in the network (which we do for all network nodes before starting the actual solution process as explained in Section~\ref{subsec:linearApprox}) takes more than 30 minutes otherwise, even though this only means solving a single linear program. While the approach by Huang et al.~\cite{DBLP:journals/corr/HuangKWW16} does not suffer from this limitation, it cannot handle general verification properties, which we believe to be important. We plan to work on tackling the network size limitation of the approach presented in this paper in the future.
\section*{Acknowledgements}
This work was partially funded by the Institutional Strategy of the University of Bremen, funded by the German Excellence Initiative.
\bibliographystyle{alpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,703 |
\section{Introduction}
The measurement of \JPsi meson pairs that are directly created in the primary
interaction (prompt) in proton-proton (pp) collisions at $\sqrt{s}=7$\TeV
provides general insight into how particles are produced during proton
collisions at the CERN LHC.
Owing to the high flux of incoming partons at the LHC energy,
it is expected that more than one parton pair will often scatter in
a pp collision~\cite{Kom:2011bd}. These multiparton scattering contributions
are difficult to address within the framework of perturbative quantum
chromodynamics (QCD), hence the need for experimental studies (see \eg,
Ref.~\cite{Ko:2010xy} and references therein). The general assumption is
that single-parton scattering (SPS) is the dominant process. Double-parton
scattering (DPS) and higher-order multiple-parton interactions are widely
invoked to account for observations that cannot be explained otherwise, such
as the rates for multiple heavy-flavor production~\cite{Berger:2009cm}.
New measurements will help the creation of
more realistic particle production models. The production of
\JPsi meson pairs provides a clean signal in a parton-parton interaction
regime that is complementary to the one probed by studies based on hadronic
jets. Multiple-parton interactions can lead to distinct differences in
event variables that probe pair-wise balancing, such as the absolute
rapidity difference $\abs{\Delta y}$ between the two \JPsi mesons~\cite{Baranov:2011ch,Baranov:2013,Kom:2011bd}. The strong correlation
of two \JPsi mesons produced via SPS interaction results in small values of
$\abs{\Delta y}$, whereas large values of $\abs{\Delta y}$ are possible for
production due to DPS.
In contrast to earlier experiments where quark-antiquark annihilation
dominated~\cite{Badier:1982ae,Badier:1985ri}, the dominant \JPsi production
process in pp collisions at the LHC is gluon-gluon
fusion~\cite{Humpert:1983yj}. At the parton level, the two \JPsi mesons
are either produced as color-singlet states or color-octet states that turn
into singlets after emitting gluons. Color-octet contributions for \JPsi pair production at transverse momentum (\pt) of a pair below 15\GeVc and low
invariant mass are considered to be negligible, but play a greater role as
\pt increases~\cite{Berezhnoy:2011xy,Qiao:2009kg}. Next-to-leading-order
QCD calculations also indicate enhanced contributions from color-singlet
heavy-quark pair production at higher \pt \cite{Campbell:2007ws,Artoisenet:2007xi,Gong:2008hk,PRL.111.122001}.
The CMS experiment provides access to \pt measurements above 15\GeVc.
Recently, the LHCb experiment measured the cross section for \JPsi pair
production in pp collisions at $\sqrt{s}=7$\TeV
to be $5.1 \pm 1.0 \pm 1.1\unit{nb}$ (where the first uncertainty is
statistical and the second systematic) within the LHCb phase space (defined
as $2 <y^{\JPsi}<4.5$ and $\pt^{\JPsi}<10\GeVc$)~\cite{Aaij:2012dz}.
Theoretical calculations of \JPsi pair production via SPS based on
leading-order color-singlet states predict a cross section of 4\unit{nb},
with an uncertainty of about 30\%~\cite{Berezhnoy:2011xy,Berezhnoy:2012xq}.
This prediction is consistent with
the measured value. The CMS experiment samples a \JPsi production regime
complementary to LHCb, with coverage at higher \pt and more central rapidity.
Hence, \JPsi pair production cross section measurements by CMS provide
new information for the development of production models that include
higher-order corrections and DPS.
Model descriptions of \JPsi pair production are also a crucial input
to quantify nonresonant contributions in the search for resonances.
States can be searched for
with CMS in a wider \JPsi pair invariant-mass range as
compared to previous experiments. For example, the bottomonium ground state
$\eta_b$ is expected to decay into two \JPsi mesons in analogy to the
$\eta_c$ charmonium ground state that decays into two $\phi$
mesons~\cite{Maltoni:2004hv}. However, explicit calculations based on
nonrelativistic QCD (NRQCD)~\cite{Braaten:2000cm,Jia:2006rx} predict
this decay mode to be highly suppressed, so any observation of this
process could indicate possible shortcomings of present NRQCD approaches.
Other predicted resonant states that could decay into two \JPsi mesons
are exotic tetraquark charm states~\cite{Berezhnoy:2011xy}. A CP-odd
Higgs boson, \eg, in the next-to-minimal supersymmetric standard
model~\cite{Dermisek:2005ar}, is predicted with a mass near
the $\eta_b$. Mixing with a CP-odd Higgs boson could alter
the behavior of the $\eta_b$ with respect to QCD
predictions~\cite{Domingo:2009tb,Domingo:2010am}. The BaBar experiment first
observed the $\eta_b$ state in radiative $\Upsilon$
transitions~\cite{Aubert:2009as} and published an upper limit on the
effective coupling of a CP-odd Higgs boson with mass below
9.3\GeVcc to b quarks~\cite{babarhiggs}.
No evidence for a CP-odd Higgs boson was
found by CMS in the $\mu^+\mu^-$ invariant-mass spectrum for masses
between 5.5 and 14\GeVcc~\cite{Chatrchyan:2012am}.
This Letter presents a measurement of the cross section for prompt
\JPsi pair production with data recorded with the CMS experiment
in pp collisions at a center-of-mass energy of 7\TeV.
Acceptance corrections are calculated based on the measured
\JPsi meson kinematics, and efficiency corrections are calculated based
on the measured decay-muon kinematics of each event thereby minimizing the
dependence on production models. Monte Carlo (MC) simulation
samples for different production models with either strongly correlated
\JPsi mesons (SPS model) or less correlated \JPsi mesons (DPS model)
are only used to define the phase-space region and validate the
correction method. They also provide guidance for the parameterization
of various kinematic distributions in the events. The SPS generator is a
color-singlet model~\cite{Berezhnoy:2011xy} implemented in
{\PYTHIA6}~\cite{Sjostrand:2006za}, and the DPS generator is implemented
in {\PYTHIA8}~\cite{Sjostrand:2007gs} using color-singlet and -octet
production models.
The cross section measurement is evaluated in a predefined region of the
\JPsi phase space that, in turn, is constrained by the muon
identification and
reconstruction capabilities of CMS. The differential cross section of
\JPsi pair production is calculated as
\begin{linenomath}
\begin{equation}
\frac{\rd\sigma ({\Pp\Pp}\to \JPsi\,\JPsi+X)}{\rd{}x} =\\ \sum_i
\frac{s_i} { a_i \cdot \epsilon_i
\cdot (BF)^2\cdot \Delta x
\cdot \mathcal{L}}.
\end{equation}
\label{eq:cross-section}
\end{linenomath}
The sum is performed over events $i$ in an interval $\Delta x$, where $x$
represents a kinematic variable describing the \JPsi pair.
In this analysis, $x$ is taken as the invariant mass of the \JPsi pair
($M_{\JPsi\,\JPsi}$), the absolute difference in \JPsi meson rapidities
($\abs{\Delta y}$), and the transverse momentum of the \JPsi pair
($\pt^{\JPsi\,\JPsi}$). The quantity $s_i$ is the signal weight per event. The
acceptance value $a_i$ calculated for each event represents the
probability that the muons resulting from the \JPsi decays pass the muon
acceptance. The detection efficiency $\epsilon_i$ is the
probability for the four muons in an event to be detected and pass the
trigger and
reconstruction quality requirements. The integrated luminosity of
the dataset is $\mathcal{L}$, and $BF$ is the branching fraction for the \JPsi decay into two muons.
The total cross section in the \JPsi phase-space window is determined by
summing over all events.
\section{CMS detector}
\label{sec:cms_detector}
A detailed description of the CMS detector can be found
elsewhere~\cite{Chatrchyan:2008zzk}. The primary components used in
this analysis are the silicon tracker and the muon systems.
The tracker operates in a 3.8\unit{T} axial magnetic field generated by a
superconducting solenoid with an internal diameter of 6\unit{m}. The
innermost part of the tracker consists of three cylindrical layers of pixel
detectors complemented by two disks in the forward and backward directions.
The radial region between 20 and 116\cm is occupied by several layers
of silicon strip detectors in barrel and disk configurations. Multiple
overlapping layers ensure a sufficient number of hits to precisely
reconstruct tracks in the pseudorapidity range $\abs{\eta} < 2.4$, where
$\eta = -\ln{[\tan{(\theta/2)}]}$\ and $\theta$\ is the polar angle of
the track measured from the positive $z$ axis. The coordinate
system is defined to have its origin at the center of the detector,
the $x$ axis pointing to the center of the LHC ring, the $y$ axis pointing
up (perpendicular to the plane of the LHC ring), and the $z$ axis aligned with
the counterclockwise-beam direction. An impact parameter resolution around
15\mum and a \pt resolution around 1.5\% are achieved for charged
particles with \pt up to 100\GeVc. Muons are identified in the
range $\abs{\eta}< 2.4$, with detection planes made of drift tubes, cathode
strip chambers, and resistive-plate chambers embedded in the steel
flux-return yoke of the solenoid. The CMS detector response is
determined with MC simulations using \GEANTfour~\cite{Agostinelli2003250}.
\section{Event selection and efficiencies}
\label{sec:reco}
This analysis uses an unprescaled muon trigger path designed to achieve
the highest possible signal-to-noise ratio and efficiency for \JPsi pair
searches during the 2011 data taking.
This trigger requires the presence of at least three
muons, two of which must be oppositely charged, have a dimuon invariant
mass in the interval between 2.8 and 3.35\GeVcc, and a vertex fit
probability greater than 0.5\%, as determined by a Kalman filter
algorithm~\cite{Fruhwirth:1987fm}. Reconstruction of muons proceeds by
associating measurements in the muon detectors with tracks found in the
silicon tracker, both called segments. A given muon segment can be
associated with more than one silicon track at the time of reconstruction,
allowing reconstructed muons to share segments in the muon system.
An arbitration algorithm then assigns each muon segment to a unique
muon track. Muons are further required to pass the following
quality criteria: (i) the associated track segment must have hits in at least
two layers of the pixel tracker and at least 11 total silicon tracker hits
(pixel and strip detectors combined), and (ii) the silicon track fit $\chi^2$ divided by the number of degrees of freedom must be less than 1.8. Three of the muons are required to fulfill the criteria
\begin{linenomath}
\begin{equation}
\begin{aligned}
\label{eq:tpselect}
&\pt^\mu > 3.5\GeVc &\qquad\text{if}\quad && \abs{\eta^\mu}<1.2,\\
&\pt^\mu > 3.5 \rightarrow 2\GeVc &\qquad\text{if}\quad && 1.2<\abs{\eta^\mu}<1.6, \\
&\pt^\mu > 2\GeVc &\qquad\text{if}\quad && 1.6<\abs{\eta^\mu}<2.4,
\end{aligned}
\end{equation}
\end{linenomath}
where the $\pt$\ threshold scales linearly downward with
$\abs{\eta^\mu}$ in the range
$1.2<\abs{\eta^{\mu}}<1.6$. They must further be matched to the muon
candidates that triggered the event. The fourth muon (not required to match to
the trigger muon candidates) is allowed to pass the looser acceptance criteria
\begin{linenomath}
\begin{equation}
\begin{aligned}
\label{eq:looseselect}
&\pt^\mu > 3\GeVc &\qquad\text{if}\quad &&\abs{\eta^\mu}<1.2,\\
& p^\mu > 3\GeVc &\qquad\text{if}\quad && 1.2<\abs{\eta^\mu}<2.4,
\end{aligned}
\end{equation}
\end{linenomath}
where $p^\mu$ is the magnitude of the total muon momentum.
Candidate events must have two pairs of opposite-sign muons each with an invariant mass close to the \JPsi mass~\cite{PDG2012}.
Each \JPsi candidate is further required to be within the phase space
\begin{linenomath}
\begin{equation}
\begin{aligned}
\label{eq:psiselect}
&\pt^{\JPsi} > 6.5\GeVc &\qquad\text{if}\quad &&\abs{y^{\JPsi}}<1.2,\\
&\pt^{\JPsi} > 6.5 \rightarrow 4.5\GeVc &\qquad\text{if}\quad && 1.2<\abs{y^{\JPsi}}<1.43, \\
&\pt^{\JPsi} > 4.5\GeVc &\qquad\text{if}\quad && 1.43<\abs{y^{\JPsi}}<2.2,
\end{aligned}
\end{equation}
\end{linenomath}
where the $\pt^{\JPsi}$\ threshold scales linearly with $\abs{y^{\JPsi}}$
in the range $1.2<\abs{y^{\JPsi}}<1.43$. The boundaries are optimized to obtain maximum coverage of the \JPsi phase space within the muon acceptance. If there are more than two \JPsi candidates in an event,
the candidates with the highest vertex fit probabilities are
selected. For signal MC simulation samples in which multiple
collision events per bunch crossing (pileup events) are included,
this selection process finds the correct dimuon combinations for
99.7\% of the selected events.
In addition to the invariant mass of each dimuon candidate, $m^{\JPsi}$,
two event variables sensitive to the prompt \JPsi pair topology are defined:
(i) the proper transverse decay length, $ct_{xy}$, of
the higher-$\pt$ \JPsi, and (ii) the separation significance, $\delta d$,
between the \JPsi mesons. Calculating the proper transverse decay
length requires identification of the primary vertex in an event,
defined as the vertex formed by charged-particle tracks with the
highest sum of \pt squared that can be fit to a common
position, excluding the muon tracks from the two
\JPsi candidates. The transverse decay length in the laboratory frame is
given as $L_{xy} = (\vec{r}_\mathrm{T} \cdot \ptvec^{\JPsi})/\pt^{\JPsi}$,
where $\vec{r}_\mathrm{T}$ is the vector pointing from the primary vertex
to the \JPsi vertex in the transverse plane.
The proper transverse decay length is then
calculated as $ct_{xy} = (m^{\JPsi}/\pt^{\JPsi}) \cdot L_{xy}$
and is required to be in the range from $-$0.05 to 0.1\cm.
The separation significance is defined as the ratio of the magnitude
of the three-dimensional vector $\Delta \vec{r}$ between the two reconstructed
\JPsi vertices
and the uncertainty of the distance measurement, $\sigma_{\Delta \vec{r}}$
(which includes the uncertainty in the vertex position, as determined by the
Kalman filter technique, and the uncertainty of the muon track fit):
$\delta d \equiv \abs{\Delta \vec{r}} / \sigma_{\Delta \vec{r}}$. The requirement
$\delta d < 8$ is imposed. From a data sample of pp collisions
corresponding to an integrated luminosity of
4.73\fbinv \cite{CMS-PAS-SMP-12-008}, 1043 candidate events
containing a \JPsi pair are found.
The kinematics of the $\JPsi\,\JPsi\to 4\mu$ final state is sensitive to
the underlying physics of production and decay, and this analysis probes
a higher-$\pt$ region of \JPsi pair production than previous experiments.
Therefore, the dependence on production model assumptions is minimized.
Given the relatively small number of events in the final-analysis
event sample it was affordable to calculate acceptance and efficiency
corrections on an event-by-event basis using the measured \JPsi and muon momenta. The procedure has the merit of not depending on
assumptions regarding correlations between production observables.
The muon acceptance is evaluated by generating a large number
of simulated decays starting from the reconstructed four momenta
of the two \JPsi mesons in an event.
The acceptance correction, $a_i$, for a given event $i$ is the
number of times all four muons survive the acceptance criteria,
listed in Eqs.~(\ref{eq:tpselect}) and (\ref{eq:looseselect}),
divided by the total number of trials for the event.
The angle of the decay muons with respect to the direction of flight
of the parent \JPsi, in the \JPsi rest frame, is assumed to be
isotropically distributed. Deviations from this assumption are
considered and discussed later.
The event-by-event acceptance-correction procedure is
evaluated with both SPS and DPS MC simulation samples.
For each sample of $N$ events within the \JPsi phase space,
the muon acceptance criteria are applied to obtain a sample of
accepted events. For each of the surviving events $i$,
the corresponding $a_i$ is obtained as described above. The corrected
number of signal events within the \JPsi phase space, $N^\prime$,
is then calculated as a sum over the survivors, $N^\prime = \sum_i 1/a_i$.
The difference between $N$ and $N^\prime$ is used to estimate
the systematic uncertainty in the method.
The efficiency correction is also determined on a per-event basis
by repeatedly generating \JPsi pair events where the
generated muon momenta are the measured muon momenta from the
reconstructed event. The event is then subjected to the complete
CMS detector simulation and reconstruction chain.
The efficiency correction, $\epsilon_i$, for a measured event $i$ is the
fraction of simulated events that pass the trigger and reconstruction
requirements.
The number of efficiency-corrected events is then given as
$\sum_i 1/\epsilon_i$, summed over the events that survive the
trigger and reconstruction requirements. An average
efficiency for the sample in bins of the observables, $\Delta x$, is
obtained as the number of events that survive the trigger and reconstruction
requirements, divided by the number of efficiency-corrected events.
The method is evaluated with samples of reconstructed SPS
and DPS \JPsi pair MC simulation events.
For comparison, the average efficiency is alternatively determined from the SPS and DPS
MC simulation samples with simulated muon momenta. The average efficiency is
then given as the number of events surviving the trigger and reconstruction
criteria, divided by the number of events originally generated in the \JPsi phase space and muon acceptance region. In contrast to the first method,
this efficiency calculation is based on true muon momenta. The difference
between these two average efficiencies is due to the resolution of the
detector and is accounted for by a scaling factor which is in close agreement
between the two production models.
\section{Signal yield}
\label{sec:fit_technique}
An extended maximum likelihood method is performed to separate the
signal from background contributions in the data sample. The signal
weights $s_i$ in Eq.~(\ref{eq:cross-section}) are derived with the
sPlot technique~\cite{sPlot}. The signal yield resulting from the
fit is equal to the sum of the $s_i$. These weights are used to obtain
the signal distribution in bins of kinematic variables that quantify
the \JPsi pair production. Correlations between fit variables and
production observables are found to be negligible from simulated samples.
Four kinematic variables are selected to discriminate the \JPsi pair signal
from the background: (i) the $\mu^+\mu^-$ invariant mass
of the higher-$\pt$ $\JPsi$, $M^{(1)}_{\mu\mu}$, (ii) the $\mu^+\mu^-$ invariant
mass of the lower-$\pt$ $\JPsi$, $M^{(2)}_{\mu\mu}$, (iii) the proper transverse decay
length of the higher-$\pt$ $\JPsi$, $ct_{xy}$, and (iv) the
separation significance, $\delta d$, between the two \JPsi candidates.
Five categories of events are identified:
\begin{enumerate}
\item events containing a real prompt \JPsi pair (sig),
\item background from at least one nonprompt
\JPsi meson, mostly from B-meson decays (nonprompt),
\item the higher-$\pt$ prompt \JPsi and two unassociated muons
that have an invariant mass within the \JPsi mass window,
\item the lower-$\pt$ prompt \JPsi and two unassociated muons
that have an invariant mass within the \JPsi mass window, and
\item four unassociated muons (combinatorial-combinatorial).
\end{enumerate}
The categories 3 and 4 have a common yield (\JPsi-combinatorial), and
the parameter $f$ is defined as their relative fraction.
The likelihood function for event $j$ is obtained
by summing the product of the yields $n_{i}$ and the probability density
functions (PDFs) for the four kinematic variables $P_i(M^{(1)}_{\mu\mu})$,
$Q_i(M^{(2)}_{\mu\mu})$,
$R_i(ct_{xy})$, $S_i(\delta d)$ with the shape parameters for each of
the five event categories $i$.
The likelihood for each event $j$ is given as:
\begin{linenomath}
\begin{equation}
\begin{aligned}
{\ell}_j & = n_\text{sig}\left[ P_1 \cdot Q_1 \cdot R_1 \cdot S_1 \right]
+ n_\text{nonprompt}\left[ P_2\cdot Q_2\cdot R_2\cdot S_2 \right] \\
& + n_{\JPsi\text{-combinatorial}}
[
f \cdot P_3 \cdot Q_3 \cdot R_3 \cdot S_3 + (1-f) \cdot P_4\cdot Q_4 \cdot R_4 \cdot S_4 ] \\
& + n_\text{combinatorial-combinatorial}\left[ P_5 \cdot Q_5 \cdot R_5\cdot S_5 \right].
\end{aligned}
\end{equation}
\end{linenomath}
The yields $n_{i}$ are determined by minimizing the quantity
$-\ln{\mathcal{L}}$~\cite{roofit}, where $\mathcal{L} = \prod_j {\ell}_j$.
According to the signal MC simulation, the invariant mass and $ct_{xy}$ of the
higher-\pt \JPsi are correlated by about 13\%. All other correlations between event
variables are below 5\%. Therefore, the parameterization for each variable is
independently determined. Several parameterizations for each distribution
are considered, and the simplest function with the least number of parameters
necessary to adequately describe the observed distribution is selected as
the PDF. For parameterizations that result in equally good descriptions of
the data (as measured by the $\chi^2$ of the fit of the distribution in data
for a given variable), the difference in signal yields is used as a measure
of the systematic uncertainty.
For the likelihood fit, the sum of two Gaussian functions with a common mean is
used to parameterize the signal \JPsi invariant mass PDFs $P_1$ and $Q_1$; the
same parameters are used to describe the nonprompt components $P_2$ and $Q_2$,
and the \JPsi part of the \JPsi-combinatorial cases $P_3$ and $Q_4$. The
widths of the Gaussian functions are fixed to the best-fit values obtained
in simulation samples. A sum of two Gaussians is also used
to describe the signal $ct_{xy}$ PDF $R_1$. The nonprompt background
distribution $R_2$ is fit by an exponential function convolved with a single
Gaussian. The separation significance PDFs for the signal and nonprompt
components, $S_1$ and $S_2$, are parameterized with a single Gaussian convolved with
an exponential function. Simulated event samples are used to parameterize
the prompt and nonprompt $ct_{xy}$ and $\delta d$ distributions.
The distributions of the signal variables as predicted by the simulation of
SPS production agree with those from DPS production.
Combinatorial background shapes are obtained directly from data.
Two $M_{\mu\mu}$ sideband regions are defined in the ranges
$[2.85,3]$ and $[3.2,3.35]\GeVcc$, adjacent to the signal region
defined as $[3,3.2]\GeVcc$, and PDF parameters are estimated
from fits to combinations of samples in data where only one or neither of
the \JPsi candidates originate from the signal region. The mass distributions
are parameterized under the assumption that they only contain contributions
from true \JPsi candidates and combinatorial background. Third-order Chebyshev
polynomial functions are used to describe the combinatorial components of each
invariant mass PDF $Q_3$ and $P_4$ in the partially combinatorial and completely
combinatorial category. In the latter case, it is required that $P_5$ equals
$P_4$ and $Q_5$ equals $Q_3$. A sum of two Gaussians is used
for $R_{3-5}$, and a Landau function plus a first-order
Chebyshev polynomial is used to parameterize $S_{3-5}$.
The final fit is performed on the full data sample. The mean values of the
central Gaussian functions of the two $\mu^+\mu^-$ invariant-mass distributions
are left free, as is the proper decay time of the nonprompt component.
The fit yields $n_\text{sig} = 446 \pm 23$ signal events.
Figure~\ref{fig:fit} shows the distributions of the event variables from
data with the fit result superimposed. The fit is validated by repeatedly
generating simulated samples from the PDFs for all
components and no bias is found. Furthermore, the robustness of the fit is
probed by adding combinations of simulated signal and background events
to the data set. To ensure that the cross section determination is insensitive
to changing conditions, the distributions of the variables used in the
likelihood fit are compared in subsets of events. Event variable
distributions from events containing six reconstructed primary vertices or
fewer agree with distributions in events containing more than six primary
vertices (within statistical uncertainties). The behavior is confirmed with
MC simulation signal samples generated with and without pileup contributions.
The variable distributions also agree between the two major 2011 data-taking periods.
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.49\textwidth]{figures/Psi1_mass_Fit_EffCut}
\label{fig:fit1}
\includegraphics[width=0.49\textwidth]{figures/Psi2_mass_Fit_EffCut}
\label{fig:fit2}
\includegraphics[width=0.49\textwidth]{figures/Psi1_CTxy_Fit_EffCut}
\label{fig:fit3}
\includegraphics[width=0.49\textwidth]{figures/Psi1To2Significance_Fit_EffCut}
\label{fig:fit4}
\caption{Distributions of $M_{\mu\mu}^{(1)}$~(top left),
$M_{\mu\mu}^{(2)}$~(top right), $ct_{xy}$~(bottom left), and distance
significance $\delta d$ (bottom right) for the candidate events and the
projections of the fit results.
The data are shown as points with the vertical error bars representing the
statistical uncertainty. The fit result to the full sample is shown as a
solid line. Individual contributions from
the various categories are shown in different line styles: signal (short dashes),
nonprompt background (long dashes), $\JPsi$-combinatorial components
(dots), and the pure combinatorial component (dashes and dots).
}
\label{fig:fit}
\end{figure*}
\section{Systematic uncertainties}
\label{sec:syst}
The uncertainty in the \JPsi dimuon {branching fraction} is taken from
the world average~\cite{PDG2012} (2\% when added
linearly). The systematic uncertainty corresponding to the integrated
{luminosity} normalization is estimated in previous studies (2.2\%)~\cite{CMS-PAS-SMP-12-008}. Simulated event samples based on SPS and DPS production models are
used to estimate the uncertainty in the event-by-event {acceptance
correction} method: $N$ simulated events are subjected to the acceptance
criteria, and the event-based acceptance correction is applied to arrive
at a corrected yield, $N^\prime$. The uncertainty is taken as half of the
relative difference between the two yields, $N$ and $N^\prime$. The larger
value among the SPS- and DPS-based samples is quoted (1.1\%). The precision
of the event-based {efficiency correction} is limited by the number of
reconstructed events, $n_{\text{reco},i}$, found after the substitution
process for each event $i$. The cross section is recalculated by
repeatedly varying $n_{\text{reco},i}$ according to Gaussian functions with
standard deviation $\sqrt{\smash[b]{n_{\text{reco},i}}}$. The standard deviation of the
resulting cross section distribution is used as an estimate of the
uncertainty in the efficiency calculation (4.4\%). The relative
{efficiency scaling factor} is determined from SPS and DPS MC
simulation samples, representing very different
scenarios of \JPsi pair kinematics. The uncertainty due to model
dependence of the scaling factor is defined as the difference in the
cross section between either model and the average of the two (0.2\%).
The small uncertainty demonstrates that there is little
overall model dependence.
The {muon track reconstruction} efficiency is derived from simulated
events. The uncertainty is estimated from data and simulation samples
that contain at least one reconstructed \JPsi. For each muon in an event, the
tracking efficiency in data and simulation is obtained as a function
of the measured muon pseudorapidity~\cite{CMS-PAS-TRK-10-002}.
The relative uncertainty is defined as the absolute difference between the
data- and simulation-based values divided by the data-based value. Individual
muon uncertainties are added linearly per event (3.0\%) since correlations
between the muons are not taken into account.
The efficiency to trigger and reconstruct \JPsi pair
events relies on {detector simulation}. To evaluate the uncertainty
event-based efficiency values are instead constructed from single-muon
efficiencies. The single-muon efficiencies are determined by applying a
``tag-and-probe'' method~\cite{MUO-10-004} to control samples in data and
simulation that contain single \JPsi decays to muons. Hence, correlations among the two \JPsi mesons in the event are neglected.
The difference in the signal yield in data when corrected with efficiencies found
from either data or simulation is used to measure the uncertainty.
The event-based efficiency correction is defined as the
product of the event's trigger efficiency, given that all muons are found
offline, and the event efficiency for reconstructing, identifying, and
selecting offline all four muons in an event. The trigger efficiency is
calculated from the single-muon trigger efficiencies and the dimuon
vertexing efficiency as the trigger requires at least three reconstruced
muons, two of which must be fit to a \JPsi vertex.
The offline reconstruction efficiency for a single muon is given as the
product of the tracking efficiency, muon identification efficiency, and the
efficiency to pass the offline quality criteria. All muon efficiencies are
obtained as a function of muon $\pt$\ and $\eta$\ from previous
studies~\cite{MUO-10-004}. The probability to successfully fit both
vertices in an event is greater than 99.9\% for SPS and 99.6\% for
DPS simulation samples. Therefore, the offline event reconstruction
efficiency is considered to be entirely a product of the muon reconstruction
efficiencies. The largest deviation of the corrected signal yield using
the single-muon efficiency values from data control samples compared
to simulation is chosen as a conservative measure of the uncertainty
(6.5\%).
All {PDF parameters} that are fixed for the maximum likelihood
fit are varied by their uncertainty, as determined from the fits to the data
sidebands and MC simulation samples. The prompt $ct_{xy}$ distribution
is also parameterized using a sum of three Gaussians.
Alternative fit shapes such as third-order polynomials or exponential
functions are used for the background models.
A Crystal Ball function~\cite{ref:crystalball} is considered as an
alternative to
the parameterization of the \JPsi invariant-mass distribution. A resolution
function convolved with an exponential function is considered for
the separation significance of the combinatorial background components. The
largest difference in signal yields between fits with different shape
parameterizations is taken as the uncertainty from the PDFs (0.6\%).
To evaluate the dependence of the PDF
parameterization on the {production model}, both reconstructed DPS and SPS
samples are used. The difference in signal yields between fits with those two
PDF sets is considered as an uncertainty (0.1\%).
The total systematic uncertainty is calculated as the sum in quadrature of
the individual uncertainties (9.0\%). The individual relative uncertainties
for the total cross section are listed in Table~\ref{tab:systematics}.
The systematic uncertainty for each differential cross section is also
evaluated on a per-bin basis for all uncertainties due to the acceptance and
efficiency corrections.
\begin{table}[htbp]
\centering
\topcaption{Summary of relative systematic uncertainties in the
\JPsi pair total cross section.}
\begin{tabular}{lc} \hline
Source & Relative uncertainty [\%] \\
\hline
Branching fraction & 2.0 \\
Integrated luminosity & 2.2 \\
Acceptance correction & 1.1 \\
Efficiency correction & 4.4 \\
Efficiency scaling factor & 0.2 \\
Muon track reconstruction & 3.0 \\
Detector simulation & 6.5 \\
PDF parameters & 0.6 \\
Production model & 0.1 \\
\hline
Total & 9.0 \\
\hline
\end{tabular}
\label{tab:systematics}
\end{table}
To study the effect of nonisotropic \JPsi decay into muons on the
measured cross section, the event-based acceptance is determined using
extreme scenarios.
Defining $\theta$ as the angle between the $\mu^+$ direction in the
\JPsi rest frame and the \JPsi direction in the pp center-of-mass
frame, the angular distribution of decay muons is parameterized as:
$f(\theta) = 1 + \lambda \cos^2\theta$,
where $\lambda$ is a polarization observable~\cite{Chao:2012iv},
with $\lambda = 0$ corresponding to an isotropic
\JPsi decay. Compared to the $\lambda = 0$ case, the total cross section
is 31\% lower for $\lambda = -1$ and 27\% higher for $\lambda = +1$.
The differential cross section measurements for $\lambda = \pm 1$ lie
within the statistical uncertainties of the $\lambda = 0$ case when scaled
to the same total cross section, indicating that different polarization
assumptions do not affect the shapes of the cross section distributions.
Once the value of $\lambda$ has been measured, it can be
used in the acceptance calculation to mitigate
this source of uncertainty.
\section{Results}
\label{sec:final}
The total cross section obtained by summing over the sample on an
event-by-event basis and assuming unpolarized prompt
\JPsi pair production is
\begin{linenomath}
\begin{equation}
\sigma({\Pp\Pp}\to \JPsi\, \JPsi+X) = 1.49 \pm 0.07 \pm 0.13\unit{nb},
\end{equation}
\end{linenomath}
with statistical and systematic uncertainties shown, respectively.
For the measurement, the values $\mathcal{L} = 4.73 \pm 0.10$\fbinv~\cite{CMS-PAS-SMP-12-008}
and $BF(\JPsi \to \Pgmp\Pgmm)$
= (5.93 $\pm$ 0.06)\% \cite{PDG2012} are used. The differential cross
section as a function of the \JPsi pair invariant mass ($M_{\JPsi\,\JPsi}$),
the absolute rapidity difference between \JPsi mesons ($\abs{\Delta y}$), and
the \JPsi pair transverse momentum ($\pt^{\JPsi\,\JPsi}$) is shown in
Fig.~\ref{fig:dsigdx}.
The observed differential cross section is not only a result of the kinematics
of \JPsi pair production, but also of the \JPsi phase-space window (given in
the figures) available for measurement.
The corresponding numerical values are summarized
in Tables~\ref{tab:dsigdmTable}, \ref{tab:dsigdyTable},
and~\ref{tab:dsigdptTable}, respectively.
\begin{table}[h!]
\centering
\topcaption{Differential cross section in bins of the \JPsi pair invariant
mass ($M_{\JPsi\,\JPsi}$). The uncertainties shown are statistical first,
then systematic. }
\begin{tabular*}{0.49\textwidth}{@{\extracolsep{\fill}}ll} \hline
$M_{\JPsi\,\JPsi}$ (\GeVccns)& $\rd\sigma / \rd{}M_{\JPsi\,\JPsi}$ (nb/(\GeVccns)) \\ \hline
6--8 & $0.208 \pm 0.018 \pm 0.069$ \\
8--13 & $0.107 \pm 0.011 \pm 0.025$ \\
13--22 & $0.019 \pm 0.002 \pm 0.001$ \\
22--35 & $0.008 \pm 0.001 \pm 0.001$ \\
35--80 & $0.007 \pm 0.001 \pm 0.001$ \\
\hline
\end{tabular*}
\label{tab:dsigdmTable}
\end{table}
\begin{table}[h!]
\centering
\topcaption{Differential cross section in bins of the
absolute rapidity difference between \JPsi mesons ($\abs{\Delta y}$).
The uncertainties shown are statistical first, then systematic.}
\begin{tabular*}{0.49\textwidth}{@{\extracolsep{\fill}}ll} \hline
$\abs{\Delta y}$ & $\rd\sigma / \rd\abs{\Delta y}$ (nb) \\ \hline
0--0.3 & $2.06 \pm 0.14 \pm 0.25$ \\
0.3--0.6 & $1.09 \pm 0.13 \pm 0.16$ \\
0.6--1 & $0.421 \pm 0.057 \pm 0.077$ \\
1--1.6 & $0.040 \pm 0.006 \pm 0.006$ \\
1.6--2.6 & $0.025 \pm 0.005 \pm 0.005$ \\
2.6--4.4 & $0.205 \pm 0.033 \pm 0.058$ \\
\hline
\end{tabular*}
\label{tab:dsigdyTable}
\end{table}
\begin{table}[h!]
\centering
\topcaption{Differential cross section in bins of the transverse momentum
of the \JPsi pair ($\pt^{\JPsi\,\JPsi}$). The uncertainties shown are
statistical first, then systematic.}
\begin{tabular*}{0.49\textwidth}{@{\extracolsep{\fill}}ll} \hline
$\pt^{\JPsi\,\JPsi}$ ($\GeVc$) & $\rd\sigma / \rd\pt^{\JPsi\,\JPsi}$~(nb/(\GeVcns)) \\ \hline
0--5 & $0.056 \pm 0.007 \pm 0.012$ \\
5--10 & $0.048 \pm 0.006 \pm 0.010$ \\
10--14 & $0.108 \pm 0.013 \pm 0.012$ \\
14--18 & $0.089 \pm 0.009 \pm 0.012$ \\
18--23 & $0.019 \pm 0.002 \pm 0.003$ \\
23--40 & $0.003 \pm 0.001 \pm 0.001$ \\
\hline
\end{tabular*}
\label{tab:dsigdptTable}
\end{table}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/XS_by_Mass_stat_syst_err_logY}
\includegraphics[width=0.49\textwidth]{figures/XS_by_dY_stat_syst_err}
\includegraphics[width=0.49\textwidth]{figures/XS_by_pT_stat_syst_err}
\caption{Differential cross section for prompt \JPsi pair production
as a function of the \JPsi pair invariant mass ($M_{\JPsi\,\JPsi}$, top left),
the absolute rapidity difference between \JPsi mesons ($\abs{\Delta y}$, top
right), and the \JPsi pair transverse momentum ($\pt^{\JPsi\,\JPsi}$, bottom),
over the \JPsi phase space given in the figure, assuming unpolarized \JPsi production. The shaded regions represent the statistical uncertainties only,
and the error bars represent the statistical and systematic uncertainties
added in quadrature.
}
\label{fig:dsigdx}
\end{center}
\end{figure}
A search for the $\eta_b$ is performed by examining
the \JPsi pair invariant-mass distribution around the
nominal $\eta_b$ mass~\cite{PDG2012}, before efficiency and acceptance
corrections. From samples of simulated \JPsi pair events produced
via SPS or DPS, the acceptance times efficiency is found to be nearly
linear in the mass interval 8.68--10.12\GeVcc. The reconstructed
Gaussian width of the $\eta_b$ is 0.08\GeVcc, as determined from a
\JPsi pair MC simulation sample generated according to a Breit--Wigner
function with the nominal $\eta_b$ mass and width~\cite{PDG2012}.
The signal search interval
9.16--9.64\GeVcc corresponds to three standard deviations on each
side of the mean mass value. Two sideband regions of the same width
as the signal region are defined as the intervals 8.68--9.16\GeVcc and 9.64--10.12\GeVcc.
A first-degree polynomial is used to fit the number of events in the
sideband regions. Extrapolating these yields to the signal region predicts
$15\pm 4$ nonresonant events. The total number of \JPsi pair events in this
region in data is 15. Hence, no significant $\eta_b$ contribution is observed.
\section{Summary}
\label{sec:interpretation}
A signal yield of $446\pm 23$ events for the production of prompt
\JPsi meson pairs has been observed with
the CMS detector in pp collisions at $\sqrt{s}=7$\TeV
from a sample corresponding to an integrated luminosity of
$4.73 \pm 0.10$\fbinv. A data-based method has been used to correct for
the acceptance and efficiency, minimizing the model dependence of the cross
section determination. The total cross section of prompt \JPsi pair
production measured within a phase-space region defined by the
individual \JPsi \pt and rapidity is found to be
$1.49 \pm 0.07\stat \pm 0.13\syst$\unit{nb},
where unpolarized production is assumed.
Differential cross sections have been obtained in bins of the
\JPsi pair invariant mass, the absolute rapidity difference between the two
\JPsi mesons, and the \JPsi pair transverse momentum.
These measurements probe \JPsi pair production at higher \JPsi \pt and more central rapidity than the LHCb measurement~\cite{Aaij:2012dz},
providing for the first time information about a kinematic region
where color-octet \JPsi states and higher-order corrections play a
greater role in production. The differential cross section in
bins of $\abs{\Delta y}$ is sensitive to DPS contributions to
prompt \JPsi pair production. The differential cross section
decreases rapidly as a function of $\abs{\Delta y}$. However, a non-zero
value is measured in the $\abs{\Delta y}$ bin between 2.6 and 4.4.
Current models predict that this region can be populated via DPS
production~\cite{Baranov:2011ch,Baranov:2013,Kom:2011bd}.
There is no evidence for the $\eta_b$ resonance in the \JPsi pair invariant-mass distribution above the background expectations derived from the $\eta_b$ sideband regions.
Since models describing the
nonresonant \JPsi pair production in the CMS \JPsi phase-space
window are not available, an upper limit on the production cross section times branching fraction for $\eta_b\to \JPsi\, \JPsi$ cannot be obtained.
Model descriptions of \JPsi pair production at higher
\pt are crucial input to quantify nonresonant contributions in the search for new states at different center-of-mass energies. The cross section measurements presented here provide significant new information
for developing improved theoretical production models.
\section*{Acknowledgements}
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMWFW and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); MoER, ERC IUT and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NIH (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); NRF and WCU (Republic of Korea); LAS (Lithuania); MOE and UM (Malaysia); CINVESTAV, CONACYT, SEP, and UASLP-FAI (Mexico); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS and RFBR (Russia); MESTD (Serbia); SEIDI and CPAN (Spain); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU and SFFR (Ukraine); STFC (United Kingdom); DOE and NSF (USA).
Individuals have received support from the Marie-Curie programme and the European Research Council and EPLANET (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation \`a la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Council of Science and Industrial Research, India; the HOMING PLUS programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund; the Compagnia di San Paolo (Torino); and the Thalis and Aristeia programmes cofinanced by EU-ESF and the Greek NSRF.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 107 |
\section{Introduction}\label{sec:Intro}
Galaxy clusters at redshift $z=0$ are the most massive virialised objects in the Universe, residing in dark matter haloes with $M_\textrm{z=0}\geq 10^{14}~\textrm{M}_\odot$. At higher redshifts, the progenitors of these objects -- which are referred to as protoclusters -- are not yet virialised and are spread out over a scale of $\sim20h^{-1}~\textrm{cMpc}$ at $z\sim2$ \citep{Chiang2013AncientProto-clusters,Muldrew2015WhatProtoclusters}. Protoclusters are some of the densest structures on this scale at $z\sim 2-3$ when the cosmic star formation rate density is greatest, quasar activity peaked, and massive galaxies assembled the majority of their mass \citep{Madau2014CosmicHistory}. Protoclusters can therefore be used to study the effects of high density environments on galaxy formation and evolution at this important epoch,
and can also be used constrain structure formation and cosmological models via their growth rate \citep{Kravtsov2012FormationClusters}.
The detection of protoclusters presents a challenge, however. Unlike mature clusters, they lack a hot X-ray emitting intra-cluster medium \citep{Overzier2016TheProtoclusters}. Therefore, they are most commonly found through the presence of large concentrations of galaxies. This a challenging observational task; protoclusters are large and diffuse objects, so they present only a small density enhancement over the field. Furthermore, protocluster galaxies are still forming, so they do not lie on a well defined red sequence \citep{Gladders2000AAlgorithm}.
Current observational techniques for finding protoclusters fall into two main categories. The first of these is to use typical galaxies as tracers. This has been performed using both photometric \citep{Daddi2009TwoRedshifts, Chiang2014DiscoveryCosmos} and spectroscopic \citep{Steidel2005SpectroscopicRedshift, Cucciati2014DiscoveryVUDS, Lemaux2014VIMOS3.3, Chiang2015SurveyingHETDEX, Lemaux2017TheZ4.57} surveys. The most massive protoclusters are rare (with number density $<10^{-6}\,\textrm{cMpc}^{-3}$) and spatially extended. Consequently, in order to find protoclusters using this method, large cosmic volumes need to be surveyed. Until recently, galaxy redshift surveys that are deep enough to probe galaxies at $z\gtrsim2$ only cover areas of $\lesssim 1$ deg$^2$, such as the COSMOS survey \citep{Scoville2007TheOverview}, resulting in only a few tens of protoclusters discovered through this method over the last decade. This is about to change with ongoing and forthcoming surveys such as the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP)\footnote{https://hsc.mtk.nao.ac.jp/ssp/} and the Large Synoptic Survey Telescope (LSST)\footnote{https://www.lsst.org}; using only a small subset of the HSC-SSP data, \citet{Toshikawa2018GOLDRUSH.Area} have already discovered over 179 protocluster candidates.
The second of these methods is to focus searches around biased tracer objects, such as high redshift radio galaxies \citep{LeFevre1996Clustering3.14, Venemans2007I.Protoclusters, Hatch2011GalaxyGalaxies, Cooke2014AEnvironments}, quasars -- or quasi-stellar objects (QSOs) -- \citep{Wold2003OverdensitiesQuasars, Zheng2006AnQuasar, Capak2011AZ5.3, Wylezalek2013GalaxySpitzer, Morselli2014PrimordialLBT, Adams2015DiscoveryTelescope} and Ly-$\alpha$\ blobs \citep{Badescu2017DiscoveryZ=2.3, Cai2017Discovery2.3}, which are thought to correlate well with high density regions at high redshift, and/or be the progenitors of local brightest cluster galaxies. These are also very bright objects, and can therefore be found using significantly shallower surveys. This approach greatly reduces the observational cost and it is how the majority of presently spectroscopically confirmed protoclusters at $z \gtrsim 1.3$ have been discovered. However, this comes at the expense of introducing a strong selection bias to the resulting sample.
Protoclusters are not only traced by overdensities of galaxies, however, but also by intergalactic hydrogen gas \citep{Adelberger2003GalaxiesOverview, Mukae2017CosmicField}. The residual neutral hydrogen (H\,\textsc{i}) in this otherwise highly ionised medium can be detected in absorption in background quasar spectra through the resulting Ly-$\alpha$\ absorption (i.e. the Ly-$\alpha$\ forest). The idea of locating protoclusters as regions overdense in H\,\textsc{i} was first proposed by \citet{Francis1993SuperclusteringTwo} with the first observational detection by \citet{Francis1996A2.38}. Recently there has been renewed interest in this approach following the advent of large QSO surveys, such as the Baryon Oscillation Spectroscopic Survey (BOSS) \citep{Dawson2013TheSDSS-III}, the Extended Baryon Oscillation Spectroscopic Survey (eBOSS) and the forthcoming Dark Energy Spectroscopic Instrument (DESI) \citep{Vargas-Magana2019UnravelingDESI} and WEAVE-QSO surveys \citep{Pieri2016WEAVE-QSO:Telescope}.
The recent work has been approached from two different directions. The first is Ly-$\alpha$\ forest tomography \citep{Lee2014Observational2,Stark2015ProtoclusterMaps,Lee2018First2.55}, where the underlying 3-D mass distribution is reconstructed using multiple Ly-$\alpha$\ forest spectra in the same patch of sky. Applying insight obtained from a $256^{3} \, h^{-3}\rm\,cMpc^{3}$ collisionless dark matter simulation, \citet{Lee2016ShadowField} have used this method to successfully detect a galaxy overdensity at $z=2.44$ in the COSMOS field, as well as cosmic voids at $z\sim2.3$ \citep{Krolewski2018DetectionField}. The second method -- which we focus on in this work -- is to search for the most massive overdensities at $z=2\textrm{--}4$ from a large survey volume using strong, coherent Ly-$\alpha$\ absorption along the line of sight \citep{Cai2016MAppingMethodology}. This approach is more effective at redshifts where the density of bright background sources is too low for tomography. Guided by mock spectra extracted from a 1 $h^{-3}\rm\,cGpc^{3}$ collisionless dark matter simulation combined with an approximate scheme for modelling Ly-$\alpha$\ absorption \citep{Peirani2014LyMAS:Field}, \citet{Cai2016MAppingMethodology} identified protoclusters as being closely associated with what they call Coherently Strong intergalactic Ly-$\alpha$\ absorption systems (CoSLAs). Groups of CoSLAs have been used to discover a massive overdensity at $z=2.32$ \citep{Cai2017MAppingZ=2.32}.
These approaches have had compelling success in locating potential protoclusters at $z\simeq 2-3$. However, fully hydrodynamical simulations that directly connect the gas distribution at $z>2$ to present-day clusters are required to establish whether coherent Ly-$\alpha$\ absorption accurately tracks the progenitors of the largest collapsed clusters at $z=0$, or if a more complex relationship between mass and Ly-$\alpha$\ absorption may disrupt this picture. Furthermore, for protocluster searches that employ individual sight lines to detect CoSLAs, high column density Ly-$\alpha$\ absorbers that possess large damping wings can be a significant contaminant \citep[cf.][]{Cai2016MAppingMethodology}.
In this work we shall address these points by investigating the properties of Ly-$\alpha$\ absorption systems within protoclusters at $z > 2.4$ using fully hydrodynamical simulations performed by the Sherwood \citep{Bolton2017The5}, \textsc{eagle} \citep{Schaye2015TheEnvironments} and Illustris \citep{Vogelsberger2014IntroducingUniverse} projects. With a typical volume of order $10^{6}\rm\,cMpc^{3}$, these simulations only contain of order $\sim 10$ clusters with masses $M_{\rm z=0}\geq 10^{14}M_{\odot}$. On the other hand, unlike larger collisionless dark matter simulations, they include a full treatment of the gas physics using a variety of different (sub-grid) feedback models. More importantly, however, these simulations also have sufficient resolution to correctly incorporate Ly-$\alpha$\ absorption features over a wide range of \hbox{H$\,\rm \scriptstyle I$}\ column densities, including absorption systems with damping wings. Our approach therefore provides a complementary perspective on the relationship between Ly-$\alpha$\ absorption on small scales and the distribution of mass in protoclusters at $z\simeq 2.4$. Furthermore, as we explicitly track the formation of structure to $z=0$ in the simulations, we are able to assess the completeness and contamination of a sample of protocluster candidates selected using coherent line of sight Ly-$\alpha$\ absorption.
This paper is structured as follows. We first introduce the hydrodynamical simulations we use in Section~\ref{sec:Sims}, and compare the \hbox{H$\,\rm \scriptstyle I$}\ column density distribution in each model to observational constraints in Section~\ref{sec:CDDF}. In Section~\ref{sec:Sample} we describe the basic properties of protoclusters in the simulations. We characterise mass overdensities on large scales by their Ly-$\alpha$\ absorption in Section~\ref{sec:Results}, and examine their connection to protoclusters in Section~\ref{ssec:PCFrac}. Finally, the effectiveness of using line of sight Ly-$\alpha$\ absorption to detect candidate protoclusters at $z\simeq 2.4$ is discussed in Section~\ref{sec:CompCont}, before we summarise and conclude in Section \ref{sec:Conclusion}. Throughout the paper, we refer to comoving distance units using the prefix \lq\lq c".
\section{Hydrodynamical simulations}\label{sec:Sims}
The hydrodynamical simulations used in this work are summarised in Table \ref{tab:Sims}. Outputs at two different redshifts were used for each model: the output closest\footnote{This corresponds to $z=2.4$ for Sherwood, $z=2.478$ for \textsc{EAGLE} and $z=2.44$ for Illustris.} to $z=2.4$, along with the corresponding output at $z=0$. Below, we briefly describe the relevant properties of each simulation in turn, although further, extensive descriptions of these simulations may be found elsewhere.
\begin{table*}
\centering
\caption{Hydrodynamical simulations used in this work. The columns list, from left to right: the simulation name, the box size in $h^{-1}\, \rm cMpc$, the total number of resolution elements, the dark matter and gas particle masses (or equivalently for Illustris, the typical hydrodynamical cell mass), and the number of objects that have a total mass at $z=0$ (as defined by a friends-of-friends halo finder) in the range $M_{\rm z=0}\geq10^{14}\rm\, M_{\odot}$ (clusters), $10^{13.75}\leq M_{\rm z=0}/M_{\odot}<10^{14}$ (large groups) and $10^{13.5}\leq M_{\rm z=0}/M_{\odot}<10^{13.75}$ (small groups).}
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
Name & Box size & $N_\textrm{tot}$ & $M_\textrm{dm}$ & $M_\textrm{gas}$ & $M_{\rm z=0}\geq10^{14}~\textrm{M}_\odot$& $10^{13.75}\leq \frac{M_{\rm z=0}}{M_{\odot}}<10^{14}$ & $10^{13.5}\leq \frac{M_{\rm z=0}}{M_{\odot}}<10^{13.75}$ \\
&[$h^{-1}\ $cMpc]& & [$\textrm{M}_\odot$] & [$\textrm{M}_\odot$] & Clusters & Large groups & Small groups \\
\hline
Sherwood & 80 & $2\times1024^3$ & $5.07\times10^7$ & $9.41\times10^6$ & 29 & 23 & 46 \\
\textsc{eagle} & 67.77 & $2\times1504^3$ & $9.70\times10^6$ & $1.81\times10^6$ & 10 & 15 & 33 \\
Illustris & 75 & $3\times1820^3$ & $6.26\times10^6$ & $1.26\times10^6$ & 14 & 18 & 29 \\
\hline
\end{tabular}
\label{tab:Sims}
\end{table*}
\subsection{Sherwood}\label{ssec:Sherwood}
The Sherwood project \citep{Bolton2017The5} consists of a set of large, high resolution simulations of the Ly-$\alpha$\ forest performed using a modified version of the smoothed particle hydrodynamics (SPH) code \textsc{P-Gadget-3}, last described in \citet{Springel2005TheGADGET-2}. In this work we predominantly use the 80-1024-ps13 simulation from \citet{Bolton2017The5}, which we shall refer to as \lq\lq Sherwood". This simulation was performed in a $80^{3}h^{-3}\ \textrm{cMpc}^3$ volume using the star formation and galactic outflow model developed by \citet{Puchwein2013ShapingWinds}. This assumes a \citet{Chabrier2003GalacticFunction} initial mass function (IMF) and a wind velocity, $v_{\rm w}$, that is proportional to the galaxy escape velocity, such that the wind mass-loading scales as $v_\textup{w}^{-2}$.
The ultraviolet (UV) background follows the spatially uniform \citet{Haardt2012RadiativeBackground} model, which quickly reionises the IGM at $z=15$. This model assumes the hydrogen is optically thin and in photo-ionisation equilibrium, and has an \hbox{H$\,\rm \scriptstyle I$}\ photo-ionisation rate of $\Gamma_{\rm HI}=9.6\times10^{-13}\rm\,s^{-1}$ at $z=2.4$. Additionally, a small boost to the \hbox{He$\,\rm \scriptstyle II$}\ photo-heating rate, $\epsilon_{\rm HeII}=1.7\epsilon_{\rm HeII}^{\rm HM12}$, has been applied at $2.2<z<3.4$ to better match observational constraints on the IGM temperature during and after \hbox{He$\,\rm \scriptstyle II$}\ reionisation \citep{Becker2011DetectionMedium}. The Sherwood models adopt a Planck 2013 consistent cosmology \citep{PlanckCollaboration2014PlanckParameters} with $\Omega_m=0.308,\ \Omega_\Lambda=0.692,\ \Omega_b=0.0481,\ \sigma_8=0.826,\ n_s=0.963\ \textrm{and}\ h=0.678$.
In addition to the fiducial simulation described above, we also use the 80-1024 simulation described in \citet{Bolton2017The5}. This does not follow a physically motivated star formation model, but is identical to our fiducial run in all other respects. Instead, gas particles with density $\Delta=\rho/\langle \rho \rangle > 1000$ and temperature $T < 10^5\,$K are converted directly into collisionless star particles. This \lq\lq quick Ly-$\alpha$"\ approach increases computational speed while having a minimal effect on the low column density absorbers in the Ly-$\alpha$\ forest \citep{Viel2004InferringSpectra}. As this removes all the cold, dense gas in the simulation, the incidence of \hbox{H$\,\rm \scriptstyle I$}\ absorbers with column densities $N_{\rm HI} \gtrsim 10^{17}\rm\,cm^{-2}$ will be underpredicted. We include this approach here, however, as it will more closely approximate results from earlier work using post-processed dark matter simulations that have insufficient resolution for modelling dense gas. We refer to this model as Sherwood ``QLy-$\alpha$".
\subsection{EAGLE}\label{ssec:EAGLE}
The \textsc{EAGLE} (Evolution and Assembly of GaLaxies and their Environments) simulation \citep{Schaye2015TheEnvironments, Crain2015TheVariations, McAlpine2016TheCatalogues} was performed with a customised version of \textsc{P-Gadget-3}, where the standard SPH approach has been modified following the \textsc{anarchy} scheme described by \citet{Schaye2015TheEnvironments}. The simulation we use in this work is the Ref-L0100N1504 model. This has a smaller box size than Sherwood, corresponding to $67.77^{3}\ h^{-3}\ \textrm{cMpc}^3$, but with a factor of $\simeq 5$ better mass resolution (see Table \ref{tab:Sims}). Star formation is modelled using the approach of \citet{Schaye2008OnSimulations} assuming a Chabrier IMF, while stellar feedback follows the stochastic, thermal scheme described in \citet{DallaVecchia2012SimulatingFeedback}. In addition, EAGLE follows gas accretion onto black holes; feedback from active galactic nuclei (AGN) is included using the methodology of \citet{Booth2009CosmologicalTests}.
The spatially uniform, optically thin UV background used in EAGLE follows \citet{Haardt2001ModellingCUBA}; this quickly reionises the hydrogen at $z=9$. An additional $2\rm\,eV$ per proton of energy from \hbox{He$\,\rm \scriptstyle II$}\ reionisation is also added at $z\simeq 3.5$, resulting in gas temperatures at mean density that are $\sim 4000\rm\,K$ larger compared to Sherwood by $z=2.4$. The \hbox{H$\,\rm \scriptstyle I$}\ photo-ionisation rate at $z=2.478$ in the \citet{Haardt2001ModellingCUBA} model is $\Gamma_{\rm HI}=1.4\times 10^{-12}\rm\,s^{-1}$, a factor of $1.5$ larger than the more recent \citet{Haardt2012RadiativeBackground} UV background used in Sherwood. \textsc{EAGLE} assumes a $\Lambda$CDM cosmology consistent with \citet{PlanckCollaboration2014PlanckParameters}, where $\Omega_m=0.307,\ \Omega_\Lambda=0.693,\ \Omega_b=0.04825,\ \sigma_8=0.8288,\ n_s=0.9611 \ \textrm{and}\ h=0.677$.
\subsection{Illustris} \label{ssec:Illustris}
Finally, we also use the Illustris-1 simulation (herafter referred to as ``Illustris") in our analysis \citep{Vogelsberger2014IntroducingUniverse, Nelson2015TheRelease}. Unlike Sherwood and \textsc{EAGLE}, Illustris is performed with the moving-mesh hydrodynamics code \textsc{arepo} \citep{Springel2010EMesh}. Illustris has a slightly smaller volume than Sherwood ($75^{3}\ h^{-3}\ \textrm{cMpc}^3$), but it has the highest mass resolution of the three simulations we consider. The star formation, stellar feedback and AGN feedback models are described in detail by \citet{Vogelsberger2013APhysics}. Star formation in Illustris also uses a Chabrier IMF and is based upon the \citet{Springel2003CosmologicalFormation} model. The stellar feedback uses a variable winds approach, where the wind velocity, $v_\textup{w}$, is scaled to the local dark matter velocity dispersion.
The spatially uniform UV background follows \citet{Faucher-Giguere2009AReionization}, which quickly reionises the hydrogen in the simulation at $z=10.5$. This UV background model has a \hbox{H$\,\rm \scriptstyle I$}\ photo-ionisation rate $\Gamma_{\rm HI}=6.1\times 10^{-13}\rm\,s^{-1}$ at $z=2.44$ -- a factor of $0.6$ smaller than \citet{Haardt2012RadiativeBackground} -- and produces gas temperatures at mean density around $1500\rm\,K$ lower at $z=2.4$ compared to Sherwood.
Furthermore, unlike Sherwood and EAGLE, Illustris uses an on-the-fly prescription for the self-shielding of hydrogen from Lyman continuum photons, following the approach of \citet{Rahmati2013OnSimulations}. As we will discuss below, incorporating self-shielding is necessary for correctly capturing the incidence of absorption systems with column densities $N_{\rm HI}\geq 10^{17.2}\rm\,cm^{-2}$ (i.e. absorption systems that are optically thick to Lyman continuum photons). The Illustris simulations assume a WMAP-9 consistent cosmology \citep{Hinshaw2013Nine-yearResults}, with $\Omega_m=0.2726,\ \Omega_\Lambda=0.7274,\ \Omega_b=0.0456,\ \sigma_8=0.809,\ n_s=0.963\ \textrm{and}\ h=0.704$.
\subsection{Generation of mock Ly-$\alpha$\ absorption sight-lines} \label{ssec:spectra}
Mock Ly-$\alpha$\ absorption spectra were extracted along sight lines drawn from the Sherwood simulation using the SPH interpolation scheme described by \citet{Theuns1998P3M-SPHForest}, combined with the Voigt profile approximation from \citet{Tepper-Garcia2006VoigtFunction}. Each sight line consists of $1024$ pixels, and is drawn in a direction parallel to the simulation boundaries, starting from a position selected at random on the projection axis. A total of $30,000$ sight lines ($10,000$ along each projection axis) were extracted from Sherwood, corresponding to an average transverse separation of $0.8h^{-1} \rm \,cMpc$. The transmitted flux in each pixel is given by $F = e^{-\tau}$, where $\tau$ is the Ly-$\alpha$\ optical depth.
Sight lines were extracted with the same average transverse separation from EAGLE and Illustris, although there are some small differences in the methodology due to the different hydrodynamics schemes employed by these models. For EAGLE, the $M_{4}$ cubic spline kernel \citep{Monaghan1985AProblems} used in the standard version of \textsc{P-Gadget-3} was replaced with the $C_{2}$ \citet{Wendland1995PiecewiseDegree} kernel when performing the SPH interpolation. In Illustris there are no smoothing lengths, $h_{\rm i}$, associated with the hydrodynamic cells. Instead, we assign these based on the volume, $V_{\rm i}$, of each Voronoi cell, where
\begin{equation} h_{\rm i} = \left(\frac{3N_{\rm sph} V_{\rm i}}{4\pi}\right)^{1/3}, \end{equation}
and we adopt $N_{\rm sph}=64$ for the number of smoothing neighbours. We furthermore set all star-forming hydrogen gas with $n_{\rm H}>0.13\rm\,cm^{-3}$ to be fully neutral in Illustris, correcting for the unphysical neutral hydrogen fractions produced by the sub-grid star formation model. In addition, since Illustris already incorporates self-shielded hydrogen on-the-fly, the \citet{Rahmati2013OnSimulations} prescription for self-shielding was applied in post-processing to both Sherwood and EAGLE.
Lastly, in order to correct for the approximately factor of two uncertainty in the UV background \hbox{H$\,\rm \scriptstyle I$}\ photo-ionisation rate $\Gamma_{\rm HI}$ \citep{Bolton2005The2-4}, we rescale the optical depths in each pixel of the mock spectra by a constant, such that the Ly-$\alpha$\ forest effective optical depth obtained from all the mock sight lines, $\tau_\textrm{eff} = -\ln\left<F\right>$, where $\left<F\right>$ is the mean transmitted flux, matches observational constraints \citep{Theuns1998P3M-SPHForest, Lukic2014TheSimulations}. We use the $\tau_\textrm{eff}$\ measurements from \citet{Becker2013ASpectra} for this purpose. At $z=2.4$, these data correspond to $\tau_{\rm eff}=0.20$.
\section{The \hbox{H$\,\rm \scriptstyle I$}\ column density distribution function}\label{sec:CDDF}
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/CDDF.pdf}
\vspace{-8mm}
\caption{The \hbox{H$\,\rm \scriptstyle I$}\ column density distribution function (CDDF) at $z\simeq 2.4$ obtained from Sherwood (solid blue curve), EAGLE (orange curve) and Illustris (brown curve). For comparison, the dotted blue line represents the CDDF from the QLy-$\alpha$\ simulation. Observational data from \citet{Kim2013The3.2} at $\left<z\right>=2.7$ and $\left<z\right>=2.13$ have been added at $N_{\rm HI}<10^{17}\rm\,cm^{-2}$, while data points from \citet{Noterdaeme2012Column9} at $\left<z\right>=2.5$ and \citet{Prochaska2009OnTime} $\left<z\right>=3$ are displayed at $N_{\rm HI}>10^{20}\rm\,cm^{-2}$.
\label{fig:CDDF}
\end{figure}
Before proceeding to analyse the properties of protoclusters in Ly-$\alpha$\ absorption, we must first verify if the simulations reproduce the observed distribution of H\,\textsc{i} column densities\footnote{We will refer to Ly-$\alpha$\ absorbers in four groups based on their column densities: Ly-$\alpha$\ forest ($N_{\rm HI} < 10^{17.2}~\textrm{cm}^{-2}$), Lyman-limit systems (LLSs, $10^{17.2}\leq N_{\rm HI}/\textrm{cm}^{-2}<10^{19}$), super Lyman-limit systems (SLLSs, $10^{19}\leq N_{\rm HI}/\textrm{cm}^{-2} < 10^{20.3}$) and damped Ly-$\alpha$\ absorbers (DLAs, $N_\textrm{HI}\geq 10^{20.3}~\textrm{cm}^{-2}$). We will also refer to SLLSs and DLAs collectively as \lq \lq damped systems".}
at $z\simeq 2.4$. The column density distribution function (CDDF) obtained from the three simulations are displayed Figure \ref{fig:CDDF}, along with observational measurements from \citet{Kim2013The3.2}, \citet{Noterdaeme2012Column9} and \citet{Prochaska2009OnTime}. The simulated CDDFs at $N_{\rm HI}<10^{17}\rm\,cm^{-2}$ are calculated by integrating the \hbox{H$\,\rm \scriptstyle I$}\ number density in each pixel in the mock sight lines over $50\rm\,km\,s^{-1}$ windows \citep{Gurvich2017TheForest}. However, as absorption systems with $N_{\rm HI}>10^{17}\rm\,cm^{-2}$ are comparatively rare, we instead compute the CDDF by projecting the H\,\textsc{i} density for the entire simulation box onto a 2-D grid consisting of $20000^2$ pixels \citep{Altay2011ThroughSimulations,Rahmati2013OnSimulations,Bird2014DampedFeedback,Villaescusa-Navarro2018IngredientsMapping}. The discontinuity seen at $N_{\rm HI}=10^{17}\rm\,cm^{-2}$ in Figure \ref{fig:CDDF} represents the point we switch from the integration method to the projection method.
The fiducial Sherwood simulation (solid blue curve), as well as the EAGLE (orange curve) and Illustris (brown curve) are in good agreement with observational data over a wide range of column densities, although the level of agreement at $N_{\rm HI}>10^{21.3}\rm\,cm^{-2}$ is possibly fortuitous given that a treatment of molecular hydrogen is not included in our analysis \citep{Altay2013TheAbsorbers, Crain2017TheGalaxies}. The Illustris simulation also predicts a greater incidence of systems with $10^{17}\leq N_\textrm{\rm HI}/\rm cm^{-2} \leq 10^{20}$, relative to EAGLE and Sherwood. The reason for this difference is unclear, although we speculate this may be in part because the self-shielding correction is included on-the-fly in Illustris. The photo-heating (and hence pressure smoothing) experienced by the high column density, self-shielded gas will therefore differ from the post-processed EAGLE and Sherwood runs.
For comparison, we also show the CDDF from the QLy-$\alpha$\ model in Figure \ref{fig:CDDF} (blue dotted curve). The QLy-$\alpha$\ model is only a good match to the observational data for the Ly-$\alpha$\ forest at $N_{\rm HI} \lesssim 10^{17}\rm\,cm^{-2}$. This is because most of the overdense gas that forms the strongest absorption systems has been converted into collisionless star particles (see Section \ref{ssec:Sherwood}). Overall, this comparison demonstrates that Sherwood, EAGLE and Illustris will adequately capture the incidence of Ly-$\alpha$\ absorbers at $z \simeq 2.4$ over a wide range of \hbox{H$\,\rm \scriptstyle I$}\ column densities.
\section{Simulated protoclusters}\label{sec:Sample}
\subsection{Protocluster definitions}\label{ssec:defs}
\begin{figure*}
\includegraphics[width=\textwidth]{Figures/SummaryAllLS_z2_4.pdf}
\vspace{-1.0cm}
\caption{Top: Distribution of $R_{95}$ for the protoclusters (blue) and large and small protogroups (orange and brown) in Sherwood (left), EAGLE (centre) and Illustris (right) at $z\sim2.4$. Bottom: Scatter plots showing the triaxiality (where $T=1$ and $T=0$ are prolate and oblate spheroids, respectively) against the sphericity (where $s=1$ and $s=0$ are spherical and aspherical, respectively) of the protoclusters and protogroups, where $a$, $b$ and $c$ are the principal semi-axes and $a\geq b\geq c$. Contours show the kernel density estimate for all points.}
\label{fig:Summary}
\end{figure*}
\begin{figure*}
\includegraphics[width=\textwidth]{Figures/9PannelRenderAll.png}
\vspace{-0.7cm}
\caption{Projected maps of the normalised gas density, temperature and \hbox{H$\,\rm \scriptstyle I$}\ fraction for protoclusters at $z\sim 2.4$ with masses $M_{\rm z=0}=10^{14.3}\,\textrm{M}_\odot$, $M_{\rm z=0}=10^{14.4}\,\textrm{M}_\odot$ and $M_{\rm z=0}=10^{14.3}\,\textrm{M}_\odot$ in Sherwood (top row), \textsc{eagle} (middle row) and Illustris (bottom row). The projection depth along the $z$-axis is $15h^{-1}~\textrm{cMpc}$, and is centred on the centre of mass of each protocluster. The dashed circles show $R_{95}$, and the star symbols correspond to the locations of the 10 most massive FoF groups within $R_{95}$. The yellow stars correspond to the most massive FoF groups in each protocluster, which have mass $M=10^{13.6}\,\textrm{M}_\odot$, $M=10^{12.8}\,\textrm{M}_\odot$ and $M=10^{13.1}\,\textrm{M}_\odot$ in Sherwood, \textsc{eagle} and Illustris, respectively. The yellow arrows overlaid on the left column show the peculiar velocity field in the protoclusters relative to the centre of mass.}
\label{fig:3Panel}
\end{figure*}
\begin{figure*}
\includegraphics[width=\textwidth]{Figures/3PanelHistAllSimsSep.pdf}
\vspace{-0.8cm}
\caption{Volume weighted distributions of the normalised gas density, temperature and H\,\textsc{i} fraction in all protoclusters (line histograms) compared to the field (filled histograms) in Sherwood (blue), EAGLE (orange) and Illustris (brown).}
\label{fig:SimHist}
\end{figure*}
We now turn to describe the protoclusters in the simulations at $z\sim2.4$. We identify the protoclusters by tracking all particles in a friends-of-friends (FoF) group\footnote{Note this means that our quoted protocluster masses will be systematically larger than virial mass estimators such as $M_{200}$ -- the total mass enclosed within a sphere whose mean density is $200$ times the critical density. For example, \citet{White2000TheHalo} demonstrate that FoF halo masses will be approximately 10 per cent greater than $M_{200}$.} at $z=0$. We will refer to the total mass of the $z=0$ FoF group as the mass of the protocluster, $M_{z=0}$. Earlier studies \citep[e.g.][]{Muldrew2015WhatProtoclusters} have instead identified simulated protoclusters by tracing the merger tree of $z=0$ haloes back to the redshift of interest. However, as we are primarily interested in following large scale Ly-$\alpha$\ absorption, in this work we also choose to follow the mass that is not bound in haloes at $z\simeq 2.4$.
We consider three mass bins in our analysis: $M_\textrm{z=0}\geq10^{14}~\textrm{M}_\odot$ (clusters), $10^{13.75}~\textrm{M}_\odot\leq M_\textrm{z=0}<10^{14}~\textrm{M}_\odot$ (large groups) and $10^{13.5}~\textrm{M}_\odot \leq M_\textrm{z=0}<10^{13.75}~\textrm{M}_\odot$ (small groups). We refer to the cluster progenitors as protoclusters, and the group progenitors as protogroups; we include the latter to provide a comparison to lower mass systems. The number of FoF groups in each bin is summarised in Table \ref{tab:Sims}. We also define the size of each of the protoclusters/groups at $z\simeq 2.4$ using $R_{95}$,\, which corresponds to the radius of a sphere around the protocluster centre of mass that contains 95 per cent of $M_{\rm z=0}$.
Additionally, following an analysis of the Millennium simulation \citep{Springel2005SimulationsQuasars} by \citet{Lovell2018CharacterisingProtoclusters}, for each protocluster we calculate two parameters derived from the principal semi-axes of a triaxial mass distribution, where $a\geq b\geq c$. The first is the axis ratio $s=c/a$ which provides a measure of sphericity, with $s=1$ corresponding to a spherical distribution and $s\sim0$ corresponding to a highly aspherical distribution. The second is the triaxiality parameter \citep{Franx1991TheShapes} given by
\begin{equation}
T=\frac{a^2-b^2}{a^2-c^2}
\end{equation}
This quantifies whether the mass distribution resembles a prolate ($T\sim 1$) or oblate ($T\sim0$) spheroid.
\subsection{The size and shape of simulated protoclusters}
The $R_{95}$\ distributions in the three mass bins are shown in the top row of Figure \ref{fig:Summary} for each simulation. The protoclusters tend to be larger than the protogroups, and typically have $R_{95}=5$--$10\rm\,h^{-1}\,cMpc$. These sizes are broadly consistent with the 90 per cent stellar mass radii recovered from the Millennium simulation at $z=2$ by \citet{Muldrew2015WhatProtoclusters}, as well as the radial extent determined by \citet{Hatch2011GalaxyGalaxies} for overdensities around radio galaxies at $z\sim2.4$. The properties displayed in Figure \ref{fig:Summary} are also largely consistent between the different simulations, suggesting that (as expected on large scales) variations in the hydrodynamics schemes and sub-grid physics have little impact on the overall size and shape of the protoclusters. Furthermore, in all three cases, the smaller, lower mass protogroups do not differ significantly in shape from the protoclusters.
As noted by \citet{Lovell2018CharacterisingProtoclusters}, however, a simple triaxial model does not fully capture the distribution of groups and filaments within protoclusters. In Figure~\ref{fig:3Panel}, we therefore display two-dimensional projections of protoclusters selected from each of the three simulations: these have masses $M_{\rm z=0}=10^{14.3}\,M_{\odot}$ (Sherwood), $M_{\rm z=0}=10^{14.4}\,M_{\odot}$ (EAGLE) and $M_{\rm z=0}=10^{14.3}\,M_{\odot}$ (Illustris). Several different protocluster morphologies are apparent, ranging from a structure dominated by a massive central halo (Sherwood) to a more diffuse structure with multiple, lower mass haloes (e.g. EAGLE). Figure~\ref{fig:3Panel} also shows the location of the most massive FoF group (yellow stars) along with the next nine most massive FoF groups (white stars) in each protocluster. These are located in overdense regions where, in general, the gas temperatures are higher as a result of shock heating and feedback from stellar winds and/or black hole accretion. It is also apparent that regions with high H\,\textsc{i} fractions trace the overdense gas; photo-ionisation equilibrium with the UV background means the \hbox{H$\,\rm \scriptstyle I$}\ number density, $n_{\rm HI}$, scales with the square of the gas density.
Finally, in Figure \ref{fig:SimHist} we show the volume weighted distribution of the gas density, temperature and \hbox{H$\,\rm \scriptstyle I$}\ fraction for the protoclusters compared to the ``field" (i.e. regions outwith $R_{95}$) in each of the simulations. The differences in gas properties between the protoclusters and the field are small, but as might be anticipated from an inspection of Figure~\ref{fig:3Panel}, the protoclusters are slightly more dense, hotter and have a larger \hbox{H$\,\rm \scriptstyle I$}\ fraction. Figure \ref{fig:SimHist} also highlights the differences in gas properties between the three different simulations. Although the gas density distribution is similar in all three cases, there are differences in the temperature and \hbox{H$\,\rm \scriptstyle I$}\ fraction distribution that arise from the different UV background models (see Section~\ref{sec:Sims}). Illustris (EAGLE) has a slightly larger (smaller) average H\,\textsc{i} fraction compared to Sherwood. This is due to the smaller (larger) \hbox{H$\,\rm \scriptstyle I$}\ photo-ionisation rate used in the UV background model and -- to a lesser extent -- the dependence of the \hbox{H$\,\rm \scriptstyle I$}\ fraction on the lower (higher) gas temperature through the \HII\ recombination rate.
\section{Characterising mass overdensities in Ly-$\alpha$\ absorption on $15h^{-1}~\textrm{cMpc}$ scales}\label{sec:Results}
\begin{figure*}
\includegraphics[width=\textwidth]{Figures/Contour_AllHydro.pdf}
\vspace{-0.8cm}
\caption{The relationship between $\delta_m$ and $\delta_{\tau_\textrm{eff}}$\ on $15\,h^{-1}\rm\,cMpc$ scales \citep[following][]{Cai2016MAppingMethodology}. Left to right, the panels show the results from Sherwood, EAGLE and Illustris. The contours represent the number density of data points relative to the central contour, with the number density decreasing by $1/3$ dex with each contour. Black points represent every system which lies outside of these contours (i.e. with relative number density $<10^{-3}$). Additional points corresponding to the centre of mass of the protoclusters (blue circles) large protogroups (orange triangles) and small protogroups (brown squares) are also displayed.}
\label{fig:ContourPS13_All}
\end{figure*}
\begin{figure*}
\includegraphics[width=\textwidth]{Figures/columnDensityVsDeltaM.pdf}
\vspace{-0.8cm}
\caption{The fraction of $15h^{-1}~\textrm{cMpc}$ segments in bins of $\Delta\delta_\textrm{m}=0.2$ whose largest constituent \hbox{H$\,\rm \scriptstyle I$}\ column density measured on $50~\textrm{km s}^{-1}$ scales corresponds to either Ly-$\alpha$\ forest, LLSs, SLLSs or DLAs in Sherwood (left), EAGLE (centre) and Illustris (right). The upper panels display the total number of $15h^{-1}~\textrm{cMpc}$ segments in each bin.}
\label{fig:MassNHIFrac}
\end{figure*}
Observationally, protoclusters are identified as large scale, high density regions. For this reason, we first investigate the Ly-$\alpha$\ absorption properties of mass overdensities within the hydrodynamical simulations, with no consideration as to whether these eventually collapse to form a cluster by $z=0$. From this, we determine the effectiveness of using Ly-$\alpha$\ absorption to detect large scale overdense regions at $z=2.4$. It is important to note, however, that this does not address how effectively Ly-$\alpha$\ absorption probes the gas at $z\simeq 2.4$ that \emph{actually} collapses to form clusters by $z=0$ -- we consider this further in Section~\ref{ssec:PCFrac}.
\subsection{Relationship between mass and effective optical depth on $15\,h^{-1}\rm\,cMpc$ scales}\label{ssec:MassVsTeff}
In order to assess how well Ly-$\alpha$\ absorption traces large scale, high density regions, we consider the correlation between mass and Ly-$\alpha$\ effective optical depth on $15h^{-1}~\textrm{cMpc}$ scales. We choose this scale as it corresponds to the characteristic size of protoclusters at $z\geq2$ \citep{Chiang2013AncientProto-clusters, Muldrew2015WhatProtoclusters}, and also follows the scale adopted by \citet{Cai2016MAppingMethodology}.
Our procedure throughout this work is as follows. First, we sum the masses of all particles in $15^{3}\,h^{-3}\rm\,cMpc^{3}$ cubic volumes along every simulated Ly-$\alpha$\ forest sight line (i.e. 30,\,000 in total for Sherwood, each with length $80h^{-1}\rm\,cMpc$), and compute the mass overdensity, $\delta_{\rm m}=\frac{m-\left<m\right>}{\left<m\right>}$ in each volume. Hence, $\delta_{\rm m}$ is a 3D average, where $m$ is the total mass within each $15^{3}\,h^{-3}\,\rm cMpc^{3}$ volume, and $\langle m \rangle$ is the mass contained in a $15^{3}\,h^{-3}\,\rm cMpc^{3}$ volume with density equal to the mean density, $\langle \rho\rangle=\rho_{\rm crit}\Omega_{\rm m}(1+z)^{3}$, where
\begin{equation}
\left<m\right> =4.25\times 10^{14}\,M_{\odot} \left(\frac{h}{0.678}\right)^{-1} \left(\frac{\Omega_{\rm m}}{0.308}\right).
\label{eq:mass}\end{equation}
\noindent
Masses above (below) this threshold represent overdense (underdense) volumes on $15\,h^{-1}\rm\,cMpc$ scales. Next, we associate every $\delta_{\rm m}$ with the Ly-$\alpha$\ forest effective optical depth, $\tau_{\rm eff}=-\ln\langle F \rangle$, obtained from the $15\,h^{-1}\rm\,cMpc$ segments of simulated spectrum that pass directly through the centre of each $15^{3}\,h^{-3}\,\rm cMpc^{3}$ volume\footnote{That is, the average transmission for each 1D spectral segment, $\langle F \rangle=e^{-\tau_{\rm eff}}$, is obtained by averaging over all the pixels in a section of simulated spectrum with length $15\,h^{-1}\rm\,cMpc$.}. We then compute $\delta_{\tau_\textrm{eff}}=\frac{\tau_\textrm{eff}-\left<\tau_\textrm{eff}\right>}{\left<\tau_\textrm{eff}\right>}$, where $\tau_\textrm{eff}$ is the effective optical depth measured in each $15h^{-1}\rm\,cMpc$ spectral segment, and $\left<\tau_\textrm{eff}\right>=0.20$ is obtained from \citet{Becker2013ASpectra} at $z=2.4$. In this way, we associate $\delta_{\tau_\textrm{eff}}$ from every 1D $15h^{-1}\rm\,cMpc$ segment with $\delta_{\rm m}$ in the surrounding 3D $15^{3}\,h^{-3}\,\rm cMpc^{3}$ volume \citep[see also][]{Cai2016MAppingMethodology}.
The relationship between $\delta_m$ and $\delta_{\tau_\textrm{eff}}$, averaged over $15h^{-1}~\textrm{cMpc}$ scales, is displayed in Figure~\ref{fig:ContourPS13_All}. In Sherwood, EAGLE and Illustris the bulk of the simulation box corresponds to volumes close to the mean density, and there is a weak positive correlation between $\delta_m$ and $\delta_{\tau_\textrm{eff}}$. The centre of mass for all protoclusters and most protogroups (shown by the blue, orange and brown symbols) reside in overdense volumes, although note these are not usually associated with segments of high Ly-$\alpha$\ opacity; the protocluster centre of mass will not always coincide with a halo or large value of $\tau_\textrm{eff}$\ (cf. Figure~\ref{fig:3Panel}). The differences between the three simulations are minimal at $\delta_{\tau_{\rm eff}}\lesssim 2$, suggesting that variations in numerical methodology have little impact on the properties of the low density gas on large scales. By contrast, more significant differences between the simulations are apparent at $\delta_{\tau_{\rm eff}}>2$, corresponding to segments containing high column density, self-shielded absorption systems. These systems reside in both overdense and underdense volumes. The number of $\delta_{\tau_{\rm eff}}>2$ segments is greatest in Illustris, which is the simulation with the largest number of absorption systems at $10^{17}\leq N_{\rm HI}/\rm cm^{-2}\leq 10^{20}$ (see Figure~\ref{fig:CDDF}).
Haloes and galaxies are biased tracers of mass overdensity, therefore we expect a correlation between the presence of the high column density systems -- responsible for the high $\delta_{\tau_\textrm{eff}}$\ tail in Figure~\ref{fig:ContourPS13_All} -- and $\delta_m$. In Figure~\ref{fig:MassNHIFrac} the fraction of $15h^{-1}~\textrm{cMpc}$ segments containing an absorption system, classed as either a DLA, SLLS, LLS or as Ly-$\alpha$\ forest based on the largest constituent column density in each segment, is shown in bins of $\delta_m$ for each of the simulations. In all three simulations, the vast majority of the volume -- irrespective of overdensity -- is traced by Ly-$\alpha$\ forest absorption, with 97.1, 98.0 and 93.5 per cent of $15h^{-1}~\textrm{cMpc}$ segments containing systems with a maximum column density of $N_{\rm HI}<10^{17.2}~\textrm{cm}^{-2}$ in Sherwood, EAGLE and Illustris, respectively. At all $\delta_m$, Illustris has a greater fraction of segments containing SLLSs than the other simulations, with $\sim20$ per cent of all segments associated with volumes of $\delta_m>1$ containing a LLS or SLLS. Note, however, that DLAs in Illustris are only present in segments associated with volumes of $\delta_{\rm m}\leq 0.8$, implying that absorption systems in the most overdense volumes in Illustris have slightly lower \hbox{H$\,\rm \scriptstyle I$}\ fractions relative to EAGLE and Sherwood. This is broadly consistent with the \hbox{H$\,\rm \scriptstyle I$}\ column density distribution, where Illustris contains an excess of absorption systems with $10^{17}\leq N_{\rm HI}/\rm cm^{-2}\leq 10^{20}$ relative to Sherwood and EAGLE. Figure~\ref{fig:MassNHIFrac} also demonstrates that damped systems with $N_{\rm HI}>10^{19}\rm\,cm^{-2}$ are present at all overdensities on $15h^{-1}~\textrm{cMpc}$ scales, explaining the scatter with $\delta_{\rm m}$ in the high $\delta_{\tau_\textrm{eff}}$\ tail observed in Figure~\ref{fig:ContourPS13_All}.
This highlights the importance of self-consistently modelling the incidence of high column density \hbox{H$\,\rm \scriptstyle I$}\ absorption systems when identifying overdense volumes using Ly-$\alpha$\ absorption. Damped systems produce large values of $\delta_{\tau_\textrm{eff}}$\ regardless of large-scale environment, and so segments of high $\delta_{\tau_\textrm{eff}}$\ will not uniquely probe overdense volumes. In addition to this, in all three simulations there is an increase in the fraction of segments containing systems with $N_{\rm HI} \geq 10^{19} \textrm{cm}^{-2}$ with increasing $\delta_m$. This implies that highly overdense volumes, such as those that may collapse to form a cluster by $z=0$, will contain a greater fraction of sight lines that pass through a damped system.
\subsection{CoSLAs: are they mass overdensities?}
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/CoSLAMassPDF_HydroWCai.pdf}
\vspace{-0.8cm}
\caption{Probability distribution of $\delta_\textrm{m}$ for $15h^{-1}~\textrm{cMpc}$ volumes with an associated $\delta_{\tau_\textrm{eff}} > 3.5$ (i.e. CoSLAs) after removing all sight lines containing damped systems in Sherwood (blue), \textsc{EAGLE} (orange) and Illustris (brown). The corresponding distribution from the analysis of a $1\,h^{-3}\,\rm cGpc^{3}$ post-processed collisionless dark matter simulation by \citet{Cai2016MAppingMethodology} is displayed as a black dashed line. The total number of CoSLAs identified in each simulation is: Illustris (36), EAGLE (32), Sherwood (22).}
\label{fig:CoSLAMassPDF}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.1\columnwidth]{Figures/LowMassVsHighMassSC.png}
\vspace{-0.8cm}
\caption{Examples of two $15h^{-1}~\textrm{cMpc}$ volumes with an associated $\delta_{\tau_\textrm{eff}} > 3.5$ (i.e. CoSLAs) in Sherwood, one of which is drawn from an underdense volume with $\delta_\textrm{m}=-0.09$ (left), with the other corresponding to an overdense volume with $\delta_\textrm{m}=0.98$ (right). In each case, the upper panels display the normalised gas density projected over a distance of $1h^{-1}~\textrm{cMpc}$ centred on the sight line along which the Ly-$\alpha$\ absorption spectrum is extracted (white dashed line), while the lower panels show the corresponding simulated Ly-$\alpha$\ absorption. The underdense CoSLA has $\delta_{\tau_{\rm eff}}=3.76$ and a maximum $N_{\rm HI}=10^{18}~\textrm{cm}^{-2}$, while the overdense CoSLA has $\delta_{\tau_{\rm eff}}=3.68$ and a maximum $N_{\rm HI}=10^{15.3}~\textrm{cm}^{-2}$.}
\label{fig:LowMassCont}
\end{figure}
From Figure~\ref{fig:MassNHIFrac}, it is evident that the majority of overdense volumes are traced by absorption from the Ly-$\alpha$\ forest and LLSs. We now explore whether any of these overdense volumes would be classified as being associated with Coherently Strong intergalactic Ly-$\alpha$\ Absorption systems (CoSLAs) within the hydrodynamical simulations. \citet{Cai2016MAppingMethodology} (hereafter C16) associate protoclusters with CoSLAs, defining these as all segments of Ly-$\alpha$\ absorption with $\delta_{\tau_\textrm{eff}}>3.5$ on $15h^{-1}~\textrm{cMpc}$ scales after excluding any high column density absorbers with damping wings. Therefore, before selecting CoSLAs, we first remove all sight lines that contain column densities $N_{\rm HI}>10^{19}~\textrm{cm}^{-2}$ in the simulated spectra. We take this approach to ensure that we not only remove the segments contain the damped system itself, but also any neighbouring segments where extended damping wings from these systems are still present. Due to the correlation between $\delta_m$ and the fraction of damped systems (see Figure~\ref{fig:MassNHIFrac}), this process preferentially removes segments corresponding to overdense volumes.
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/CoSLA-NHIFrac_QLyA.pdf}
\vspace{-0.8cm}
\caption{The filled histogram displays the number of $15h^{-1}~\textrm{cMpc}$ volumes associated with segments of $\delta_{\tau_\textrm{eff}} > 3.5$ (i.e. CoSLAs) in bins of $\Delta\delta_\textrm{m}=0.2$ for the QLy-$\alpha$\ simulation. The colour of each stacked bar indicates the class of absorber each segment corresponds to in Sherwood, which is identical to QLy-$\alpha$\ except for the lack of any cold ($T<10^{5}\rm\,K$), dense ($\Delta>10^{3}$) gas. The corresponding distribution for Sherwood is shown as the black dotted line, where there is a greater frequency of CoSLAs with $\delta_{\rm m}<0.4$. The total number of $15h^{-1}~\textrm{cMpc}$ segments with $\delta_{\tau_\textrm{eff}} > 3.5$ in each simulation is: Sherwood (22), QLy-$\alpha$\ (16).}
\label{fig:QLyACoSLA}
\end{figure}
C16 used a collisionless dark matter simulation coupled with LyMAS \citep{Peirani2014LyMAS:Field} to show that almost all CoSLAs correspond to mass overdensities in the range $0<\delta_m<2$, with $\sim70$ per cent having $\delta_m\geq0.5$. \citet{Chiang2013AncientProto-clusters} showed that, at similar redshifts and scales ($z=2$ and $16.3h^{-1}~\textrm{cMpc}$ respectively), more than 80 per cent of volumes with $\delta_m\geq0.8$ will collapse to form clusters with $M_{z=0}>10^{14}~\textrm{M}_\odot$ by $z=0$. The expectation is therefore that most CoSLAs will also probe structures that collapse to form clusters by $z=0$.
In Figure~\ref{fig:CoSLAMassPDF}, we examine the $\delta_m$ associated with CoSLAs within each of the three hydrodynamical simulations. The distributions for Sherwood and Illustris are consistent with one another, with a median $\delta_m=0.36\pm0.07$ and $\delta_{\rm m}=0.35\pm0.08$, respectively, where we estimate the uncertainty by bootstrapping with replacement. EAGLE has a higher median $\delta_{\rm m}=0.59\pm0.11$, formally differing from Illustris and Sherwood by $1.8\sigma$. However, given the relatively small number of sight lines with coherent absorption on $15h^{-1}~\textrm{cMpc}$ scales in the simulations, this may reflect differences in the rare, massive structures within each simulation. Regardless of the median of each distribution, however, in all three simulations the CoSLAs cover a broad $\delta_m$ range including $\delta_m\leq0$, indicating that they do not exclusively probe large scale mass overdensities at $z=2.4$. As a further illustration, Figure~\ref{fig:LowMassCont} displays examples of CoSLAs from both underdense ($\delta_m=-0.09$) and overdense ($\delta_m=0.98$) volumes in Sherwood. In both cases, we observe that a CoSLA arises from the alignment of structure along the line of sight. This can be due to LLSs associated with the alignment of several haloes punctuated by voids (left panel), or multiple lower column density absorbers associated with a more extended, filamentary structure (right panel).
These results differ from C16, who found CoSLAs to have a median $\delta_m=0.75\pm0.03$ and almost exclusively correspond to overdense volumes. To explore the reason for this difference, we also consider the incidence of CoSLAs in the QLy-$\alpha$\ simulation. The QLy-$\alpha$\ model does not produce any damped systems and under produces LLSs (see Figure~\ref{fig:CDDF}) due to missing high density gas, but is otherwise identical to Sherwood. Consequently, the QLy-$\alpha$\ model should better approximate the lower resolution simulation\footnote{C16 use a collisionless dark matter simulation in a $1h^{-3}\rm\,cGpc^{3}$ volume with $1024^{3}$ particles, yielding a particle mass of around $9.6\times 10^{10}\rm\,M_{\odot}$. For comparison, the typical dark matter particle mass needed to resolve the small scale structure of the Ly-$\alpha$\ forest at $z\simeq 2$ is $\sim10^{7}M_{\odot}$ \citep{Bolton2009ResolvingSimulations}.} used in C16.
In Figure~\ref{fig:QLyACoSLA}, the $\delta_m$ distribution of CoSLAs in Sherwood is compared to the QLy-$\alpha$\ simulation (this time in terms of number in each bin). The CoSLAs identified in QLy-$\alpha$\ are coloured according to the maximum column density drawn from the matching locations within the Sherwood simulation. There are several points to note from Figure~\ref{fig:QLyACoSLA}. First, QLy-$\alpha$\ contains two CoSLAs that are in segments containing SLLSs or DLAs in Sherwood (meaning these segments were discarded), while Sherwood contains eight segments classified as CoSLAs that are not present in QLy-$\alpha$. These eight CoSLAs all contain LLSs or high column density Ly-$\alpha$\ forest systems in Sherwood, but have lower column densities -- and therefore lower values of $\delta_{\tau_\textrm{eff}}$\ -- in QLy-$\alpha$. Furthermore, all these additional systems have $\delta_m<0.4$, which acts to dilute the correlation between $\delta_m$ and $\delta_{\tau_\textrm{eff}}$. This emphasises the importance of correctly capturing LLSs in the models; not only will a fraction of the CosLAs in the QLy-$\alpha$\ model be contaminated by the presence of damped systems, but additional systems with $10^{16}\lesssim N_{\rm HI}/\rm cm^{-2}\leq 10^{19}$ are missed, resulting in an erroneously strong correlation between $\delta_m$ and $\delta_{\tau_\textrm{eff}}$\ for CoSLAs. Second, the CoSLAs in QLy-$\alpha$\ have a median $\delta_m=0.40\pm0.02$, with $\sim50$ per cent of CoSLAs having $\delta_m\geq0.5$. This median $\delta_{\rm m}$ is consistent with the Sherwood simulation, although note it is still smaller when compared with C16.
\subsection{Effect of box size and mass resolution on CoSLAs}\label{ssec:ResBox}
Another important factor to consider is the box size (and therefore maximum cluster mass) and mass resolution used in the simulations. We examine how mass resolution and box size impact on the relationship between mass overdensities and $\delta_{\tau_\textrm{eff}}$\ on $15h^{-1}~\textrm{cMpc}$ scales using four QLy-$\alpha$\ runs drawn from the Sherwood simulation suite. These simulations are summarised in Table~\ref{tab:ConvSims}. As previously, sight lines were extracted with the same average transverse separation of $0.8h^{-1}\rm\, cMpc$ from all runs.
\begin{table*}
\centering
\caption{Hydrodynamical simulations used to test the effect of box size and mass resolution on CoSLAs. All four simulations were performed using the quick-Ly-$\alpha$\ approach that excludes dense, star forming gas (see Section~\ref{sec:Sims}). The columns list, from left to right: the simulation name, box size, number of resolution elements, dark matter and gas particle masses, and the number of CoSLAs per unit volume obtained using sight lines with an average transverse separation of $0.8h^{-1}\rm\,cMpc$. All of the $80h^{-1}\rm\,cMpc$ boxes were performed with initial conditions generated using the same random seed.}
\begin{tabular}{lcccccc}
\hline
Name & Box size & $N_\textrm{tot}$ & $M_\textrm{dm}$ & $M_\textrm{gas}$ & $N_{\rm CoSLAs}/V$ \\
& [$h^{-1}\ $cMpc] & & [$\textrm{M}_\odot$] & [$\textrm{M}_\odot$] & [$ h^{3} \,\rm cMpc^{-3}$]\\
\hline
80-512 & 80 & $2\times 512^3$ & $4.06\times10^8$ & $7.52\times10^7$ & $4.9\times10^{-5}$ \\
80-1024 & 80 & $2\times 1024^3$ & $5.07\times10^7$ & $9.41\times10^6$ & $3.1\times 10^{-5}$\\
80-2048 & 80 & $2\times 2048^3$ & $6.34\times10^6$ & $1.17\times10^6$ & $1.4\times 10^{-5}$\\
160-2048 & 160 & $2\times 2048^3$ & $5.07\times10^7$ & $9.41\times10^6$ & $5.5\times 10^{-5}$\\
\hline
\end{tabular}
\label{tab:ConvSims}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Figures/CoSLA_PDF_BoxSize.pdf}
\vspace{-8mm}
\caption{The probability distribution of CoSLAs as a function of $\delta_m$ (upper panel) and $\delta_{\tau_\textrm{eff}}$\ (lower panel) for the QLy-$\alpha$\ model (80-1024) compared to a simulation with the same mass resolution but a volume eight times larger (160-2048). The $\delta_{\rm m}$ distribution from C16 is displayed as the dashed black histogram in the upper panel.}
\label{fig:CoSLA-BoxComp}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Figures/CoSLA_PDF_Resolution.pdf}
\vspace{-8mm}
\caption{The probability distribution of CoSLAs as a function of $\delta_m$ (upper panel) and $\delta_{\tau_\textrm{eff}}$\ (lower panel) for the QLy-$\alpha$\ model (80-1024) compared to a simulations with the same box size but a particle mass that is eight times larger (80-512) or smaller (80-2048). The $\delta_{\rm m}$ distribution from C16 is displayed as the dashed black histogram in the upper panel.}
\label{fig:CoSLA-ResComp}
\end{figure}
\begin{figure}
\centering
\includegraphics{Figures/CoSLA_SpecResComp.pdf}
\vspace{-4mm}
\caption{Simulated CoSLAs from the same segment drawn from the 80-512 (blue) and 80-2048 (dark red) models. This segment corresponds to $\delta_{\tau_{\rm eff}}=3.90$ in 80-512 but only $\delta_{\tau_{\rm eff}}=3.67$ in 80-2048, due to the increased transmission from the better resolved underdense gas in the higher resolution model.}
\label{fig:SpecResComp}
\end{figure}
The effect of an increase in box size on the $\delta_m$ and $\delta_{\tau_\textrm{eff}}$\ distributions for CoSLAs is displayed in Figure~\ref{fig:CoSLA-BoxComp}. The $\delta_m$ distribution in the largest volume simulation (160-2048) extends toward higher values as a consequence of the rarer, more massive systems present in the larger volume. The number of CoSLAs per unit volume increases by $77$ per cent in the 160-2048 simulation (see Table~\ref{tab:ConvSims}) relative to the fiducial 80-1024 (QLy-$\alpha$) model, although both the 80-1024 and 160-2048 models share similar median values of $\delta_{\rm m}=0.40\pm0.02$
and $\delta_{\rm m}=0.48\pm0.05$, respectively. The corresponding $\delta_{\tau_\textrm{eff}}$\ distribution displays similar behaviour, with a tail that extends to larger values of $\delta_{\tau_\textrm{eff}}$\ in the 160-2048 model. The effect of the $\sim 1000$ times larger volume used by C16 therefore likely explains some of the differences relative to this work. The CoSLAs selected in C16 (shown by the dashed black histogram in the upper panel of Figure~\ref{fig:CoSLA-BoxComp}) probe rarer, more massive systems that are not present in our small, higher resolution models. We are unable to reliably assess how effectively CoSLAs probe overdense $15^{3}h^{-3}\rm\,cMpc^{3}$ volumes with $\delta_{\rm m} > 1$ using our fiducial $80^{3}h^{-3}\rm\, cMpc^{-3}$ simulation box. However, this still does not fully explain the greater incidence of CoSLAs at overdensites of $\delta_{\rm m}<0.5$ within the hydrodynamical simulations compared to C16, even when using QLy-$\alpha$\ models that underpredict the number of CoSLAs at $\delta_{\rm m}\leq 0.4$ due to missing LLSs. A possible explanation is that the lack of high $\tau_{\rm eff}$ systems at $\delta_{\rm m}$ is even more pronounced in much larger collisionless simulations, due to unresolved high column density absorption from small scale structure, but the exact reason for this difference remains unclear.
Mass resolution also has an important -- but more subtle -- effect on the typical $\delta_{\tau_\textrm{eff}}$\ and $\delta_m$ associated with CoSLAs. The top panel of Figure~\ref{fig:CoSLA-ResComp} highlights how the $\delta_{\rm m}$ distribution of CoSLAs is shifted toward lower values for the 80-512 simulation when compared to the two higher resolution models. The median $\delta_{\rm m}=0.38\pm0.01$ for the 80-512 model, compared to median values of $\delta_{\rm m}=0.40\pm0.02$ and $\delta_{\rm m}=0.42\pm0.04$ for 80-1024 (QLy-$\alpha$) and 80-2048, respectively. However, the average number of CoSLAs per unit volume also decreases by a factor $\sim 2$ when increasing the mass resolution by a factor of 8 (see Table~\ref{tab:ConvSims}), suggesting this quantity is not yet converged with mass resolution.
The lower panel of Figure~\ref{fig:CoSLA-ResComp} elucidates the origin of this behaviour: at higher mass resolution the high $\delta_{\tau_\textrm{eff}}$\ tail of the CoSLA distribution is truncated in comparision with the lower resolution runs. This is a consequence of the failure of the lower resolution runs to correctly resolve the structure of the Ly-$\alpha$\ absorption, particularly in underdense regions, which are too opaque in low mass resolution models \citep{Bolton2009ResolvingSimulations}. The effect of this poorly resolved underdense gas is to increase the number of volumes on $15h^{-1}\rm\,cMpc$ scales that are associated with segments which exceed the CoSLA identification threshold of $\delta_{\tau_{\rm eff}}=3.5$. An illustration of this effect is shown in Figure~\ref{fig:SpecResComp}, where Ly-$\alpha$\ absorption from a randomly selected segment in the 80-2048 simulation (dark red) is compared directly to the 80-512 simulation (blue). This implies that -- in addition to not capturing absorption from high column density, self-shielded absorbers -- low mass resolution models that do not adequately resolve the structure of the Ly-$\alpha$\ forest on small scales will overpredict the incidence of CoSLAs above a fixed $\delta_{\tau_\textrm{eff}}$\ threshold.
\section{Characterising protoclusters in Ly-$\alpha$\ absorption}\label{ssec:PCFrac}
We now address the question of how protoclusters, as opposed to overdensities, are characterised by Ly-$\alpha$\ absorption. Whilst -- observationally -- protoclusters are identified as overdense regions at $z\sim2$, when using simulations we have access to \emph{a priori} knowledge of which structures will collapse to form a cluster along with the associated cluster mass. We now use this simulation based \lq\lq true" definition of a protocluster to examine the opacity of protoclusters at $z=2.4$ on $15h^{-1}~\textrm{cMpc}$ scales. We do not expect variations in the numerical methodology used by Sherwood, EAGLE and Illustris to significantly change these results, so from this point onward we focus on analysing only the Sherwood simulation.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Figures/PC_Hist_wNHI_Sherwood.pdf}
\vspace{-0.8cm}
\caption{Top: Total number of $15h^{-1}\rm\,cMpc$ segments in bins of $\Delta \delta_{\tau_{\rm eff}}=0.3$.
Middle: The fraction of $15h^{-1}\rm\,cMpc$ segments in bins of $\Delta \delta_{\tau_{\rm eff}}=0.3$ associated with protoclusters or protogroups. The colour of each stacked bar indicates the mass of the protocluster or protogroup associated with the absorption.
Bottom: probability distribution for all $15h^{-1}\rm\,cMpc$ segments that are associated with protoclusters. The colour of each stacked bar indicates the maximum column density within each segment.}
\label{fig:PCFrac}
\end{figure}
\begin{figure*}
\includegraphics[width=\textwidth]{Figures/PcTeffExampleRenders.png}
\vspace{-0.8cm}
\caption{Example of four different $15h^{-1}\rm\,cMpc$ segments that pass through protoclusters in Sherwood. The top panels show projected maps of the normalised gas density in a $15h^{-2}\rm\,cMpc^{2}$ slice with a projection depth of $1h^{-1}~\textrm{cMpc}$. The white dashed line shows the direction in which the Ly-$\alpha$\ absorption spectrum was extracted, while the yellow circles indicate the cross section of all protoclusters that intersect with the slice. The bottom panels show the corresponding mock Ly-$\alpha$\ absorption spectra. From left to right, each segment corresponds to: a low opacity segment with $\delta_{\tau_{\rm eff}}=-0.20$ containing a maximum \hbox{H$\,\rm \scriptstyle I$}\ column density $N_{\rm HI}=10^{13.8}~\textrm{cm}^{-2}$, a higher opacity segment with $\delta_{\tau_{\rm eff}}=1.72$ and maximum $N_{\rm HI}=10^{14.8}~\textrm{cm}^{-2}$, a CoSLA with $\delta_{\tau_{\rm eff}}=4.57$ and maximum $N_{\rm HI}=10^{16.9}~\textrm{cm}^{-2}$, and a DLA with $\delta_{\tau_{\rm eff}}=4.65$ and maximum $N_{\rm HI}=10^{20.5}~\textrm{cm}^{-2}$.}
\label{fig:PCTeffEx}
\end{figure*}
In Figure~\ref{fig:PCFrac}, the fraction of all $15h^{-1}\rm\,cMpc$ segments in bins of $\Delta$$\delta_{\tau_\textrm{eff}}$~$=0.3$ that are associated\footnote{A $15h^{-1}~\textrm{cMpc}$ segment is associated with a protocluster or protogroup if at least one third of the segment passes within $R_{95}$\ of any protocluster/group.} with protoclusters (blue shading) or protogroups (orange and brown shading) are displayed in the central panel. In the event that a system is associated with more than one protocluster, it is associated with the most massive one only. The corresponding number of $15h^{-1}~\textrm{cMpc}$ segments in each bin is shown in the upper panel, with most segments close to the mean, $\delta_{\tau_{\rm eff}}=0$.
Note that in almost all bins, the majority of the ``associated" segments are in the protoclusters with $M_{\rm z=0}\geq 10^{14}\,M_{\odot}$. This in part reflects the fact that the protoclusters occupy a larger fraction of the simulation volume relative to the protogroups\footnote{Protoclusters occupy 7.7 per cent of the Sherwood simulation volume at $z=2.4$, whereas the large and small protogroups together only occupy 5.0 per cent.)}. Typically fewer than 10 per cent of segments with $\delta_{\tau_{\rm eff}} \lesssim 0$ are associated with protoclusters, increasing to $\sim 40$ per cent at $\delta_{\tau_{\rm eff}}=2$. At $\delta_{\tau_{\rm eff}}\gtrsim2$, however, this fraction declines, where on average only $\sim25$ per cent of segments are associated with protoclusters. Figure~\ref{fig:PCFrac} therefore demonstrates that for $\delta_{\tau_{\rm eff}}<2$ there is a positive, albeit weak, correlation between $\delta_{\tau_\textrm{eff}}$\ and the likelihood of the associated volume collapsing to form a cluster by $z=0$.
In the lower panel of Fig.~\ref{fig:PCFrac}, the probability distribution for $15h^{-1}\rm\,cMpc$ segments residing within the protoclusters is displayed as a function of $\delta_{\tau_\textrm{eff}}$, with the shading indicating the fraction of segments in each bin that contains a given maximum \hbox{H$\,\rm \scriptstyle I$}\ column density. The majority of segments passing through protoclusters only contain Ly-$\alpha$\ forest absorption systems and have $\delta_{\tau_\textrm{eff}}\sim0$: 28 per cent of the segments which are associated with protoclusters are Ly-$\alpha$\ forest with $\delta_{\tau_{\rm eff}}<0$, and 84 per cent are entirely Ly-$\alpha$\ forest with $\delta_{\tau_{\rm eff}}<1$. Thus, while the majority of segments with $\delta_{\tau_\textrm{eff}}$$<1$ do not correspond to protoclusters, but rather to ``field'' regions, the majority of
segments associated \emph{with} protoclusters lie in the same $\delta_{\tau_\textrm{eff}}$\ range and share identical characteristics. This reflects the fact that protoclusters at $z\simeq 2.4$ are extended over large scales \citep{Muldrew2015WhatProtoclusters};
this means that it is very difficult -- or impossible -- to disentangle most protocluster sight lines from the field on $15h^{-1}~\textrm{cMpc}$ scales.
At larger values of $\delta_{\tau_\textrm{eff}}$\ the segments associated with protoclusters become dominated by SLLSs and DLAs. At $\delta_{\tau_{\rm eff}}>2.9$, the majority of segments associated with protocluster contain damped systems, and 96 per cent of the segments with $\delta_{\tau_{\rm eff}} \gtrsim 3.5$ (i.e. the threshold for CoSLAs) that are associated with protoclusters arise from high density gas producing damped Ly-$\alpha$\ absorption. At $\delta_{\tau_{\rm eff}}>3.5$ SLLSs and DLAs are the dominant probe of gas that forms clusters by $z=0$, but -- critically for protocluster selection by Ly-$\alpha$\ absorption -- the middle panel of Figure.~\ref{fig:PCFrac} shows that these are not uniquely associated with protoclusters at $z\simeq 2.4$, but are more likely to be in the field.
Figure~\ref{fig:PCTeffEx} displays selected examples of segments that pass through protoclusters that cover the full range of $\delta_{\tau_\textrm{eff}}$\ observed in the simulation. Most sight lines passing through protoclusters with $\delta_{\tau_{\rm eff}}<2$ are exemplified by the two left-hand panels in Figure~\ref{fig:PCTeffEx}. In these cases the sight line passes through two filamentary structures perpendicular to the sight line (left panel, $\delta_{\tau_{\rm eff}}=-0.20$) or through a small region of overdense gas within a protocluster (second left panel, $\delta_{\tau_{\rm eff}}=1.72$), but in both cases do not exhibit extended regions of saturated Ly-$\alpha$\ absorption.
By contrast, in the two right-hand panels of Figure~\ref{fig:PCTeffEx}, two examples of segments that pass through protoclusters and have $\delta_{\tau_{\rm eff}}>3.5$ (i.e. the CoSLA threshold) are displayed. The right-most panel ($\delta_{\tau_{\rm eff}}=4.65$) displays strong absorption due to the presence of an extended damping wing arising from a DLA; there is no large scale gas overdensity present along the sight line. The segment displayed on the second right ($\delta_{\tau_{\rm eff}}=4.57$) has no damped absorbers, but instead the absorption arises from haloes and a filament that are aligned with the sight line and are associated with two intersecting protoclusters. The high value of $\delta_{\tau_\textrm{eff}}$\ in this case results from extended Ly-$\alpha$\ forest absorption. This implies that it is the orientation of dense gas with respect to the sight line that has the greatest impact on the value of $\delta_{\tau_\textrm{eff}}$, rather than either the overdensity $\delta_{\rm m}$ (see also Figure~\ref{fig:LowMassCont}) or the presence of a protocluster.
\section{Selecting protoclusters with line of sight Ly-$\alpha$\ absorption}\label{sec:CompCont}
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/PCVsFieldMassPDF_Stacked.pdf}
\vspace{-0.8cm}
\caption{The probability distribution of CoSLAS identified in Sherwood as a function of $\delta_\textrm{m}$, coloured by whether the CoSLA is associated with a protocluster (12; blue) or associated with the field (10; orange).}
\label{fig:PCVsFldPDF}
\end{figure}
Now that we have characterised the Ly-$\alpha$\ absorption associated with protoclusters, we finally turn to examine the effectiveness with which one can select protocluster regions using line of sight Ly-$\alpha$\ absorption. We have established that most of the sight lines that pass through protoclusters with $M_{\rm z=0}\sim 10^{14}\,M_{\odot}$ exhibit low values of $\delta_{\tau_\textrm{eff}}$, and it is impossible to distinguish these from the field. On the other hand, for large values of $\delta_{\tau_\textrm{eff}}$\ most of the segments associated with protoclusters are the result of damped absorption systems with $N_{\rm HI}>10^{19}~\textrm{cm}^{-2}$, but these systems do not uniquely trace protoclusters.
In addition, in the simulations there are a small subset of high opacity segments within protoclusters that do not exhibit damped absorption (e.g. the example CoSLA shown in the second from right panel in Figure~\ref{fig:PCTeffEx}). This is demonstrated further in Figure~\ref{fig:PCVsFldPDF}, which shows the distribution of CoSLAs in Sherwood split into two groups as a function of $\delta_{\rm m}$: those that are associated with a protocluster and those that are not. We find 55 per cent of CoSLAs are associated with protoclusters with a median $\delta_m = 0.4\pm0.1$, in comparison to CoSLAs that probe the field, with median $\delta_m = 0.06\pm0.05$. This indicates that, while the two populations cannot be separated by their Ly-$\alpha$\ absorption spectra alone, approximately half of CoSLAs -- which in general arise from the alignment of overdense structure along the line of sight -- are indeed associated with protoclusters. Thus we find that CoSLAs are not a unique tracer of protoclusters, even when assuming perfect removal of damped systems from the mock data.
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/Completeness_and_Contamination_80_1024_ps13+SS_z2p400.pdf}
\vspace{-0.8cm}
\caption{Contamination (dashed) and \lq\lq completeness" (solid) for protoclusters (with $M_{\rm z=0}\geq 10^{14}M_{\odot}$) when selecting all segments above a fixed threshold of $\delta_{\tau_\textrm{eff}}$\ on $15h^{-1}\rm\,cMpc$ scales (see text for details). The blue curves correspond to the case where all absorption is considered, orange curves to the case where sight lines containing DLAs are removed, and brown curves to the case where SLLSs and DLAs are removed. Results are displayed only where there is more than one $15h^{-1}\rm\,cMpc$ segment in each bin.}
\label{fig:ComContPs13+SS}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/PCTauEffSurFuncSherwood_NewMasks.pdf}
\vspace{-0.8cm}
\caption{Reverse cumulative distribution functions, $P(>\delta_{\tau_\textrm{eff}})$, for all $15h^{-1}\rm\,cMpc$ segments that are associated with each of the 29 protoclusters in Sherwood. Each line is coloured according to the mass of the cluster at $z=0$. The grey shaded region indicates the area above the CoSLA selection threshold of $\delta_{\tau_\textrm{eff}} > 3.5$.}
\label{fig:SurFunc}
\end{figure}
\begin{figure*}
\centering
\includegraphics{Figures/4SelectedPcExampleRenders.png}
\vspace{-5mm}
\caption{Example of four different CoSLAs that pass through protoclusters in Sherwood. The top panels show projected maps of the normalised gas density in a $15h^{-2}\rm\,cMpc^{2}$ slice with a projection depth of $1h^{-1}~\textrm{cMpc}$. The white dashed line shows the direction in which the Ly-$\alpha$\ absorption spectrum was extracted. The bottom panels show the corresponding mock Ly-$\alpha$\ absorption spectra. The numbers in the upper left of each panel correspond to the CoSLAs listed in Table~\ref{tab:PC_CoSLAs}. From left to right, the maximum \hbox{H$\,\rm \scriptstyle I$}\ column density associated with each CoSLA is $N_{\rm HI}=10^{16.5}~\textrm{cm}^{-2}$, $10^{14.9}~\textrm{cm}^{-2}$, $10^{19.0}~\textrm{cm}^{-2}$ and $10^{14.7}~\textrm{cm}^{-2}$.}
\label{fig:PCCosLARenders}
\end{figure*}
The effective optical depth threshold that defines a CoSLA as a segment exhibiting $\delta_{\tau_{\rm eff}}>3.5$ on $15h^{-1}\rm\,cMpc$ scales is to some extent arbitrary. We may therefore consider whether a fixed threshold in $\delta_{\tau_{\rm eff}}$ can be chosen that minimises the contamination within a sample of protoclusters selected from the mock spectra. We define the contamination as $N(\geq \delta_{\tau_{\rm eff}})/N_{\rm tot}(\geq \delta_{\tau_{\rm eff}})$, where $N(\geq \delta_{\tau_{\rm eff}})$ is the number of $15h^{-1}\rm\,cMpc$ segments above the $\delta_{\tau_\textrm{eff}}$\ threshold that do not pass through at least 5$h^{-1}\rm\,cMpc$ of a protocluster volume, and $N_{\rm tot}(\geq \delta_{\tau_{\rm eff}})$ is the total number of $15h^{-1}\rm\,cMpc$ segments above the $\delta_{\tau_\textrm{eff}}$\ threshold in the entire simulation volume. We also define the sample \lq\lq completeness" for a given $\delta_{\tau_\textrm{eff}}$\ threshold as the fraction of protoclusters with at least one sight line that exhibits absorption on $15h^{-1}\rm\,cMpc$ scales above the $\delta_{\tau_{\rm eff}}$ threshold. This is defined as $N_{\rm pc}(\geq \delta_{\tau_{\rm eff}})/N_{\rm pc, tot}$, where $N_{\rm pc}(\geq \delta_{\tau_{\rm eff}})$ is the number of unique protoclusters probed by $15h^{-1}\rm\,cMpc$ segments above the $\delta_{\tau_\textrm{eff}}$\ threshold, and $N_{\rm pc,tot}$ is the total number of protoclusters in the simulation (i.e. 29 for Sherwood, see Table~\ref{tab:Sims}).
These values are plotted as a function of the $\delta_{\tau_\textrm{eff}}$\ threshold in Figure~\ref{fig:ComContPs13+SS} for three different cases: where absorption in all the sight lines is considered (blue curves), where sight lines containing DLAs are removed from the sample (orange curves), and the case where both SLLSs and DLAs are removed (brown curves). When considering all sight lines, the protocluster sample remains 100 per cent complete using a threshold up to $\delta_{\tau_\textrm{eff}} \sim 6$ because all protoclusters contain DLA systems that produce such high $\delta_{\tau_\textrm{eff}}$. The contamination also remains very high at $\sim 75$ per cent, which reflects the fact that the damped Ly-$\alpha$\ absorbers do not uniquely probe protocluster environments. If we instead assume perfect removal of all sight lines containing DLAs, the contamination rate falls slightly to around 65 per cent for the segments with the strongest absorption. Finally, removing all the $15h^{-1}\rm\,cMpc$ segments containing SLLSs as well as DLAs causes the contamination to drop to zero at $\delta_{\tau_\textrm{eff}}=4.1$, i.e. all the remaining segments with this $\delta_{\tau_\textrm{eff}}$\ uniquely trace protocluster gas. Consequently, if we assume perfect removal of damped absorption systems, it is possible to obtain a perfectly clean sample of protoclusters by applying a sufficiently high threshold in $\delta_{\tau_\textrm{eff}}$.
This clean sample of protoclusters is, however, highly incomplete as not all protoclusters will exhibit strong coherent Ly-$\alpha$\ absorption on $15h^{-1}\rm\,cMpc$ scales. This can be seen in Figure~\ref{fig:ComContPs13+SS}, where the drop in contamination is accompanied by the completeness falling to only $17$ per cent. This means that less than a fifth\footnote{The completeness is likely to be strongly dependent on the protocluster mass. Since we do not probe massive protoclusters -- due to our small box size -- this completeness is likely to be a lower limit.} of protoclusters have any sight lines with $\delta_{\tau_{\rm eff}}>4.1$. This is further exemplified in Figure~\ref{fig:SurFunc}, which shows the reverse cumulative distribution function of $\delta_{\tau_\textrm{eff}}$\ for each of the 29 protoclusters in Sherwood. The colour of each line corresponds to the mass of the resulting cluster at $z=0$. This demonstrates that high $\delta_{\tau_{\rm eff}}$ segments that pass through protoclusters are rare: less than 0.1 per cent of segments that pass through $M_{\rm z=0}\sim 10^{14}\rm\,M_{\odot}$ protoclusters have $\delta_{\tau_{\rm eff}}>3.5$ -- corresponding to the CoSLA theshold defined by C16, shown by the grey shading.
All the CoSLAs associated with protoclusters in Sherwood are listed in detail in Table~\ref{tab:PC_CoSLAs}: a total of 14 unique protoclusters over a mass range of $10^{14.0}$--$10^{14.7}~\textrm{M}_\odot$ are probed by 12 unique CoSLAs, and 42 per cent of the CoSLAs associated with protoclusters pass through more than one protocluster. A sample of four of the CoSLAS associated with protoclusters in Sherwood are displayed in Figure~\ref{fig:PCCosLARenders}. We observe again that typically there is an alignment of structure along the sight line that causes the coherent Ly-$\alpha$\ absorption. Orientation of structure to the line of sight, rather than association with a protocluster of a given mass or an overdensity, appears to be a critical factor that determines the extended nature of the Ly-$\alpha$\ absorption. In this context, we briefly note that \citet{Finley2014AstrophysicsProtocluster} argued for the detection of an intergalactic filament based on observations of multiple LLSs and SLLSs with $N_{\rm HI}>10^{18}\rm\,cm^{-2}$ along two closely separated quasar sight lines at $z=2.69$. The seven strong \hbox{H$\,\rm \scriptstyle I$}\ absorption systems observed by \citet{Finley2014AstrophysicsProtocluster} span $\sim 1700\rm\,km\,s^{-1}$, corresponding to $16.6h^{-1}\rm\,cMpc$ at $z=2.69$. While \citet{Finley2014AstrophysicsProtocluster} could not definitively rule out association of the \hbox{H$\,\rm \scriptstyle I$}\ absorbers with a protocluster, their favoured interpretation is broadly consistent with our analysis. Taken a step further, this suggests that CoSLAs may in fact be a tracer of extended filamentary structure in the early Universe. An intriguing possibility is that CoSLAs, and their association (if any) with galaxies and/or metal absorption lines, may therefore provide a route to identifying and studying filamentary environments at $z>2$.
\begin{table}
\centering
\caption{All CoSLAs (defined as $15h^{-1}\rm\,cMpc$ segments with $\delta_{\tau_{\rm eff}}>3.5$ that do not contain a damped Ly-$\alpha$\ absorber) in Sherwood that are associated with protoclusters. A total of 14 protoclusters are probed by 12 unique CoSLAs. Note that each CoSLA may probe multiple protoclusters.}
\begin{tabular}{c|c|c|c}
\hline
CoSLA & $\delta_{\tau_\textrm{eff}}$ & $\delta_m$ & Protocluster $M_{z=0}$ \\
& & & $\left[\log_{10}(\textrm{M}_\odot)\right]$\\
\hline
1 & 4.57 & 0.56 & 14.54, 14.04\\
2 & 4.46 & 0.80 & 14.72, 14.28\\
3 & 4.18 & 0.38 & 14.19\\
4 & 4.05 & 0.37 & 14.25\\
5 & 4.05 & 0.42 & 14.41\\
6 & 3.99 & 0.12 & 14.05 \\
7 & 3.93 & 0.35 & 14.22\\
8 & 3.80 & 0.72 & 14.35, 14.18\\
9 & 3.68 & 0.98 & 14.59, 14.53\\
10 & 3.67 & 0.70 & 14.72, 14.53\\
11 & 3.56 & 0.62 & 14.59\\
12 & 3.53 & 0.12 & 14.06\\
\hline
\end{tabular}
\label{tab:PC_CoSLAs}
\end{table}
\section{Conclusions}\label{sec:Conclusion}
In this work we have used state of the art hydrodyamical simulations from the Sherwood \citep{Bolton2017The5}, EAGLE \citep{Schaye2015TheEnvironments, Crain2015TheVariations, McAlpine2016TheCatalogues} and Illustris \citep{Vogelsberger2014IntroducingUniverse, Nelson2015TheRelease} projects to examine the signature of protoclusters observed in Ly-$\alpha$\ absorption at $z\simeq 2.4$. Building upon earlier work using low resolution collisionless dark matter simulations \citep[e.g.][]{Stark2015ProtoclusterMaps,Cai2016MAppingMethodology}, here we use models that resolve small scale structure in the IGM, correctly reproduce the incidence of \hbox{H$\,\rm \scriptstyle I$}\ absorption systems (including high column densities that are self-shielded to Lyman continuum photons), and track the formation of structure to $z=0$.
We examine the impact of small scale gas structure on the signature of large scale overdensities on $15h^{-1}\rm\,cMpc$ scales, finding that the simulation mass resolution required for resolving Ly-$\alpha$\ absorption on small scales is also a requirement for correctly modelling the average $\tau_{\rm eff}$ on $15h^{-1}~\textrm{cMpc}$ scales. This is necessary for capturing the incidence of \hbox{H$\,\rm \scriptstyle I$}\ absorption systems over a wide range of column densities, including damped systems and LLSs. At the same time, however, adequate mass resolution is necessary for correctly capturing the opacity of underdense regions in the IGM. A mass resolution that is too low will overpredict the typical Ly-$\alpha$\ effective optical depth on $15h^{-1}~\textrm{cMpc}$ scales.
We furthermore assess the prevalance of coherent Ly-$\alpha$\ absorption within protoclusters at high redshift. Our main conclusions may be summarised as follows:
\begin{itemize}
\item We confirm there is a weak correlation between the mass overdensity, $\delta_m$, and the effective optical depth relative to the mean, $\delta_{\tau_\textrm{eff}}$, on a $15~h^{-1}~\textrm{cMpc}$ scales, in the simulations, although there is a large amount of scatter that, particularly at large values of $\delta_{\tau_\textrm{eff}}$, means it is not possible to uniquely identify large scale overdensities with strong Ly-$\alpha$\ absorption. This remains true even if first removing all damped Ly-$\alpha$\ absorption systems that arise from dense, neutral gas on small scales.
\item We examine the properties of coherently strong intergalactic Ly-$\alpha$\ absorption systems (CoSLAs) in the simulations. CoSLAs -- defined by C16 as regions on $15h^{-1}\rm\,cMpc$ scales with $4.5$ times the average Ly-$\alpha$\ effective optical depth after excluding damped absorption systems -- are rare objects, only accounting for 0.1 per cent of all $15h^{-1}~\textrm{cMpc}$ spectral segments drawn from the models. They probe a wide range in mass overdensity, $\delta_{\rm m}$, including underdense regions on $15h^{-1}\rm\,cMpc$ scales, and so do not uniquely trace significant mass overdensities.
\item Protoclusters with $M_{\rm z=0}\simeq 10^{14}\,M_{\odot}$ exhibit a broad range of signatures in Ly-$\alpha$\ absorption, with $\delta_{\tau_{\rm eff}}$ ranging from $-0.5$ to $>8$. However, the vast majority (84 per cent) of sight lines passing through what we define as protoclusters in the simulations contain only low column density Ly-$\alpha$\ forest absorption and have $\delta_{\tau_{\rm eff}}<1$. This signature is identical to the field, so the majority of sight lines through protoclusters do not bear a telltale signature in Ly-$\alpha$\ absorption. Most sight lines with high $\delta_{\tau_{\rm eff}}$ are a result of passing through a damped absorption system. A small subset of sight lines through protoclusters do, however, exhibit high $\delta_{\tau_{\rm eff}}$ due to coherently strong intergalactic Ly-$\alpha$\ absorption systems i.e. CoSLAs. LLSs and high column density Ly-$\alpha$\ forest absorbers are typically responsible for this absorption.
\item Assuming the perfect removal of damped Ly-$\alpha$\ absorption systems, CoSLAs are a good but non-unique probe of protoclusters at $z\simeq 2.4$. In the Sherwood simulation, approximately half of CoSLAs with $\delta_{\tau_{\rm eff}}>3.5$ trace protoclusters with $10^{14} \leq M_{\rm z=0}/M_{\odot} \leq 10^{14.7}$. The other 46 per cent of CoSLAs arise from LLSs that are aligned along the line of sight.
\item We find that threshold of $\delta_{\tau_{\rm eff}}>4.1$ -- corresponding to regions on $15h^{-1}\rm\,cMpc$ scales with $5.1$ times the average Ly-$\alpha$\ effective optical depth after excluding damped absorption systems -- enables us to select a completely pure sample of protoclusters from simulated spectra. However, CoSLAs are rare within the volumes that protoclusters occupy: less than $0.1$ per cent per cent of sight lines that pass through at least $5h^{-1}\rm\,cMpc$ of a protocluster volume exhibit $\delta_{\tau_{\rm eff}}>4.1$, excluding absorption caused by damped systems. This means that any sample of protoclusters selected with the CoSLA technique will be incomplete. We stress, however, that throughout this work we are limited by the box size of the hydrodynamical simulations. In particular, there are no $10^{15}~\textrm{M}_\odot$ cluster progenitors in any of the models considered here; it is possible that more massive protoclusters have a stronger association with CoSLAs.
\end{itemize}
\noindent
Finally, we note that visual inspection of CoSLAs suggests that coherent Ly-$\alpha$\ absorption typically selects structure orientated along the line of sight to the observer, regardless of whether or not this is associated with a protocluster. With the advent of large spectroscopic QSO surveys such DESI \citep{Vargas-Magana2019UnravelingDESI} and WEAVE-QSO \citep{Pieri2016WEAVE-QSO:Telescope} in the next few years, further investigation of the potential of CoSLAs for identifying intergalactic filaments in the high redshift Universe may be a worthwhile endeavour.
\section*{Acknowledgements}
We thank Zheng Cai for comments on a draft version of this paper. The Sherwood simulations were performed with supercomputer time awarded by the Partnership for Advanced Computing in Europe (PRACE) 8th Call. We acknowledge PRACE for awarding us access to the Curie supercomputer, based in France at the Tre Grand Centre de Calcul (TGCC). This work also made use of the DiRAC Data Analytic system at the University of Cambridge, operated by the University of Cambridge High Performance Computing Service on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant (ST/K001590/1), STFC capital grants ST/H008861/1 and ST/H00887X/1, and STFC DiRAC Operations grant ST/K00333X/1. DiRAC is part of the National E-Infrastructure. We thank Volker Springel for making \textsc{P-GADGET-3} available, and Simeon Bird for helpful advice regarding the incidence of DLAs in Illustris. We acknowledge the Virgo Consortium for making their simulation data available. The \textsc{EAGLE} simulations were performed using the DiRAC-2 facility at Durham, managed by the ICC, and the PRACE facility Curie in France at TGCC. We also thank the Illustris collaboration for making their data publicly available. JSAM is supported by an STFC postgraduate studentship. JSB acknowledges the support of a Royal Society University Research Fellowship.
\bibliographystyle{mnras}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,034 |
using Microsoft.SPOT.Debugger;
using PropertyChanged;
using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using Template10.Common;
using Windows.Storage;
using Windows.Storage.FileProperties;
using Windows.System.Threading;
namespace MFDeploy.Models
{
[ImplementPropertyChanged]
public class DeployFile
{
public StorageFile DFile { get; set; }
//public string FileName { get; set; }
//public string FilePath { get; set; }
public string FileBaseAddress { get; set; }
public string FileSize { get; set; }
public string FileTimeStamp { get; set; }
public bool Selected { get; set; }
// public DeployFile(string fileName, string filePath, ulong fileBaseAddress, ulong fileSize, string fileTimeStamp, bool selected = true)
public DeployFile(StorageFile dFile, bool selected = true)
{
DFile = dFile;
Selected = selected;
// get file property, we need the file date creation
LaunchThreadToGetFileProperties();
// get file memory address and size
LaunchThreadToGetBaseAddressAndSize(dFile);
}
private void LaunchThreadToGetBaseAddressAndSize(StorageFile file)
{
WindowWrapper.Current().Dispatcher.Dispatch(async () =>
{
var r = await SRecordFile.ParseAsync(file, null);
List<SRecordFile.Block> lb = r.Item2;
FileSize = String.Format("0x{0}", lb[0].data.Length.ToString("x4"));
FileBaseAddress = String.Format("0x{0}", lb[0].address.ToString("x4"));
});
}
private void LaunchThreadToGetFileProperties()
{
WindowWrapper.Current().Dispatcher.Dispatch(async () =>
{
var prop = await DFile.GetBasicPropertiesAsync();
FileTimeStamp = prop.ItemDate.ToString("g");
});
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,856 |
Q: Sort an array[(int, int)] in order to a list in scala I am currently working on Spark framework using Scala.
I have an array C: Array[(int, int)] to which i want to sort according to order of a map A:Map[int,int]. The code is given below:
line 1:import scala.util.Sorting
line 2: object Test {
line 3: def main(args: Array[String]){
line 4: val A = Map(5 -> 41, 1 -> 43, 2 -> 41, 3 -> 59,4 -> 51 )
line 5: val B= A.toList.sortBy(x => x._2)
line 6: B.foreach(println)
line 7: val C=Array((1,15),(2,9),(3,6),(4,3),(5,4))
line 8: Sorting.quickSort(C)(Ordering[(Int)].on(k => (A.get(k._1).get)))
line 9: for(j <- 0 to C.length-1){
line 10: println(C(j))
line 11: }
line 12: }
line 13: }
When I sort the A according the value part (line 5) and print, i get the output (line 6) like this:
(5,41)
(2,41)
(1,43)
(4,51)
(3,59)
But when i sort C according the key part of A (line 8), i get the following output (line 9-11)
(2,9)
(5,4)
(1,15)
(4,3)
(3,6)
As both (5,41) and (2,41) have same value and, in B (5,41) appears before (2,41). But after sorting C according to key part of A, why 2 is appearing before 5. Why is the result not uniform in both B and C?
Please suggest me the changes in the code to get a uniform result in both B and C. I want the C as: (5,4) (2,9) (1,15) (4,3) (3,6).
A: Replace A.get(k._1).get which makes your ordering non deterministic as you see with
B.map(_._1).indexOf(k._1) which will correctly order based on the index of the key in B, which is what you want.
A: Map is by-design unordered hence sorting by value alone may result in nondeterministic order. If your business logic allows changing of the sorting criteria, consider making it consistent by ordering by value + key on both the Map and Array:
import scala.util.Sorting
val A = Map(5 -> 41, 1 -> 43, 2 -> 41, 3 -> 59, 4 -> 51 )
val B = A.toList.sortBy(x => (x._2, x._1))
B
// res1: List[(Int, Int)] = List((2,41), (5,41), (1,43), (4,51), (3,59))
val C = Array((1,15),(2,9),(3,6),(4,3),(5,4))
Sorting.quickSort(C)(Ordering[(Int, Int)].on( k => (A.get(k._1).get, k._1) ))
C
// res23: Array[(Int, Int)] = Array((2,9), (5,4), (1,15), (4,3), (3,6))
As a side note, to avoid crash due to key not found, consider using gerOrElse with a default (e.g. Int.MaxValue, etc):
Sorting.quickSort(C)(Ordering[(Int, Int)].on( k => (A.getOrElse(k._1, Int.MaxValue), k._1) ))
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,334 |
Q: Why does apt-get install a newer dependency than supported by the specified package I have encountered a problem where a package, at a specific version, requires another package, also at a specific version, but Apt-Get selects a newer version of the dependency and then fails.
I have experienced this with Puppet Lab's MCollective and the Nginx Mainline PPAs and likely other packages so my question is about the general approach to Apt-Get dependency resolution but I will use Nginx as my example.
I have a package mirror (built with Aptly) containing the v1.7.5 of the nginx-full package and all its dependencies and also the newer v1.7.6 package and all its dependencies.
If I execute apt-get install nginx-full=1.7.5-1+trusty1 then the install fails with the message:
The following packages have unmet dependencies:
nginx-full : Depends: nginx-common (= 1.7.5-1+trusty1) but 1.7.6-1+trusty1 is to be installed
However, if I execute apt-get install nginx-full=1.7.5-1+trusty1 nginx-common=1.7.5-1+trusty1 then installation succeeds.
When I have both versions 1.7.5 and 1.7.6 of the nginx-common package on the mirror and the nginx-full package explicitly states it requires 1.7.5 of nginx-common, and nginx-full is the package I've requested, why does apt-get still select the incompatible 1.7.6 version of nginx-common?
Here is the output of dpkg -s nginx-full after installing 1.7.5 showing the exact version dependency constraint:
Version: 1.7.5-1+trusty1
Depends: nginx-common (= 1.7.5-1+trusty1), libc6 (>= 2.14), ...
In this instance, the chain of exact versions required is short so the workaround is easy but there are at least two problems for me:
*
*Other packages have much longer chains of dependencies that are tedious to discover and then append to the apt-get command line.
*Until a newer version of a dependency is published to a package mirror it is easy to be unaware of the impending issue.
What I can't comprehend is why the dependency resolution is apparently ignoring the exact version constraint on the specified package. More importantly I'd like to know how I can ask Apt-Get to honour the constraints without having to manually replicate the package metadata onto my apt-get parameters.
A: What you're experiencing is the problem with apt / apt-get not being as smart as you think it is.
This problem happens when trying to downgrade your package(s) or install an older package version than the version the repositories have as the latest candidate (with regard to your apt priority pinning and other policies regarding repository priorities). When you downgrade your package, you actually have to specify for each individual dependency which version you're downgrading to, or in this case which specific version you actually want to install.
In the case of the nginx packages, where nginx-full and nginx-common depend on each other, you have to explicitly tell apt to install each of the packages of the specified version(s). This is because 1.7.6-1+trusty1 supersedes 1.7.5-1+trusty1 by version number. As a result, you have to specifically say "Only install the package of this specific version" because of the superseded version(s) existing, i.e. apt-get install nginx-full=1.7.5-1+trusty1 nginx-common=1.7.5-1+trusty1
Not relevant to your question, but this also happens when you install from a repository that has a lower apt pinning priority than another version, in which case you have to specify the versions and/or source(s) to install from manually, i.e. sudo apt-get install nginx-full/trusty-proposed nginx-common/trusty-proposed being a prime example of trying to install a package and dependencies from the proposed repository, which has a much lower apt priority than PPAs or the main repositories.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,647 |
namespace Computers.Logic.VideoCards
{
using System;
public abstract class VideoCard
{
public void Draw(string a)
{
ConsoleColor color = this.GetColor();
Console.ForegroundColor = ConsoleColor.Gray;
Console.WriteLine(a);
Console.ResetColor();
}
protected abstract ConsoleColor GetColor();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,912 |
\section*{}
\vspace{-1cm}
\footnotetext{\textit{$^{a}$~Department of Physics, Aksaray University, 68100 Aksaray, Turkey. E-mail: sozkaya@aksaray.edu.tr}}
\footnotetext{\textit{$^{b}$~ Department of Atomic, Molecular and Nuclear Physics \& Instituto Carlos I de F\'isica Te\'orica y Computacional, Faculty of Science, Campus de Fuente Nueva, University of Granada, 18071 Granada, Spain. }}
\footnotetext{\dag~Electronic Supplementary Information (ESI) available: [details of any supplementary information available should be included here]. See DOI: 10.1039/cXCP00000x/}
\footnotetext{\ddag~Additional footnotes to the title and authors can be included \textit{e.g.}\ `Present address:' or `These authors contributed equally to this work' as above using the symbols: \ddag, \textsection, and \P. Please place the appropriate symbol next to the author's name and include a \texttt{\textbackslash footnotetext} entry in the the correct place in the list.}
\section{Introduction}
The unique properties triggered by the reduced
dimensionality~\cite{Chen-2015,Jin-2016,Mak-2010,Jing-2013} of two-dimensional (2D) electronic materials have been object of exhaustive investigations since
the discovery of graphene by Novoselov et al.~\cite{Novoselov-2004}.
Several families of 2D materials, such as transition metal
dichalcogenides (TMDs)~\cite{Coleman-2011}, hexagonal boron
nitrides~\cite{Novoselov-2005}, few-layer metal oxides
~\cite{Sreedhara-2013}, or metal chalcogenides (e.g. MoS$_2$~\cite{Joensen-1986},
WS$_2$~\cite{Seo-2007}) have been recently identified and investigated
to be used in semiconductor devices, rechargeable ion batteries,
and catalysis devices. Recently, a new family of 2D transition metal
carbides and carbonitrides (MXenes) have received special attention
due to their potential applications in sensors, electronic devices,
catalysis, storage systems, energy conversion, and spintronics
~\cite{Anasori-2015,Gao-2016,Zhang2017,Pandey-2017,Zha-2016,Liang-2017,Fu-2017,Akgenc2021,Abdullahi2022,Siriwardane2021}.
MXenes are generally synthesized by selectively etching
"A" layers from M$_{n+1}$AX$_n$ phases
~\cite{Naguib-2012,Naguib2013,Lukatskaya-2013,Mashtalir-2013,Tang-2016},
where M is an early transition metal, A is an element from the
IIIA or IVA groups, and X is either carbon or nitrogen (n =1, 2, or 3).
"A" atoms are removed from the MAX phase during the etching process.
For MXenes, surface termination with functional groups (-O, -OH, or -F) passivate
the outer-layer metal atoms. Many theoretical studies show that most of
the MXenes are metallic and that the majority of them still remain metallic after
surface passivation. Exceptionally, while pristine monolayer Sc$_2$C is
metallic, monolayer Sc$_2$CT$_2$ (T = OH, F, and O) are semiconductors with
a bandgap of 0.45–-1.80 eV after surface functionalization (-O, -OH, or -F)
~\cite{Yorulmaz-2016,Wang-2017}, which opens an exciting path towards the controlled design of 2D MXenes with selected properties. Besides the impact on the electronic structure, surface functionalization might
also effect other properties such as the thermodynamic stability, elasticity, and electronic and thermal transport
of the MXenes ~\cite{Zhang-2017,Zha-2016,Eames-2014}. For instance, S-terminated
MXenes have lower diffusion barriers (related to the electronic mobility and charge/discharge rate of the electrode material)
than their O-functionalized counterparts ~\cite{Zhu-2016,Li-2020}, hence providing new candidates for the selection of the most adequate electrode material for batteries. Shao {\it et al.} investigated the stability and electronic properties of nitrogen-functionalized MXenes with the help of a detailed
\textsl{ab initio} pseudopotential calculation ~\cite{Shao-2017}. Their results indicate that Nb$_2$CN$_2$
and Ta$_2$CN$_2$ Mxenes are direct bandgap semiconductors with ultrahigh carrier mobility. Following this result, Xiao {\it et al.} \cite{Xiao-2019} presented first-principles total-energy
calculations that proved that Ti$_2$C and its derivatives,Ti$_2$CC$_2$, Ti$_2$CO$_2$,
and Ti$_2$CS$_2$ have excellent potential for their use as anodes in Na-ion batteries.
Following these enticing results, Janus materials created from MXenes with different surface termination (i. e., different functional groups on each of the two surfaces) have already been investigated for varied applications
\cite{Jin-2018,Wang-2019,Akgen-2020,Xiong-2020}. Due to the two different
chemical surfaces, the mirror asymmetry of Janus materials introduces an out-of-plane structural symmetry, which can lead to new properties such as an intrinsic out-of-plane electric field or spin-orbit-induced spin splitting. These exciting possibilities have generated a surge of interest in these materials \cite{Khazaei-2017}.
Structural symmetry-breaking, as the one found in 2D Janus MXenes, which present different terminations on each of their two surfaces, is key in defining the electronic properties of two-dimensional materials. Many efforts have been taken, for instance, to breaking the in-plane symmetry of graphene (see, for example, Zhang {\it et al.} ~\cite{Zhang-09} and references therein), and van der Waals heterostructures (see, for instance, Hunt {\it et al.} ~\cite{Hunt-13}). The direct electronic band gaps of transition metal dichalcogenide monolayers are due indeed to this in-plane inversion asymmetry. Additionally, it has been shown that, by breaking the out-of-plane mirror symmetry with either external electric fields ~\cite{Yuan-13, Wu-13}, or with an asymmetric out-of-plane structural configuration ~\cite{Cheng-13}, similar to the one that exists in 2D Janus MXenes, an additional degree of freedom allowing spin manipulation can be obtained. Hence, creating such asymmetry can lead to a wealth of physical phenomena not present in standard MXenes. For instance, while silicene lacks a sizeable energy gap, thus preventing it from being used as a semiconductor in 2D nanodevices, Sun {\it et al}. reported that the Janus silicene, a silicene monolayer asymmetrically functionalized with hydrogen on one side and halogen atoms on the other, exhibits good kinetic stability, with a band gap that can be tuned between 1.91 and 2.66 eV ~\cite{Sun-16}.
Based on Density Functional Theory (DFT) calculations, it has been reported that the band gap of other asymmetrically functionalized MXenes such as Janus Cr$_2$C – Cr$_2$CXX' (X, X' = H, F, Cl, Br, OH) can be effectively tuned by a selection of a suitable pair of chemical elements (functional groups) ending the upper and the lower surfaces ~\cite{He2016}. State-of-the-art DFT calculations found that the monolayer Janus transition metal dichalcogenides (JTMDs) WSSe and WSeTe exhibit superior mobilities than conventional TMD monolayers such as MoS$_2$ ~\cite{Wang2018}.
Some of these Janus 2D materials have already been synthesized. For instance, in 2013, Janus graphene was successfully fabricated by Zhang {\it et al.} \cite{Zhang-2013}. In subsequent works,Janus-functionalized graphene (asymmetrically terminated by H, F, O) was synthesized by several groups \cite{Karlicky-2013,Holm-2018}. Recently, Zhang {\it et al.} ~\cite{Zhang-2017} developed a Janus MoSSe by substituting the top Se layer of MoSe$_2$ with S atoms. This approach, which breaks the out-of-plane structural symmetry of transition metal dichalcogenide monolayers, was also followed by Lu {\it et al}\cite{Lu17}, who found that this material presents an optically active vertical dipole, making it a 2D platform to study light–matter interactions where dipole orientation is critical, such as dipole–dipole correlations and strong coupling with plasmonic structures.
Recently, thermodynamically stable 2D Janus monolayers SWSe, SNbSe, and SMoSe were synthesized for the first time by Qin {\it et al.}~\cite{Qin-22}, showing excellent excitonic and structural quality and making them very valuable for optoelectronic applications.
Inspired by the experimental achievements of Janus 2D materials, the synthesis of Janus MXenes with asymmetrically surface-functionalized groups is expected in the near future.
As the functionalization of quality surfaces plays an essential role in many applications, such as high-speed electronic devices ~\cite{Pang-19}, the search for new functional groups or atoms to enable the design of Janus MXenes for next-generation devices becomes mandatory. Besides, experimentally obtained MXenes are usually covered with functional groups such as O, -OH, and F. Indeed, sulfur is a promising functional group for MXenes owing to its earth abundance, low cost, and environmental benignity, and S-terminated MXenes could be thus employed as anode material in Na- and K-based batteries. The aim of this work is hence to provide a comprehensive study of all possible 2D Janus materials based on the Sc$_2$CX MXene (X=O, F, OH) when functionalized with some of its most common functional groups (T=C, S, N) on both surfaces. We must note, however, that structures with mixed passivation might also be obtained experimentally. Further work is necessary to assess their relative stability with respect to the structures studied here and to determine their electronic, mechanical and magnetic properties. Our results will help to identify those structurally stable materials with the desired electronic properties based on their electronic character, gap size, and magnetic properties.
\section{Method}
All the calculations were carried out with the use of the {\it Vienna Ab initio}
Simulation Package (VASP)~\cite{vasp1,htt,Kre-99,Kres-96}, based on
Density Functional Theory (DFT). In this method, the Kohn--Sham single particle functions are
expanded based on plane waves up to the cut-off energy of 51 Ry. The lattice constants were
optimized and atoms were relaxed without any constraint until the Hellmann-Feynman forces became less than 0.003 eV/ {\AA}.
The Fermi level Gaussian smearing factor was set to 0.05 eV.
The electron--ion interactions were described by using the projector
augmented-wave (PAW) method~\cite{Kre-99,Bl-94}. For electron
exchange and correlation terms, a Perdew--Zunger-type
functional~\cite{Per-81,Per-92} was used within the generalized
gradient approximation (GGA)~\cite{Bl-94} in its PBE parameterization
~\cite{Perdew-1996}. To overcome the well-known underestimation of the band gap values by this functional, we have refined our calculations by using the Heyd–Scuseria–Ernzerhof (HSE06) functional, which has been proven to provide more accurate band gaps and band edge positions than the PBE. For metallic MXenes, we found that their HSE06 band structures are similar to the ones found using the GGA-PBE ~\cite{Heyd-2003, Heyd-2006, Krukau-2006} (Fig. S1 in the
supplementary material). However, for semiconductor MXenes (Sc$_2$COS, Sc$_2$CFN, and Sc$_2$COHN) the band gap obtained when using the hybrid (HSE06) functional increases significantly compared to that obtained by GGA-PBE. The HSE06 functional was also used to calculate the spin-polarized band structure of the MXenes.
The self-consistent solutions were obtained by employing a ($50\times50\times1$) Monkhorst--Pack~\cite{Mon-76}
grid of k-points for the integration over the Brillouin zone.
To prevent spurious interaction between isolated layers, a vacuum layer of at least 15 {\AA} was included
along the direction normal to the surface. The elastic tensor
of each system is derived from the stress–strain approach ~\cite{Le-02}.
\section{Results and Discussions}
Firstly, we fully relaxed the pristine Sc$_2$C. The ground-state structure of pristine Sc$_2$C is found to be a hexagonal crystal structure with equilibrium lattice parameters {\it a}={\it b}=3.29 Å.
Once the ground-state crystal structure of pristine Sc$_2$C was obtained, we explored different functionalizations of the Sc$_2$C
MXene monolayer with X= O, F, OH; T= C, N, S.
\begin{figure}[h] \centering
\includegraphics[width=10cm,clip=true]{Fig1.jpg}
\caption{Side views of the four possible models for Sc$_2$CXT (X = O, F, OH; T = C, S, N) MXenes. Sc atoms are shown in purple, C atoms in brown, X (X = O, F, OH) atoms in red, and T (T = C, S, N) atoms in yellow. Please note that stoichiometry is the same for the 4 structures: the seemingly duplicated atoms are an artifact of the periodic boundary conditions in the figure.}
\label{models}
\end{figure}
MXT Janus MXenes have four possible functional group models. As presented in Fig. 1, we label them as FCC, HCP, FCC+HCP, HCP+FCC. In the FCC model, each of the X and T atoms is aligned with one of the Sc atoms, whereas in the HCP model these surface atoms are positioned directly below and above the C atoms. As for the FCC+HCP model, the X atoms are placed in the same position as in the FCC model (aligned with the Sc atoms) and the T occupies the same position as in the HCP model (below the C). Reversely, in the HCP+FCC model T atoms are located as in the FCC scheme and the X atoms are placed as in the HCP.
The calculated lattice parameters ($a$), relative total energies with respect to the most stable model structure for each material ($\Delta$E),
and formation energies ($\Delta$E$_f$) of these four configuration models are listed in Table 1. Formation energies are calculated to get the most stable configuration based on equation 1:
\begin{equation}
\label{eq1}
\Delta E_{\mathrm{f}}=E_{\mathrm{tot}}(MXT)-E_{\mathrm{tot}}(M)-
E_{\mathrm{tot}}(X)-E_{\mathrm{tot}}(T)
\end{equation}
\begin{table*}
\centering
\caption{Lattice parameters $a$ (in {\AA}), relative energies with respect to the minimum energy for each model $\Delta$E (in eV), formation energies $E_f$ (in eV),
and band gaps (in eV) for the Sc$_2$CXT (X= O, F, OH; T= C, N, S) in the four possible models.
Here, M, D, and ID indicate the metallic, direct or indirect bandgap semiconductor character, respectively. $E_f$ and $E_g$ are shown only for the most stable model of each structure.}
\centering \vspace{2mm} \centering
\footnotesize\setlength{\tabcolsep}{8pt}
\begin{tabular}{c| c c| c c| c c| c c| c c }
\hline
& \multicolumn{2}{c}{FCC} & \multicolumn{2}{c}{HCP}& \multicolumn{2}{c}{FCC+HCP}& \multicolumn{2}{c}{HCP+FCC} \\
Material & ~$a~$ & $\Delta$E &~$ a ~$& $\Delta$E &~$ a~$ & $\Delta$E &~$ a~$ & $\Delta$E &$E_f$ & $E_g$ \\
\hline
Sc$_2$COC & 3.33 & 3.15 & 3.54 & 2.66 & 3.58 & {\bf 0.00} & 3.35 & 3.45 & -4.50 & M \\[0ex]
Sc$_2$COS & 3.32& 0.98& 3.51& 0.48& 3.62& {\bf 0.00}& 3.55& 0.40& -7.88& 2.45 (ID) \\[0ex]\raisebox{1.5ex}
Sc$_2$CON & 3.36 & 2.87& 3.45& 0.55& 3.49& {\bf 0.00}& 3.51& 2.03& -9.85& M \\[0ex]
\hline
\raisebox{1.5ex}
Sc$_2$CFC &3.29& 1.83& 3.56& {\bf 0.00}& 3.62& 0.87& 3.25& 2.39 & -3.83& M \\[0ex]\raisebox{1.5ex}
Sc$_2$CFS &3.28& {\bf 0.00}& 3.43& 1.11& 3.25& 0.40& 3.24& 0.64& -7.77& M \\[0ex]\raisebox{1.5ex}
Sc$_2$CFN & 3.36 & 2.87& 3.45& 0.55& 3.49& {\bf 0.00}& 3.51& 2.03& -8.95& 1.67 (ID) \\[0ex]
\hline
\raisebox{1.5ex}\
Sc$_2$COHC &3.30 & 1.98& 3.57& {\bf 0.00}& 3.68& 0.26 & 3.24& 3.86& -4.49& M \\[0ex]\raisebox{1.5ex}
Sc$_2$COHS &3.30& {\bf 0.00}& 3.45& 0.90 & 3.31 & 2.79 & 3.28 & 1.59 & -4.36& M \\[0ex]\raisebox{1.5ex}
Sc$_2$COHN &3.47& 1.96& 3.48& {\bf 0.00}& 3.29 & 5.80 & 3.26& 5.29 &-10.43& 1.06 (D) \\
\hline
\end{tabular}
\end{table*}
where E$_{tot}$(MXT), E$_{tot}$(M), E$_{tot}$(X), E$_{tot}$(T)
are the total energy of the fully functionalized MXT (f-MXT), the total
energy of pristine Sc$_2$C, and the total energy of (1/2) (O$_2$, F$_2$, (OH)$_2$ in the gas phase), C, N, and S (their stable bulk forms), respectively. For the most stable configurations of each material, the thickness of the monolayers (L) and the bond lengths between the surface Sc and the functional group ($d_{Sc-X}$ and $d_{Sc-T}$) have been calculated and are presented in Table 2. From now on we will focus only on the 9 most stable geometries out of the 36 studied potential monolayers.
\begin{figure}[h] \centering
\includegraphics*[width=9cm,clip=true]{Fig2.jpg}
\caption{Band structures for the most stable models of (a) Sc$_2$COT, (b) Sc$_2$CFT and (c) Sc$_2$COHT monolayers,
(T= C, S, N). The Fermi energy is set at zero.}
\label{models}
\end{figure}
As seen in Table 1, all the formation
energies (E$_f$) of the MXT under study indicate that the synthesis of these MXenes is
highly possible (in our convention, more negative formation energy indicates a more
stable structure.) This stability is further confirmed by the fulfillment of all the materials of the Born criteria for 2D hexagonal materials, as we describe in more detail below. The most favored model is the FCC+HCP, with 4 of the 9 possible structures presenting their most stable geometry in that model. It is followed by the HCP (3 structures) and the FCC (2 structures) models.
\begin{figure}[h] \centering
\includegraphics*[width=9 cm,clip=true]{Fig3.jpg}
\caption{The charge-density isosurfaces of VBM and CBM states of (a) Sc$_2$COS, (b) Sc$_2$CFN, and (c) Sc$_2$COHN. Sc atoms are shown in purple, C atoms in brown, O atoms in red, H atoms in white, F atoms in orange, S atoms in yellow, and N atoms in blue.}
\label{models}
\end{figure}
Our results show that, for any given X = O, F, OH, the further functionalization of the opposite surface with nitrogen makes the MXT monolayers more stable than the functionalization with C or S. This
can be explained by looking at the electronegativity of these species. Indeed, E$_f$ becomes
more negative with the increase of the electronegativity of T (C, S, or N)
atoms. The Sc-T bond is formed by the electrons transitioning
from the Sc to the T atoms, which causes the T atoms to gain
more electrons, hence further strengthening the Sc-T bond and thus resulting in a
more negative E$_f$. Due to the stronger bond of Sc-N, the $d_{Sc-N}$ bond length of MXN is also shorter than that obtained for the other functional species $d_{Sc-T}$ (T=C, S), as seen in Table 2. The oxygen functionalization indicates a strong Sc–O interaction, which leads to the shortest $d_{Sc-O}$ bond length.
\begin{table*}
\centering
\caption{Calculated monolayer thickness (L) and bond lengths between the
Sc and the functional groups ($d_{Sc–-X}$, $d_{Sc-–T}$) and between
the Sc and C atom ($d_{Sc-–C}$) of Sc$_2$CXT (X= O, F, OH; T = C, S, N) MXenes.}
\centering \vspace{2mm} \centering
\begin{tabular}{l|ccccc}
\hline &&&&& \\[-3ex]
{Material} & ~$L$({\AA}) ~& ~$d_{Sc–-C}$({\AA})~&~$d_{Sc–-X}$({\AA})~&$d_{Sc–-T}$({\AA})~\\[0.5ex]
\hline &&&&& \\[-3ex]
~Sc$_2$COC~ &3.54& 2.24& 2.12& 2.14& ~ \\[0.5ex]
~Sc$_2$COS~&3.97 & 2.20 & 2.14 & 2.46 & ~\\[0.5ex]
~Sc$_2$CON &3.66& 2.25 & 2.09 & 2.06 & ~\\[0.5ex]
\hline &&&&& \\[-3ex]
~Sc$_2$CFC &3.98& 2.50& 2.35 & 2.08 & ~\\[0.5ex]
~Sc$_2$CFS & 5.47& 2.32& 2.20& 2.58& ~\\[0.5ex]
~Sc$_2$CFN &3.84& 2.34& 2.27& 2.03& ~\\[0.5ex]
\hline &&&&& \\[-3ex]
~Sc$_2$COHC & 5.00 &2.35& 2.39 & 2.08 & ~\\[0.5ex]
~Sc$_2$COHS &6.51& 2.33& 2.26 & 2.58 & ~\\[0.5ex]
~Sc$_2$COHN &5.00& 2.31 &2.36 & 2.04 & ~\\[0.5ex]
\hline
\hline
\end{tabular}
\end{table*}
\begin{figure}[h] \centering
\includegraphics*[width=9 cm,clip=true]{Fig4.jpg}
\caption{Projected Density of States (PDOS) of the most relevant orbitals for different elements for the (a) Sc$_2$COT, (b) Sc$_2$CFT and (c) Sc$_2$COHT (T=C, S, N) Janus monolayers. The Fermi level is set to zero.}
\label{models}
\end{figure}
\begin{figure}[h] \centering
\includegraphics*[width=10cm,clip=true]{Fig5.jpg}
\caption{Electron localization function isosurfaces of the (a) Sc$_2$C, (b)Sc$_2$CON, (c) Sc$_2$CFT and (d) Sc$_2$COHT (T = C, S, N) Janus monolayers. The magnitude bar for the ELF is shown in the top left panel. The insert panel at the top right shows the structural models for the Janus monolayers.}
\label{models}
\end{figure}
The electronic properties of bare MXenes will be as well affected by surface
functionalization. While the Sc$_2$C monolayer has a metallic character, previous
reports ~\cite{Yorulmaz-2016,Wang-2017} show that surface
functionalization of the Sc$_2$CO$_2$, Sc$_2$CF$_2$ and Sc$_2$C(OH)$_2$
monolayers transform them in semiconductors with a band gap energy (E$_g$) of 1.76 eV (indirect band),
0.94 eV (indirect band), and 0.51 eV (direct band), respectively. To analyze the potential change in the electronic properties of the Janus Sc$_2$CXT studied here, we present in Fig 2. the corresponding energy band
structures, where the conduction band is shown in yellow in those cases where a change from the metallic to a semiconductor character is found. Of the 9 Sc$_2$CXT MXenes, six retain their metallic character, while three of them become semiconducting upon functionalization: both Sc$_2$OS and Sc$_2$FN are semiconductors with indirect band gaps of 2.45 and 1.67 eV,
respectively, while Sc$_2$COHN is a direct band gap semiconductor with a band gap of 1.06 eV. This gap value makes
Sc$_2$COHN a valuable candidate for its use in solar cells, whereas the metallic Sc$_2$CXT monolayers would be much better candidates to act as high-performance electrode materials ~\cite{Lebegue-2009},
than, for instance, the semiconducting MoS$_2$ monolayer ~\cite{Wu-2015, Du-2010}.
To understand the charge distribution of the electronic states near the gap, we have calculated the charge-density isosurfaces of the valence band maximum (VBM) and conduction band maximum (CBM) states of the semiconducting Janus monolayers. As seen in Fig.3, the VBM is primarily derived from the $d$ orbital of the Sc atoms. The bonds between the d orbitals of the Sc atom and the sublayer atoms (C, S, or N) present a $\pi$ bonding character. The CBM is primarily derived from the $p_z$ orbitals of the O atom for Sc$_2$COHN, the Sc $d_z^2$ states and the $p_z$ orbitals of O and F atoms for Sc$_2$COS and Sc$_2$CFN, respectively.
To gain further insight into the electronic
properties we have calculated the partial
density of states (PDOS) for the nine Janus MXenes. As seen from Fig. 4, the valence states are dominated by the Sc (d) states and C (p) states, while the conduction band is dominated by the Sc (d) states for all structures. The strong hybridization between the Sc 3d states and C 2p states is responsible for the mixed
covalent/metallic/ionic character of the Sc–C bonds ~\cite{Sun-04}.
To analyze the electron distributions, we examined the electron localization functions (ELF) ~\cite{Silvi-94} within the (110) plane of the Sc$_2$CXT Mxenes. Since the ELF is similar for both the Sc$_2$COT and Sc$_2$CON MXenes we only show the results obtained from the Sc$_2$CON analysis. From the ELF map in Fig.5 we can see that both the Sc$_2$C and the Sc$_2$CXT surfaces have a cloud of lone-pair electrons. For Sc$_2$CFS and Sc$_2$COHS (the FCC structures) the electrons are mainly located around the F, OH, and S atoms.
In addition to the electronic properties, the mechanical properties of 2D materials play also a critical role regarding their potential technological and practical applications.
We have then investigated the mechanical stability of Janus MXenes according to the Born criteria ~\cite{Born-08}. Based on the Born stability criteria, 2D hexagonal crystals should obey the following conditions:
\begin{equation}
\label{eq2}
\ C_{11}>0, \ C_{66}>0, \ 2*C_{66}=C_{11}-C_{12}, \ and \ C_{11}> |C_{12}|
\end{equation}
\begin{table*}
\centering
\caption{Elastic constants ($C_{ij}$ in N/m), Young's moduli (Y in N/m),
in-plane stiffnesses (B in N/m), shear moduli (G in N/m), and Poisson's ratios ($\nu$).}
\centering \vspace{2mm} \centering
\footnotesize\setlength{\tabcolsep}{8pt}
\begin{tabular}{l|ccccccc}
\hline &&&&& \\[-3ex]
~Material~ & ~$C_{11}$~ & ~$C_{12}$~&~$C_{66}$~&$Y$~ & ~$B$~ & ~$G$~ & ~$\nu$~\\[0.5ex]
\hline &&&&& \\[-3ex]
~Sc$_2$COC~ &~135.180& 45.921 &44.629&119.580&90.590& 44.630& 0.340 ~ \\[0.5ex]
~Sc$_2$COS~ & 137.358& 41.169& 48.094 &125.018& 89.300& 48.094& 0.300~\\[0.5ex]
~Sc$_2$CON &157.733& 53.363& 52.185& 139.680& 105.500& 52.185& 0.338~\\[0.5ex]
\hline &&&&& \\[-3ex]
~Sc$_2$CFC &117.189 &40.012& 38.588& 103.527& 78.550& 38.588& 0.341~\\[0.5ex]
~Sc$_2$CFS& 129.088 &36.981& 46.054& 118.494& 82.980& 46.054& 0.286~\\[0.5ex]
~Sc$_2$CFN &155.300 &38.294& 58.503 &145.857& 96.850& 58.503& 0.247~\\[0.5ex]
\hline &&&&& \\[-3ex]
~Sc$_2$COHC& 119.908& 39.593 &40.158& 106.835 &79.730& 40.158& 0.330~\\[0.5ex]
~Sc$_2$COHS &127.867& 37.777& 45.045 &116.706& 82.770& 45.045& 0.295~\\[0.5ex]
~Sc$_2$COHN &152.177& 39.448 &56.364& 141.951 &95.780& 56.364& 0.259~\\[0.5ex]
\hline
\end{tabular}
\end{table*}
The elastic constants $C_{66}, C_{11}, C_{12} $ are tabulated in Table 3. As we can see,
all Janus MXT MXenes satisfy the mechanical stability criteria, which shows that they are mechanically stable. Due to the hexagonal symmetry, the elastic constant $C_{11}$ is higher than $C_{12}$. Specifically, we have found that the elastic constant $C_{11}$ increases for N-terminated surface MXenes. The $d_{Sc–-N}$ bond lengths of these Sc$_2$CXN (X:O, F, OH)
MXenes are smaller than the other bond lengths, given in Table 2. The
stronger bonds between Sc and N atoms hence resulting in larger $C_{11}$ elastic constants.
To further characterize the mechanical properties of these materials, we calculated Young's modulus $Y$, Poisson's ratio $\nu$, the
shear modulus G, and the in-plane stiffness $B$, derived from the elastic constants as ~\cite{Thomas-2018,Wei-2009,Andrew-2012},
\begin{equation}
\label{eq3}
\ Y=(C_{11}-C_{12})/C_{11}, \ \nu =C_{12}/C_{11},
G=(C_{11}-C_{12})/2, \ B=Y/2*(1-\nu)
\end{equation}
While Poisson's ratio $\nu$ reflects the mechanical ductility
and flexibility, Young's modulus (Y) reflects the stiffness of materials. If Young's modulus value is high, the material is stiffer. The stiffest
crystal found here is Sc$_2$FN. To put in context these results, we compare our calculated $Y$ with that of other well-known 2D materials. The $Y$ values calculated here for the Janus MXenes family are within the range of 103.527 and 145.857 N/m, much less stiff than that of graphene (341 N/m) ~\cite{Cak-2014} and comparable with the values obtained for the TMDC family ~\cite{Cak-2014}. On the other hand, the $Y$ values of the Janus MXenes studied here are much larger than that of silicene (62 N/m)~\cite{Mortazavi-2017} and germanene (44N/m)~\cite{Mortazavi-2017}. The Poisson's ratios of the Sc$_2$CXT MXenes studied here are in a range between 0.286 and 0.341 N/m, comparable with those for MoS$_2$ (0.3 N/m) ~\cite{Cak-2014}. This demonstrates that f-MXenes can maintain mechanical flexibility under large strain, making them excellent materials for applications where strain is going to be an issue.
Finally, we have analyzed the magnetic properties of this novel class of Janus MXenes. Since the recent discovery of magnetic order in 2D materials \cite{Gong-2017,Huang-2017,Wang-2016}, the possibility of exploring new physical phenomena and designing truly novel 2D devices has become a reality that is already being explored experimentally \cite{Zhong-2017}. In particular, the search for half-metallic magnets (HMMs) and spin-gapless semiconductors (SGSs) \cite{Wang2008} has been very intense, due to their intriguing tunable physical properties, and we have explored this possibility in this work. It is worth mentioning that the Sc$_2$CO$_2$ monolayer actually exhibits ferroelectric polarization \cite{Chandrasekaran2017}, and the exploration of the ferroelectric properties of the Janus Sc$_2$CXT (X = O, F, OH; T = C, S, N) family needs to be addressed in the near future.
To explore the magnetic character of the Sc$_2$CXT (X = O, F, OH; T = C, S, N), we have calculated the total energies of the ferromagnetic (FM), non-magnetic (NM), and antiferromagnetic (AFM1 , AFM2, and AFM3) spin ordering of the nine most stable materials studied in this work in a 2$x$2 supercell. This is the minimum supercell size needed to properly study all the possible AFM configurations, as depicted in Figure 6.
Our results are tabulated in Table 4 for the magnetic structures and Table S1 for the NM ones. The ground state of all structures is NM except for Sc$_2$CFC and Sc$_2$COHC. For Sc$_2$CON, the total energy difference between the FM and NM state is very small, which means that the ground state could easily be unstable in a specific environment. Then, the transition to an FM ground state might occur under certain conditions, and for this reason we have also explored the magnetic properties of Sc$_2$CON (presented in SI). Our results show that Sc$_2$CFC is FM with a supercell magnetic moment of 3.99 $\mu$B, while the ground state of Sc$_2$COHC is AFM1 with a supercell magnetic moment of 3.98 $\mu$B. If the ground state of Sc$_2$CON became magnetic, it would be FM with a supercell magnetic moment of 2.24 $\mu$B. The magnetic moments of the individual atoms for Sc$_2$CFC and Sc$_2$COHC are also given in Fig S2.
\begin{table*}
\centering
\caption{The total energies (in eV) of non-magnetic E$_{NM}$, ferromagnetic E$_{FM}$ and antimagnetic E$_{AFM1}$, E$_{AFM2}$ and E$_{AFM3}$ states (eV), Curie or Néel temperatures (K), and Magnetic Anisotropy Energy (MAE) ($\mu$ eV per Sc atom ) for the high symmetry directions of Sc$_2$CFC and Sc$_2$COHC for a
2x2x1 supercell.}
\centering \vspace{2mm} \centering
\begin{tabular}{l|ccccccccccc}
\hline &&&&&&&&&&& \\%[-2ex]
~Material~& E$_{FM}$ &E$_{NM}$
& E$_{AFM1}$
& E$_{AFM2}$ &E$_{AFM3}$
&T$_{C/N}$ & (100)& (010)& (110)& (001)& (111) \\[0.5ex]
\hline &&&&&&&&&&& \\%[-3ex]
~Sc$_2$CFC~ &-147.578& -147.435& -147.576& -147.576& -147.576& 22 & 32 & 32 & 32 & 24 & 0 ~\\[0.5ex]
~Sc$_2$COHC & -171.452 & -171.284 & -171.453 & -171.452 & -171.453 & 9 & 8 & 8 & 8 & 0 & 5 ~\\[0.5ex]
\hline
\hline
\end{tabular}
\end{table*}
\begin{figure}[h] \centering
\includegraphics*[width=9 cm,clip=true]{Fig6.jpg}
\caption{The non-magnetic and four different magnetic configuration models: NM, FM, AFM1, AFM2, AFM3 considered in the calculation.
Magnetic state configurations. In Sc atoms, "$+$" represents spin-up, and "$-$" represents spin-down.}
\label{models}
\end{figure}
\begin{figure}[h] \centering
\includegraphics*[width=8cm,clip=true]{Fig7.jpg}
\caption{The band and spin-resolved density of states structures of (a)Sc$_2$CFC, and (b)Sc$_2$COHC. The Fermi level is set at 0 eV. Bands and DOS for spin-down (up) are shown on the left (right) panels. The black and blue arrows show the HSE06 band gap and the fundamental band gap, respectively.}
\label{models}
\end{figure}
The spin-dependent electronic band structure and DOS for the ground states of Sc$_2$CFC (FM) and Sc$_2$COHC (AFM) are shown in Fig. 7 ( the spin-polarized bandstructure and DOS of Sc$_2$CON in the FM state are shown in Fig. S3.). Remarkably, their band structures present a half-metallic character, where the majority (spin-up) electrons show semiconducting behavior with an HSE06 half-metallic (or fundamental) energy gap of 0.60 eV and 0.48 eV, respectively, while the minority (spin-down) electrons present a metallic behavior. Half-metallicity is a very much searched for quality in nanomaterials, since, provided the half-metallic energy gap of the majority electrons is wide enough, the electronic transport in this material would be totally spin-polarized, dominated by the minority-spin electrons. Similar results have been found for other Janus materials \cite{Zhang2021,He2019,He2016,Akgenc2020, Zhong-20, Wang-19, Zuntu-21, Wang-16}. In our case, the values of the fundamental bandgaps found for Sc$_2$CFC and Sc$_2$COHC are larger than those typically found for chalcogenides and some types of perovskites, making them much more suitable for spintronics applications. A small fundamental band gap means that the half-metallic character could disappear under certain conditions, such as strain, so having a half-metallic gap larger than at least half an eV is essential to preserve this important property.
The Curie or Néel temperature (T$_C$ or T$_N$) of Sc$_2$CFC and Sc$_2$COHC was calculated based on the mean-field theory approximation (MFA), which is estimated by k$_B$T$_{(MFA)}$= (2/3) $\Delta({E})$ ~\cite{Jin-09}. Here, k$_B$ and $\Delta({E})$ are, respectively, the Boltzmann constant and, for Sc$_2$CFC (which has a FM ground state), the energy difference between FM and AFM (all the AFM energies are equal), or, for Sc$_2$COHC (which has a AFM ground state), the energy difference between FM and AFM1 (the energy of AFM1 is the same as the AFM3, and smaller than AFM2). The T$_C$ of Sc$_2$CFC, and T$_N$ of Sc$_2$COHC were calculated to be 22 K, and 9 K, respectively. Although these values are low, the energy difference between the magnetic and non-magnetic states in both systems is large enough to likely maintain the magnetic features at a temperature close to room temperature, as in the case of
Sc$_2$NO$_2$ ~\cite{Liu-17, Khazaei-13}.
The Magnetic Anisotropy Energy (MAE) determines the FM ordering in 2D materials.
We have examined the impact of
spin-orbit
coupling
(SOC)
and electron
localization on the magnetic properties of Sc$_2$CFC, and Sc$_2$COHC. The MAE is defined as the difference in energy between the system with a given spin orientation, and the energy of the most stable spin orientation, which is called an easy axis. To understand the nature of FM ordering in these materials, we consider the in-plane a (100), b (010), and a+b (110), and the out-of-plane c (001) and (111) high-symmetry axes.
Since the changes in energies are very low, we used a dense k-mesh of 21 × 21 × 1 for more accurate energy convergence. As we can see in Table 4, both Sc$_2$CFC and Sc$_2$COHC exhibit an out-of-plane easy axis, which are the (111) and (001), respectively. The calculated MAEs are 32 and 8 $\mu$eV (per Sc atom) for (100), (010), and (110) directions. These values are similar to previous calculated results for 2D
materials ~\cite{Hu-16}, but larger than those found for Fe (1.4 $\mu$eV per Fe atom) and Ni (2.7 $\mu$eV per Ni atom)
bulks ~\cite{Daalderop-90}.
These features make Sc$_2$COHC, and especially Sc$_2$CFC, very promising candidates for spintronic applications.
\section{Summary}
We have presented a full
{\it ab initio} study of the atomic geometry, stability, electronic and magnetic properties of a new family of functionalized Janus MXenes, Sc$_2$CXT (X:O, F, OH; T:C, N, S). Following energetic and mechanical criteria, we have shown the high stability of these materials. In some cases, functionalization with the appropriate groups changes the metallic character of the original Sc$_2$C MXene into a semiconducting one, with either a direct or indirect band gap, depending on the actual surface termination. In addition, the analysis of the mechanical properties proves the excellent mechanical flexibility of f-MXenes under large
strain. Finally, we found that three of the Janus MXenes under study in this work present ferro- or antiferromagnetic characteristics, particularly a half-metallic character with a significant fundamental energy gap.
All these remarkable properties make the Sc$_2$CXT (X:O, F, OH; T:C, N, S) Janus MXene family a valuable candidate for future optoelectronic and spintronic nanodevices.
\section*{Author Contributions}
Sibel Özcan: Conceptualization (lead); formal analysis (lead); writing – original draft (lead); writing – review and editing (equal). Blanca Biel: Conceptualization (supporting); writing – review and editing (equal); funding acquisition (lead).
\section*{Conflicts of interest}
There are no conflicts to declare.
\section*{Acknowledgments}
S.Ö. and B.B. kindly acknowledge financial support by the Junta de Andalucía under the Programa Operativo FEDER P18-FR-4834. B.B. also acknowledges financial support from AEI under project PID2021-125604NB-I00. The Albaicín supercomputer of the University of Granada and TUBITAK ULAKBIM, High Performance and Grid Computing Center (TRUBA resources) are also acknowledged for providing computational time and facilities.
\section*{References}
| {
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William R. Allan (born 1970) is a Scottish classicist specializing in Greek epic and tragedy, particularly the plays of Euripides. He is currently McConnell Laing Fellow and Tutor in Greek and Latin Languages and Literature at University College, Oxford and Professor of Greek, Faculty of Classics, University of Oxford. He was formerly Assistant Professor of Classics at Harvard University.
Background
He was educated at Glenrothes High School in Fife, then studied at the University of Edinburgh and the University of Oxford, receiving an MA and DPhil, respectively.
Works
The Andromache and Euripidean Tragedy (Oxford University Press, 2000; paperback edn. 2003)
Euripides: The Children of Heracles (Aris and Phillips, 2001)
Euripides: Medea (Duckworth, 2002)
Helen (Cambridge University Press, 2008). commentary
References
Alumni of the University of Edinburgh
People educated at Glenrothes High School
Classical scholars of Harvard University
Scottish classical scholars
1970 births
Living people
Fellows of University College, Oxford
People from Glenrothes | {
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{"url":"https:\/\/www.sparrho.com\/item\/fourier-transform-of-rauzy-fractals-and-point-spectrum-of-1d-pisot-inflation-tilings\/233f2e1\/","text":"# Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings\n\nResearch paper by Michael Baake, Uwe Grimm\n\nIndexed on: 25 Jun '20Published on: 25 Jul '19Published in: arXiv - Mathematics - Metric Geometry\n\n#### Abstract\n\nPrimitive inflation tilings of the real line with finitely many tiles of natural length and a Pisot-Vijayaraghavan unit as inflation factor are considered. We present an approach to the pure point part of their diffraction spectrum on the basis of a Fourier matrix cocycle in internal space. This cocycle leads to a transfer matrix equation and thus to a closed expression of matrix Riesz product type for the Fourier transforms of the windows for the covering model sets. In general, these windows are complicated Rauzy fractals and thus difficult to handle. Equivalently, this approach permits a construction of the (always continuous) eigenfunctions for the translation dynamical system induced by the inflation rule. We review and further develop the underlying theory, and apply it to the family of Pisa substitutions, with special emphasis on the Tribonacci case.","date":"2021-04-19 09:53:46","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8089389801025391, \"perplexity\": 875.0420342844689}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038879305.68\/warc\/CC-MAIN-20210419080654-20210419110654-00251.warc.gz\"}"} | null | null |
Božidara je ženské křestní jméno slovanského původu. Znamená "dar boží, bohem daný". Její variantou jsou Theodora, Božena a Bohdana.
Známé nositelky jména
Božidara Turzonovová – slovenská herečka srbsko-makedonského původu
Literatura
Miloslava Knappová, Jak se bude vaše dítě jmenovat?
Robert Altman, Osud podle jména
Ženská jména
Ženská jména slovanského původu | {
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Q: Complexity. Why dont constants matter? Can someone please explain to me in a simple way why constants don't matter when it comes to big O notation? Why does the complexity stay the same when you add a constant. This is not a homework question I just want to understand this better. Let me get this straight big O is for seeing the behavior of a function as it approaches infinity right?
I got it. Thanks so much everyone.
A: In practicality, sometimes constants do matter. But, when we speak of Big O notation, we're looking at asymptotic behavior. The reason a constant doesn't affect the asymptotic behavior is because a function with a faster growing curve will always overtake a function with a slower growing curve even in the presence of a huge constant (though it will take longer to get there, of course).
Thus we say that the constant "doesn't matter", because it can never change the asymptotic relationship between the curves.
A: The answer is... by definition. If some function f(x) is big-O of some function g(x), it just means that f(x) is "eventually" smaller than some constant times g(x). (i.e. for large enough x). The actual value of the constant doesn't matter; nor does the position of the "eventually" behavior - as long as it's true for a large enough x and a large enough constant, you're covered.
You can add in constants, or anything with a smaller O, and it won't matter - it's all about which term grows the fastest, and which term dominates as x grows.
A: It does not matter for complexity theory, which is interested only in how the function scales as the input size grows.
A constant does not affect how the function behaves as input sizes grow towards infinity at all.
However, if you are interested in actually running a certain piece of code, you may very well be interested in a large constant overhead and how the function performs for smaller input sizes.
Difference between complexity theory and practice.
A: Big O explains how the complexity changes as the input gets large. The larger your input gets the less important constants are. For example, multiplying something by 10 is significantly less important than squaring something when n gets to one million or one billion. Big O is not an exact measurement, it's a way to put your algorithm in a rough class of complexity so you can round off the constants because they are not meaningful with huge n values.
A: Constants are not important in big-O notation because the concern is scalability.
A: Big-O notation is about how the usage of resources (time, memory) changes when the number of items changes. So, for example, if my computer can sort 10 items in 1 second, 20 items in 4 seconds, and 30 items in 9 seconds, then it is O(n²).
If I can sort those same items by hand in 10 seconds, 40 seconds, and 90 seconds, respectively, then I am still O(n²), but my constant factor is 10 times larger: The same size of problem takes me ten times as long as the computer.
A: Let say we are computing "big" factorials all the day.
*
*function 1 compute the factorial of 50, then the factorial of N and return the factorial of N.
*function 2 compute the factorial of N and return the result.
Complexity of both functions is O(N).
*
*function 1 will complete in, let say 2 hours + 1 ms
*function 2 will complete in 2 hours
Because the constant 50 is not related to N (we can ignore that gap when N is close to +inf), in theory we can first compute the factorial of 50 it does not matter... in real life better avoid to compute the factorial of 50 before returning the answer.
So happy with our results that tomorrow we will compute bigger factorials.
| {
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Empire in Africa
This series of publications on Africa, Latin America, Southeast Asia, and Global and Comparative Studies is designed to present significant research, translation, and opinion to area specialists and to a wide community of persons interested in world affairs. The editor seeks manuscripts of quality on any subject and can usually make a decision regarding publication within three months of receipt of the original work. Production methods generally permit a work to appear within one year of acceptance. The editor works closely with authors to produce a high-quality book. The series appears in a paperback format and is distributed worldwide. For more information, contact the executive editor at Ohio University Press, 19 Circle Drive, The Ridges, Athens, Ohio 45701.
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AREA CONSULTANTS
Africa: Diane M. Ciekawy
Latin America: Thomas Walker
Southeast Asia: William H. Frederick
Global and Comparative Studies: Ann R. Tickamyer
The Ohio University Research in International Studies series is published for the Center for International Studies by Ohio University Press. The views expressed in individual volumes are those of the authors and should not be considered to represent the policies or beliefs of the Center for International Studies, Ohio University Press, or Ohio University.
Empire in Africa
ANGOLA AND
ITS NEIGHBORS
David Birmingham
_Ohio University Research in International Studies_
Africa Series No. 84
Ohio University Press
Athens
© 2006 by the
Center for International Studies
Ohio University
www.ohio.edu/oupress
Printed in the United States of America
All rights reserved
15 14 13 12 11 10 09 08 07 06 5 4 3 2 1
The books in the Ohio University Research in International Studies Series
are printed on acid-free paper ™
_Library of Congress Cataloging-in-Publication Data_
Birmingham, David.
Empire in Africa : Angola and its neighbors / David Birmingham.
p. cm. — (Ohio University research in international studies. Africa series ; no. 84)
Includes bibliographical references and index.
ISBN 0-89680-248-5 (pbk. : alk. paper)
1. Angola—History. 2. Portugal—Colonies—Africa—History. I. Title. II.
Series: Research in international studies. Africa series ; no. 84.
DT1325.B55 2006
325'.340967—dc22
2005032141
## Contents
Preface
1. The Idea of Empire,
2. Wine, Women, and War
3. Merchants and Missionaries
4. A Swiss Community in Highland Angola
5. The Case of Belgium and Portugal
6. Race and Class in a "Fascist" Colony
7. The Death Throes of Empire
8. Destabilizing the Neighborhood
9. Carnival at Luanda
10. The Struggle for Power
11. A Journey through Angola
Notes
Further Reading
Index
## Preface
Angola's imperial age spanned five centuries, from 1500 to 2000. In some respects the experience resembled Hispanic colonization in America, with battalions of conquistadores and cohorts of mission friars and pastors bringing European customs and beliefs to Angola's peoples. In other respects, as outlined in the eleven essays presented here, Angola's experience was closer to that of its African neighbors. The Portuguese influence spreading out from the cities of Luanda and Benguela paralleled the Dutch influence in the south, where settlers created urban enclaves that rippled out into the countryside, coopting local herdsmen, craftsmen, and concubines. Two centuries later, Portuguese colonial practice came to resemble the practices of French and Belgian neighbors in the north as financiers were given a free hand in carving out giant land concessions complete with untrammeled rights over the resident populations of potential conscript workers. On the eastern shore of Africa, imperial experiences in Mozambique linked the Portuguese closely to the British as the railway age opened up an era of mine prospecting, which saw ambitious foreigners and teams of migrant navvies flock to Angola.
Among the themes of modern imperial colonization, one of the most influential and lasting was the impact of Christianity on societies that—unlike some of those in northern, western, and eastern Africa—had little or no previous experience of external world religions. In this book, reference will be found not only to the way empire-builders used the Catholicism of the Portuguese colonizers of Angola and of their Belgian neighbors in Congo, but also to the role of American Methodists, British Brethren, and Swiss Presbyterians in creating empires. Each of these evangelizing traditions inspired and educated some of the African men and women who inherited the colonial territories from the imperial conquerors and carried forward the religious legacy of empire.
A second overarching theme of empire was the introduction of currency transactions, including coins and banknotes, into a broad range of economic activity that had previously been governed by barter exchanges measured in assortments of trade goods. The changes affected not only the old long-distance marketing of salt and iron, but also the new commerce in coffee and sugar, in rubber and ivory, that linked Central Africa to Europe and America and became the economic grounding of empire. The exchange principles affecting commodities were, however, slow to affect the African labor market. In Angola and the neighboring territories, labor recruitment took forms of compulsion, coercion, and conscription that were distressingly similar to the practices utilized by precolonial slave traders. Violence as a means of driving men and women to work ceaselessly for foreigners continued to be prevalent throughout most of the colonial period.
Military might was another means by which empires in Africa were created. During the last years of the nineteenth century and the first decades of the twentieth, columns of white imperial soldiers and black colonial conscripts imposed foreign rule. The legacy of warfare was particularly lasting in the Central African empires, and it was soldiers who captured the nationalist agenda in the 1960s. Thereafter, for a full generation, their officers profited from neocolonial extractive economies. Even where the departing empire builders had created integrated economies, with urban consumers supplied by local industries, the new postimperial rulers tended to revert to the old colonial practice of raw extractive plunder. They concentrated their attention on diamond washing, copper mining, and petroleum drilling.
Empire in Angola and in the neighboring territories was essentially the creation of men—entrepreneurs, engineers, colonels, and merchants. A few colonial settlers sent home for white brides, and pioneering preachers were often wedded to an ascetic lifestyle. Most other imperial immigrants adopted a predatory masculine attitude to the ill-defended female half of the conquered population. In Angola, white migrants fathered mixed-race children whose ambiguous identities were akin to those of the "coloured" population of South Africa and the "mestizo" population of South America, and Afro-Europeans in the Central African territories experienced the same contradictory mix of racial disadvantages and racial privileges experienced by the Anglo-Indian population of the British Empire in Asia.
The most potent legacy of empire—to use the term in its loosest sense—was the oil revolution of 1973. In Angola, much as in Nigeria and other Atlantic-basin territories, this revolution made the country's soldier-politicians the dependent clients of the Texan oil giants. The corrupting lure of petroleum made an equitable and democratic future hard to grasp when independence arrived in 1975. Warfare, both civil and intrusive, was the first bitter harvest of the imperial age. In the longer term, however, the new wealth may hold out a promise of hope for Angola's vibrant people. Ingenuity and energy have enabled some of each rising generation to survive during Angola's hundred years of foreign plunder and thirty years of civil conflict. With better education, better sanitation, better statesmanship, better friends, Angola's people have rich potential in an impoverished land.
## 1
## The Idea of Empire
_This book is mainly about the nineteenth- and twentieth-century impact of empire on the Atlantic fringe of Central Africa, on the peoples whose homes spread across Angola and its neighbors from the Congo forests to the Namibian deserts. To put the story into its wider African context, this introduction, loosely drawn from a presidential address to the African Studies Association of Great Britain, provides a deeper chronological background_.
The concept of empire and the associated spread of cultural and economic influence is a very old one. In Africa imperial ideas have flowed and ebbed for more than two thousand years, linking African peoples with their fellows all the way from China in the far east to Brazil in the far west. In some cases the flow has been accompanied by the migration of peoples and in others by the dissemination of religious ideas. Trade has commonly been a vehicle for bringing societies into contact with one another, but in some cases military expeditions have played their role too. Cultural contacts have affected many aspects of people's daily lives, from the adoption of the camel by pastoral peoples in northern Africa to the introduction of the cassava root as one of the main crops of Africa's tropical farmers. Imperial connections have commonly been linked to the search for rare forms of wealth such as gemstones and precious metals. Imperial ideas have deeply affected the ways in which people live, often establishing new concepts of town dwelling or innovative forms of municipal government. Empires have also penetrated the home, creating new patterns of marriage, modifying domestic patterns of architecture, and widening culinary tastes. The footprints of empire have also left behind radical changes of language, innovative concepts of literacy, and new renderings of music.
Two thousand years ago, seafarers from Indonesia were among the earliest and most influential of migrant colonizers to settle the islands and mainlands of Africa. Whether their small vessels, with outriggers and sails, brought them as far as the Atlantic shore is not yet known, but their cultures did reach many tropical parts of the continent over land. Fish trapping and rice growing remained quintessentially east coast activities, but the playing of great wooden xylophones, with resonance chambers made from large calabash gourds, became popular in Angola, and Indonesia's rhythmic tunes invigorated the lives of young and old alike. In the wetter regions of Africa, vegeculture, based on the old African yam, was enhanced by the adoption of a thousand varieties of banana, which enriched carbohydrate dishes and extended the range of Africa's alcoholic beverages. While enjoying their banana beer, forest peoples played board games which had spread across Africa from the Indonesian colonies of Mozambique and Madagascar. On a less cheerful note, this early "empire," like many later ones, spread previously unknown weeds to the farms of Africa and diseases to its farmers.
The Asian dimension of cultural and commercial influence in Africa was varied. The links between India and Africa are illustrated in the wealthy material culture of Axum, an empire that flourished in the Ethiopian highlands in classical times. The lending and borrowing of domestic animals such as humped cattle and cereal crops such as tropical millet enriched both Africa and Asia. The great Persian Empire sent expeditions to Africa from their trading harbors on the shores of the Gulf. It is likely that textiles, the mainstay of many commercial empires, were part of the produce that they shipped down the African coast in exchange for the timbers required by desert architects whose local resources were limited to stone and clay. Cotton planting may have reached Africa by the Persian route, and in later centuries cotton became a key factor affecting the rise and fall of several empires. By the thirteenth century, textiles were reaching Africa from as far away as China and wealthy mine magnates in Zimbabwe, as well as wholesalers on the entrepôt islands of the Tanzanian coast, ate their meals off Chinese porcelain. Chinese mandarins at the imperial court at Nanjing even admired a live giraffe as it disembarked on a quayside several hundred miles up the Yangtze River. Direct trade on the giant Chinese merchant vessels did not last long, however, and in the sixteenth century the Indian Ocean harbors fell into the hands of European colonizers with much smaller ships. Even so, Arab trade between Africa and the Middle East continued to thrive, and on the banks of the Euphrates exotic brides from Ethiopia were greatly admired in the palace gardens of Baghdad.
A more military tradition of empire, quite distinct from that of the Indonesian settlers and Asian traders, developed along the northern coast of Africa with the rise of the Roman Empire. Roman legal traditions relating to citizenship, town governance, slave status, and land tenure were adopted by the Lusitanian peoples of what later became Portugal and were later transferred by Portuguese colonizers to parts of tropical Africa, including Angola. Roman imperial customs had not emerged in the Mediterranean overnight but had grown from deep roots in the earlier imperial civilizations of Egypt, Syria, and Greece. In giving their emperors god-like status, the Romans were following the Egyptians and their worship of the pharaoh. Like Egypt's landed aristocracy, Rome's depended on large cohorts of slaves, and the servile system was later adopted by Portuguese imperialists on both shores of the Atlantic. Phoenician merchants from the harbor towns of Syria created colonies across North Africa, most notably at Carthage, and these later became the southern bastions of Rome's empire. The Greeks not only created colonies in such fertile havens as Cyrenaica but also sent teams of adventurers into the dry Sahara to search for rare trade goods. In the fourth century before the Common Era Alexander's Greek armies conquered Egypt before setting off for Persia and India. Three centuries later one of his successors, the Greek empress Cleopatra, surrendered the fertile valley of the Nile to the Roman legions.
The blending of peoples and of cultures became a distinctive feature of empires in Africa. One fully creolized African from Libya, Septimius Severus, adopted the Roman style of living and rose through the ranks of military and political service to become the god-emperor of all that Rome had conquered.
In the Arab empire, which erupted into Africa at the end of the first Muslim century thirteen hundred years ago, conquered peoples accommodated themselves to the customs of their rulers and adopted new cultural identities in Creole communities. In a pattern of blending later witnessed in other empires, Arab culture, the Arabic language, the Arabic script, and the Islamic faith were readily accepted by many peoples of the towns. The Arabization of northern Africa went further than the converting of townsmen and the recruiting of local military regiments when rural Bedouin migrations began to have an impact on pastoral communities across the great African plains. Only in the mountains of northern Africa did traditional Berber customs, languages, and religions retain their distinctive traits.
In modern times three empires have had an impact on Africa comparable to that of the great waves of Romanization and Arabization: the sixteenth-century Ottoman incorporation of northern Africa into the Turkish empire; the seventeenth-century creation of a Dutch empire that linked southern Africa to the spice colonies of India and Indonesia; and the creation by the Portuguese of an empire on the western and eastern seashores. All three imperial powers, after staking claims to urban footholds along the coast, were eager to gain access to the deep interior of Africa. Portuguese investors penetrated the inland commercial empires of West Africa, sending embassies to the court of Mali on the Niger and capturing a part of gold trade. Turks entered the tropics from Cairo bringing firearms and chain mail to the empire of Kanem on Lake Chad. The Dutch frontier was opened up by hunters, and when ivory became one of Africa's most desirable exports, all three empires attempted to increase their hold on the elephant country of Central Africa.
One feature of the early-modern empires that has not attracted adequate attention is the role of the great Jewish diasporas of Europe and the Near East. Jewish accountants, clerics, and merchants apparently played a vital role in the economic well-being of the Ottoman empire, and Jews were well protected by Islamic traditions of tolerance. Beyond the western reach of the Ottomans, the city of Tetuan in Morocco thrived on Jewish professionalism and enterprise. Out in the Atlantic, Dutch imperial dynamism owed much to Jewish initiatives, backed by Jewish capital, which had taken refuge in Amsterdam after being driven out of the Iberian kingdoms by Christian intolerance and persecution. A covert but indispensable feature of the Portuguese empire was nevertheless the surviving economic strength of its Jewish communities. Portuguese colonial governors could not permit Jews to practice their religion openly, but they could not afford to deprive themselves of Jewish commercial and industrial services. The first Portuguese lord-proprietor sent to govern Angola, a grandson of the Atlantic explorer Bartholomeu Dias, was explicitly permitted to take Jewish craftsmen with him and may himself have been descended from Jewish ancestors. Little attempt has been made to assess the Jewish contribution to the multicultural Creole communities of Angola, but it is suspected that in the seventeenth century the Portuguese royal inspector of taxes at Luanda had a Friday night job as the rabbi of a clandestine synagogue. In the nineteenth century the street names in the up-country trade fair of Dondo still hinted at a persistent Jewish folk memory.
This book is not about empires at large but about empire in Central Africa. In addition to Turkish encroachment on the north and Dutch encroachment on the south, the eastern fringes of Central Africa saw migrants from the shores of the Indian Ocean bring their cultural baggage inland and create Creole towns of mixed race and heritage on both the Zambezi and the upper Congo. Swahili-speaking Muslims from Mozambique were trading bolts of cotton textiles in exchange for ropes of copper ingots by the fifteenth century, and their new commercial towns were linked to ancient mines that had been exploited since the eighth century. Trading was linked into networks that specialized in the local production and distribution of salt and of dried fish. In the nineteenth century another wave of colonization spread inland from Zanzibar with the Swahili search for ivory. The traders and their local agents built towns with shady boulevards where status-conscious Creoles flourished and Swahili became the language of trade and political command. The Creole towns of the upper Congo, like the older Zambezi towns, were in regular contact with great caravans of Portuguese Creole merchants that had crossed the continent from Angola. So influential were the Swahili Creole traditions in eastern Congo that when a European empire encroached, driven by Leopold of Saxe-Coburg in Brussels, the language of authority of the multinational white mercenaries was Swahili rather than French or Dutch, and after 1908 it remained in use in the Belgian colonial army.
The Creole societies of other protocolonial communities along the Atlantic seaboard bear interesting comparison to those of Angola. While Angola's closest neighbors were the settlements created in the nineteenth century at Libreville in Gabon and Boma in Congo, the communities on the upper coast of Guinea in West Africa had much deeper and older parallels. French Creoles formed the merchant aristocracy of Saint Louis on the Senegal River and the slave entrepôt on Gorée Island. Portuguese Creoles became the imperial agents of the Bissau rice-growing rivers and the salt islands off Cape Verde. British Creoles were descended from surplus slaves repatriated from eighteenth-century Canada, who had been given an Anglo-Scottish homeland on the Sierra Leone estuary. American Creoles were liberated slaves who called their home Liberia and named their administrative town after the fifth American president, James Monroe. The Creole communities in Angola and in Guinea had recognizably similar features; each blended the cultural customs of visiting merchant sailors from Atlantic Europe with those of their African hosts. Attempts at colonization and cultural hybridization began on Africa's islands, stretching from the Canaries to the tropical offshore island of São Tomé. By 1575 a bridgehead had been established on the shoreline island of Luanda, separated from the mainland by a magnificent palm-fringed bay. From there settlers moved to the mainland and established diplomatic relations with the kingdoms of Kongo and Ndongo. In the seventeenth century these kingdoms would be overrun by conquistadores who established the colony of Angola.
The western frontier of Central Africa, which had been opened in the late fifteenth century by Portuguese pioneers, experienced several distinctive phases of cultural change. The first involved the introduction of Christianity. In the sixteenth century the rising kingdom of Kongo was riven by armed factionalism. One of the smaller, and possibly less legitimate, factions hit on the idea of incorporating the Christian deity, along with a host of angels borrowed from Portugal, into its pantheon. The royal pretender adopted the westernized name of Afonso, and in 1506 he soundly routed his rivals. He established a royal court in which black scribes corresponded with Rome. The process of westernization proceeded apace, and one black prince visited Europe, where the pope accorded him the style of bishop. Under the auspices of a black Christian king, immigrant merchants—many of them "New Christians," whose ancestors had been Portuguese Muslims and Jews—established large African families. The clusters of merchant compounds were thronged with slaves, clients, retainers, children, and concubines. Close commercial relations were maintained with the sugar islands of São Tomé, where the ideological system of tropical slave plantations was perfected. In the sixteenth and seventeenth centuries the plantation system was transferred to the Americas, where it took root first in Brazil, then in the Caribbean, and finally in North America. Creoles from the islands and town dwellers throughout the kingdom thrived on their association with the Kongo court and its provincial aristocracy, but they did so increasingly at the expense of an impoverished rural peasantry. After two generations of exploitation, revolution broke out. The sansculottes from the countryside rose up and swept away the court, the princes, and the Afro-European commercial elite with its urban culture. The king and his entourage cowered on the fevered hippopotamus islands of the Congo estuary.
To restore the old kingdom and to maintain an imperial trading base in Central Africa, the united Hapsburg crown of Spain and Portugal decided to send conquistadores to Africa to recover the trading grounds that had supplied slaves not only to the Portuguese plantations of Brazil but to the Spanish mines of Colombia and Peru. Six hundred Portuguese imperial soldiers arrived in Central Africa in 1568 with matchlocks, flintlocks, and other muzzle-loading firearms and were joined by conquistadores from Spain and the Netherlands. Many military men married into the local society and joined the ranks of the Creole bourgeoisie. Black genes soon overwhelmed white ones, and by 1681 Cadornega, an old soldier from Portugal, bemoaned the fact that in his beloved Angola sons were swarthy, grandsons were dusky, and all else was blackness. This black military class of African Creoles with Portuguese culture survives to the present day. One military commander, scion of a long-since black family of early seventeenth-century Dutch origin, almost succeeded in overthrowing the first independent government of Angola in 1977. Other members of the Creole elite became part of the establishment, the _nomenklatura_ , of postimperial Angola.
In the nineteenth century Creole life focused as much on commerce as on military rank. In Luanda a flourishing high society astonished visitors who witnessed the elegant balls held in the governor's palace. Among this elite a sophisticated local capitalism had begun to emerge. Dona Ana Joaquina, the grande dame of Luanda black society, began to set up plantations of her own in Africa and to use slaves in local enterprises rather than sell them abroad to the Americas. In preference to buying Brazilian firewater, she made her own trade rum. She also owned a small trans-atlantic sailing fleet that enabled her to import her horses from Uruguay. One sophisticated member of society who moved easily among Creoles as well as expatriates was the young Swiss adventurer Héli Chatelain, whose career features in later chapters of this book. Chatelain was a watch seller (he came from near the city of La Chaux-de-Fond, where 380,000 gold watches were made each year), a schoolteacher, a choirmaster, a compiler of dictionaries, a collector of artifacts for the Smithsonian museum, a commercial attaché with consular duties, and a campaigner against the practice of slavery. After years of protesting at the inhumanity of slave hunting, Chatelain left Angola in 1907, fearing that his enemies might poison his food or set fire to his thatched Christian village. Soon afterwards, Angola's four-hundred-year-old Creole civilization faced a series of rude shocks. The rise of racist attitudes spuriously associated with Darwin now provided a new intellectual context for empires. Portugal, after a violent republican revolution, became less tolerant of multiracial fraternization and began to reject the racial pragmatism of the old Creole tradition. By the 1920s the racism that gripped South Africa was almost matched by the segregationist behavior of republicans in Angola.
Empires that had become racist and segregationist had ambivalent attitudes toward the great variety of Christian missions that spread into Central Africa in the late nineteenth century. The relationship between empire and mission had always been a fraught one, but the rulers of empire tolerated missionaries in the hope that they would instill "civilized" European values and inculcate in Africans a "loyalty" to the conquerors. Missions were expected to teach colonial languages, create consumer demands that would benefit home industries, train clerical workers or artisans, and remove Christian converts from the influence of village headmen empowered by magic rituals. Missions attempted some of these tasks, but the consequences were not always as expected and networks of old-school scholars disseminated new ideas and ambitions. Communities that had lost their old cohesion and their respect for village elders regrouped in chapels whose leaders reflected many varieties of Christian charisma. The roots of the anticolonial nationalism that brought an end to empire can often be found in religious congregations.
The implosion of empire that transformed Central Africa began in 1960, inadvertently triggered by the prime minister of Great Britain, Harold Macmillan. Having witnessed the failure in Egypt of one last attempt to use old-fashioned gunboats to stem the tide of decolonization, Macmillan became convinced that the days of empire were over and the remnants needed to be tied up in new colonial-type partnerships between the industrial countries of the north and the suppliers of raw material in the south. Now that Egypt was lost, the most important imperial-type territory in Africa was South Africa, which had been self-governing since 1910. Although it was a virtually independent part of the British Empire, Macmillan still took a paternalistic, almost patronizing interest in the country's economic stability. When touring black Africa to acknowledge the reality of the new postcolonial order with speeches about "the wind of change" blowing through Africa, he also called in at Cape Town on February 3, 1960. The image of a wind of change had become such a banal commonplace in West Africa that Macmillan's speechwriter, a normally shrewd former secretary to both Clement Attlee and Winston Churchill, failed to realize that the words would play very differently in South Africa. The speech was designed gently to encourage South Africa to move ideologically into a postcolonial world devoid of prejudice or segregation, a world in which neocolonial market relations would be able to flourish unthreatened by the prospect of rebellion. Sensitive Dutch-speaking nationalists in Cape Town's white parliament heard the speech quite differently, as a threat to their dignity and autonomy, a warning to soften the laws of apartheid or risk dire consequences. People in the black townships of South Africa, however, heard the words almost as a promise that Great Britain, having systematically betrayed black aspirations in South Africa for a hundred years, was now prepared to give a voice to African people. The two interpretations of Macmillan's speech met in headlong confrontation only six weeks after he delivered it. The Pan African Congress, hoping to steal a march on the African National Congress, organized a celebratory demonstration aimed at accelerating the demand for new freedoms and the abolition of pass-law controls. White society panicked at the PAC's audacity and met the crowd at Sharpeville with live police ammunition. Singing and dancing stopped abruptly and five dozen fleeing demonstrators were killed by bullets fired at their backs. Macmillan's agenda for managed change in South Africa was dead. But his speech fueled the stirrings of rebellion in Central Africa, and the chain reaction took only twelve months to reach Angola.
The first Sharpeville-type massacre of the forthcoming Central African revolution occurred in June 1960 in a remote northern community of Mozambique. The people of Mueda, who had not been paid for their hard labor on the colonial plantations, went cap in hand to the district office to ask for their arrears of salary. Portugal's fascist-style empire was not accustomed to such popular protest, and the police pulled out their revolvers to quell the impertinence. How much blood was shed has never been measured, but the initial effect, as in South Africa after Sharpeville, was to silence black protest in Mozambique for several years. In the neighboring colonies of Malawi, Zambia, and Zimbabwe, however, rebelliousness continued to grow, and the London rulers of empire had to acknowledge that their grand experiment in creating an imperial partnership in the British corner of Central Africa between white settlers and black peasants had failed. Black governments came to power in the two northern territories, while white settlers mounted a counterrebellion and held on to power in the south.
A different sequence took place in the Belgian sphere of influence in Congo. The rebellions of July 1960 postdated, rather than predated, the Brussels grant of independence. Belgians expected the transition from conventional colonialism to neocolonialism to be free of any ripples of disturbance. Life, the white settlers believed, would carry on as before. Some black nationalists, however, expected immediately to inherit the earth, complete with motor cars, piped water, smart uniforms, and generous paychecks. Conflicting expectations led to unrest, and within a week of independence Belgium had reinvaded Congo and seized its capital city with armed parachute commandos. Worse was to follow when, with the connivance of Europe's industrial corporations, Belgian investors decided to insulate the rich mining province of Katanga from national disturbances by creating a puppet regime led by collaborating black secessionists. Several years of civil war were to follow.
In Angola the decolonizing process, though similar to the process in the neighboring territories, had distinctive features that made it unexpectedly protracted and violent. The violence began in January 1961 when Angola's peasants began to despair of ever getting paid for the cotton that they had been "growing for the governor." Children, they said, were hungry because their mothers had spent too much time picking cotton and not enough time harvesting maize. The Portuguese imperial response to the protests was even more violent than it had been in Mozambique six months earlier, as a rudimentary air force firebombed recalcitrant Angolan villagers, who fled by the thousands to seek safety in the forests of the newly independent Congo. Two months later, farm conscripts working on Angola's northern coffee estates also tried to obtain payment of overdue wages. They too were harshly treated when gangs of white vigilantes were issued with weapons and encouraged to hunt down "troublemakers." In between the two rural uprisings, violence had also broken out in Luanda in February 1961, when groups of young hotheads attempted to free their nationalist heroes from the local prison. The reaction in the city, connived at by officialdom, was to root out any westernized Africans who might advocate social or economic change to the colonial order. The year 1961 saw the first of the great orgies of bloodshed which were to mar the history of Luanda again in 1977 and also in 1992. Violence became the lasting legacy of empire.
## 2
## Wine, Women, and War
_This chapter on the Portuguese and the Dutch in Africa was originally delivered as an "entertainment" in Portuguese at a summer school for historians held at the New University in Lisbon in 1999. The themes of alcohol and miscegenation during the wars of conquest are key to the imperial experience in Africa. Exploring them in the context of Portuguese and Dutch colonialism in southern Africa brings out not only the tragedies that beset African life in the imperial age, but also the ironies that one encounters when analyzing colonial policies and European attempts at justifying them. The Portuguese, for instance, tried to limit the alcoholism that so undermined work schedules on their plantations by curtailing the colonial production of rum, but they replaced the rum with large-scale imports of Portuguese wine. Their puritan Dutch neighbors, meanwhile, turned surplus agricultural produce into local gin to sell to mine workers, though mine owners gradually came to realize that they would have preferred their black labor to stay sober. Both Dutch and Portuguese colonial traditions were very largely male affairs, and when not drowning their sorrows in drink or fighting each other for the control of wealth, colonial men in both spheres were fathering semiblack families with whom they maintained changing and ambiguous relations_.
In the early 1880s a small band of Dutch-speaking farmers, the Boers, made their way across the Kalahari Desert to settle in the southern highlands of Angola. Over the next fifty years they provided one of the links, and also comparisons, between Portuguese West Africa and British South Africa. The trekkers also created a colonial model for some of the Madeiran, Brazilian, and Portuguese settlers who struggled to scratch a living, agrarian or commercial, from the soils and peoples of south Angola. Transport was one field in which the Boers became important entrepreneurs; they created a network of wagon trails across the plateau and as far as the Zambezi before Portugal's royalist colonizers, financed by foreigners, built railways into the highlands and before Portugal's republican colonizers, driven by the dynamic high commissioner Norton de Matos, built roads for automobiles through the interior. The Boers brought the ox plow to tsetse-free areas of southern Angola and extended the cultivation of wheat and maize. They were also great hunters who followed the dwindling stocks of Angola's elephants into the remote fastnesses of the interior. In Angola, however, one important source of early colonial wealth was derived neither from farming nor from hunting but from the distilling of strong alcohol. Back in the Transvaal, successful Boer farmers turned their maize into gin. In Angola, on the other hand, planters grew sugar cane in the Angolan lowlands and turned it into _aguardente_ (rum)—the driving force of colonial penetration.
The Angolan rum trade had deep historic roots. Firewater from Brazil had been an important item in the Atlantic trade in slaves before slave exports were outlawed in 1850. Rum remained a bargaining commodity of a new trade that exported indentured workers from Angola. Rum as a means of purchasing the slave-like men and women for shipment to the cocoa islands of São Tomé had many advantages for the traders and disadvantages for the local society. A more beneficial trade commodity would have been cotton cloth. Textiles were valuable and permanent goods that benefited women and children, whereas alcohol was drunk largely by men. Cloth could be stored and exchanged as part of a commercial system of barter, and it did not create a physical craving or addiction but stimulated market activity as fashions changed. For Portuguese traders, however, opportunities for trading in cloth and clothing were limited by the small, inefficient scale of the textile industry in Lisbon and Oporto. Portugal had to buy many of its bales of cotton prints from England and pay for them by the yard in foreign exchange. Alcohol, by contrast, was readily available at soft currency prices.
In the colonial trade a lucrative alternative to cotton textiles and fortified alcohol was gunpowder. Between January 1909 and September 1912, Angola officially imported nearly two million kilograms of loose gunpowder and sold it at considerable profit to Africans, who owned hundreds of thousands of muzzle-loading guns. In 1906 a child slave could be bought in eastern Angola for a single small keg of gunpowder. The gun trade had its downside, however: as colonial occupation spread across the Angolan plateau, armed black resistance increased. In 1913 Norton de Matos decided—despite howls of dismay from the traders—to ban the sale of gunpowder. With gunpowder banned and cloth expensive, the trade in alcohol became ever more attractive. Chiefs who sold slaves to caravan leaders became so addicted to distilled liquor that they offered ever more slaves for ever less rum. The victims themselves were quietened by tots of rum as they were led away for overseas shipment. At the coast, when the new slaves put their cross—in place of a signature—on an indenture paper that signed away all economic freedom, they did so with less reluctance when in a state of alcoholic exhilaration. The recruiting of indentured workers in Africa came to resemble the historic recruiting of mercenaries for the armies of Europe, where press gangs used alcohol as an inducement to enlist.
In Angola the alcohol trade enriched the traders but impoverished other colonizers. Workers who had a taste for rum did not work as hard as those who were sober. The trade became so ubiquitous, however, that the functioning of the whole colonial economy seemed to depend on the rum connection. As a consumer commodity rum created a thirsty demand but brought political problems as well as having moral, psychological, and physical disadvantages. Rum production in Africa did not suit the fiscal requirements of metropolitan authorities in Portugal. Colonial rum undermined the trade in the Portuguese wines traditionally sold to African consumers, and the rulers of empire therefore attempted ineffectually to limit the distilling of rum on the Angolan sugar estates. In 1911 the newly established Portuguese republic took more energetic measures to ban local liquor. To encourage metropolitan producers to seek colonial markets for their salvation, export duty on Portuguese wines, both common and fortified, was reduced. Angola's imports rose to eight million liters of "native" wine a year—possibly ten times more per capita than even wine-producing France was able to sell to its conquered subjects.
One of the ironically amusing documents on the colonial wine trade concerns the way in which the Portuguese wine lobby presented its sales pitch at a meeting held in 1901 in the prestigious premises of the Lisbon Geographical Society. The wine growers recommended that settlers should drink green wine ( _vinho verde_ ) as a refreshing drink for a hot climate. For the "natives," on the other hand, proclaiming their great patriotism and in the name of the campaign against alcoholism, they advocated the increased consumption of fortified wines. It would be useless, they said, to offer Africans wines with an alcoholic content of only 19 percent; to wean them off rum, fortified wines of at least 23 percent should be offered to them. The directors of the Real Companhia Vinicula rejoiced that, by good fortune, brandy was cheap in Portugal and the company could offer strong wines at reasonable prices to help with the national export drive. Through such an offer they could help to combat the pernicious sale of alcohol distilled by planters in Angola.
The amazing audacity of the wine merchants went even further. They also pleaded for an abolition of government controls on the quality of such wines, and instead of any verification of the wines being sold for African consumption, they wanted an unfettered hand to develop their colonial outlets. The market principle, they argued, would be perfectly effective in limiting the levels of adulteration by profiteers. It would be intolerable to colonial wine merchants if inspectors were permitted to examine export wines or impose standards that might bankrupt Portuguese producers. Equally intolerable would be any check on quality made by the customs officers when the supplies arrived in the colonies. In one final plea for preferential treatment, the alcohol traders demanded that colonial currencies be made fully convertible to facilitate the repatriation of profits.
Twenty years later, Angolan officials were still advocating an increase in Portuguese alcoholic imports in order to reduce alcoholism. The patriotic high commissioner, Norton de Matos, back in Africa for a second term of office, proposed a token duty on fortified Portuguese wine and punitive duties on Scottish whisky and French brandy. Wine imports into Angola continued to grow, reaching 35 million liters per year after the Great Depression and World War II. Imported wine was used not to recruit indentured workers for the plantations of São Tomé, but to pay conscripted black migrants who were sent to work on the coffee estates inside Angola. The Angolan taste for Portuguese wines did not end with the fall of empire—ten years after decolonization, Portugal's famous green wine was still available in the bars of Luanda.
The Dutch colonizers of the Transvaal were conducting a quite different set of political debates on the role of alcohol in colonial society. One method of attracting black workers to the South African mines was to offer them stronger beverages than they were able to brew in the villages. The role of alcohol in South African business increased when colonial rulers and settler republics decided that selling guns and gunpowder was strategically dangerous and threatened their political hold on conquered territories. The labor touts tried to switch from payment in guns to payment in woolen blankets. Although blankets were valued in the cold highlands of the Transvaal, they were not sufficient to attract migrant workers to the mining camps. Alcohol was therefore adopted as an inducement to work in the Transvaal mines and in many other South African enterprises.
There was one major difference between labor recruitment in Angola and in the Transvaal: while plantation workers from Angola could never escape from São Tomé island and so remained exiles and prisoners for life, Transvaal miners could always walk home, and frequently did so when it was time to bring in the harvest, settle a village dispute, or find a wife. One way of limiting any desire by black workers to leave mine employment was to create an addiction to beverages that were available in the gold rush camps. One such beverage was the potent gin made cheaply from surplus German potatoes and smuggled into the Transvaal via Mozambique as though it were a produce of Portugal. An alternative inducement, appreciated by Boer farmers and Afrikaner politicians, was Transvaal gin, distilled from local maize.
As has been demonstrated by Charles van Onselen in his social and economic studies of the Witwatersrand, the Boers who ruled the Transvaal made significant profits out of the trade in alcohol for the mines. The price offered for "mealies" for the miners' porridge was usually low, and maize could not be stored until prices improved. Maize that was converted into gin, on the other hand, could be safely stored and sold when prices were advantageous. The profits on gin were higher than the profits on cornmeal, and Boer farmers liked to sell their harvest to the Transvaal distilleries. Although the Dutch resented the intrusion by "foreign" industrialists into their agricultural paradise, they did like the new market for alcohol that Johannesburg opened up. At the same time, they perversely resented industrialization when black agricultural labor was siphoned off the farms by mine owners who offered competitive wages and strong drink to former field hands who were willing to go down the gold mines. Distillery profits spoke loudly to the farmers, however, and despite their Calvinist puritanism, Boers sold maize to the gin mills and ignored the moral consequences of creating alcoholic addiction among their black Transvaal brethren.
It was British colonial policy, rather than Calvinist moral philosophy, that challenged the Transvaal distilling industry. Opposition to the supply of gin to black Transvaal mine workers, like opposition to the supply of rum to black Angolan field hands, was based on the negative impact of inebriation on worker productivity. Angola tried to forbid the sale of rum within three kilometers of a plantation. In the Transvaal cheap gin created such a pattern of inebriation that little mine work could be carried out on a Monday. (A similar hangover day in gin-swilling industrial England was called Saint Monday.) Having used gin to attract migrants to Johannesburg, the mine-owning capitalists, pragmatic Englishmen, soon began to campaign against supplying alcohol to workers at the mines. Their proposal to ban the sale of alcohol to black Transvaal consumers immediately ran into opposition from Afrikaner farmers concerned lest they lose the lucrative market for distillery maize. The irony of Transvaal Puritans campaigning for the continuation of alcohol sales to black workers almost matches the absurdity of Norton de Matos advocating the sale of fortified wines to combat alcoholism in Angola.
Political differences between British mine owners and Afrikaner land owners grew more intense through the 1890s and spread from alcohol policy to other matters of high politics. The Afrikaners wanted to preserve their monopoly on transport by continuing to supply the mines with coal brought in on ox wagons from a railhead on the Transvaal border. Transvaal politicians had been reluctant to cause large-scale unemployment among Boer ox breeders and wagon drivers by allowing the British—or the Portuguese—to build a railway through to Johannesburg. Although the Transvaal farmers were eventually persuaded to accept the extension of the mining industry, and even the building of the railways, they wanted to replace lost revenue with new duties and caused much offense to the British by imposing heavy customs tariffs on dynamite needed to open up the deep-level mines. All these disputes—over ox riding, dynamite duty, and alcohol sales—were used by the British Empire as excuses to launch a war in which they aspired to conquer for themselves the world's largest and richest reef of gold mines. Whatever concessions President Kruger's South African Republic made, still the English demanded more. In 1896 hotheaded imperialists attempted to take over the Transvaal in Jameson's conspicuously unsuccessful coup d'état. Three years later Britain invaded.
Once the British, by means of the longest and most expensive war they had fought since the defeat of Napoleon, had established themselves in Southern Africa, and once they had decided that they were not going to use alcohol as the means of buying, recruiting, or attracting workers for their colonial enterprise, they had to find other ways of obtaining labor. Their normal method, like that of the Portuguese, was to recruit _serviçais_ , known in English as "indentured laborers." Labor inside South Africa was relatively scarce and therefore expensive and was needed by white Afrikaner farmers, who, unlike black Transvaal farmers, soon recovered the vote and a political voice in the newly conquered colonial Africa. In order to find mine labor, therefore, the British looked outside South Africa and to the globalized market for labor. The three cheapest sources of labor in 1900 were British India, northern China, and Portuguese Africa. The British recruited labor from all three sources in a pattern of oscillation governed by market forces and political expediency. China was initially their main source of migrants. By 1906, however, the working conditions of Chinese "coolies" in the Transvaal mines so closely resembled the working conditions of old-fashioned plantation slaves that the traffic between China and South Africa caused a moral outcry in Britain. A newly elected Liberal government in London decided to end Chinese migration to Africa and repatriated the Chinese workers.
Indian labor had a much longer history of service in Africa than Chinese labor. When slave labor was no longer legitimate, Indians had been used to expand South Africa's sugar plantations. Indian navvies, who were cheaper than African workers, were used extensively to lay railway lines in Africa. Unlike Africans, Indians were unlikely to escape from building sites, since they could not expect to be welcomed as refugees in African villages. The contracting of indentured serviçais from India prompted complaints both from the colonial government in British India and from the Indian middle class. One of India's most famous lawyers, Mohandas Gandhi, went to South Africa to lobby against the exploitation of, and discrimination against, Indian labor. His campaigning coincided with the rise of a new labor policy as the British Empire turned to its third source of ultracheap labor, Portuguese Africa.
The most dynamic branch of the Portuguese trade in African indentured servants, still flourishing four centuries after its origin in the 1480s, was the one that took serviçais from Angola to São Tomé. On the island the so-called servants grew cocoa, which was sold primarily to English chocolate manufacturers. The Portuguese planters had tried to use both Chinese and Indian indentured workers but had had little success. Indentured workers from Angola—sometimes purchased with alcohol—were much cheaper. A domestic slave in Africa could be bought for as little as two small kegs of rum, and by a legal fiction could then become an indentured serviçal available for export. Shipping costs were also much lower for indentured servants bought in Angola than for contract workers shipped from India or China. Angolan serviçais only had to be fed and watered for a few days while manacled to the decks of the boats to prevent their being swept overboard in rough weather. On arrival the conditions under which cocoa workers labored were little better than those endured by the Chinese in the Transvaal mines, and by 1906 agitation against the São Tomé slave trade was growing in parallel with agitation against the Chinese coolie trade. To add to the colonial ferment, humanitarian lobbies were publicizing severe atrocities being committed by the agents of King Leopold in the interior of the Congo.
Britain was reluctant to take the lead in creating a moral agenda in colonial policy. The Swiss-born missionary Héli Chatelain pointed out that it would have been difficult for the British consul in Angola to complain openly about the Portuguese export of serviçais when Britain itself was importing Indian serviçais to built a railway from Benguela to the Congo copper mines. British firms that bought São Tomé cocoa could not, however, close their eyes to the horrors of the trade in kidnapped workers. Many of these had been brought out of the Congo wearing heavy wooden shackles and sold to the Benguela "slave" dealers for little more than ten pounds each. The British manufacturer William Cadbury visited both São Tomé and Benguela in person before accepting that the trade was so inhuman that he should recommend to the chocolate industry at large that an alternative source of cocoa be sought. The mortality rate among São Tomé serviçais—mostly young workers who should have been at the peak of their health—was particularly horrendous, and despite the health care offered on the best plantations, one person in ten died each year. The first year of service, while the imported workers grew accustomed to the lowland tropical climate, the new disease environment, and the bare diet of cassava flour, took a particularly heavy toll. The factor that ultimately sealed the fate of the plantation system, however, was not disease but the failure to repatriate contracted migrants. Recruits were never given the opportunity to return to the land of their birth, or even the land of their purchase. To all intents and purposes they were slaves whose term of service would never come to an end.
One of the ways in which the trade in contract labor to São Tomé differed from other patterns of indentured labor movement in southern Africa was in the proportion of women workers sold by the Angolan caravaneers. The attraction of women workers was that they were cheaper to buy and that they received a smaller monthly allowance than men—less than one thousand _reis_ (five English shillings) per month, which was sometimes withheld as punishment for alleged insubordination. Women were not normally hired for mining or railway building but were regularly used on agricultural plantations as in the days of the old slave trade, when one-third of the victims were women. In Angola it was not only exporters of farm labor who bought women. Boers, who lived in married family units with Dutch wives, bought or captured female slaves to do domestic work. Some of the women who accompanied the Boer wagons were Khoisan (Bushmen) women who had been captured during the crossing of the Kalahari Desert or seized from the desert nomads of southwestern Angola. Other servants were bought from caravans coming out of Congo on their way down to Benguela.
The fact that colonization was a largely male affair had widespread and lasting consequences, especially in Portuguese Angola and at the Dutch Cape. Soldiers, traders, bureaucrats, and technicians, who did not take their wives to Africa, recruited substitute women among their defeated opponents. A population of mixed race—and mixed culture—developed in both the Portuguese and the Dutch settlements. This creolized population had very ambiguous class affiliations within the complex structure of colonial society. In South Africa the so-called coloured population was gradually marginalized and finally excluded from mainstream society and politics, although its members spoke Dutch as their only language and belonged to the Calvinist Church, albeit a segregated branch of that church. In Angola the _assimilado_ Creoles of mixed race were not systematically excluded from colonial society. They often spoke Portuguese in preference to Kimbundu or Umbundu, and many belonged to the Catholic Church. Their status rose and fell throughout the twentieth century. In that century the Cape coloured Creoles gradually evolved into a separate, racially defined community that regularly received dark-skinned Afrikaners who had been excluded from white society and lost light-skinned coloureds who managed to pass for white.
In Angola colonial society continued to absorb conquered women into at least the fringes of settler society long after that had ceased to be possible in South Africa. South Africa passed a series of increasingly severe "morality laws" to try to prevent interracial association and inhibit the growth of any new generation of mixed-race population. Angola's assimilado population, both old and new, black and _mestiço_ , underwent more fluctuating fortunes. The republic of 1910 had brought in a period of harsh repression for the nonwhite peoples of the colony, even those who had previously had high status. Old assimilados were driven from their semiprivileged positions in government by office seekers from Portugal who wanted salaried jobs, to which they thought their white skin entitled them. The republic's great dictatorial governor, Norton de Matos, was keen to enhance white prestige in Angola and strongly discouraged matrimonial mixing. He demanded that any representative of the state in a district office take a white wife and drive out the hitherto accepted concubines and their mestiço children. Racial theory, however, was quite impractical in Angola. The republican carpetbaggers soon fathered a new generation of mixed-race Angolans who aspired to at least some of the privileges their white procreators had enjoyed. The fall of the Portuguese republic in 1926 and the creation of the new monetarist colonial order by the accountant-dictator António Salazar did little to change the social order. Angola remained a society in which black men were compulsorily condemned to work for the state (or sometimes for private but favored colonial enterprises) while black women were liable to be condemned to sexual slavery.
In the 1930s both Portugal and South Africa became mesmerized by the racial theories of Nazi Germany. Nazi ideology was particularly critical of any mixed racial ancestry and exalted an idealized, mythical set of "pure" racial characteristics described as "Nordic." This ideology suited South Africa well. The descendants of ten generations of Dutch men who had fathered their children with Khoi women or Asian slaves were driven lower down the social scale when an Afrikaner "master race" led by General Hertzog replaced the pragmatic British overrule espoused by General Smuts. In Portugal the appeal of German fascism had as much to do with the dictatorial ordering of society and the crushing of democratic debate as with the ideology of racial domination. In a climate of authoritarian colonial exploitation, no easing of racial inequalities occurred and black people remained the workers, white people remained the managers, and mestiço people struggled to retain what privileges they could. The Nazi ideologies of racial purity did little to protect the women of Angola from sexual depredations. When fascism fell out of favor in 1945, Salazar's spin doctors pretended that interracial breeding proved that the Portuguese had never been racists in the Nazi or Afrikaner mold. The argument was far from convincing, however, and it was disdain for black skins that made Angola's women such ready targets for male exploitation or abuse. White women, by contrast, were protected by strong conventions of chastity.
In his memoirs General Delgado, who led the opposition to the Salazarian dictatorship, succeeding General Norton de Matos, until the secret police murdered him in 1958, reported on his increasing awareness of the racial arrogance of the colonial system and its cruelly casual attitude towards women. He witnessed the activities of a commission that toured Angola to investigate conditions in the empire and report back to the great dictator. During the commission's regal progress around the colony, one local officer was so fearful that his distinguished guests might catch some "social disease" that he conscripted young girls from the novitiate of a nearby Catholic convent to serve as "hostesses," trusting that their virginal cleanliness would have been ensured by the supervision of the nuns. When Delgado expressed his profound horror at the racist denial of elementary human dignity and at the concept of a state-sponsored rape of the most pious of imperial pupils, he was told in surprised tones that no one in Angola would be in the least concerned if a mere "nigger girl" became pregnant, even if she were a trainee nun. Tragically for the country, such predatory attitudes proved highly contagious, and over the next half century most of the armies that sprang up in Angola tolerated a greater or lesser level of sexual violence against black women. Only when the new epidemics of sexually transmitted disease made fornication a threat to the immune systems of men, rather than merely an assault on the piety and dignity of women, did some armies begin to take action to protect their men.
The effect of colonization on the women of Mozambique was even more severe than on the women of Angola. Mozambique, especially southern Mozambique, had become the chief source of migrant labor for the mining towns of industrial South Africa. Since the mines primarily recruited male workers—60,000 by 1906 and three times as many a generation later—the Portuguese effectively had to run their colonial economy on female labor. In rural societies women whose men were absent farmed the village plots, raised the children, and cared for burned-out old people discarded by the mines. Men who might have stayed on the farm were liable to be conscripted into compulsory labor service on the chain gang and so preferred to return to the mines. Only a few rare men were successful enough as entrepreneurs to bring back money from the mines, buy a plow, and set themselves up as modernizing farmers, protected by wealth and status from the labor razzias of the state.
The removal of men from the Mozambique economy depressed the peasant sector of production and made labor scarce in colonial government service as well as in private enterprise. When men were not available to uproot the weeds and cut the branches along the footpaths used by colonial soldiers patrolling Mozambique, it was women who were forcibly rounded up to do the public works corvée. And when the plantation companies could not find men to plant, harvest, and carry their rice, it was women who were compelled to work on the company farms. Mozambique also developed sugar estates, copra groves, and sisal plantations, and to make these profitable women were taken away from subsistence farming and their domestic responsibilities in single-parent mine-migrant families and compelled to work for the international investors at menial rates of pay. Under the powerful persuasion of French, Belgian, and British capital and the dictatorial force of Portuguese colonialism, Mozambique extracted more labor from its subjected peoples than any other colony in the twentieth century. Women bore much of the brunt.
In South Africa, although the incorporation of women into the system of colonial extraction was perhaps a little slower to tighten its harsh grip, women were nevertheless taken into the wage sector quite early. When the mining industry reopened after the Anglo-Boer War, many black men were still employed in towns as domestic servants. In order to release this supply of male labor for harder forms of service, strategies were adopted to replace male cooks and stewards with female servants. One strategy was to encourage working-class female immigration to South Africa. The white urban proletariat, hitherto cared for by houseboys in boarding houses, would become a settled and married workforce with wives to do their domestic work. White housewives, however, continued to employ steward boys, and a second strategy had to be devised. Reluctance to train girl servants was overcome by a whispering campaign that suggested that white housewives would not be safe in their homes if they hired male black servants. Panic became endemic, black men were diverted into industry, and black women were squeezed out of their village homes to take up the lonely, segregated lives of urban scullery maids. Many had to care for the white children of their employers while their own children were left behind with grandparents on the reservations. In the second half of the twentieth century apartheid conscripted even more of South Africa's black women, both through migration to the towns and through requiring them to feed retired workers sent to live out their days in the rural Bantustans.
Back in Angola, while many men continued to be addicted to alcohol, women carried out much of the work that enabled society to survive the wars of conquest, the era of colonial exploitation, the wars of national liberation and foreign intervention, and the civil wars of the 1990s. While sustaining the subsistence economy, women also bore children, but even at the end of the twentieth century one-quarter of their offspring died before the age of five.
## 3
## Merchants and Missionaries
_One of the most enduring legacies of empire, in Africa as in the Americas, has been Christianity. A generation after the colonial governors and their plumed retinues had withdrawn from the tropics, white-robed priests and sober-suited catechists—both expatriate and indigenous—continued to be central figures in African society. In Angola the mission tradition was varied in origin, in style, and in social impact_. _The colonial state was officially Roman Catholic, but many of the missionaries were Protestant. Without state support Protestant missions had to seek commercial ways of financing their chapels, schools, and health clinics. One dramatic experiment brought forty American evangelists to Luanda in the 1880s in an endeavor to set up a self-financing Christian community. The experiment failed, but the survivors were adopted by a Methodist missionary society and became the Methodist Church of Angola. Some of its scions were the founders of modern nationalism. This chapter, which originally appeared in 1998 in the journal_ Lusotopie, _has been expanded with additional material presented to a conference held in Berlin in 2003 to celebrate the retirement of Beatrix Heintze, who spent her academic career meticulously exploring the cultural and anthropological records that illuminate Angola's history_.
The assumption is often made that Protestants in the Portuguese world were subversive of the imperial agenda. This assumption is not invariably correct; some foreign missionaries supported the imperial agenda. Conflict and mutual suspicion did sometimes erupt in the early days of formal colonialism, as when the Baptist Missionary Society spilled over from King Leopold's Congo and began proselytizing in Portuguese Congo. So worried was the Portuguese state at the intrusion of foreigners that it lent a gunboat to a pioneering Catholic priest, Father Barroso, in the hope that he might establish the first modern Portuguese mission in northern Angola. During the "scramble for Africa," royal imperialists in Lisbon aspired to preserve a Portuguese national identity in their sphere of influence. They were willing to use Catholic persuasion and propaganda and feared that Protestants would be less patriotic in their loyalty to the empire. (See the discussion in chapter 5 of the role of the Protestant missions in Angola.)
The simple thesis of Catholic support for the imperial cause and Protestant antagonism toward it has two defects. First, the idea that there was perpetual harmony between Catholics and the Portuguese state is not sustained by the evidence; the church was not able to generate patriotic acclaim for the empire in Angola. Gunboat or no gunboat, the Barroso mission to Portuguese Congo was not a great success. Nor indeed was the subsequent progress of the Catholic Church in Portugal itself. In 1910 the church was persecuted almost to the point of being outlawed by Portugal's republican government. First threatened by anticlerical freemasonry, the church was later also held in check by fascist-style nationalism. Foreign Protestants in the colonies were not the primary worry of the Portuguese church. Most Catholic missionaries in Angola came from France, Italy, Spain, or Germany, not from Portugal itself. Not until 1940, sixteen years after the coup d'état that brought "fascist" Catholic soldiers to power in Portugal, did the imperial government agree to a concordat with the Vatican that preserved Lusitanian patriotism while accommodating the demands of the church. Thereafter a secular clergy that came from Portugal to serve in white settler parishes was supportive of the imperial design. The members of the regular monastic clergy who served the black missions did not come from Portugal and were as ambivalent about some Portuguese colonizing activities as any "subversive" Protestant clergyman.
The second problem with the thesis is that the idea of a deep, persistent hostility between Protestants and Portuguese is also flawed. It is certainly not valid for the 1880s, when Portugal's most patriotic heroes began the long process of exploring and conquering Angola. At least some of the foreign missionaries seem to have been positively keen to celebrate the prowess of the explorers and cheer the planting of the flag. But the cooperation between Portuguese and Protestants goes back further. Although the seventeenth-century Dutch who settled in Angola and fathered great families, such as the Van Dunems, were Roman Catholics, many of the Dutch merchants and seamen of the time were Calvinists. Yet they too were so integrated into the colonial fabric that some Capuchin priests who converted Angolans to Catholicism traveled to and from Rome via Amsterdam using Calvinist ships. In the nineteenth century cooperation between Portuguese colonizers and Protestants was the norm. The Scottish Presbyterian traveler David Livingstone was a particularly vivid witness to the mutual respect that Portuguese and Protestants showed for one another in Angola. While writing up his diaries, Livingstone enjoyed the comfortable hospitality of a great colonial planter of the Luanda highlands and admired his estates without once mentioning that he was one of Angola's leading dealers in slaves. Pragmatic cohabitation epitomized dealings between whites in protocolonial Angola. It enabled missionaries of the 1880s to use the network of merchant footpaths along which Portuguese colonization spread. Cooperation among missionaries, merchants, and the slowly encroaching colonial state was often close and harmonious.
The connection between missionaries and merchants predates the scramble for Africa by almost one hundred years. Campaigners against the abomination of slavery long believed that the way to end it was to introduce into Africa new forms of wealth, new modes of production, and new types of labor incentive. David Livingstone, whose visit to Luanda occurred in 1854, believed that "legitimate trade" would bring an end to the trade in slaves and open up the continent to new opportunities. Traders were less convinced of the benefits that missionaries might bring to their profit-making ventures. Mary Kingsley, who visited Luanda in 1893 and thought it the most beautiful city in West Africa, was sure, unlike Livingstone, that missionaries and merchants had incompatible objectives. Coastal sea captains had convinced her that mission work only spoiled the activities of merchants. Her polemical writings portray the merchants as the true bearers of civilization before missionaries ruined the natives. In Luanda, however, the evidence did not bear out her thesis. Harmony between missionaries and merchants had been enhanced in 1885 by the activities of a remarkable young Swiss of twenty-six, Héli Chatelain, who was at home in both the mission and the merchant environment. The sea captains who plied up and down the coast were as much his friends as they were Mary Kingsley's.
Nearly all the Protestant missionaries who arrived in Angola during the 1880s found themselves as dependent on the hospitality of merchants as Livingstone had been. The long-distance traders who acted as agents of coastal merchants had created a reliable network of communications reaching across Angola to the Zambezi and Congo basins and to the inland markets of the East African Swahili. The trade routes also spread south into the sphere of influence of Angola's Boer community. These merchant paths enabled Protestants to cover hundreds of miles in relative comfort and safety, all the while enjoying the hospitality of Portugal's commercial pioneers, not the least of them being Antônio da Silva Porto. Protestant missions spread into Angola from the north, east, and south, as well as from the Atlantic coast. In the west the mission to which Héli Chatelain was initially attached started at Luanda Bay before quickly heading for the more fertile and less pestilential highlands.
The Luanda mission was nondenominational, initially supported not by Portuguese traders but by the benevolence of a British trading and planting firm, Newton Carnegie and Co. The mission was extraordinary in its size and eccentric in its ideology. With no mother church in the industrialized north, it expected to survive in the old mendicant tradition. The members, an ill-assorted band of forty beggar-evangelists mainly from America, landed in Angola in 1885. Their leader was an irrepressible American Methodist, Bishop William Taylor, whose dynamic energy, persuasive powers, and dictatorial management style enabled him to establish mission posts along the west coast of Africa in American Liberia, French Equatorial Africa, King Leopold's Congo, and Portuguese Angola. The principle underlying the enterprise, in theory if not always in practice, was that each mission was to be self-sustaining. The missionaries would have to live off the land rather than expect handouts from their home congregations in Europe or the United States.
The advance scout of the Taylor mission, and later its deputy station manager at Luanda, was Héli Chatelain, a Swiss immigrant to America who had survived the vicissitudes of his life by developing a sharp eye for business deals. Chatelain's wry account of Taylor's arrival in Angola sheds light on both the colony and the mission. The American band brought no less than forty tons of provisions to set themselves up before their "self-sustaining" ideology could take root. They naturally feared that they would experience great difficulty in clearing their stock of trade goods through Portuguese customs, and it was young Chatelain's task to overcome any bureaucratic hurdles. He did so by quickly establishing close friendships with anyone of importance in the tiny administrative and business world of Luanda, including customs officers and policemen. He soon discovered that everyone in Luanda was more or less dependent on Newton Carnegie and Co., and Mr. Newton himself became Chatelain's guardian angel. When the much-heralded expedition arrived, it was Mr. Newton who waved his wand over the crates and got them through customs with minimal difficulty.
The merchants of Luanda not only helped the Taylor mission negotiate its way through the colonial bureaucracy. They also facilitated relations between the Protestants and the established Catholic hierarchy. Chatelain's closest ally was a Catholic priest who had lost his preaching license, but in good Swiss style he also managed to establish adequately businesslike relations with the official church. When Chatelain advised the secretary to the Roman Catholic bishop of the impending arrival of nineteen Protestant evangelists together with their wives and children, the news was received with phlegmatic acceptance. The Catholic Church, like everyone else in Luanda, was partially dependent on Newton Carnegie and Co. for its material well-being, and the bishop's office had little option but to express a moderately cordial welcome to an enterprise apparently blessed by Mr. Newton himself. The young Swiss Protestant was offered a glass of the bishop's wine. It was, he said, the best that he had ever tasted. Relations between Catholics and Protestants were not hostile, and both were associated with, and to a degree dependent on, the colonial merchants, including the wine merchants, who were one foundation of Portugal's raison d'être in Africa.
The merchant community in this commercial city soon recognized that the mission expected to live off generosity and credit. The self-sufficient Protestants resembled mendicant friars more than worker-priests. Chatelain himself noted in his copious diaries all the houses where he was able to obtain free lunches and dinners from merchants who enjoyed his cultured company. An expedition of forty expatriates that the Methodist Church in America had refused to endorse or finance was a much greater burden on the host colony than a single, rather charming young Swiss bachelor. The merchants quickly rescinded their welcome, and Newton Carnegie and Co. refused to grant the Taylor mission even enough credit to sustain the Spartan lifestyle of an evangelist. Chatelain himself slept on a bare rented floor until he became so ill and bony that the expedition's doctor felt compelled to lend him his personal camp bed on which to convalesce. Many of the missionary party suffered from acute dysentery, and unable to afford medicine, they claimed stoically that to take medicine was to challenge the will of God and cheat the death that he might have chosen for them. Chatelain came near to death, but his merchant friends secured a bed for him in the Maria Pia hospital and even persuaded the governor-general to waive the hospital bills since Chatelain had no money and lived off charity. The Luanda merchants were unable, however, to raise the price of a steamer passage to Europe for Chatelain when he fell seriously ill.
The concept of an industrial mission of craftsmen and traders who could be independent of foreign subsidies and could build a chain of self-reliant Christian communities across Angola did not work well. The greatest problem was alcohol, which was the basis of so much of Africa's trade, particularly in the Latin colonies. When Bishop Taylor presented his project to the authorities, the governor-general of Angola immediately told him that he would be unable to recruit bearers to carry his stores inland unless he offered to pay them with spirits, the cheapest of which would probably have been illicitly distilled cane "brandy" with a potentially high level of noxious alcohol. Dealing in spirits as a necessary means of survival would probably have offended even Taylor's pragmatic attitude to commerce. Moreover, when Taylor arrived in Angola he decided to announce that all of his followers, with two exceptions, would become Methodists at the stroke of a pen, without the laborious teaching and testing that normally preceded conversion. The exceptions were two Quakers, who were unlikely to have been more accepting than the Methodists of the idea of trading in liquor. The problem anticipated by the governor did not recede, however, and for Portuguese trading stations in the bush self-reliance meant distilling local firewater, the type of raw spirits that had done so much to destroy indigenous culture and society on the North American frontier. As the Methodist stations struggled to survive, accusations arose over reports that the Pungo Andongo mission, near the old royal capital of the Ndongo kingdom, had a whiskey distillery on its premises. The mission store was also alleged to have offended against Victorian morality by introducing games and installing a billiard table.
The evangelists, unable to survive in Luanda, set out to find the fertile country in the hinterland that Livingstone had so lyrically described. The two legitimate activities that kept the self-sufficient mission stations functioning for the next ten years were teaching and gardening. The evangelists offered school lessons to the children of traders, but the children who were enrolled, even in the great market town of Dondo, could usually be counted on the fingers of two hands, and when it was time to pay school fees the children were liable to absent themselves. Gardening therefore became the basis of self-reliance. The evangelists, however, were not robust, their diet was seriously inadequate, their health was often poor, and several of them died or saw their children die. Like other colonists they tried to hire labor to work their plots for them. To attract workers the missions had to offer payment in good American calico. Mission gardens could not escape the merchant nexus or the need for foreign support, and the great experiment failed. Bishop Taylor retired in 1896, the concept of self-reliance was dropped, and the Methodist Church of America formally adopted the Protestant mission stations of the Luanda hinterland. With this foreign sponsorship the mission communities in the Malange district began to flourish, to win converts, and eventually to establish a prestigious high school that trained some members of Angola's modernizing elite.
During his years with the self-reliant mission Héli Chatelain became a great traveler. Despite being very lame from childhood illness, he did much of his traveling on foot and was affectionately known in Angola as "Long-leg and Short-leg." Although he was a quintessentially humane man and remarkably free of the exploitative racial prejudices that were normal in both America and Africa during his lifetime, he did avail himself of a traveling hammock for his long journeys into the Kimbundu-speaking hinterland of Luanda. Hammocks required stout porters, and on his travels through the high forest and tall grass toward Malange he was commonly frustrated at the difficulty in getting his caravan crew underway on cold, dew-laden mornings. Where the country was steep, he did ease the burden on his bearers by undertaking the most trying stretches on foot. Etiquette among fellow travelers, however, was a mystery to him; white men, lounging back on their pillows en route to the coast, scarcely raised a hand to greet the eccentric Swiss when their paths crossed. When Chatelain reached his final destination near Malange, he set about recruiting the services of a cobbler's son, Jeremiah, to be his linguistic informant. Over the next dozen years the two men produced grammars and dictionaries of the Kimbundu language, which were used not only in Christian missions but in the Portuguese colonial administration.
Historically, water transport has usually been easier and cheaper than land transport, and Chatelain, who had grown up among the great Alpine lakes, tried to avail himself of boats where possible. In Africa, however, small-scale shipping along the rivers met with great frustration. The Kwanza River, on the route between Luanda and the port of Dondo, downstream from Malange, had coastal sand bars that were dangerous to negotiate. Once boats did enter the river, the light winds and strong currents made progress slow. Overnight shelters were primitive. Most traders, notably those responsible for exporting coffee, found that head loading their sacks to the coast was safer, faster, and more reliable than river transport. Attempts to use small steam-driven boats on the river also met with poor results, as the ever-ambitious Chatelain discovered on a frustrating, mosquito-plagued trip to visit a slave-worked sugar plantation owned by one of his city friends. The late-Victorian belief in technological modernity was not checked by handicaps on the river, however, and colonial visionaries dreamed of building a steam railway that would run from Luanda into the deep interior beyond Malange. The pretentiously named Royal Transafrica Railway Company was designed to link the magnificent harbor of Luanda with the interior of Congo and even with the east coast of Africa. The scheme, however, was a catastrophic failure. The line was built so cheaply that it could not withstand storms and floods. Financially, it was such a poor risk that investors had to be promised a profitable return by the government whether or not they ran any trains or generated any operating revenue. And economically the line was ill conceived in aiming to tap a coffee zone whose produce had such a high ratio of value to weight that head porterage was a perfectly viable transport option, while bulk freight from the mines remained inaccessible on the far side of the Belgian colonial frontier. Chatelain's one attempt to travel by rail as a passenger was fraught with mishaps.
As transport officer for the so-called Methodist mission, Chatelain became familiar with all the lightermen who ferried cargoes from ships anchored in Luanda bay to the city beaches. Persuading the crews of these small boats to hurry when the great steamers were getting ready to depart was a fine art. The risk of being separated from one's cabin trunks was a frightening prospect for any white person who had reached the farthest ends of western Africa. The boatmen learned how to drive hard bargains to frustrate Chatelain's Swiss prudence in matters of finance. During his service the mission transport manager got to know many of the officers who staffed the steamships that served the coast. When Chatelain was taken seriously ill, a ship's captain took him on board for a rest cruise down the coast to the slightly healthier port of Benguela. The trip widened Chatelain's horizons, and during the second half of his African career he worked on the Benguela plateau rather than in the Luanda hinterland.
While Bishop Taylor and his acolyte Chatelain established worker missions in western Angola, alternative variants of Protestantism took root in eastern Angola. In the fifty years to 1930, the largest Protestant mission in the country was that of the Plymouth Brethren, or Darbyites, or _frères larges_ , as they were sometimes called in Europe. The Brethren arrived in Angola along the trails from the south that had once brought Livingstone up from South Africa. Their journeys were facilitated by Silva Porto, the great Portuguese transcontinental merchant, who helped the missionaries organize transport and find hospitality. In the far interior, however, he was unable to find a cobbler when the young English missionary Frederick Arnot's boots wore out. Arnot tried walking barefoot like everyone else on the African trade paths, but the hot sand blistered his feet. Old Silva Porto was in time able to rent him a riding ox that could plod its way through a thousand miles of thorn bush and even swim rivers. Arnot and the Brethren eventually reached the eastern highland of Angola and set up their headquarters close to Silva Porto's trading emporium.
The Plymouth Brethren were perfectly aware that their necessary association with traders involved adopting a muted attitude toward the slave trade. Arnot's friend Silva Porto had been a supplier of slaves to the west coast for half a century. The young missionary persuaded himself, however, that the trade in alcohol had been an even worse evil than the trade in slaves. He even saw glimmers of hope when Silva Porto claimed that he planned to grant freedom to his personal household slaves. The ancient merchant also repeated the old claim that he was rescuing slaves from cannibalism and even converting them to Christianity by putting "holy salt," blessed by a priest, on their tongues. "I, too, am a missionary," said Silva Porto beguilingly. The centuries-old traditions of justifying slavery as Christian redemption and providing wholesale Catholic "baptism" by giving magic salt were still being practiced in the last quarter of the nineteenth century. Neither the old Portuguese nor the young Englishman seemed much concerned about the differences of Christian belief that later caused conflict between Angola's Catholics and Protestants.
Salt was not only a source of religious power but a means of economic survival in the Angolan interior. The Plymouth Brethren tried to use mules, rather than bearers, to ensure that their supplies of trade salt got through to the highland stations from the great West Coast salt pans of Benguela. When one caravan of ten mules failed to complete the journey, the missionaries were compelled to hire human porters as the merchants did. The missionaries' porters were free men to whom they paid the standard rate for carrying a headload up from the coast. This standard rate, however, had become much depressed as the great slave raids of the late nineteenth century swept through the highlands. Men regularly allowed themselves to be conscripted as porters in order to avoid being sold to plantation contractors and exported for the remainder of their natural lives. The porters attempted to bargain with the missionaries for better wages, but even though the Brethren were probably the best endowed of all the Angolan missions, the labor market did not allow much humanitarian generosity.
A pragmatic association with merchants enabled missionaries to survive economically in Angola's deep interior. Merchants sometimes helped missionaries to survive politically as well. In 1890 a newly elected king of Bihé decided that he would expel all aliens from his territory, Portuguese invaders and British missionaries alike. It was Silva Porto, the old settler-merchant, who warned the Brethren of the impending war and enabled them to negotiate a treaty of friendship with the king. While the missionaries stayed on, the Portuguese were driven out and even Silva Porto, who had spent his entire trading life in Bihé, felt that his merchant network was at risk. Rather than leave his highland home, he attempted to blow himself up spectacularly by igniting a few kegs of gunpowder. His friends at the mission made valiant attempts to save his life, but his burns were too severe. The loyalty of the Protestant missionaries to their Portuguese patron did not protect them from the wrath of the colonial invaders when they returned. The Portuguese state sent a whole army up from the coast to recover raiding grounds where slaves were traditionally captured. In November 1890 a thousand colonial foot soldiers and ninety Boer horse commandos arrived in Bihé to wreak revenge against the people who had caused them such grievous losses. The Brethren wisely negotiated with the posse of Boers, offering them excellent meals cooked by a European missionary wife. As tempers cooled, settlers and missionaries were able to orchestrate a peace plan and persuade the king of Bihé to come out of hiding and surrender. Led away by the Portuguese commander of the punitive expedition, the king entrusted his gun to Frederick Arnot, the missionary peacemaker. Arnot, like other missionaries in Angola, did not question the right of the colonial powers to impose their rule over Africa.
Cooperation between merchants and missionaries, and mutual toleration between Portuguese and Protestants, were in evidence even during the spectacular conflict that occurred in the kingdom of Bailundu, next door to the Brethren's host kingdom of Bihé. A mission of American Congregationalists had maintained reasonably good relations with the Methodists, the Swiss, and the Brethren, though the local Catholic Church viewed it with suspicion. When the anti-Portuguese rebellion broke out in 1902, the Protestant missionaries, good, loyal believers in the rights of the colonizing powers, did their best to help restore imperial law and order. They even supplied military intelligence to the commanders of the invading colonial army concerning the movements of the Bailundu regiments. When a besieged Portuguese unit ran short of supplies, the missionaries provided it with food and trade goods from the mission warehouse. Such cooperation, however, was no longer enough to convince the official Portuguese mind that Protestants were anything but subversive foreigners. The era of pragmatic collaboration began to give way to one of suspicion, and relations among missionaries, merchants, and administrators became more brittle than they had been through the nineteenth century. Chatelain nevertheless returned to Africa in 1897 with a new mandate to create a self-sustaining mission in Angola's southern highland.
## 4
## A Swiss Community in Highland Angola
_This chapter, which was first published in the proceedings of a conference on African history held in Lisbon in September 1999,_ _explores the last ten years of Héli Chatelain's entrepreneurial endeavors as he tried to sustain his small Christian community on the edge of the southern highlands. Although his village of local peasants and emancipated slaves was isolated and remote, Chatelain was able to maintain a worldwide correspondence in English, French, German, Dutch, and Portuguese with his theological sponsors and commercial suppliers in Switzerland and America. He also understood the people of the local Ovimbundu kingdoms and wrote hymns for them in their own vernaculars. After his death the Swiss churches took over Chatelain's venture, and some of the leading politicians of modern Angola went to school at the Swiss mission before being sent to Switzerland for further training_.
The concept of establishing self-sufficient Christian communities in Angola did not die out immediately when Bishop Taylor's Malange mission, which Héli Chatelain had helped to create, was converted into an officially sponsored Methodist field of proselytizing. Ten years after he had visited Benguela as a convalescent, Chatelain returned to Angola's southern harbor city determined to try out for himself the ideals that had inspired the now-retired Bishop Taylor. Resolving the problem of how to maintain a balance between missionary idealism and merchant pragmatism had not become any easier. When Chatelain arrived in Benguela in 1897, the slave trade had revived so vigorously that the city was shipping out some 4,000 men and women each year. The aim of the new mission was to stem the flow of this trade and create a chain of hostels for escaped slaves all the way from Benguela to the great raiding grounds among the Nganguela people of the upper Cunene and Zambezi rivers. Such was the dependence of missionaries on merchants, and of merchants on the sale of slaves, however, that no such chain of safe havens could be established. Without ox trails and bush stores, any mission penetration was virtually impossible. Chatelain's Swiss-American mission only ever set up one station. It was located not in the remote hunting grounds of the slave catchers, but in settler territory on the high plateau, and its best customers were not free Africans, but Boer immigrants from South Africa. The Dutch settler community at Caconda with which the mission traded was an offshoot of a larger Boer colony at Humpata on Angola's southernmost plateau.
The dependence of the Swiss mission on the Boer colony began from the moment when Chatelain landed at Benguela and found that rinderpest fever had decimated Angola's stock of oxen. He had a long wait before he could negotiate the hire of Boer wagons to haul his equipment through the coastal scrub and up the escarpment to Kalukembe, the site near Caconda that he had chosen for his station. Once established, the mission became a trading post that depended on its Boer customers for its economic viability. Had the mission agreed to harbor slave refugees from the Boer slave farms, the commercial side of the enterprise would have lost its most lucrative business clients. Although Boer customers did not like the mission's social ideology, they had little direct access to suppliers in Europe and no network of international credit that would enable them to order goods from abroad. They therefore welcomed Chatelain's business acumen, his ability to make credit deals with overseas suppliers, and his familiarity with the necessary settler crafts. His workshop installed anvils and forges with which Boer carts could be repaired, and his mission artisans kept the transport system of southern Angola running. Chatelain's import-and-export business underpinned the semi-self-sufficient mission's finances. To remain solvent, the mission had to play down any antislaving ideals its sponsoring League of Friends had dreamt of. These sponsors, to the great chagrin of the Portuguese, had even named the station "Lincoln," after the American president who had outlawed slavery in the United States.
Although Chatelain had little success in protecting captive Angolans from slavery, he did develop commercial relations with his free black neighbors. He traveled round the villages with his own ox cart buying beans and maize each season as the crop was harvested. The wares that he peddled in exchange were those that would have been seen at any roadside fair in his native Switzerland. Unlike all other traders, he refused to sell wine and brandy, but his mobile wagon shop carried sugar, salt, cooking oil, dried meat, and soap as basic necessities, as well as ironmongery, padlocks, spades, hoes, wire, traps, penknives, and cutlery. His stock of crockery included cups, plates, casseroles, and bowls. Vanity was catered for with ten different kinds of glass beads, shirt and coat buttons, bracelets, earrings, belts, and colored kerchiefs. Textiles ranged from cotton prints and woolen blankets to shirts, trousers, coats, and caps. The traveling bazaar was complete with supplies of flint, lead, gunpowder, medicines, sewing needles, matchsticks, mirrors, writing paper, and mouth organs.
A perambulating bean merchant with a forty-acre small holding staffed by casual black labor from neighboring villages was no great threat to the Portuguese merchant community. But Chatelain was more than that. He was a missionary with world connections who still aspired to end the Angolan slave trade. In 1903 Chatelain feared that at any time an "accident" might happen to him or to his mission station. He let it be known that should any violence occur, or should his mission be burnt down by the slave raiders, a report that he had lodged with the Swiss consul at Lisbon would be made public. This report would give full details of the violence affecting local trading conditions and specify the manner in which colonial and military officials in Angola not only tolerated the slave trade but personally benefited from it. Creaming off a share of the slave profits could make an officer's tour of duty in a harsh colony staffed by convicts economically attractive.
In between bartering maize for crockery and corresponding with foreign businessmen about the slave trade, Chatelain developed an almost limitless range of commercial sidelines in his efforts to create a self-sustaining Christian community. Among his papers is a manuscript catalogue of postage stamps issued by the Angolan post office. It describes the different colors, reigns, denominations, embossments, perforations, printing errors, and overprintings found on the stamps in his commercial collection. Each stamp bears a retail price in Swiss francs at which Chatelain would be able to supply specimens to collectors. Stamp collecting seems a far cry from Chatelain's initial crusade to challenge polygamy, witchcraft, alcoholism, and all the running sores that missionary societies had tolerated for too long. His great ambition, however, was to sanctify the practice of commerce as the economic mainstay of mission. This was a much harder challenge than sanctifying the practice of farming, craftwork, teaching, or nursing. Chatelain found it difficult to attract shopkeepers to come from Switzerland and accept his own bachelor asceticism, or to recruit mission workers willing to acknowledge his belief that it was colonization that represented the road to liberty for Africa's people. One had to be patient and pragmatic when waiting for the benefits of "civilization" to trickle down. Chatelain's commercial path on the Angolan highlands proved to be a much stonier one than that of the Swiss missionaries from Basel who created an integrated network of country stores and village chapels on the Gold Coast in modern Ghana.
If transport had been a problem in the Kwanza valley when Chatelain was helping establish the Malange mission, it proved to be an even greater problem on the southern highlands. The coastal rivers tumbled down into the sea, and the inland ones created unpredictable flood plains across which ferry services were jealously guarded by boatmen licensed and taxed by an embryonic colonial state based at the old eighteenth-century trading fortress of Caconda. Rather than use the expensive and somewhat dehumanizing services of hammock bearers, Chatelain traveled around the highlands on a donkey. He presented a bizarre appearance: a short-sighted, bespectacled figure wearing a three-piece suit decorated with a gold watch chain and shod in long, elasticized boots that almost dragged along the ground beneath him. The donkey, however, had its limitations as a pack animal, and in 1881 a revolution in highland transport had occurred. Several dozen extended families of South African Boers had crossed the Namibian "thirstlands" and reached Angola thanks to the novel technology of the ox wagon. Similar wagons were used on the plateau farms of Switzerland, and at least one ambitious expedition had used them to cross the breadth of Europe and settle a Swiss colony on the shores of the Black Sea under the patronage of Tsar Alexander. Wagons, however, had three major requirements: large-scale financial credit for the capital outlay, veterinary knowledge to keep the teams of twenty oxen healthy, and carpenters or blacksmiths to repair the wagon beds and refit the great metal hoops that rimmed the wooden wheels. None of these services could be adequately supplied by African merchants, stockmen, and artisans, nor indeed by Portuguese-speaking settlers moving up from the Benguela coast. Chatelain therefore aspired to create a Christian community in the highland that would be financed by a wagon workshop to serve the late-nineteenth-century transport riders.
Setting up an ox wagon business in early colonial Angola was not easy, and the economics of transport were even more problematic than the fraught relations with slave-owning clients. The steep, ever-shifting, deeply rutted tracks leading up from the coast to the highland were far from ideally suited to establishing a transport system that could rival, let alone replace, the footpaths used by runners and bearers. When his wagons were not in service, Chatelain advised American visitors that they could expect to pay porters three thousand reis per thirty-kilogram head load to bring luggage up to his station from Benguela. This method was often faster and cheaper than bringing goods up by wagon, but porters were hard to recruit and were sometimes less trustworthy than wagons, so travelers and merchants often paid the wagon rate for each ton per league. On a good trip Chatelain could take his wagon down to Benguela from Kalukembe, a hundred miles, in ten days. Returning uphill was likely to take thirty days. The journey was not possible when there was no grass for the oxen to eat along the way or when the holes in the river beds had dried up and the animals were left gasping with thirst. When the rains began, travel was equally difficult. Although some streams had small bridges for porters to walk across, the wagons had to go to the bottom of each ravine to a suitable ford before being hauled up on the other side. It was often necessary to unload the ton-and-a-half of boxes, sacks, and bales of merchandise before each river crossing, lest the vehicle sink so deep into the mud that even extra oxen could not pull it out. Inland from Kalukembe the journey could be equally slow, not so much because of drought or flood as because of the constant need to cut down trees to open a new trail. When Chatelain's Swiss assistant Ali Pieren took the mission wagon from Kalukembe to Bihé in search of business, the journey of 150 miles took him thirty-three days. A column of porters could have done it in half the time, but a wagon with a full payload could carry as much as fifty men could carry on foot.
One key to the wagon trade was the quality of the wagons. Before the advent of the motor car, one of the most prestigious manufacturers of carriages and wagons in the United States was the firm of Studebaker, which had workshops in South Bend, Indiana. On April 11, 1906, Chatelain, in order to improve his credentials as a dealer in transport equipment, wrote to Studebaker. His letter was very specific about the needs of the local transport industry and was therefore addressed to the manufacturers rather than to the company's sales agent in New York. He wanted an estimate for a wagon similar to one he had ordered in 1904 but with two-and-a-half-inch concord steel axles capable of withstanding encounters with tree stumps, rocks, and termite hills. A wagon that Chatelain had previously bought from California had only had one-and-a-half-inch axles, which had not proved robust enough. Wagons from Portugal, although competitively priced, were not as sturdy as the ones developed in America for the transcontinental wagon trails. The Indiana Studebaker was equipped with wheels five inches thick and had massive rear-screw brakes. For the African trade the bed of each wagon had to be sixteen feet long with sideboards two-and-a-half inches thick. Chatelain wanted his wagons equipped with side boxes rather than a high front box and with twenty yokes and twenty sets of ox chains instead of the more usual ten yokes supplied for work on the American plains.
Chatelain had not been satisfied with his first Studebaker purchase. That wagon had broken down three days out of Benguela. Its sideboards had cracked under heavy loads on steep trails and its paint had peeled in the tropical sun. Chatelain, who had sold the wagon on credit, had been forced to grant his customer a discount for poor workmanship, and the invoice had remained long in dispute. Despite his diminished profit margin on this first order, Chatelain hoped that if he placed regular orders Studebaker might offer him favorable terms. Competition in the wagon trade was growing, and a Cape Town firm of wagon builders had established an agency in Benguela and poached two of Chatelain's customers. Equally worrying was the rumor that the firm of Marques Pires and Company had acquired exclusive rights to the sale of any Studebaker landed at Benguela. Chatelain pointed out to his Indiana correspondent that Marques Pires did not have a good reputation and he therefore hoped that Studebaker would continue to supply the mission with high-class wagons.
Some of the larger sums in Chatelain's business accounts concern the trade not only in wagons but also in oxen. The station sometimes held sixty or more bullocks and oxen on its land as well as its ten cows and an assortment of goats, sheep, ducks, hens, and pigeons. On February 5, 1906, Chatelain wrote in Afrikaans to Susanna Behan, a widowed neighbor, about payments she owed him on six oxen that were worth 210 milreis. Such a sum, he said, was too large to be entrusted to an "ordinary Kaffir," so he suggested that she make the payment to the Benguela Railway Company and ask the company office to issue her a money order, which she could safely send by messenger. Still the debt was not paid, and Chatelain had constant cause to grumble about the large sums his clients owed him. He maintained close social contacts with his neighbors and spoke fluent Afrikaans, but credit was a slippery business and on one frustrated occasion he wrote that a "Boer's word of honour is as tangible as the wind." Relations with his Swiss colleagues were not always harmonious either, and he had difficulty recruiting catechists and artisans willing to turn their hand to any task, however menial. Alfred Balmer, with help from casual black laborers, did much of the station's plowing, hoeing, weeding, and harvesting. Ali Pieren, who was more mechanically inclined, serviced, repaired, and improved the water wheel that drove the corn mill. On one memorable morning the Swiss farmhands were thrilled to see the mill doing 82 revolutions per minute. They hoped that with more rain to fill the mill-stream they might achieve 100 revolutions and grind one whole sack of meal each day. Frederic Leuba, the most intellectual of the mission staff, wrote poetry, fiction, and plays when not engaged in heavy farm work. The temperamental mission lad, Girod, was a trial to his elders and was liable to beat laborers who displeased him, a colonial practice that the mission aspired to eradicate.
Chatelain was also the district pharmacist and a medical adviser with a cosmopolitan clientele on the highland. When Apol-linario Pereira wrote that he was feeling ill, Chatelain diagnosed a poor condition of the liver and sold him two mild types of purgative. One was a powder also used by menopausal women; the other was a liquid, of which he was to take 30 drops along with a dose of the powder, wrapped in fine paper and swallowed like a pill, the treatment to be repeated daily as necessary. He also recommended that Mr. Pereira take quinine and sent him a flask of good-quality quinine powder, for which he charged 2,000 reis. He recommended four or five measures of quinine per day until the fever subsided and then one a day for a week afterwards. It was not only local Portuguese who consulted Chatelain about their ailments. When Mr. Wilde, at the Benguela railhead, thought he had a tapeworm, Chatelain told him to eat nothing but bread and milk for a whole day and then, on an empty stomach, to take the powder that Chatelain sold him. He also recommended a strong dose of Epsom salts and a diet of pumpkin seeds. If his spleen was the cause of the problem, he was to take bottled medicine three times a day and rub ointment on the pain before wrapping himself in a warm bandage. For good measure Chatelain also sold his patient some antidyspepsia tablets to be taken twice daily before meals. Another client, the widow Behan who had so chronically failed to pay for her oxen, was also a patient of the mission. Although Chatelain did not succeed in curing her husband of his chronic syphilis, he was able to treat the widow so effectively that she lived on to marry again. On such amateur foundations was the great Swiss hospital at Kalukembe founded.
Chatelain was also the district photographer. He developed film for other missions and took portrait photographs of local personalities. In January 1906 he at last found the time and materials to develop and print a batch of seventy-two photographs of life on the Caconda plateau. In 1897 he had captured his first meeting with the elders of Kalukembe on film, and ten years later many of them were still his friends and neighbors. One picture showed the all-important mission warehouse, with his own bedroom, office, and pharmacy on the upper storey. He photographed a South African prospector with a Welsh assistant who passed through with their pack mules. He photographed the commandant of the local fortress with his slave concubine and their tame monkey. Chatelain took pictures of clay potters, of water carriers, of women threshing wheat and pounding corn, of men drinking beer and taking snuff, of chiefs and kings posing with their stool bearers, of a hunter with a valuable white seashell on his wig, of girls wearing strings of beads and leg rings. Pictures of Boer houses and wagons were accompanied by pictures of a "Bushman" mother, captured in the Kalahari, raising children fathered by a native Boer drover. The village also had four "Hottentot" drovers brought up from Damaraland. One fifteen-year-old Boer woman whom Chatelain met was married to a Portuguese Jew from Morocco. He also photographed a young Boer widow with several children who possessed her own Studebaker wagon, ordered, Chatelain mentioned regretfully, from the factory in America rather than through his mission agency. When making photographs for Otto Wilde of the Benguela Railway, the photographer charged two milreis for six paper prints, or a little more for those printed as postcards. One picture taken with his camera was of Chatelain himself, wearing his three-piece suit with the gold watch chain—recalling his youth in the watchmaking canton of Neuchâtel in Switzerland—and a large-brimmed hat.
Chatelain also traded in firearms. When touring the plateau buying maize, beans, sweet potatoes, and cassava to feed the mission or sell to the colonial administration, he paid with gunpowder and gun flints stocked in his traveling bazaar. The mission itself was poorly equipped with hunting guns but did possess one Winchester. Attempts to make their own cartridges for such a modern weapon nearly led to the death of both Chatelain and his assistant. When they tried to remove a jammed cartridge, it exploded and made Chatelain temporarily deaf. He asked that the next Swiss worker coming out to Africa bring with him 500 manufactured cartridges, though clearing war materials through Portuguese customs was to prove difficult. Good-quality guns using cartridges were a major item sold by the bush traders in Angola. From 1899, however, foreigners such as the Swiss and the Boers needed a government license before they could hold such weapons, and Chatelain spent much effort obtaining the necessary documents for his customers. Some of the modern guns being sold in 1906 were old breech-loaders left over from the American Civil War. The trade guns sold to the natives, by contrast, were ancient muzzle-loading muskets, which could be sold in huge numbers without a license. These ubiquitous guns provided the traders with good profits on the sale of loose gunpowder to be poured down the barrel. Musket balls were made by local artisans to suit the needs of both warfare and hunting. Muzzle-loaders were not very effective in elephant hunting, and much of the ivory trade gradually fell into the hands of Boers who used high-velocity rifles to kill their prey and then carried their trophies to the coast on ox wagons. Muzzle-loaders were, however, useful to slave raiders or caravan guards who herded captives along the trade paths to the ports. Only in 1913, after international pressure had curtailed the sale of captives, was the trade in loose gunpowder outlawed. The administration simultaneously confiscated a quarter of a million guns held in native hands.
Social disruption in early colonial Angola was fueled as much by the ubiquitous sale of cheap rum as by the sale of muzzle-loaders. Chatelain steadfastly refused to trade in alcohol and preached vainly against the lack of sobriety all around. The trade in wine and rum, together with the equally addictive trade in tobacco and snuff, was the mainstay of trading posts scattered across the Caconda highland, but Chatelain refused to allow such commodities on his premises. He accepted that his staff surreptitiously smoked tobacco, and he was even told that one of his expatriate employees sold rum behind his back, but the mission store refrained from dealing with chiefs who could be induced to sell cattle for as little as a couple of bottles of rum per bullock. Tobacco did become a bone of contention between moralizing missionaries and struggling merchants. In the days of the Brazilian slave trade, tobacco had been a key currency commodity used for trading transactions and wage payments. In the twentieth century tobacco was one of the crops that the Portuguese administration would have liked to grow in Angola, after the British success in growing it in Zimbabwe and Malawi. The tobacco industry, however, did not take root in Angola, and settlers improbably blamed the missions. The missionaries' moral scruples, it was said, had so infected the peasant population that Angolans refused to buy, plant, or sell tobacco. Such a belief in the power of the missions would have astonished Chatelain. The idea that mission morality, rather than good, rational African business sense, had discouraged black farmers from growing tobacco was based on growing Portuguese xenophobia. By the early twentieth century this xenophobia was shared by the cultured Portuguese members of colonial high society who had once been Chatelain's intimate friends, as well as by petty-minded administrators and isolated traders in the bush.
The Swiss mission campaigned in vain against the widespread abuse of alcohol and tobacco, but its primary raison d'être was to stem the practice of slavery. Slave owning remained almost universal in the early twentieth century, as Chatelain was constantly reminded, and virtually all his Boer customers were slave holders. In May 1906 he was woken one morning before dawn by cries of distress coming from his compound. Under cover of darkness the son of Jan Vermaak, a local Boer settler, had ridden over to the mission with two black employees, one a Kwamato and the other a Nganguela, and stealthily entered the compound to seize a woman whom he claimed to be his slave. To stop the woman from screaming, Vermaak had hit her so hard that she fell as though dead, with a large head wound. His men tied her up and were attempting to carry her off when the alarm was raised by women in neighboring sleeping quarters. Chatelain, in Afrikaans, informed young Vermaak that slavery was no longer accepted under Portuguese colonial law. The woman turned out to be a Luba from Katanga who had been captured by rebel soldiers in Leopold's Congo and sold to a black slave trader from Bihé. She was then caught up in the Huambo War and captured as a war trophy by a Portuguese soldier, who sold her to a white dealer. He in turn sold her for cash to Vermaak senior, from whose compound she had fled to seek refuge in the mission. Prevented from recovering her, young Vermaak threatened to call up the Boer commando unit and have the interfering wagon dealer assassinated. Chatelain reported the incident to the district commissioner at Caconda, pointing out that the 1890 Brussels Conference Act, to which Portugal was party, outlawed Angola's slave trade and that the act required the commissioner to punish anyone holding slaves within Portuguese territory. This was the third time, Chatelain said, that Boers had tried to kidnap people from the mission claiming them to be their legally owned slaves.
Chatelain's protest about the illegality of slave holding generated hostility among some of his best customers and closest neighbors, who resented the refuge given by the mission to abused natives who had escaped their masters. One such refugee was a child called Domingo, who arrived at the station one Sunday morning during the hour of worship. His father was a goldsmith who had moved with his son from Luanda to Benguela and had bought a female slave as his concubine. When Domingo was about six years old, his father died and his aunt, not knowing what to do with him, gave him to the Belgian trading agency in the town. The Casa Belga apparently sold him to a Belgian planter, Raes, who had married a Boer widow and settled on the Caconda plateau. Raes had a particularly violent reputation—it was said that he had killed one of his black mistresses and burned her hut to hide the evidence—but it was his wife who took to beating young Domingo with a hippopotamus-hide whip when she was displeased with the performance of his domestic duties. When the brutalized child arrived at the Swiss mission, Raes flew into such a rage that he threatened to kill Chatelain. He reminded him that Lincoln, after whom the Swiss mission was named, had been murdered for freeing slaves. Raes warned him that within a year there would be nothing left of the Swiss station but smoldering ashes. Since both Raes and his wife owed significant trading debts to the Swiss store, burning the mission account books might have been financially advantageous to them. Chatelain handed young Domingo over to the district commissioner at Caconda. There, however, the only justice he received was another beating to deter him from running away again. Such beatings were normal, as one slave holder explained to the investigative journalist Henry Nevinson. It was his practice, he said, to flog any escapee very soundly for a first offense and, if he or she tried to escape a second time, to sell the escapee immediately to a slave dealer. Once shipped to the island plantations of São Tomé, a slave would have no chance of escape. The fate of little Domingo after the authorities had returned him to his Belgian owner is not recorded.
One of the horrors experienced in slave-owning societies was the physical exploitation of defenseless women, who had no right to privacy or protection from unbridled male lust. Abuses that had been rife in the slave-based Roman empire of southern Europe, the Arab empire of the Near East, and the Hispanic empire of the Americas survived into the twentieth century in the Portuguese empire in Africa. Chatelain, himself a confirmed bachelor, began only slowly to recognize the corrosion that colonial sexual practices brought to the highland society in which he lived. He worried that his Swiss colleagues might, unlike himself, fall under the tempting influence of the unmarried girls who lived at the station. More seriously, he worried that when he hired out his Swiss craftsmen to work in the encampments of English railway engineers and mineral prospectors, they might adopt the colonial custom of inviting young slave concubines to share their cabins. His letters to his sister, who became his main confidante after the deaths of his mother and grandmother, are full of forebodings about the ubiquitous temptations of tobacco, alcohol, and above all sexual immorality. He was particularly disturbed by the behavior of one mission-educated adolescent who offered her sexual services to an engineer who camped near the mission while installing a telegraph cable up to Caconda from the coast. The young "whore" was a protégée of a former colleague and Chatelain felt himself in loco parentis, yet he was deeply reluctant to condone her behavior or receive her back at the station when her white lover repudiated her to move on up the telegraph line.
The ubiquitous exploitation of black people by white rulers caused Chatelain almost as much grief as drunkenness. Chatelain had known Commissioner Valdez since Valdez had arrived in Luanda as a beardless youth embarking on a colonial career and Chatelain had been the cultured man-about-town whom every educated Portuguese liked to invite to dinner to entertain the guests. Twenty years later, when Valdez was transferred from his provincial posting at Caconda and his removal caravan passed in front of the mission, he barely stopped to pass the time of day with his old companion of city days. He was now a portly man with a grizzled beard who was carried in a hammock. Behind him another set of bearers carried another hammock containing his _mestiço_ mistress, and behind her was a string of young black women whom Chatelain took to be the commissioner's harem. The coolness of the commissioner's greeting may have reflected the fact that Valdez, like many other Portuguese officers in the highlands, was in deep financial debt to Chatelain's country store. The exploitation practiced by expatriates in Angola in 1906 was not confined to the Roman-style opulence of the Portuguese proconsuls. The English were no better, in Chatelain's view, since they bought not only men to work in their copper mines by day but women to serve their pleasure by night.
One of the surprising revelations in Chatelain's letter books is how cosmopolitan the expatriate community on the Caconda plateau had become. When the fourth Swiss laborer, Girod, arrived at the mission station, he was accompanied by a German boy of eighteen whom he had met on the slow boat from Antwerp as it steamed down the entire African coast picking up deck passengers along the route and depositing them where jobs were available. The German boy, without any experience or qualifications, was hoping to make his living in Angola simply by virtue of being European. But 1906 was not a good year for adventurers. Work on the railway was not going well, the transport business was in such deep recession that Chatelain's wagons were traveling half empty, and Angola was full of footloose whites seeking their fortune. Most recently, four Boer brothers, all with wives, families, and wagon trains of oxen, had taken eight months to cross the thirstlands via Lake Ngami. Chatelain welcomed more Boer settlers, especially when they opened a new wagon trail to Benguela, and hoped they would enhance his business. Expectations of new English custom were disappointed, however, when it became clear that the Benguela railway would bypass Caconda and take a more northerly route to the interior via Huambo and Bihé. This disappointment was matched by the pain caused to Chatelain by wilder sorts of adventures like those of a brilliant, violent young Scottish aristocrat, Captain Cunningham, who had served the British during their invasion of the Tranvaal and then managed to get himself murdered in the little Angolan kingdom of Kasanje during an unnecessary dispute over the price of goat's meat.
Although Chatelain spent much of his trading time dealing with foreigners and even, apparently, recruiting their contract laborers, he saw his main purpose in Angola as mission work. Others did not see him in quite that light. The Union Castle shipping line refused to grant the Swiss Mission Philafricaine, which had replaced the American League of Friends as Chantelain's prime sponsor, the 10 percent discount that it allowed to bona fide missionary passengers on its steamers. Chatelain wrote to the company giving the British and Foreign Bible Society as a reference that could certify that he had been selling Bibles in Angola for twenty years. Bibles and religious tracts were one of his commercial lines, and he wrote to other mission stations offering these in different qualities, colors, bindings, and letterings, always carefully stressing the price. His greatest difficulty was in obtaining religious tracts (other than the Psalms) in Angola's vernacular languages. His letters to the mission at Bailundu seemed to go astray when he ordered a catalogue of their Umbundu hymnbooks. If, he said plaintively, the missionaries were really too busy to deal with this Swiss request, he wondered if one of the mission ladies might reply to his letter. In his depressed isolation he even wondered how soon the new telegraph might reach the American mission and how much per word it would cost him to order his books by Morse code. He was not optimistic, however, remembering that in Luanda twenty years before, runners with cleft sticks delivered messages more quickly than the electric telegraph.
Chatelain was so meticulous about chasing up his trading debts, calculating his profits on corn milling and transport riding, and living a frugal life that toward the end of his career he could write to R. G. Moffat at the American station in Bihé that he had accumulated enough money to open a second Swiss station. The Mission Philafricaine would not, he admitted, finance a lifestyle comparable to that of an American missionary, but his own workers accepted a much more Spartan subsistence. The new Swiss mission, he said, was to be "self-helping," though not self supporting. It would have to be in a region in which it would be possible to trade in rubber, the only commodity that was both legitimate and profitable. The Lincoln station, he admitted, had so far made a loss on both cattle raising and agriculture. In 1906, when the rains were poor, Chatelain took cartloads of maize and beans to afflicted villages to sell in exchange for cattle, which he brought back to the mission, where water and fodder were still available. Despite his best efforts, however, no less than twenty large animals died of fever, as well as eighty goats. Chatelain's days were filled with tending sick but valuable animals, especially those he had sold on credit to customers who refused to pay for them until their survival was ensured. Under this stress the mission entrepreneur dreamt of using Lincoln as a stepping-stone to more promising and profitable regions in the deep interior.
Chatelain's Swiss business methods are clearly spelled out in his papers. His religious beliefs, by contrast, are something of a mystery. At Bishop Taylor's multidenominational mission in Luanda he was often intolerant of all but his Quaker colleagues. His relations with the churches in Switzerland were also complex, and he seems to have been linked neither to the theological school of the Free Church of Vaud nor to the established State Church of Vaud. His small support committee, in which the dominant figure was his sister (later his biographer), included a few ordained men, but it had no wide popular appeal. He corresponded with the Zion City religious community in America, which supplied him with a periodical called _Leaves of Healing,_ and he hoped that his Boer customers would have their hearts healed by reading the Dutch edition. He particularly craved the healing prayers of Dr. Dowie of Zion City, Illinois. He considered it a miracle that despite his persistent fevers, rheumatism, consumption, catarrh, and nearsightedness, and a lifelong limp caused by one shortened leg and one stiff hip, he was still alive. His friends in America tried to persuade him to give up using drugs and to trust in faith healing, but when malarial delirium struck him down he never quite dared to give up quinine. These friends did not approve of his laying on of hands to cure illnesses, and Chatelain admitted that none of those whom he had cured had seriously repented of their sins. In addition to trying to eradicate drinking and smoking, he tried to discourage the eating of pork. Despite his piety, the devil still sent Chatelain "locusts, lice, jiggers, rats, mildew, drought, moths, rust, termites and cattle sickness which destroyed everything that rogues and burglars of every race had not already robbed." The Lincoln station, he told a correspondent in Zion City, was not exactly the lion's den, but it was full of serpents.
During his last year in Angola Chatelain became weary, confiding to C. E. Welsh that the missionary path was a stony one. Whenever an African trainee seemed ready to become a church member, he became puffed up and unruly. All too often, stealing and immorality forced Chatelain to dismiss his acolytes. Even when men did progress toward a Christian understanding, their wives pulled them back "into a semi-heathen state." Some of Chatelain's pupils were slaves who had run away from their Boer and Portuguese owners, but they were not grateful for the protection they had received and were unready, in his eyes, to join the church. Recruiting Europeans willing to accept Chatelain's austere standard of simple living was difficult, and he fell out with most of his expatriate followers, bitterly comparing some of them to Judas. Chatelain nonetheless dreamed of recruiting further mission workers that they might break the fetters of thousands of slaves and cause the whole colonial system to collapse.
Such subversive sentiments did not go unobserved in the Portuguese imperial administration, although Chatelain tried to be discreet in his comments about the political situation in Angola. In 1906, in a letter to Brother Welsh in America, he asked him not to publicize the letter or to speak of it except in very guarded terms. Persecution of Protestants was on the increase, and an American missionary in Bihé had been expelled. Worse still, missionaries of all nationalities were ordered to hand over any refugee slave who might be living in their compounds and to forbid entry to any new refugees who might have fled from their owners. Even more immediately, the district commissioner in Caconda informed Chatelain that all missions were forbidden to teach reading and writing to anyone, even household members living in the mission compounds. The authorities were particularly frightened of missionaries who taught literacy through the vernacular languages. Outraged, Chatelain appealed to higher authority against this form of persecution. He predicted that if such an incredible ban on education were to be applied to the Boers, who cherished the right to teach their own children in their own language, the ensuing violence would threaten the whole Portuguese presence in southern Angola.
Despite his desire to see the slave-trading system "totter and fall like the walls of Jericho," Chatelain was aware that imperial overrule by Portugal did provide him with some protection that he could ill afford to lose. Threats to the mission were a constant fact of life. Chatelain had been warned by several sympathizers about the possibility of his being poisoned. The fears that Chatelain might be murdered by his deeply indebted customers, or that his station might be burnt down to remove all traces of the monies that were owed to him, were not entirely fanciful. When Henry Nevinson followed the slave trail from Bailundu to Catumbela, he too feared that he might be murdered because of any publicity he might give to the practices of the slave traders. Chatelain read Nevinson's articles in _Harper's_ magazine with great care and told his correspondents that as far as he could detect they were an entirely accurate record of the conduct of the slave trade. He himself remained anxious to protect his own position and avoid giving publicity to the crimes of the Portuguese.
The last years of the old Portuguese monarchy's imperial rule in Angola were characterized by a sense of impending doom. Everyone, Chatelain said, was expecting the revolution to break out in Lisbon, but the national finances of Portugal were in such a state that that the republicans were reluctant to take on the economic crises of the empire. One of Chatelain's friends from his Luanda days was a Freemason who gladly anticipated the sweeping away of the old Catholic order. A district commissioner appointed to Caconda in 1907 was also a covert Freemason, as he discreetly admitted to Chatelain, but his revolutionary credentials did not lead him to be lenient in the matter of petty regulations and mindless bureaucratic form filling. It was this plague of bureaucracy that finally drove Chatelain out of Angola. Shortly before he left, Chatelain traveled to Caconda to confront the district commissioner. The mission had been fined 25 milreis for illegally selling rum. Chatelain pointed out that this was preposterous since he was known to be a ferocious opponent of the rum trade. The commissioner explained that the fine was not for selling rum, but for failing to fill in tax returns for the months of January, February, and March 1907, in which he was expected to declare how much rum, if any, he had sold. Chatelain thereupon pointed out, with perfect logic, that the forms had only been issued in April and that it would therefore not have been possible for him to submit them in January, February, or March. The district tax collector resolutely refused to accept any excuses, and rather than face prison and the confiscation of his trading stock, Chatelain paid the fine. When he appealed to the provincial governor at Benguela, the fine was confirmed, and it gradually became clear to Chatelain that not only did the local authorities wish to drive him out of the highland, but the imperial authorities wished to drive him out of Angola. The governor-general in Luanda allegedly claimed that he had never heard of Chatelain or his Protestant trading station, but everyone else in the colonial establishment knew Chatelain, and many were pleased to see him go.
The Caconda district records, now somewhat water stained but safely housed in Angola's National Archive, reveal that the authorities were well aware that Chatelain had been publishing "defamatory" stories about Angola in the United States. His accounts of labor conditions were so damaging to Portuguese imperial pride that his removal was deemed desirable. The imperial government in Luanda, however, was more aware than any magistrate in the provinces that Chatelain was an internationally renowned personage and that any high-handed deportation might have even worse consequences than his journalistic activities. Hence the attempt to drive him out by legal chicanery and the use of bureaucratic forms that no one else had ever been required to complete. What neither the local magistrates nor the central government realized, however, was that Chatelain's subversive reports were sent not merely to Christian congregations in America but also to business houses in Europe. Portugal was about to be engulfed in a revolution with wide international ramifications. Before the storm broke, however, Chatelain sailed for Switzerland, and in 1908 he died in his bed in Lausanne.
## 5
## The Case of Belgium and Portugal
_Belgium and Portugal, the two small colonial powers in Africa, are not normally juxtaposed for comparative purposes. Instead the Belgian sphere is normally contrasted with the neighboring French colonies of equatorial Africa and the Portuguese sphere is seen to have more in common with the white-settler colonies of southern Africa. In this chapter, a longer version of which was originally published in French, it is shown that links and similarities between the two Roman Catholic imperial mini-nations are in fact illuminating Both countries, for instance, had strong anticlerical traditions, which played significant roles in the empires, sometimes at the inspiration of Masonic lodges. Both modern empires were born in the trauma of 1908, when Belgium confiscated the Congo state from its Saxe-Coburg king, Leopold, while Portuguese revolutionaries assassinated their Saxe-Coburg king, Carlos, and introduced republicanism to their colonies. Thereafter the links and comparisons involve financial ties, attitudes to race, the management of plantation policies, the transport of industrial metals, and the marketing of gemstones_.
In the year 1908 the colonial empires in Central Africa underwent a severe crisis brought on by the growth of a humanitarian movement which challenged the way in which some European powers had disregarded the human rights of their colonial subjects. The forces behind the humanitarian protest included the press, the missionary churches, diplomats in consular services, and some members of the business community. The two prime targets of the outcry against excessive exploitation were the rich plantation island of São Tomé, administered under the authority of the Portuguese crown, and the struggling Congo state, personally ruled by the king of the Belgians. The transformation of both the Portuguese empire in Africa and the soon-to-be Belgian territories in Africa went hand in hand during the years 1908 and 1910. The repercussions of change and reform were felt not only in São Tomé and Congo, but also in Angola, the mainland colony from which São Tomé had been drawing most of its servile labor.
Of the two great colonial crises that brought the minor powers to world attention in the years preceding the First World War, the Congo crisis was the first to break. Leopold II of Saxe-Coburg, king of the Belgians and cousin to the royal houses of both Britain and Portugal, had had a long career in business adventures. The Congo "Free" State, which his commercial and military associates had carved out of the heart of Africa in the 1880s, was very much his personal fief. It was neither a free independent state, governed by its own people, nor a colony governed on behalf of the people of Belgium. The Belgian national exchequer did not provide any tax revenue to overcome the constant fiscal crises that afflicted the management of the king's African estates. In order to recover the costs of his personal investment in the colonial adventure, Leopold therefore rented out subcolonies to free-enterprise corporations, which paid him a license fee in return for an unfettered hand in extracting natural wealth and exploiting human resources. The king also retained a crown domain under his personal control. The king's soldiers, administrators, and traders were of mixed national origin, coming from Britain, Scandinavia, and America as well as from Belgium. The best-known, not to say most notorious, of Leopold's associates was the Welsh-born American journalist Henry Morton Stanley, who, after "discovering" Livingstone, had used the firepower of his modern rifles to shoot his way across Africa and down the Congo River to the Atlantic. Another of Leopold's powerful agents was the great Swahili slave hunter Tippu Tip, whose domains in eastern Congo became part of Leopold's fief. The king assured the international community that his objectives were humane and that he aimed to suppress, rather than encourage, the capturing of slaves in Central Africa. The violent methods practiced by Stanley and Tippu Tip nevertheless survived under Leopold's rule.
The Congo crisis of 1908 arose essentially because Leopold, after twenty years of effort, found it increasingly difficult to meet the cost of administering his territory. Profits for himself and his associates were extracted by the tactics of coercion and kidnapping that had been brought into the Congo by earlier Christian and Muslim slave catchers from Angola in the west and from Zanzibar in the east. Leopold's agents also used violent compulsion to force villagers into the dangerous pursuit of ivory elephants, thus emulating the armed gangs of "Turkish" ivory traders from Egypt who terrorized the populations of northern Congo. The kidnapping of women and children, who were then held prisoner until their male relatives had hunted enough ivory to pay a suitable ransom, was officially condemned but covertly practiced by the Leopoldian state. As ivory, confined to the most inaccessible heartlands of the continent, became scarce, violence rose to new heights. New terror was added to the Leopoldian system of extraction when rubber was recognized as a valuable Congo resource.
In the late nineteenth century the motor industry and electrical industry began to create a demand for wild rubber with which to manufacture vehicle tires and insulate copper cables. Leopold's agents found that rubber could be tapped in the deep equatorial forest. They drove not only men but women and children to explore the most dangerous recesses of the wilderness in search of any plant, creeper, or root that could yield rubber. At first the agents made easy profits, but by the turn of the century the price of rubber began to fall as regular plantations were established in Southeast Asia and Latin America. Leopold's revenue shrank and his agents pressed their conscripts to travel ever deeper into the forests to bring ever more rubber to the riverside depots. Villagers who failed to meet the quota arbitrarily imposed on them were severely beaten. In the extreme cases that brought the Congo to world attention, collectors who failed to find surviving rubber-yielding plants were viciously mutilated. Hands were publicly amputated, spreading mortal dread among the foragers and forcing them to neglect their family crops in a desperate search to find rubber or be maimed for life. These Congo atrocities were brought to light and given credence by the British consul in Congo, Roger Casement. A Congo Reform Movement successfully mobilized international opinion against the tyrannical exploitations that Leopold had failed to curb. In 1908 the Belgian parliament decided that the king was not a fit person to be entrusted with a colony and Congo was annexed by the government to become the Belgian Congo. Thereafter a European ministry rather than a royal entrepreneur was the ultimate arbiter of colonial affairs. The extraction of colonial wealth continued, but a minimal level of supervision was introduced.
The Portuguese dimension to the colonial crisis of 1908 was in some respects similar to the Belgian one. The colony of Angola had been carved out of Central Africa by Portuguese soldiers in the last quarter of the nineteenth century, just as the Congo state had been carved out of adjacent territory by Leopold's mercenaries. To make the territory economically self-financing, conscripts were forced to clear footpaths so that long-distance caravans to the interior could fetch rubber and ivory gathered under conditions that were probably little better than those imposed on the conquered peoples of Congo. Unwilling porters, underpaid and underfed, constantly tried to flee from their harsh servitude. Some found refuge in stockaded havens built by kingdoms that resisted colonial domination, and others sought protection in mission stations that helped them to escape from a life that was little better than slavery. The invading colonials used armed force to recapture runaway conscripts as though they were slaves. African resistance, however, created such difficulty in establishing a plantation economy in Angola that Portugal preferred to revitalize the old plantation complex on the offshore islands of São Tomé and Príncipe. Their policy restored the export trade of slaves shipped out of Angola.
The revival of the plantations on São Tomé was based on a change from coffee growing in the highland to cocoa growing on the warmer slopes of the islands. At their height the plantations employed 40,000 workers and needed to import about 4,000 new ones each year to replace those who died of exhaustion, disease, or despair. The majority were slaves who had been captured in the far interior of Central Africa, brought to the Atlantic coast in gangs chained together with wooden shackles, and sold to dealers who shipped them to the islands. Nominally each slave was freed before embarkation and issued a contract to serve on a plantation for a fixed term of five years. In practice no worker ever returned to mainland Africa. The opportunities for escape and even survival were limited since the islands had neither hospitable chiefs nor sympathetic missionaries to shelter the victims of abuse. At the end of each five-year period survivors were automatically assumed, without consultation, to have enrolled for a further five years.
The abuses in the Atlantic colonies became so notorious that Portugal feared that without minimal reform it would find its colonies subjected to confiscation like those of King Leopold. This fear was not entirely illusory. Another member of the Saxe-Coburg family became involved in the reshaping of Africa in the years preceding the First World War. Kaiser Wilhelm of Germany, the grandson of Queen Victoria of England, may not have been closely involved in his government's confidential diplomacy, but his chancellery did aspire to partition the colonies of Portugal. It was aided by the connivance of Great Britain, where a negative appreciation of Portugal's colonial practices was widely publicized by the journalist Henry Nevinson in his articles on the "modern slavery" of Angola. Portugal's standing as an imperial power was further damaged in 1908 by the assassination of King Carlos, the member of the Saxe-Coburg family who reigned in Lisbon. Two years later his son was driven into exile by a conspiracy of anarchists and republicans, and thereafter the way was open for Britain and Germany to plot the partition of the Portuguese empire. Their plan was simple. Southern Angola, adjacent to German Namibia, and northern Mozambique, adjacent to German Tanzania, would be handed to Germany on the ground that the Portuguese administration was incapable of dealing efficiently with the needs of modern business entrepreneurs. Britain would then help itself to the remainder of the two territories. Portugal responded to the threat by sending one of its most robust republican politicians, General Norton de Matos, to Angola as a high commissioner with a virtually dictatorial free hand. Norton was forced by foreign opinion to moderate some traditional abuses, and Angolan workers sold on contract to the cocoa islands were henceforth allowed to return home after ten years' service, a reform proposed in 1908 but intemperately rejected by the then-royal government of Portugal as an impertinent attempt by Britain to interfere with its proud sovereign autonomy. But the republicans wasted no time in finding new methods of extracting service from the huge mainland provinces that they had inherited in Angola. Norton began, in particular, to conscript women into forced labor gangs to build a network of earth roads that would meet the needs of the newly introduced automobiles and trucks.
Transport was one problem faced by all colonizers in Central Africa; King Leopold himself recognized it as the essence of colonizing. The Congo faced particular problems in that its richest mineral deposits, the ones requiring the heaviest machinery and producing the bulkiest exports, were in Katanga, a thousand miles from any seaport. The shortest route to the sea led across Lake Tanganyika, through German East Africa, and down to the traditional but remote and not very convenient old Swahili port of Dar es Salaam. An alternative "national" route to the western sea ran through Belgian territory but involved the use of several railways linked by stretches of navigable river, thus requiring multiple trans-shipments. A longer but more continuous route to the sea wended its way across several boundaries and over the Zambezi bridge to South Africa's remote ports. To overcome the handicaps British investors designed a short route that would lead out of Katanga, across the Angola plateau, and down to the Atlantic at Benguela, where a deep-water harbor could be built at Lobito Bay. This Benguela railway was not completed until the 1920s, but from then until its destruction in the 1970s it was one of the key arteries of Africa, carrying Congo minerals to a deep-sea port.
Carrying minerals from the Belgian mines could never be as profitable as extracting them from Portuguese territory. The centuries-old search for minerals in Angola, however, had never produced great bounty. Despite bitter wars over mineral rights fought between Portugal and Africa's old Atlantic kingdoms, only small quantities of copper had ever been found. In the eighteenth century a Basque-run iron foundry was built by a Portuguese dictator, the marquis of Pombal, on the site of the old mines of the Ndongo kingdom, but the ambitious venture proved abortive. In the twentieth century an expensive railway was built to open iron mines in Angola's deep south, but they failed when the price of ore collapsed. The one success in mineral extraction began in the 1920s when Angola discovered good-quality gem diamonds for the jewelry trade. The diamond fields were staked out in newly conquered territories in the northeast of the colony. A semiclosed concessionary territory was defined, financed, and administered by an extraction company linked to Britain, Belgium, and South Africa. A marketing cartel controlled by De Beers ensured that Portugal obtained good prices for Angola's gems on the highly regulated diamond market. The Lunda-speaking peoples of the northeast became the captive subjects of a state-within-the-state. They were compelled to serve long tours of duty in the open-cast mines and were held under closed conditions designed to limit pilferage. Under the authoritarian regime that governed Portugal between 1926 and 1974, this remote colonial mining complex became an enclave of expatriate exiles to which some dissident Portuguese engineers were banished as punishment for their liberal views. By the time the fascist-style regime fell, another mineral enclave had been opened up in Angola's offshore oil field of Cabinda. This too was under foreign control, though the capital and technology came from America rather than Belgium. In the 1960s oil outstripped diamonds as Angola's main mineral export. Diamonds, however, made a comeback of considerable political significance during the wars that followed Angola's independence. Some of the old diamond diggings, and many new ones, fell into the hands of rebel forces in the 1980s. These rebels smuggled their diamonds out through Congo marketing rings to pay for their supplies of food, fuel, and weapons.
In Angola minerals became the dominant export only after independence. In the Belgian Congo, by contrast, mining was important from the colonial outset. Congo mined large quantities of industrial diamonds along the rivers downstream from Angola's fields of gemstones. It also developed significant gold washing in the muddy streams of the eastern zones. Diamonds and gold both required the conscription of large numbers of subjects who were compelled to work under harsh conditions. Despite the glitter of diamonds and gold, the Belgian colonial economy became primarily dependent on industrial metals rather than rare minerals. The Congo copper mines in Katanga became the basis of the largest industrial complex in tropical Africa. The new industrial society incorporated an African workforce that became semiskilled and permanently urbanized. The miners of Katanga, unlike most colonial miners, did not form an all-male community of migrants whose economic and cultural roots remained in the countryside, but became a stabilized proletariat dependent on its wages. This Congo proletariat reproduced itself within the industrial complex, but Belgian policy did not allow these urban Africans to move far up the ladder of skill and opportunity, though it did give them significantly more literacy and technical competence than was available in Portugal's empire. Management, however, remained firmly in white hands. In between Katanga's white managers and the black workers a commercial class of expatriates included a service community from Greece. Elsewhere in Congo services and retail trading were often in the hands of Indian shopkeepers in the north and Portuguese artisans in the west.
One question in colonial history concerns the extent to which the formation of classes was influenced by racial attitudes. In the Belgian Congo, where mining and manufacturing saw the growth of a significant urban population, social mobility was hindered in two ways: first, and most obviously, by the white population's reserving for itself many of the tasks that carried status and a generous salary. The second way in which class formation was hindered, intentionally or unintentionally, was by the dividing of the African population into ethnic groups. Belgian policy on ethnicity fluctuated. Sometimes ethnic groups were seen as a safe means of dividing the native population into small segments that could be easily controlled. At other times policy favored the mixing of peoples so that no large block of workers—with an autonomous ethnic solidarity—might emerge.
The development of a stabilized labor force occurred at different times in the different regions of the Belgian Congo. Temporary labor camps initially had poor housing, poor sanitation, limited water supplies, and high levels of mortality. In sparsely populated Katanga the recruitment of labor was difficult, but the profits on mining were sufficient for companies to improve living conditions from the 1920s and so form a working-class core of employees supplemented by temporary migrant laborers. Industrial workers in the Belgian Congo gained craft skills and responsibilities that were not entrusted to Africans in the Rhodesias or South Africa. As a result they worked more effectively and so enhanced the profits of their imperial employers. The medical services provided for the stabilized workforce were improved, but mortality remained high among casual and migrant laborers. As the permanently urbanized workforce grew, the methods of social control changed. Instead of the brutal personal violence that rained down on early colonial workers, new forms of group organization were manipulated by the mining companies. They coopted the assistance of Catholic missions in creating "boy scout" solidarity and discipline within the work gangs. Social engineering was so pervasive that missions arranged marriages for urban employees by bringing brides to the city from appropriate ethnic communities. When paternalism failed, however, company managers quickly turned to violent repression. The Katanga mine strike of 1941 led to forty-eight men being shot and killed and many more wounded.
The Congo employees who came closest to forming a social class were the clerical workers, some of whom became leaders of political movements in the last years before independence. Clerical workers were divided, however, as to whether they should be the spokespersons of African society as a whole or should concentrate on the protecting of their own semiprivileged status in the colonial world, located midway between the white artisans and the black proletariat. Members of the black secretarial class were known as the _évolués_. Their ambivalent attitudes fluctuated between collaboration and protest, between enjoying salaries with status and resenting the disdain they suffered at the hands of petty white supervisors. In Belgium even liberal-minded politicians and working-class socialists did not have much sympathy toward upwardly mobile black aspirations. Racial prejudice was so strong in Congo that black resentment became the driving force behind the 1950s radicalization of colonial politics. Racial antagonism, however, was not the only emotive force to reach fever pitch in late colonial Congo, and ethnic fragmentation was the harbinger of the violent confrontations of decolonization and its postcolonial aftermath. The utopian aspirations and bitter social disappointments of Belgium's colonial subjects led to half a decade of confrontation after 1960 as the large and apparently powerful empire that the Belgians had built around their financial and industrial interests crumbled. From the ashes, however, the key institutions of colonialism—the army, the church, and the mines—were to recover their authority.
Mining, as the central activity of colonial enterprise in Congo, had wide repercussions on rural society. Despite anxious resistance, labor recruiters and food procurers reached and transformed many diverse, distinct, and remote farming communities. Villages were broken up and their inhabitants resettled along lines of road, river, and rail communication. Peasants were set to work on plantations where local crops were grown to feed miners with their accustomed diet. Compulsion and punishment remained the constant refrain after the Belgian annexation of Congo, even if the extreme horrors of mutilation had been replaced by the lash. Slow working practices may have been the result of poor incentives and inadequate nutrition, but disease also spread lethargy through the newly concentrated communities. Health officials had to learn that moving workers from one disease environment to another could have serious consequences on mortality rates in compounds. Psychiatric illness also affected colonial subjects who had been captured, roped together, and marched away in terrified columns. In semideserted villages the catastrophe of colonialism was often attributed to witchcraft. As communities became divided against themselves, misery, poverty, and recrimination spread. When the villages could no longer supply the mines with food, the Belgians opened up new-style colonial plantations.
The old colonial world had been dominated by planters, who often preferred to settle on tropical islands and relied on slave labor to produce their cotton, sugar, and tobacco. In the twentieth century some colonial powers adopted a rather different policy. Instead of moving people to islands, they conquered whole communities on the African mainland and forcibly encouraged local farmers to grow crops chosen by colonizers. These crops were not necessarily ones that would bring farmers good profits, but they provided colonizers with viable internal or export markets. The incentives used by Belgium as well as Portugal to change peasant practices involved only limited economic rewards alongside the compulsion used in agricultural policy. On traditional plantations laborers were more or less unwillingly conscripted to work for wages that did little to compensate them for their loss of freedom and the denial of their right to make rational economic choices in their agrarian strategies. Where plantations were not established, African farmers were compelled to grow crops on their own land, using their own families as labor, according to choices made by a commissioner or governor. The colonizers of Portuguese East Africa drew their revenue from sisal, coconut, sugar, and rice; in Portuguese West Africa and the Belgian Congo, where these crops were not viable, the main crops were coffee, maize, and palm oil. The most controversial crop in all areas, however, was cotton (see discussion below).
Each of the colonial crops had a distinctive effect on conquered societies. This effect was partly influenced by money, the introduction of which into rural society was one of the most far-reaching of the changes brought by colonization. In Congo the change from barter to coinage began soon after the ending of the Leopoldian system of extortion, when money was introduced to pay villagers for crops. The colonizers did not build a balanced economy based on consumer supply and demand, and cash did not long remain in peasant hands. Instead the new colonial coins were soon clawed back in taxation, which was intended not to pay for welfare services but rather to compel farmers to earn cash by growing colonial export crops. During the First World War the system was used to feed Belgian colonial troops sent to invade the German territories of Rwanda and Burundi on the Great Lake fringes of Congo. As colonial rule became entrenched, agricultural companies were set up to produce or process tropical crops. Palm oil production came to be dominated by the Unilever corporation, while peasant-grown cotton was processed by a state-run company. State intervention increased during the world Depression of the 1930s, and compulsory crop growing replaced the free-market and cooperative initiatives. Buying agencies became accustomed to purchasing crops at little more than a quarter of their free-market price. The state encouraged such extortion by building feeder roads that channeled cheap crops to the railway yards by truck. Not until the very end of the colonial period did an African productive class emerge that could buy land, select markets, and respond sensitively to a choice of economic opportunities. Protests about compulsion and an absence of free incentive had been the hallmarks of Congo farming in the Belgian period.
Colonial farming in Angola bore some similarities to Belgian agriculture in Congo. Indeed, some of the capital that was invested in Angolan plantations was Belgian, and some of the farmland that was used to grow colonial crops straddled the Atlantic railway linking Congo to Benguela. Angola's most important crop, however, was coffee grown in the north, where it had been introduced by Brazilian traders in the nineteenth century. The first "prototype" plantations used slave labor supervised by white convicts. Similar estates momentarily achieved a high level of prosperity in the 1890s, when accessible fertile land was violently seized from its African owners. Production soon declined, however, when epidemics of smallpox and sleeping sickness devastated northern Angola, spread across the region by columns of conscript porters hired not only by Portugal and Belgium but by French colonizers on the north bank of the Congo River. Once the epidemics had burned themselves out, returning colonial farmers attempted to diversify their production and avoid dependence on a single crop in a restricted zone. Sugar cane was planted on the south coast of Angola, and when distilled as rum it sometimes gave an adequate return. When failure occurred, however, the industry came to be dominated by Portugal's colonial bank, which reluctantly acquired many sugar estates in exchange for bad debts. Another innovation was sisal, brought by German migrants who left Tanzania after the First World War and established plantations on the dry edges of the Angolan highlands. In the wetter lowlands Portuguese settlers developed palm oil plantations in the hope that palm oil would compete in price, if not in quality, with the olive oil produced by Portugal itself. A later form of oil-seed production was attempted with the cultivation of sunflowers. A further diversification was cattle ranching, introduced with limited regard for the land rights of the indigenous population. The crop that caused the greatest distress, however, and the one that triggered off an anticolonial rebellion, was cotton. Already in 1945 compulsory cotton growing was reported to be causing peasants to neglect their food crops to such an extent that serious famine had spread across central Angola. The dictators of Portugal, Prime Minister Salazar and his colonial minister Caetano, refused to heed the warnings of their local advisers, and the compulsory growing of cotton on poor, dry land continued until the end of the colonial period.
Both of the small colonial powers in tropical Africa were officially Roman Catholic when they began their colonizing practices. In Congo the Catholic Church came to be an important institution among the industrial workforce. The church enhanced obedience, discipline, and African acceptance of colonial hierarchy and authority. The Portuguese Catholic Church had a similar role in Angola, where it was seen as the church of the white expatriates and settlers rather than of the common people. Curiously, however, in both imperial nations there was also strong antagonism to the church, which was sometimes reflected in the attitudes of colonizers. In Belgium an anticlerical freemasonry was one significant strand of middle-class opposition to the dominance of the church. In French-speaking areas of Belgium there was also a working-class socialism that had atheistic tendencies. In Portugal the socialism was more muted, but the republican ideology that dominated the country for a generation after 1910 was strongly anticlerical, and many of the republican leaders were anti-Catholic Freemasons. The greatest antagonism to the established colonial church hierarchy came, however, from foreign missions run by Protestants who had no loyalty to the colonial metropolis (see chapter 3).
One of the earliest of the Protestant mission churches to take root in West Central Africa was the Baptist church from Britain, which became a dominant religious, educational, and cultural force in both Congo and Angola. Missionaries were seen by many as the handmaids of colonization, instilling a respect for European culture and an obedience to Christian doctrine that facilitated governance by white foreigners. The missions in Africa did some of the work that in Europe was being undertaken by progressive states, providing schools and hospitals for those who could not afford to pay privately for medicine or education. Yet while better health and better skills made colonization more profitable, mission activities were not always to the liking of colonial administrators. The missions accepted the need for obedience to the state, but they did not instill unalloyed Belgian or Portuguese patriotism into their congregations. When protest broke out against harsh forms of colonial exploitation, administrators tended to blame missionaries for fomenting rebellion. Although the Portuguese were offended by the liberalism of Protestants, they did not wish to offend Great Britain and allowed them to evangelize. The Portuguese church was too poor, and until 1926 too persecuted, to recruit missionaries of its own who might entrench Portuguese values in the tropics by teaching and precept. The Portuguese state therefore relied on foreign-funded missions, Catholic as well as Protestant. In south Angola much proselytizing was undertaken by Holy Ghost fathers from France.
In the Belgian section of the lower Congo region the Baptist Church was caught in the same trap as the missionaries in Angola. Baptist missionaries were dependent on the state for their preaching licenses, but they did not approve of colonial profiteering at the expense of African workers. Subversion took hold among Baptist congregations, and one African preacher, Simon Kimbangu, began to speak of the rights of black peoples and even of the power of Christ to speak to Africans unaided by white intermediaries. Kimbangu was rapidly captured and imprisoned, but his message could not be quelled. Although confined to a remote region of eastern Congo, his followers remembered him, and their number multiplied in the west. Small independent Kimbanguist churches sprang up in Angola as well as in Congo, and in the 1950s the prophet's son took command of the new far-reaching church. Eventually the Kimbanguist church, freed from the trammels of the state and independent of any white theological influence, was invited to become a full member of the World Council of Churches at Geneva.
Carried by new missions, fundamentalist Protestant traditions took root and spread to villages that had not been touched by the Flemish Catholic fathers or British Baptist pastors. The new missionaries were Americans financed by suburban congregations of a broadly Pentecostal persuasion. They created their own self-reliant networks of river communication and became largely independent of the state. The new missions, along with some old Catholic and Baptist ones, became one of the lasting legacies of the colonial order.
The colonial system of Portuguese Africa evolved a pragmatic compromise between racial segregation and racial assimilation. Before 1910, under the Portuguese monarchy, a significant number of black subjects had been given posts of influence and responsibility in the colonies because there were insufficient white immigrants to run the bureaucracy, command the militia, and staff the trading houses. A Creole community blended African customs with the lifestyle of the colonizer. After the establishment of the republic in 1910, an influx of white functionaries came to the colonies to seek the fortunes that had eluded them in Europe and drive black and brown Creoles out of their work stations. Discriminatory practices were not racially enforced as they were in Belgian or British Africa, however, and assimilated Africans could gain status if they were literate, wore European clothing, spoke metropolitan Portuguese, practiced Catholic worship, and demonstrated to an inspectorate of morals that they were loyal to the "mother country" in Europe. Angola's mixed-race population, whose fathers were white and whose mothers were black, was painfully caught up in a pattern of racial legislation so complex that by the end of the colonial period a mere 2 percent of Angolans had achieved assimilated status. Even the _assimilados_ who achieved equality before the law soon discovered that under the dictatorial system of Salazar's so-called fascism legal status did not give citizens, white or black, the automatic right to participate in politics. What assimilation did provide was exemption from the forced labor regime, which by 1960 conscripted 175,000 of Angola's one million able-bodied men to work each year for colonial enterprises owned by whites. _Contratados_ (compulsory recruits) worked for twelve months at a stretch for a minimal wage that did not allow them to contribute adequately to the living expenses of their dependents. The system provided a powerful downward pressure on wages since men "volunteered" for work at very low pay rather than risk being captured as forced labor. Forced labor practices drove as many as a quarter of a million Angolans to flee across the borders, where work for meager salaries in British, French, and Belgian Africa was preferable to conscription in Angola. Those who sought to evade rural labor raids without going abroad tried to find jobs in the cities, but influx controls similar to those used in South Africa authorized the police to raid the city slums and return rural peoples to their villages, ready to be seized by recruiting agents. Expulsion from the city not only discouraged the growth of a black urban proletariat but kept opportunities open for illiterate white migrants from Portugal. In Angola merchants, artisans, and truck drivers were predominantly white and filled niches held by Indians or Syrians in East or West Africa. Many working-class whites were so poor that they lived in multiracial slums and worked as market gardeners and street vendors. The old-established African fishing industry saw successful white hawkers take over the marketing of fish and later invest in new boats and new nets, which competed with traditional fishing methods. Angola's petty merchants created a class different from any found in Belgian or British colonies. A decline in African skills in the private commercial sector was matched by a decline in skills in the public sector. Whereas the Congo _évolués_ were staffing government agencies and post offices by the 1950s, in Angola the reverse was happening: white immigrants were replacing black counter clerks and bookkeepers at the lowest levels. When independence came to Angola, there was a chronic shortage of personnel not only to maintain the commercial economy but also to staff essential administrative services.
One of the distinctive features of late colonialism in the Portuguese territories was the proud belief by the rulers that they had developed a lusotropical social class that blended African roots with Portuguese culture. The concept of lusotropicalism, imported from the mixed-race societies of Brazil, was used to justify the continuance of foreign overrule after the 1960 wave of decolonization in tropical Africa and to deny the charges of racism that critics directed at the Portuguese. The propaganda campaign failed, and in their social and racial composition Portuguese colonies came to resemble South Africa rather more than the new postcolonial societies of French and British Africa. Attempts to present the African colonies as overseas provinces of Portugal, equal in constitutional law to the domestic provinces, were economically and politically fraudulent. The racial prejudices introduced to Angola in the early twentieth century intensified during the 1950s, especially when white women began to arrive in larger numbers and dismissed the black concubines who had customarily provided settlers with marriage partners. The new colonial women also turned their wrath on the illegitimate offspring of casual unions, and no amount of government propaganda proclaiming that blacks were equal to whites in Portugal's harmonious empire could overcome the reality of an intense institutional racism. It was this racism that made the outbreak of the Angolan revolution in 1961 so violent and caused the shedding of so much blood on the streets of Luanda.
## 6
## Race and Class in a "Fascist" Colony
_The term "fascist" has been much misused by historians, and it is not at all clear that the term can appropriately be employed in relation to even the most authoritarian of the imperial powers. Although Salazar adopted some "corporatist" ideas from Mussolini, Portugal's dictator never swayed a mass movement in the way that Hitler could. Portuguese authoritarianism did, however, bite deeply and cruelly into the lives of African subjects, just as fascism bit deeply into the lives of many European peoples; Polish conscript laborers of the early 1940s in Europe might have been well placed to comprehend the agonies that Angola's forced migrants suffered when dragged off to the mines and plantations of Africa. This essay was written in 2004 as a preface to the forthcoming published version of Christine Messiant's 1983 doctoral dissertation_. _Messiant's work examines and explains the deep background to the Angolan revolution of 1961, providing a key foundation on which new generations of Angolan scholars can build a fresh understanding of their society, their culture, and their history. It remains a pioneering study in grasping the complexity, diversity, and tenacity of the social identities that molded private and public behavior in Angola_.
Angola has its own very specific amalgam of social forces derived from Portugal's Atlantic empire as well as from Central Africa's deep past. Several centuries of cultural, religious, and genetic blending created a social nucleus around the twin cities of Luanda and Benguela. In the Angolan interior, distinctive creolized communities grew up in Mbanza Kongo, Ambaka, and Caconda. The Creole enclaves enjoyed some international contacts and might bear comparison with colonized societies of the Americas, or with the British and Dutch in Asia. These societies were created during the centuries of the slave trade and modified by the advent of twentieth-century white immigrants. The new colonialism opened up an ideological debate as to whether black and brown subjects should be incorporated—assimilated—into colonial society and encouraged to acquire European skills and customs, or whether Africans should merely be hewers of wood and drawers of water. The spin doctors of imperial propaganda dwelt on the liberalism of assimilation and on the virtues of Angolans who had acquired civic status. The economic managers of empire, meanwhile, preferred to perfect a system of compulsory labor service that would underpin public and private enterprise and generate colonial revenue. In 1961 the violent strain of coercion caused the system to collapse in bloodshed. Colonial managers, migrant workers, and assimilated elites were all victims of a wave of killings that spread across Angola.
A shortage of whites had forced the early colonial state in Angola to conscript "social mestiços," wives and mixed-race sons, into the administrative class. The pragmatic incorporation of nonwhites into the colonial nucleus continued into the twentieth century despite the ferocious denunciation of miscegenation by the republican high commissioner, Norton de Matos. The gender imbalance continued after the 1926 creation of the "fascist" New State, and as late as 1950 Angola still had little more than half as many white women as white men. Whether the new generations of mixed-race children would enhance cultural loyalty to Portugal or would sow the seeds of disaffection and nationalism was a question anxiously debated among colonizers. One peculiarity was the status of mixed-race families descended from the 2,000 white convicts from Portugal who had lived in Angola in 1900 but were not listed in the colonial census. The tide of convicts continued to rise until the outbreak of the Second World War and may have accounted for 20 percent of Angola's immigrants. By that time another distinctive white community, the 2,000 Boer settlers in the south, had returned to South Africa. Proposals to replace them with Jewish settlers from Russia or white settlers from Algeria came to naught.
In attracting settlers to Angola, the key policy of colonization concerned land ownership. The colony had prospected unsuccessfully for petroleum after the First World War and had granted a mining concession that made diamonds the leading export until the Second World War. Settler wealth, however, derived from the alienation of fertile pockets of African farmland. Ninety percent of Angola's land was either uncultivated or unsuitable for cultivation. When whites took possession of three million hectares, they gained a mere 2 percent of the country's total land surface but 20 percent of good soils. The white seizure of land drove African producers onto less productive soils, where the colonial state compelled them to grow cotton and other revenue crops for white merchants. Increased pressure on poor soils did nothing to enhance land fertilization, and the cycle of fallow seasons traditionally allowed for soil regeneration grew shorter. The new, white landowners used the fertile, well-watered soils to grow coffee and other profitable export crops. The politics of land alienation were to haunt Angola throughout the twentieth century and remain the country's most controversial political issue in the twenty-first century. By then it was the African urban elite, including the generals of the two armies who had fought the civil wars, who aspired to become plantation owners. They wished to return to the agricultural agenda of the old white settlers and restore Angola's black peasants to their colonial-type status as farm laborers.
Portugal's interwar dictatorship aspired to create a colonial system sufficiently robust to survive the Great Depression of the 1930s. Portugal had been thrown into crisis by the closing of the doors to Brazilian emigration and the abrupt ending of South American wage remittances. Under Salazar the colonial office was pressed to generate alternative imperial revenues for the impoverished Iberian homeland. The policy, however much the fascists pretended otherwise, depended on allowing foreign capital to stimulate economic activity. The dictator minimized the cost of administering the colony by cutting all forms of social expenditure and allowing private investors to operate unfettered by moral or legal constraint. The new empire even restricted the old republican aspirations of white settlers in the civil service by coopting ill-paid but "assimilated" nonwhites into the bureaucracy. Assimilation in Angola was related partly to whiteness of ancestry and partly to association with long-established urban families. Access to the assimilated urban nucleus was not easy to achieve. One road to acceptance was through Protestant schooling, a process that distinguished newly acculturated Angolans from the traditional Catholic families. Despite the windows of opportunity for a few assimilados, the colonial tradition expected whites to be managers and blacks to supply labor. Attempts by international agencies to investigate Angola's labor practices were strongly resisted. Before joining the United Nations in 1955, Portugal declared that Angola was an "overseas territory," a province of Portugal, and not a colony at all. The attempt to avoid world scrutiny, as the French did in their Algerian départements, did not succeed, and the International Labour Organization resolutely condemned labor conditions in Angola during the years leading up to the coffee and cotton rebellions of 1961.
In 1960, 90 percent of Angola's sparse population lived in rural areas, on plantations, around scattered farmsteads, and in small villages. Only the other 10 percent lived in sixteen officially designated colonial towns. Of these townsmen a quarter of a million lived in provincial cities such as New Lisbon and Benguela or in remote administrative posts with civic status and architecture. The other quarter of a million urban Angolans lived in the city of Luanda. The late colonial population of Angola consisted of two million black adults, two million children, and a rising tide of white immigrants. Initially new immigrants came from unlettered peasant communities in northern Portugal and Madeira. When wives and children are included, total immigration rose to around 200,000 people, a figure comparable to the settler population of Southern Rhodesia and twice that of the expatriate population in Belgium's Congo. Most new immigrants chose to settle in towns and deemed themselves superior to the old "bush" settlers who ran small trading posts in the provinces. The number included several thousand educated, though often nonwhite, immigrants from Portugal's tropical islands. Management in both the private and public sectors came to be dominated by whites or by a few "colored" Cape Verdeans. Poor immigrants took up petty trading, and some African market women in Luanda were driven from their established street stalls by white rivals. Angola, unlike Mozambique, welcomed unskilled migrants. Their aspirations, however, caused distress, which was the background to the great nationalist awakening that virulently shook Angola's people in 1961.
The pain of white colonization was felt particularly harshly by conscripted black workers in the countryside. Conscripts were legally bound over to plantation employers for a whole year at a time, or, if they were posted to desert-coast fishing banks and frontier diamond mines, for eighteen months. Such schedules made it impossible for migrant workers to return home for seasonal planting and harvesting. Even the compulsory six-month home furlough to which migrants were entitled between contracts was not always observed. Although some laborers were recruited to work in their home districts, their daily shifts were so long that tending their own crops was well-nigh impossible. Abolishing "idleness, indolence, and vagrancy" was the watchword of the provincial governors who administered the cruel activities of the labor catchers. In 1946 Henrique Galvão, a colonial inspector who later became a prominent Portuguese politician, thought that the recruitment system was "in some respects worse than slavery." Ten years later Basil Davidson suspected that conditions had continued to deteriorate. Only in 1959 was an official at tempt made to enhance the wages of conscripts. A norm of 200 escudos a month was suggested, but this represented only about 2 British shillings a day (28 U.S. cents) at a time when laborers on the colonial Gold Coast had already negotiated 6 shillings a day. Even this tiny norm was not enforced, and white farmers continued to deduct more than half a conscript's wages for medicine, shelter, transport, work clothes, food, and other real or imaginary payments in kind. In effect wages remained at around 70 escudos a month (9 U.S. cents a day) for migrants and less for those conscripted to work in their own locality. At the end of each year matters were made worse when the colonial state tried to collect income tax, a tax which, incidentally, was rarely spent on building schools or clinics for native peoples.
The lack of educational facilities was particularly notable in Angola. Even neglected "Cinderella" colonies like Northern Rhodesia and Mozambique spent more on educating their black children, and Southern Rhodesia provided education for most of its black infants when only 5 percent of "indigenous" children attended school in Angola. When access to the modern world through education was denied, Angolans turned to modern religion, and communities of Protestants in the north and Catholics in the south grew in scope. In a country as fragmented, exploited, and deprived of labor unions and cultural organizations as Angola, missions, although not heralds of the revolution, became the unexpected focus of protest movements.
The net income derived from compulsory labor by Angola's uneducated rural masses was so meager that enterprising Angolans who had been registered as indígenas (natives) tried to find freelance contracts with employers of their own choice. Proof of such service could in theory be used to fend off the demands of recruiters seeking migrant labor. Farm work was never lucrative, and the better-paid alternative to local laboring or plantation migrancy was a job in town. In the northern town of Uíge wages could be twice the notional legal wage and in Luanda three times as much. Family dependents in town could further support the household by shining shoes or selling lottery tickets. Men with skills as nurses or artisans might upgrade their wages to 2,000 escudos a month during the urban boom, though barely 3 percent of common black folk made it to the city. In the north, east, and south of the country the best survival strategy was emigration. In neighboring countries French, Belgian, and British wages were higher, and even in Namibia a pair of boots cost a quarter what it cost in Angola. Some 100,000 of Angola's economic migrants worked for the 6,000 white Portuguese who lived in the Belgian Congo. Altogether, probably half a million Angolans lived abroad during the 1950s, and on the southern plateau it was said that everyone had heard of the fabled gold mines of Johannesburg.
Inside Angola class and race relations remained complex, and teasing out the various strands of non-native society has been one of Christine Messiant's most rewarding contributions to Angolan studies. Mixed-race assimilados were more numerous than black assimilados in schools, colleges, and seminaries, and they rose to higher ranks of responsibility in the church or the provincial administration. The highest flyers, in medicine and the law for instance, were commonly not Angola-born but were nonwhites from Portuguese India or São Tomé. Even locally born mestiços gained advantages over black assimilados and found it easier to obtain good jobs and to prove their status when rounded up in police raids. Mestiço children were not so obviously ostracized by white classmates as were black children who entered secondary school. Racism grew in the 1950s, however, and mestiço children tended to be considered the "children of blacks" rather than recognized as the "sons of whites." The number of excluded mixed-race children brought up as "native" by their unsupported mothers began to increase both in the towns and in the provinces, where they received no recognized status.
The number of registered "coloured" inhabitants in Angola, to use a South African term, never exceeded 1 or 2 percent of the population, as compared to 10 percent in South Africa. Despite its small numbers, Angola's mixed-race population had played a significant colonial role in the past. It had been distributed throughout the provincial towns at a ratio of one mestiço to every three whites, with notable clusters in Luanda, Benguela, Huambo, and Lubango. Mestiços were legally entitled to set up businesses and borrow capital. Their commercial activities spread from owning bars in the black townships of Luanda to trading in the coffee plantations of the north and the maize fields of the south. Most mestiços remained modest salary earners in government service, but in the 1950s assimilated Angolans were ever more sternly excluded from semiprivileged positions by an authoritarian political system and by the influx of unskilled whites who competed to fill the social layer between privilege and poverty that mestiços had historically occupied. New immigrants instilled a sort of petty apartheid into Angola, where previously social manners rather than skin pigmentation had divided society. This radicalization of social discrimination drove "civilized" Angolans back into association with the "natives" from whom they had so painfully struggled to distinguish themselves by language, eating habits, and mental outlook.
The controversial role played by the assimilated population in the emergence of a "nationalist" leadership among Angolans makes Messiant's search for the factors that defined the rival strands of class and status important. She identifies three categories of assimilados. Coloured Angolans, unlike coloured people in South Africa, were usually descended not from two coloured parents but from a white father who provided status to his children and a nonwhite mother. In the Angolan census of 1950 (but not in those of 1940 or of 1960) several thousand assimilados were enumerated as "white." The second category, "civilized" Angolans, mostly black but some coloured, belonged to the old high-status, Roman Catholic colonial nucleus, with ten generations of urban history behind it. The children and grandchildren of this ancient category of "aristocratic" assimilados provided one of the strands of the ruling elite that governed Angola during the civil wars of the later twentieth century. The third segment of assimilated colonial society emerged among the "native" population in the provinces. With mission support they gained sufficient education to win entry into society and sometimes, as in the case of the medical doctor Agostinho Neto, into the professions. After independence these old social categories inherited from a colonial age were challenged by new Angolan elites whose advancement was achieved through the army and the universities.
The history of Angola's cultural mestiços focuses initially on their real estate in downtown Luanda. When challenged by the 1910 wave of republican immigrants, many assimilados moved into marginal townships but retained their bureaucratic employment in the city. Their cultural association, the Liga Nacional Africana, continued to organize society balls and cultural lectures. A few were persecuted, or even sent into exile, where they established contact with assimilados living in the provinces. They mounted campaigns of protest over the erosion of the rights of nonwhites, but when the nationalist confrontation of 1961 broke out most adult assimilados preferred to lie low and protect such social advantages as they had managed to retain. Their children, however, were sometimes tempted to throw in their lot with the campaign for independence and join a hundred colonial students in Portuguese universities who abandoned their studies and fled to western Europe. The young radicals who abandoned their haughty historic families joined members of the new assimilated group, the children of artisans, catechists, traders, teachers, nurses, and office workers who had acquired modest status if not wealth. The new assimilados spoke Portuguese at work but not at home and did not belong to the same social clubs as the old assimilados. Instead they lived cheek-by-jowl with neighbors who had retained up-country contacts and African legacies. In politics, though old assimilados claimed the authenticity of history, it was the new assimilados who could most clearly hear the voice of the people. Both old and new came to recognize that after 1961 Portuguese colonialism, unlike British or French, was digging in for the long term.
The dream of colonizers in Angola had always been to find easy wealth in mineral extraction. In the sixteenth century silver had been the mythical lure, and a century later Salvador de Sà fought to conquer the small northern copper mines. In the eighteenth century the marquis of Pombal built an iron foundry, which he named Nova Oeiras after his Lisbon country seat and staffed with Basque foundry men, but it had no lasting success. After 1926 minerals brought significant rather than merely symbolic change to Angola. British engineers and financiers finally completed the building of a railway from the Benguela harbor at Lobito to the great Congo-Zambian copper mines of the Anglo-Belgian borderlands. The copper-carrying railway stimulated several ancillary branches of colonial production. Huge eucalyptus forests were planted, for instance, to fuel the locomotives with firewood. Down on the desert coast a fishing industry expanded to almost half a million tonnes a year, enabling it to export dried fish to the mining cities at the head of the line of rail. Highland peasants who were not marched to the coast to gut and dry fish and were not needed to clear the shrubs on either side of the railway were compelled to grow maize as a tax crop, their soils being too cold for compulsory cotton. New canteens were opened at the railway sidings by petty traders from Portugal. A railway required more than the produce of farmers and fishermen to fill the trucks with payloads. It also required personnel to repair the track, shovel the ash, operate the signals, sell the tickets, and staff the telegraph machines. Many of the necessary African railway personnel had been trained in the Congregationalist missions that the Swiss, Canadians, and others had set up in the highland kingdoms. The uniformed staff on the Benguela railway were part of an old-boy network that transcended the ethnic rivalries between the Ovimbundu kingdoms. Cooperation among a modernizing elite of petty functionaries replaced historic antagonism and became one of the bases of a political party created in the middle 1960s.
Angola was partially insulated from the ideological debates that led to the postwar transformation of Africa. Britain, France, and Belgium decided over the course of the 1950s to establish self-governing regimes in the African colonies. A new black bourgeoisie that had been trained in the administrative practices of the colonizers would take over responsibility for local politics. In the matter of commerce it would work with the European powers (and to a lesser extent with the United States and the Soviet Union) in a neocolonial association. Each postcolonial partnership would ensure economic and military stability in fragmented republics, helping the flow of wealth from the third world in the south to the first world in the north to continue uninterrupted. The neocolonial transformation of Africa in 1960 presented Portugal with particular problems. The black bourgeoisie in Angola was relatively small, and many of its middle-class leaders lived in semi-exile as social-democratic students in Lisbon or freemarketeering businessmen in Kinshasa. Those in Lisbon aspired to employment in the service industries of the state as doctors, teachers, agronomists, or civil servants. Some colonial students maintained covert links with white Communists who aspired to transform conservative Portugal into a radical socialist society. The socialism of Angolan students, however, was linked to a Protestant Christianity that many had acquired in the Methodist schools of Luanda. The exiles in Kinshasa had different experiences and aspirations. Their ethos was a capitalist one; they did not seek entry into state service or employment alongside colonial bureaucrats but aspired to replace white Portuguese immigrants in Angola's commercial sector. Their solidarity as a group was not only employment oriented but also ideologically based on a common religious culture derived from their schooling in the Baptist missions of both Angola and colonial Congo. The contrasted traditions of upwardly mobile middle-class exiles led to the creation of two rival political parties that were to compete with each other, as well as with the Portuguese, in the forthcoming struggle for independence.
The idea of black self-government caused acute anxiety to the colonial settlers in Angola and the imperial regime in Lisbon. The government's fears were increased by the sudden and unexpected loss of Portugal's tiny but historically prestigious colony in India. The settlers' fears were increased by the disorder that accompanied the transfer of power in Congo. Rumors of black noncommissioned officers refusing to obey their white superiors and of frightened white nuns being jostled on the street by exuberant black crowds caused unease in Angola. Congolese were observed to expect that independence would lead to an immediate transfer of wealth and status from white to black, and exaggerated rumors of conflict and disorder were relayed into Angola. As a result, all signs of Angolan insubordination were met with excessive repression, panicking both white and black communities. Early in 1961 the Portuguese government flew its tiny air force to Africa to bomb the villages of starving cotton farmers, while settlers in the city armed themselves with knives, cudgels, and even firearms to kill town Africans who enjoyed enough education to threaten the expatriate hold on white-collar jobs. The greatest colonial explosion of 1961 occurred in Angola's coffee belt. Coffee pickers paraded relatively peacefully up to a farmhouse to ask for arrears of unpaid wages. Their gesture was seen as a threat to white prestige and imperial authority and the demonstration was fired upon. Revolution immediately broke out as white expatriates killed black functionaries, dispossessed local farmers killed foreign estate owners, and day laborers killed contract migrants. The bloodshed exceeded anything that the British had witnessed on their coffee estates in Kenya during the Mau Mau war. The quasi-fascist empire armed settlers as instant vigilantes and then conscripted an expeditionary force in Europe to restore colonial authority. Almost one hundred thousand north Angolans fled across the border into Congo, where they remained for the last thirteen years of the colonial era.
The aftermath of the revolution of 1961 was dramatic and unexpected. Far from bringing a transfer of power from colonial officials to African politicians, it brought a new phase of active colonization with new foreign investment, new white immigration, and new education and training. The development of radical innovation in what had been thought of as a moribund Portuguese dictatorship was surprising. Portugal skillfully blackmailed President Kennedy into supplying American weapons to repress the rebellion. The extractive economic nationalism was replaced by an investment climate that created a manufacturing sector. Portuguese entrepreneurs in Angola expected to replace the Belgians in Central Africa and export textiles, plastics, household goods, cement, and beer to Congo. Portuguese army officers, modeling themselves on the French in Algeria and the Afrikaners in South Africa, found new status, prestige, and wealth. Profits made from war were invested in high-rise buildings in Lisbon. The coffee industry flourished with rewarding prices in the new European Community. Oil wells opened by American technology generated royalties that transformed Luanda into a thriving city. The old-style British colonial businesses were partially replaced by new Portuguese financiers and technocrats from industrial families supporting Salazar's New State in its great leap forward.
The economic boom that lifted Angola in the early 1960s had several important consequences. The first was the arrival of large contingents of willing rather than reluctant white migrants, who came as expatriates seeking wages in the cities, not as settlers seeking a permanent future in the colonies. They came to escape from the mournful culture of fado songs and Catholic pilgrimages to enjoy the open climate of palm trees, motor scooters, and beat music in a socially emancipated Africa. The rising young colonials were not the only ones who benefited from the economic revival. Some Africans gained skills and escaped from poverty, forced labor, compulsory planting, and rural indebtedness. The muddy, diseased, and dangerous black slums of Luanda had schools, jobs, and wages that were a magnet to rural migrants. Black migrants to the towns lived cheek by jowl with illiterate whites who had begun their colonial careers as hawkers and chambermaids before gaining status positions as stall holders and supervisors. White and black both competed for city jobs with the mixed-race children of the 1950s. On the frontiers of the white asphalt suburbs and the marshy black slums the distinguished assimilado families hovered on the social borders of white, black, and brown communities. It was in these twilight communities that modern politics began to take root.
The revolution of 1961 caught the exiled politicians of Angola by surprise. In Kinshasa exiles, seeing themselves as the true nationalists who deserved to inherit power in an independent republic of Angola, scrambled to take credit for the challenge to colonial power that the coffee rebellion represented. They gained international support from some of the political factions in Congo and also from the United States and China, both of which valued their strongly anti-Soviet ideology at a time when the Soviet Union was apparently gaining ground in Africa. Their attempt to enroll new refugees into guerrilla regiments that could roll back the Portuguese army and claim northern Angola as their own was not a great success. Northern guerrillas occasionally made life in the coffee belt dangerous and forced some planters to build security fences round their estates, but the politicians remained in exile until 1974, when a colonial ceasefire enabled them to move into Angola. Some settled in the coffee belt, where a new class of black capitalists supported by peasant cooperatives had emerged during the last colonial years. Returnees with financial and merchant skills aspired to replace fleeing white storekeepers and artisans. The great magnet, however, was the city of Luanda. Although northerners failed to capture the city politically, they did so economically and built up a flourishing informal sector whose enormous markets fed and clothed Luanda.
The rivals to the northern capitalists from Kinshasa were the bureaucratic socialists from the student clubs in colonial Lisbon. They called themselves the Popular Movement for the Liberation of Angola (MPLA), and it was they who captured the government of Luanda in 1974. Their long march back to the city from whence they had originally come was more traumatic than that of their rivals across the Congo border. When war broke out in 1961 they had had difficulty in finding a haven of exile in which to set up a government-in-waiting. In Brazzaville they trained a guerrilla force to penetrate the forest of Cabinda but did not seriously dent the security of the colony. In Zambia they were handicapped by their hosts' dependence on the Benguela railway, which they were not permitted to disrupt with guerrilla sabotage. In Tanzania their seaport headquarters was two thousand kilometers from home along very unreliable roads. Frustration led to factionalism, rivalry, and despondency, resulting in the withdrawal of support by the Soviet Union. In addition to being harried by Portuguese security forces and their black scouts, the MPLA freedom fighters were challenged by a third political movement that sprang up in southern Angola in the mid-1960s. Neither this southern movement nor the northern movement that it gradually eclipsed, nor indeed the MPLA itself, was in command of events when the Portuguese empire collapsed. The transformation of Angola in 1974 took all exiled politicians by surprise just as the revolution of 1961 had done.
By the beginning of 1974 Angola had reached military, political, and economic stalemate. Militarily there was little movement. The war of liberation had already acquired undertones of civil war as the Portuguese made secret agreements with rival guerrilla factions to encourage them to fight each other. Politically the country was run by unelected officials appointed by Lisbon, and very little consultation took place even with the white population, let alone the black one. Economically the boom years had come to an end as debts mounted and the coffee revenues did not keep pace with either consumer expectation or government expenditure. The oil industry was not yet controlled by its producers, as it was to be after the OPEC initiatives of 1973 and 1979, and Angola could not anticipate the huge mineral royalties that it was later to gain. A significant factor in the recession was the financial disappointment of the officer corps in the colonial army, which could no longer envisage long-term benefits for itself in Africa. Army officers began to consider the option of colonial withdrawal. At the same time Portuguese industry began to measure the potential benefits of joining the European Community, especially since Europe had attracted twice as many Portuguese migrants as the empire had done. The benefits of holding Angola by force involved levels of military taxation that industry resented. In April army captains mounted a coup, to which the captains of industry acquiesced. In Lisbon they established a series of provisional postfascist governments that explored the road to decolonization.
Angolan decolonization brought many unexpected twists and turns. Most settlers and expatriates initially assumed that little economic change would occur and that their administrative and technical expertise would guarantee them continued employment. When civil conflict between black political rivals broke out, whites continued to assume that "native restlessness" would not affect them any more than the colonial war in the bush had done. When shooting matches moved into the cities, however, settler optimism gave way to anxiety and, fifteen months after the colonial ceasefire, a sudden panic seized Angola's white population. Ninety percent of the Portuguese in Angola abruptly left the country, packing all the colonial wealth they could carry into wooden crates. Black servants gained some pickings from the stampede, but much property was wantonly destroyed. Three parties emerged from the three guerrilla armies. The northern party turned to Congo and borrowed its military regiments in an unsuccessful attempt to capture Luanda. The southern party turned to South Africa in an equally unsuccessful attempt to capture Luanda with a flying column of white commandos. The MPLA, which took some time to settle its internal differences, turned to Cuba for help, and Cuban troops flew to Africa to hold Luanda with weapons bought from Yugoslavia. Portugal decided to abandon Angola to the "people" rather than to any one political party. The three armies of intervention pounded each other around the perimeter of the capital as the last Portuguese governor slipped away on a darkened gunboat. Fourteen days later, on November 25, 1975, Portugal's own revolution was terminated by a second military coup d'état. The new government showed little interest in Angola's affairs, and civil wars accompanied by foreign interventions continued to break out. The colonial legacy, however, continued to grow, and Angola retained Portugal's heritage of language and literature, of gastronomy and culture.
## 7
## The Death Throes of Empire
_This first part of this chapter was written in 1980 as the world was trying to come to terms with the complex nature of Angolan politics and the unexpected violence of the aftermath of empire. At the time, and for some years afterwards, the international print and broadcasting media had a simplistic tendency to assume that if the violence was in Africa, the cause must surely be tribal. This painless form of analysis was taken up by diplomats and politicians. Even the great Nigerian civil war was readily explained in terms of tribalism, without any attempt to explore the deeper concepts of modernization, acculturation, education, conversion, commercialization, and professional and personal ambition in civilian and military life. Such issues later became familiar to Angola watchers. The last section of the chapter was written in 2004 and draws attention to the new source material and new scholarship of the last 25 years, which have shed fresh light on the death throes of the Portuguese empire_.
The normal explanation for Angola's fractured nationalist movement and subsequent civil wars is that they arise from ethnic divisions rooted in a thousand years of incompatible linguistic, cultural, and political evolution. This "tribal" explanation, much favored by the media at the time, can be called into question. Did not the politics of confrontation arise out of rival traditions of urban modernization rather than from the old rural base? Should not the solidarity of the Europeanized leadership be attributed to old-school networks? There is a need to investigate how far Angola's three civil war armies instilled regional patriotism through the use of Protestant—Congregationalist, Baptist, and Methodist—church hierarchies and church hymns. Was the march to war caused by modernizing forces even shallower in their time depth than the hundred-year-old Protestant churches? In the last colonial quarter century business interests proliferated, and it cannot be entirely coincidence that the two major African nations to undergo severe postcolonial civil wars, Nigeria and Angola, both did so at a moment when oil prospecting was at the point of reaching significant production. Alternative explanations can also be sought in international strategy, in the tensions of the cold war, in the western protection of South African investments, in the frustrations of Cuban revolutionary failure in the New World. Each level of investigation presents the historian with difficult tasks in approaching the sources. Within Angola political mobilization occurred not only in church congregations but in cultural associations and football clubs. Military success depended on arms shipments from Soviet satellites, financial backing from Texan oil companies, and combat training by experts in South Africa, China, and Algeria. Mercenaries came from as far away as Bradford in England. But always the trail leads to the scrambled files that the CIA tried to protect from America's Freedom of Information Act. The United States, in 1975, was seeking enemies in Africa.
On Monday, April 7, 1975, the night sky in the city of Luanda was illuminated by gunfire. Earlier that day there had been rumblings in one of the black shantytowns. Jeeploads of four-party patrols had swiftly quelled the confrontation, but not before some bloodshed had occurred. The night shooting appeared to involve heavy weapons, and the target—intended or accidental—was a South African passenger jet flying low over the northern suburbs. Only minor damage was caused to the plane, and after refueling it returned to Johannesburg. A Portuguese aircraft, allegedly carrying the country's future president, later landed unhindered.
The bullets in the fuselage of a South African plane represented a turning point in the history of southern decolonization in Africa. But were they the last shots of an old war or the first shots of a new one? Was this the culminating achievement of armed black nationalism, which had gained power in Katanga and independence in Mozambique and was now about to decolonize Angola? Or did those shots relate to the militant return of white power and a failed attempt to assassinate the future leader? The questions raised by the events of April 7 all require historical explanation, but there is much disagreement over the historical depth that it is necessary to plumb. To the average journalist the cause was "tribal": ethnic antipathy, rooted in deep tradition, impeded the growth of a national purpose and set people against people with "inevitable savagery." Nothing else was expected of "barbarous" Africa and therefore no effort was required to explain the incidents of the day as they edged Angola closer to civil war. The country's uncomfortable complexity and rapid social change were ignored. Instead the country was rationalized into three static "tribes" for the benefit of world newspaper readers.
The concept of tribalism in Africa cannot, however, easily be justified as the long-term growth of a common cultural consciousness. Small village societies were commonly more important in popular awareness than later ethnic ones. Precolonial cohesion was not a marked trait even among peoples speaking a common language. The political unity of northern Angola, which had been so striking in the sixteenth century, had given way to intense warlord rivalries in the eighteenth century. The interaction of theology and commerce, of politics and family, that had once strengthened the institutions of the old Kongo kingdom had turned to undermining and destroying them. By the nineteenth century no observer saw northern Angola as being peopled by a single Kongo tribe. The historian must therefore seek alternative roots for the common political purpose motivating the northern armed movement that took part in the events of 1975. The presumption of an ancient Kongo ethnic solidarity cannot hold.
The argument in favor of historic unity in the two other broad language zones of Angola is even less compelling. The great highland populations of the south never achieved any political cohesion in precolonial times. A dozen trading kingdoms competed sharply for control of markets, caravan routes, ferry crossings, iron mines, clay pits, farmland, and the Atlantic trading harbors. Cooperation was based on fierce commercial bargaining, armed military deterrence, and royal marriage alliances, not on a common Ovimbundu national purpose. Such a purpose had emerged only in the face of external threat and the intensification of colonial penetration in the terminal phase of empire.
The third media-inspired "tribal nation" was said to be descended from the late-medieval kingdom of Ndongo, whose king, the Ngola, gave Angola its name. This kingdom had been overwhelmed three centuries ago by European conquest, and any "traditional" ethnicity in Angola's middle zone had been overlaid by social groups that had responded to new colonial opportunity to create Luso-African Creole communities. The hybrid culture of Portugal and Africa influenced diplomacy and international relations far into the interior of Africa. A literate class of soldiers, clerks, and commercial agents, known as the Ambaquistas, became the representatives of Portuguese enterprise and the precursors of empire. The proto-imperial social culture did not foster a Kim-bundu ethnic solidarity; to the contrary, it divided society into collaborators and resisters, laborers and labor recruiters. Colonial conquest scattered the Kimbundu into pockets of isolated refugees in forest fastnesses where fighters held out against the unifying force of colonial conquest.
If ethnic loyalties and tribal animosities cannot be seen as the basis of war, a different alternative should be found to explain the rift in Angolan society that led to the shootings of April 1975. This alternative may be the division of the colonial presence in Angola. In most colonies in Africa the modern sector of society, and its attendant educational and employment opportunities, were based in a single colonial capital. Angola was different. Throughout the twentieth century it had three focal points. One was indeed the capital, the Portuguese city of Luanda. Another, across the northern border, was the Belgian city of Kinshasa, known for some decades as Léopoldville. The third was the southern port of Lobito. This division, it might be contended, was far more fundamental socially and economically than the old vernacular frontier that separated the hinterland of the three cities. Emerging twentieth-century politics revolved around leaders whose identities were formed in the modern sector of society. Traditional leaders, with a few exceptions, had limited importance in this polarized late-colonial society. In Luanda government bureaucracy was the traditional black first step on the ladder of economic opportunity. This opportunity was always fragile and could be threatened by the spasmodic immigration of poor whites from rural Portugal, who gained preference over black migrants from rural Africa. Skill, literacy, and experience counted for less than color, and competition over urban employment fostered a black political leadership in Luanda in the 1950s. Competition led to conflict and an outburst of city warfare in February 1961. Those black town dwellers who escaped into exile formed a new generation of leaders and created a quasi-Marxist political party, the MPLA.
The second city to which Angolans looked for opportunity was Kinshasa. It was a city of refuge and of exile as well as of education and employment. Those who went there learned French as their passport to status and achievement. It might be argued that literacy and a difference of modern language among city dwellers were more significant in politics and in war than a difference of vernacular and popular speech. The Kinshasa Angolans found work in small African and expatriate business firms more readily than in the Belgian bureaucracy. Their early history, however, was overlaid by the huge influx of 100,000 refugees that swept into Congo in 1961. The new escapees tried to get to the city to find employment, or in a few cases to join the incipient armed groups that exiled political leaders tried to mobilize. These Kinshasa leaders remained quite distinct from those whose cultural roots were in the rival city of Luanda. Their party became the FNLA.
The third city to which rural Angolans drifted, or were compulsorily drafted by forced labor practices, was the southern port of Lobito. An even newer town than Belgian Kinshasa, it began to replace the old roadstead of Benguela in the 1930s after the great Copper Belt railway was finally completed. Any explanation of the divisions of Angolan society needs to examine the rise of the dock-workers of Lobito. The old port of Benguela had been founded in 1615, and its _mestiço_ bourgeoisie identified fairly easily with the politicians of Luanda. The brash new city of Lobito had no such cultural roots, and its population of highland immigrants proved a fertile recruiting ground for a third political party, UNITA, whose leaders lived along the line of rail. To attribute the distinctiveness of UNITA to ethnicity before seriously attempting any urban social history would seem premature.
Angola became a nominally Catholic country during the course of the Portuguese conquest, which gained momentum from 1875. The Catholic Church was not, however, a mission church with a mandate to proselytize, heal, and educate. It was a branch of the metropolitan church and therefore primarily concerned with the white expatriate and settler community and with the assimilated elite of black and mixed-race peoples who associated with them in commercial and administrative centers. Only in the far south did a Catholic mission church of French and Alsatian parentage make a predominant impact. In the rest of the country the colonial regime left a partial educational, spiritual, and medical vacuum that Protestants sought to fill. The three Protestant churches that took root in the rural areas of Angola before 1961 provided features that affected the distinctiveness of the three political parties that grew up after 1961.
The Baptist Church had chosen the north as its field of activity in the 1870s. From the outset its activity, although missionaries did their utmost to avoid conflict with the civil authority and to instill concepts of law and order among their converts, was seen as a threat by Portugal. Despite their caution Baptists were seen by Africans as a safeguard against commercial or government exploitation. In 1913 the northern faith in Baptist benevolence was so strong that when rebellion broke out in villages from which men were being seized and taken to the logging forests of Cabinda, the colonial administration felt sure that the Baptists had fomented revolution. The British mission minister, the Reverend Bowskill, was arrested and accused of seditiously undermining the loyalty of Portugal's colonial subjects. For the next half century, until the revolution of 1961, relations between mission stations, village deacons, and the colonial state remained tense. During the thirteen-year colonial war after 1961, although officially Portugal had declared all northerners to be Catholic, African loyalty to the Baptist Church surreptitiously grew. A Baptist underground movement became the symbol of resistance. Toward the end of the war the dragooning policies that repressed Protestants gave way to a political campaign for hearts and minds, with a tacit recognition that negotiations would succeed only if conducted through recognized Baptist elders. The colonial army, though not the settlers, even recognized that economic liberalization would reduce the revolutionary nationalist temperature, and so soldiers facilitated the creation of farmers' cooperatives approved by the church.
In central Angola the Methodist Church, complete with its American-style bishops, served a similar role to that of the northern Baptists. It created an integrated leadership of high school graduates, lay preachers, and medical orderlies that was not under official supervision even in the capital. The social columns of the mission newspapers provide an unrivaled insight into the kinship web of modern colonial society in the twentieth century. By the late 1950s this structure became linked to initial attempts at mobilizing embryonic political movements. One question that arises is why it was Methodist families who were most prone to adopt Marxist ideals when contemplating a liberation struggle. Methodists, unlike Baptists, were predominantly urban and were often employed in bureaucratic positions by the state. A planned economic order offered them better prospects than a free-market one. They could idealize peasants, but their own economic interests related to salaried service rather than to commercial production. Baptists in the rich coffee zones, by contrast, held tightly to capitalist concepts of land ownership and the hiring of wage labor. Out of Methodist aspirations a black preacher's son rose to become a Marxist president.
The Congregationalists, whose churches emerged among the Ovimbundu people, were even more successful than Methodists or Baptists in facilitating the mobilization of a mass party, though the process started much later on their high plateau. Television footage from 1975 shows huge rallies singing political hymns to Presbyterian tunes. The suddenness with which UNITA mobilized highland peoples into a new common loyalty was surprising. Their leaders were the first to achieve a ceasefire agreement with the Portuguese, and cooperation began even before the Lisbon revolution of 1974. In Jonas Savimbi UNITA had the most charismatic of the political leaders. The highlands had not suffered the long history of military failure and recriminatory schisms that had so bitterly affected the rival parties farther north. Southerners, threatened by land hunger, land alienation, and labor conscription, developed a tradition of hostility both to city government in Luanda and to city entrepreneurs from Kinshasa. For a short while the UNITA leadership even studied Mao's theories of rural politicization.
Studying the social impact of semiclandestine churches in a warring colony involves more than understanding the ambitions of farmers and office workers. The role of business activity in Angola's politics became increasingly significant during the colonial war. In 1963 Portugal liberalized its financial policy of selfreliance and allowed new foreign investment to generate new forms of wealth with which to wage the war of colonial reconquest. When the Lisbon revolution of 1974 overthrew the "fascist" rulers of empire, Portugal abruptly turned away from Africa and toward Europe. International business interests in Angola had to reassess their position. In 1975 the world suddenly became convinced that Angola possessed hidden riches. In the style of the original scramble for Africa, foreigners began to safeguard their resources for fear of rival claimants. They gave political levies, mineral royalties, and voluntary contributions to political movements and guerrilla armies. The FNLA, for instance, was given funds to buy Luanda's leading newspaper, published in the heart of its opponents' MPLA territory. The financial links between Angola's exiled politicians and the personal copper fortune of President Mobutu of Congo may never be unraveled. Meanwhile, the biggest of Angola's oil companies, Cabinda Gulf, decided to gamble both ways on the outcome of the decolonization conflict and to match U.S. funding for the FNLA with comparable royalty funding for the MPLA. African gunrunners such as those that Frederick Forsythe portrayed in _The Dogs of War_ may already have been active in Angola, as they had previously been in Nigeria, and may have supplied the weapons used in the aircraft incident of April 1975. But whoever did supply those weapons, it seems likely that the partisans shot the wrong plane.
Thirty years after the shooting episode of 1975, one of the legacies of the past that most blights the lives of twenty-first-century Angolans is the range of discrepancies in wealth and status. These discrepancies make harmonious nation building particularly arduous. Women, for instance, who bore so much of the burden of African economic survival throughout the second half of the twentieth century, are largely excluded from real power in modern society in much the same way that they were marginalized in both colonial and liberation politics. In the same way village communities are excluded from the respect and status offered to affluent city gentlemen, whether in suits or in uniforms, much as they were excluded in the 1960s. The power of education is also used, and abused, in a country where few ordinary children are given even the rudiments of schooling while the children of the ruling families obtain international diplomas in prestigious foreign boarding schools. In Angola the life of a worker is seen as cheap in the eyes of a manager; while one drinks river water bought from a truck, the other drinks Evian water, which costs nearly as much as whiskey. All these discrepancies cry out for explanation, and a good social history of Angola's people during their long years of struggle is needed by all—by politicians, by social and medical workers, by schoolmistresses, by business managers, by university students, by the international white Land Rover brigades. The Brazilian scholar Marcelo Bittencourt has provided some basic tools with which to make such studies and modestly hopes that his work on contemporary history may stimulate others to publish new memoirs, letters, poems, plays that will stimulate the imagination of Angola's people and shed light on the halfremembered past, the death throes of empire, with which all of them will have to come to terms.
Many of the attempts to understand Angola's history in the second half of the twentieth century have been undertaken by foreign scholars, who, while deeply sympathetic to Angola's protracted traumas, come from another world and often write in another language. The initial work depended very much on political analysis, on diplomatic documents, on interviews with leaders of the many factions and fractions that made up the nationalist kaleidoscope. Gradually, however, the voice of the voiceless began to be heard through such dramatic initiatives as the Jaime and Barber interviews. Hard on their heels comes the Bittencourt volume, which digests and integrates the works of the international scholars, highlights and expands the range of the interview material, and, most strikingly of all, makes the first extended use of the archives of the Portuguese political police, the secret and much-feared PIDE. The police records enable serious scholars to get a better grasp than before on the factionalism that marked Angola's political history in that era.
The study of factionalism in the Angolan liberation struggles follows cycles of fashion, at one moment emphasizing ideology; at another, ethnicity; at another, foreign sponsorship; at another, educational culture. Always the skills, ambitions, and weaknesses of individual militants, politicians, diplomats, and entrepreneurs play a role. So too do the minor irritants or comforts of life on the move, in exile, in rudimentary camps, in rented mud huts. Who has food, cigarettes, the right to steal a kiss from a pretty girl, a smart new uniform, or an old-boy friendship with a party treasurer? And always race is part of the colonial mind-set, so prevalent that foot soldiers making little progress on the ground wonder if Agostinho Neto or Daniel Chipenda, both of whom have white spouses, might not have made a secret deal with the "tribe" of his in-laws. Being even half white in Angola may lead to accusations of petty bourgeois snobbery and neglect of the welfare of the two or three thousand men in the MPLA's various miniregiments of guerrillas.
Bittencourt's convincing understanding of the interaction between daily life and liberation politics is based on his admirable range of new source materials, including his own interviews conducted on visits to Luanda between 1995 and 2000. The timing of the interviews is felicitous since the founding fathers—Angola has very few founding mothers—now are elderly or have died. Those who survived were often able to respond to long and detailed interviews that sought to clarify, from various vantage points, the new questions that archival papers have raised. But writing the history of a guerrilla war presents several severe challenges. The first of these is the frequent absence of documents written during the decision-making meetings held on the ground. Political movements that formed, splintered, and re-formed in cities of exile or rudimentary forest camps did not have the resources to keep many archival papers. The very creation of the MPLA is subject to rival memories and interpretations. As Christine Messiant has said with humorous perspicacity, "Amongst ourselves even the past is unpredictable." But some documentary fossils do turn up in unexpected places, such as American university libraries, where some dispassionate scholars have been able to make good use of them. Bittencourt, however, has been able to make excellent use of an unexpected set of archival treasures: the propaganda papers, and even some of the internal policy documents, that Portugal's secret police captured from nationalist sources. Ironic though it may seem, the defeated colonizing enemy preserved a better set of records than the victorious liberation movements. The documents frozen in PIDE files have not undergone the process of adaptation that oral records undergo as each survivor tries to interpret his or her personal trajectory in the light of rapidly changing circumstances.
The records of the political police, which began its activities in Angola in 1957, contain a second type of information that is much more difficult to interpret than captured papers: the transcripts of interviews conducted with Angolan people who had been seized by military and police action. In assessing these highly tendentious data Bittencourt has the advantage of a Brazilian background with a historic memory of the effects of government-orchestrated terrorism on the people of Brazil from the mid-1960s to the late 1970s. As Umberto Eco graphically tells us in _The Name of the Rose,_ the use of torture to obtain information can have dramatically distorting effects on the data extracted from the screaming victim. Further distortions occur when prisoners of war reinvent their own positions and roles in order to gain advantage out of unintentional misfortune or treasonable aspiration. It is to Bittencourt's credit that his book is fully sensitive to the care that must be taken in using police reports. In Angola torture was used not only in dark cells to extract information, but also in open displays of army terrorism as village leaders were publicly decapitated to dissuade colonial subjects from even thinking about independence.
Some of Bittencourt's best insights weigh the factors that so deeply marked Angola's political activists in the months of transition before and after the fall of Portugal's dictatorship on April 25, 1974. In these years lassitude and disenchantment overcame optimism and ambition after ten years of ineffectual struggle. Blame and counterblame, leadership bids and counterbids, solidarity and fragmentation, class rivalry and ethnic fraternity all affected both the FNLA and the MPLA and even the then-minuscule UNITA, with its 300 isolated militants covertly attempting to make military alliances with Portuguese troops in the deep highland interior. The rumblings, accusations, arrests, and executions that accompanied the slow evolution and later rapid dissipation of the eastern revolt are lucidly laid out. The debates, subversions, and plottings of the intellectuals of Congo-Brazzaville, who became the Active Revolt in May 1974, are also explained in convincing terms. Beneath both of these tendencies, and the abortive leadership bids of Daniel Chipenda and Joachim de Pinto Andrade, lies an interesting discussion of the readjustment movement, which came from China and affected both northern and eastern wings of the MPLA. Throughout Angola's independence struggle China was a distant, little-known force with unpredictable effects on the nationalist scene. One of Angola's founding fathers, Viriato da Cruz, retired from active politics to settle in China, and both the MPLA and FNLA, at varying times and to varying degrees, were dependent on the assistance that China gave both to the socialist republic of Tanzania, where Neto had his headquarters, and to the capitalist dictatorship of Zaire, which sheltered Roberto's home base. In the early 1970s, however, it was the Chinese fashion for self-criticism that took hold in the Angolan camps of both Congo-Brazzaville and Zambia. Open seminars attended by large numbers of middle cadres and common foot soldiers discussed the predicaments of exile, the wisdom and failings of the leadership, and the injustices and inequalities that separated the "civilized" guerrillas (from Angola's towns) from the "indigenous" guerrillas (from the countryside). None of the leaders was comfortable with Chinese-style open debate or grassroots criticism, and in interviews in the 1990s both Lúcio Lara, from the Congo front, and Daniel Chipenda, from the Zambia front, remembered the unease they had felt. Agostinho Neto was even more bitter in his condemnation of any dissidence, but as absolute ruler of his movement he controlled the money and was able to move back and forth using stick and carrot to punish disloyalty and reward the faithful—a style of political management still much in evidence in present-day Angola.
The importance of Bittencourt's book is in the light it sheds on experiences that Angola's people have to understand and absorb as they seek their way toward modernity and social justice. Both the colonial and the anticolonial experiences were sometimes cruel and autocratic, riddled with prejudice and inequality. Current struggles to create a climate of open debate in Angola, with a vibrant civil society and a choice of political directions, are hampered by the weight of a past that needs to be sympathetically understood. It is encouraging that a brave Brazilian has given excellent signposts toward that sympathy and understanding.
## 8
## Destabilizing the Neighborhood
_The 1980s were a peculiarly distressing time for the peoples of Central Africa, including Angola. Wars of destabilization, partly orchestrated by the superpowers, broke out between South Africa and its neighbors. This chapter is a modification of the penultimate chapter of_ Frontline Nationalism in Angola and Mozambique, _a small book written at the request of Unesco and published in 1992_.
Nation building would have presented quite enough difficulties in Angola and its sister colony of Mozambique if the new generation of black leaders had been left to attend to their task unhindered by outside influence. Instead, outside powers became increasingly involved in forcing their choices and undermining their actions. The oldest foreign influence in the region (apart from colonial Portugal itself) was the Republic of South Africa, which had long cast its shadow over its northern neighbors. When the two new lusophone nations gained independence, they almost immediately accentuated their ideological and economic differences with South Africa by assisting the struggles for independence in Zimbabwe and Namibia. Spasmodic South African interference in the affairs of its neighbors gradually became a systematic policy of destabilization. Intervention increased after the fall of John Vorster when the army became the key political actor inside South Africa. Foreign hostility to Angola and Mozambique brought together an unholy alliance of enemies from the Congo River to the Cape of Good Hope. The forces of destabilization were recruited among black exiles as well as white settlers, among commandos from the former colonial armies, and among disappointed politicians from the nationalist movements. South Africa's policy brought damaging foreign intervention from outside Africa. Superpower involvement in Mozambique was predominantly covert, but in Angola a war by proxy between the United States and the Soviet Union replaced Vietnam as a focal point of cold war confrontation. The victims were now African rather than Asian, and they were deliberately hindered from building proud, independent nations with the freedom to choose their own development strategies.
During the old colonial war South African military intervention in the affairs of its neighbors had been limited. Some military equipment of American origin was sold or leased to the Portuguese at nominal prices. Counterinsurgency experts made strategic recommendations on guerrilla warfare, though Portugal was too proud to accept South African advice readily. South African assassination experts might have had a hand in making the parcel bomb that killed Eduardo Mondlane, the first president of the Front for the Liberation of Mozambique (Frelimo), though internal conflicts within the movement may have created the opportunity. South Africa was not above the use of murder, as demonstrated by the 1982 killing of Ruth First in her university office in Mozambique. It is symptomatic of the climate that when President Machel of Mozambique was killed in a plane crash in 1986, observers were predisposed to suspect foul play and blame the South African policy of intervention in Mozambique.
The military origins of intervention concerned the war of liberation in Rhodesia, which had escalated in 1972. When Mozambique won its independence two years later, it decided to support the freedom fighters in Rhodesia and help impose economic sanctions on the white regime of Ian Smith. When Mozambique, one of the poorest nations in the world, followed UN requests to suspend communications and transport services to the Rhodesians, it immediately became a victim of Smith's economic reprisals. The political cost of imposing sanctions on Rhodesia was even higher than the economic loss of earnings. Mozambique provided a haven for refugees escaping from the Rhodesian war and offered training grounds for soldiers going into the war. This black solidarity with black Zimbabweans brought commando reprisals by white Rhodesians and air raids by white South Africans. By 1979, however, South Africa, fearing that any prolongation of the Rhodesian war might spill over its own border, brought the antagonists to the negotiating table, and a cease-fire opened the way for a black government to rule Zimbabwe. The government that Zimbabweans elected was not quite as pliant as South Africa had expected, and Pretoria began plotting further military interventions in the frontline states.
In 1978 a sea change had taken place in the politics of South Africa. The long-serving prime minister, John Vorster, had based his effective strategies of black repression on the work of the police force and the intelligence services. A cowed home front had enabled him, despite the fact that the doctrine of apartheid prevailed at home, to reach out and build bridges with tropical Africa in countries where South African goods could be sold. This strategy came under threat when Mozambique and Angola moved toward more radical regional policies than those of South Africa's pragmatic partners in Malawi or Zambia or Congo. Radical liberation movements beyond the border led to the growth of a black consciousness movement that revived the hopes of South Africans. Demonstrators began to boycott segregated buses, and a trade union movement even attempted strikes despite the threat of reprisals. Most dramatically, in 1976 the children of Soweto rose in rebellion after years of quiescence by their parents and challenged apartheid with unbowed courage. The Mozambican slogan, _a luta_ _continua_ (the struggle continues), was proclaimed with clenched-fist salutes by black South Africans.
The white South African response to the black revival was to dump Vorster as prime minister and appoint an army man in his place. P. W. Botha gradually brought the army to the center of South African life. Soldiers were increasingly used to maintain so-called law and order and soldiers were invited to sit on sensitive political committees. Above all—from the point of view of Africa's nationalists—soldiers now intervened forcefully in foreign policy. The policies of détente and dialogue were systematically replaced by policies of confrontation and destabilization. Botha wished to eliminate the external wing of the African National Congress while at the same time suggesting to the world that nationalists such as the black rulers of Mozambique were incapable of governing themselves responsibly. His regime's antinationalist propaganda made effective use of the Soviet presence in Angola to portray the South African army as the defender of "civilized values" against the spreading threat of communism. A forward foreign policy won the army an increased share of the budget, provided its officers with more rapid promotion, permitted modernization of weapons systems, and gave soldiers power at home. An old guard of white industrialists protested quietly at the rapid rise in taxation that the militarization of South African politics entailed and regretted the loss of export markets in independent Africa. Such protests were not strong enough to override the army's preferred policies, not even when civilian politicians were ready to negotiate compromises with black nationalists in Mozambique or Angola.
The South African army's decision to destabilize Mozambique proved to be distressingly easy. First, they sought out disaffected Mozambican leaders whose ambitions had not been satisfied in the nationalist framework. Secondly, they recruited and supplied with guns white Rhodesian commandos who had fled south at the independence of Zimbabwe. These Rhodesians were experienced in fostering terror not only in Zimbabwe but also during cross-border operations inside Mozambique. A third line of support came from Portuguese settlers who had fled to South Africa or Portugal and sought to re-create the colonial El Dorado they had lost. Mozambique's antinationalist coalition sought sympathizers among black exiles in America and mounted an effective public relations campaign in Washington to persuade both the U.S. government and black American opinion that it was a legitimate political force rather than a front for South African military operations. Finally, Frelimo's opponents, the Mozambique National Resistance (Renamo), approached the new generation of British neocolonial tycoons that was being groomed to move into Africa in the Thatcher era. The Lonrho Corporation, whose head had been described by a previous Conservative prime minister as representing "the unacceptable face of capitalism," now became the accepted vanguard of the new British capitalism in Africa. Businessmen waited for the time to be ripe for Mozambique to abandon its idealistic radical notions and turn to the West for help in rebuilding an old colonial economy that had provided cheap sugar, cotton, and rice for industrialists in the world's rich north. Fighting a war on two fronts, against poverty at home and enemies abroad, proved too much for Mozambique, which had neither the economic resources nor the managerial talent to take on another long war.
In 1984 President Machel asked for peace and met President Botha on the banks of the Inkomati River. The accords they signed should have reassured South Africa that Mozambique would not be used as a base for attacks on South Africa by black exiles, and should have reassured Mozambique that South Africa would stop arming and transporting the Renamo guerrillas, who had by then made virtually the whole country ungovernable. The accords suited many of the parties to destabilization. They suited the South African government, which was bearing the cost of the attack. They suited international business, which was convinced that Mozambique was now ready to cooperate along the archtraditional economic lines laid down by the International Monetary Fund. They suited the European Community, which admitted Mozambique to the Lomé club of its tame third-world suppliers. They suited Britain, which was anxious to reopen the railway to Zimbabwe and accelerate the development of economic partnership with its former settler colony. But they did not suit the South African army, for which peace would involve retrenchment and redundancies, and they did not suit the Mozambique guerrilla leaders, who sought to replace the nationalists in government. The war therefore continued.
The second stage of the Mozambique war of destabilization was significantly different. It is true that for most of the rural people the insecurity was as terrifying as ever. Village women going to the stream for water were still liable to be captured by terrorists and taken away to serve as porters and provide sexual services for the soldiers. Young men were still press-ganged into the opposition's regiments of irregulars. Barnyards were still raided nightly for food. Refugees still streamed across the borders to Malawi and Zimbabwe, where they received a bleak welcome from host populations themselves suffering from drought, food shortages, and underemployment. But at the strategic level the terms of the war had changed. Britain no longer supported South Africa wholeheartedly in its Mozambique policy; now it offered the nationalists help in training a new officer corps for the Mozambique army in exchange for economic access to the country. Even more dramatically, Zimbabwe sent 10,000 troops into Mozambique to protect the railway from South African attack. International business even began to estimate the cost of rebuilding the hundreds of electricity pylons that had been destroyed by commandos to prevent the use of the great Cabora Bassa power supply for regional development and integration. By the end of the 1980s the new climate seemed to be in the ascendant. Peace seemed possible by the time South African politicians drove President Botha out of office in September 1989. Meanwhile the international focus of destabilization had shifted to the long-running war in Angola.
Mozambique had won its independence as a result of military victory and embarked on nation building with some degree of euphoria before the realities of poverty and the enmity of its industrial neighbors soured the celebrations. In Angola independence was gained by compromise after the confrontation of three intervening armies that tried to replace the colonial forces. The population of the capital city may have felt a spasm of euphoria when the guns were held at bay, though not beyond earshot, throughout independence day on November 11, 1975. The rest of the country had little to celebrate under the weight of Zairean and South African occupation. Gradually, however, in the months after independence the forces from the capital regained authority over the provinces. It was therefore with some surprise that the victorious nationalist party, the MPLA, discovered that the first challenge to its rule came from its very own base in Luanda city.
After the war of intervention, frustration rose rapidly among the urban poor to reach a revolutionary peak in 1977, two years after the transfer of power. Political cells had grown up in the mushrooming shantytowns of Luanda. Former employees of colonists who had fled now scraped along below the bread line. The withdrawal of industrial capital had led to the closure of manufacturing plants and the creation of serious unemployment. Insecurity in the countryside and the vision of a city paved with gold led to an uncontrolled influx from the rural areas. MPLA party cells began mobilizing the underprivileged quarters of the city and establishing discussion groups that drank in the revolutionary dogmas of Mao and Enver Hoxha. Radical ideologues emerged from both the guerrilla movement and its expatriate camp followers. The mobilization of the underprivileged took place under the guise of organizing football clubs and other innocuous activities, but the head of the black radicals, Nito Alves, was a member of the central committee of the MPLA. On May 27, 1977, he attempted to mount a coup d'état but was foiled, captured, and shot along with his closest associates. The episode cast a clear light, however, on the difficulties that the nationalists faced in creating an independent stable nation in Angola.
The attempted coup of May 1977 came as a surprise to many observers, though the evidence subsequently published shows that it should not have. Confrontation had been building up within the central committee of the MPLA since the previous October, yet it was not in the capital that the MPLA expected to experience difficulties. In this respect the politics of Angola after the Portuguese cease-fire were markedly different from those of Mozambique. In Mozambique Frelimo was a rural-based movement that faced its greatest initial problems in accommodating to the entrenched economic systems of the industrial cities, where expectations were at variance with those of long-standing party recruits from the north. It was therefore not surprising that Mozambique experienced patches of turbulence in the capital during the transition of power. In Angola, by contrast, the MPLA did have city experience in Luanda, Benguela, and Malange, and its difficulties, highlighted by the war of intervention, were in accommodating to the rural populations of the north and the plateau. As the search for accommodation went on, the attempted coup illustrated how deeply the stresses of the war had bitten into the MPLA's urban constituency, which it had previously taken for granted.
If the first challenge to the power of the MPLA nationalists came from their own backyard in Luanda, the second came from a much broader front in the provinces. The heroes of the revolution had been the workers and the peasants. The workers in the capital had discovered that their expectations had been pitched too high and independence was not all golden. The peasants learned the same lesson even more bitterly. They had been painfully incorporated into the market economy by the colonial power in the 1960s. Now in the 1970s and 1980s they were even more painfully torn away from market opportunities. By the 1980s peasants in almost all areas of the country were disenchanted with their government and open to the suggestion that alternative political movements might offer them a better range of opportunities. The short-lived rebellion of the town was followed by the long-lasting rebellion of the countryside.
Three factors made the rural rebellion in postindependence Angola particularly tenacious. First, it suited the South African government to follow up the civil war, within a very few years, with a war of destabilization comparable to the one that it was conducting in Mozambique. Secondly, it suited the United States not to recognize the MPLA government, which had eluded its control and, still worse, had sought help from the most sensitive of America's opponents, the Cubans. Thirdly, the provincial rebellion had a political focus that began with highland political mobilization by UNITA and continued with the militarization of UNITA in the far lands of southeastern Angola, where troops could be protected by South African air cover. The result was a new war with fierce international involvement.
The Angolan war gained in intensity during the 1980s and in the process tore the Ovimbundu highlands apart. By 1990 it was estimated by Africa Watch, the human rights organization, that thousands of people had been captured by UNITA guerrillas from the highlands and taken to the south to create an internal colony. The Ovimbundu, who represented some 40 percent of the total Angolan population, found themselves attacked by both the forces of UNITA and the people's army of the MPLA government.
The objective of UNITA was to impoverish zones under government control by indiscriminate robbing, killing, and burning. Survivors who failed to flee were then marched to the "liberated zones" beyond the reach of government forces. Women were taken as well as men, and children were especially prized as future guerrilla recruits. To hinder normal economic development further, plow oxen were stolen and seed grain eaten. Worse still, the farm paths were strewn with hidden land mines so that peasants attempting to return to their fields risked being maimed or killed. The UNITA policy of bringing the government to its knees by starving the peasants caused the death of half a million children in the 1980s. Malnutrition in the towns to which the people fled was accentuated by ambushes on the convoys that tried to supply them by road. Even medical convoys were ambushed without mercy. Nearly two million vulnerable people were further weakened by drought at the end of the decade.
The atrocities of war were not confined to antigovernment action. On the contrary, the MPLA strategy of confining UNITA to the southeast and preventing it from returning to the highlands as an effective political force involved equal hostility towards the highland villages. To prevent the insurgents from developing a social base, the government determined, by removing the people who might sustain enemy operations, to "drain the sea" in which guerrillas might swim like fish. The MPLA government, assuming all free peasants to be potential UNITA sympathizers, adopted the same arbitrary and inhumane strategy as its colonial forebears and herded people into barbed-wire villages under armed guard to deny them any possibility of contact with opposition forces. The moving of hundreds of thousands of farmers from their familiar terrain intensified the shortfall in food production in the 1980s. Even the soldiers often went hungry and had to loot food convoys. Young men who tried to evade conscription were subject to violent reprisals. Village leaders suspected of disloyalty to their harsh government were detained by officials of the Ministry of State Security.
Many highland peoples tacitly surmised that a change of government would bring an end to the hardships, and UNITA remained surprisingly popular. But as the situation deteriorated, the war spread to other provinces and the leadership of UNITA became increasingly authoritarian. Jonas Savimbi had once been a charismatic leader with grassroots support and foreign admirers. Some of his lieutenants were men from the business world with a foreign education and international connections. As the war dragged on, however, interpersonal violence increased, and a once-sympathetic biographer of Savimbi began to recognize the scale of the torture and killing that went on inside the guerrilla organization. In the "land at the end of the earth," the politics of semi-exiles in the 1980s were as brutal as had been the international strife of the 1970s, when it was the MPLA that tore itself apart on the remote margins of Angola.
Although the MPLA gained relatively firm command of the city in the 1980s, it was eventually forced by the ongoing war in the provinces to change its ideology and government practice. With gentle encouragement from Cuba, attempts were made to seek a policy of harmonization and economic reform. The scale of corruption and inefficiency was such that even the huge oil revenues could not compensate for the decline into austerity. Before any internal political reforms could reach the top of the national agenda, however, the foreign war with South Africa had to be resolved.
The war of destabilization in Angola was no less devastating than the one in Mozambique, though it was very different in kind and in effect. A nationalist government in Angola was less of a military threat to South Africa than one in Mozambique, whose capital lay only a few miles from the Transvaal border. Ideologically, however, radical nationalism in Angola was potentially inflaming to black opinion in South Africa and thus, according to white thinking, needed to be curbed. A campaign of destabilization was therefore undertaken in Angola. Although the war of the 1980s was in some respects similar to that of the 1970s, the balance between the "civil" and the "intervention" elements had altered. In the 1970s the fractions of the nationalist movement were still powerful enough to compete with each other for central authority. In the early 1980s the domestic opposition forces had largely withered, confined to the UNITA enclave and its highland sympathizers, but the foreign presence had grown stronger. Regional opposition strategy, orchestrated by South Africa, and intercontinental opposition strategy, orchestrated by the United States, dominated the Angolan war of the 1980s.
The first international change to affect Angola was the defeat of President Carter in the U.S. election of 1980 and his replacement by Ronald Reagan. Carter had restricted the level of U.S. involvement in local superpower confrontations and reduced the support that the United States afforded to South African strategic planners. Reagan came to power determined to stem the third-world search for national autonomy. His regime armed and financed client groups in what were euphemistically called "low-intensity conflicts." Although of low intensity by the standards of the Pentagon, they were devastating in their toll on human life and their disruption of social and economic development.
In Angola the chosen vehicle for the low-intensity campaign to arrest nationalist aspirations was the old liberation movement, which had failed to gain power in the civil war. UNITA, the former client of China, was rebuilt, rearmed, and given a protected military base as the new client of America. This base was far outside the movement's old political constituency, but it benefited from a powerful South African air shield administered in the nearby Caprivi territory of Namibia. While South Africa protected the rebel movement, the United States orchestrated a propaganda campaign to challenge the legitimacy of the government. Collaborating with South Africa caused some embarrassment to the U.S. administration, but the alliance nevertheless survived throughout the 1980s, partly because the West wanted to protect Namibian uranium supplies for its nuclear energy industries and was anxious to deter Angola from providing a haven to Namibian freedom fighters. When oil prices dropped and nuclear energy began to appear ecologically risky, the need to hold Namibia weakened, but by then South Africa was committed to destabilizing Angola for geopolitical reasons.
The South African invasions of Angola in the 1980s had a rather different effect on world opinion from those of the 1970s, when moral revulsion at seeing the armies of apartheid marching into independent Africa brought some sympathy to the struggling government of Angola. In the 1980s South Africa was able to manage its public relations better. The Angolan government was now supported by a Cuban expeditionary force of 30,000 or more troops, which South Africa represented to the anxious West as "Moscow's Gurkhas." South Africa's attempts to destabilize a potential Soviet satellite in Africa attracted political support from Europe and financial support from the Middle East. By 1988, however, South Africa began to feel that the military and political cost of trying to overthrow the Angolan government outweighed the prospective gain. Sophisticated Soviet technological resistance supported an ever more experienced Angolan army until a cease-fire could be negotiated with the encouragement of a new U.S. president, George Bush Senior. Although the war had scarcely interrupted the flow of Angolan oil to Texas, the United States was ready to explore new economic opportunities in Africa and no longer deemed it necessary to challenge the disintegrating Soviet Union or preserve the colonial status of Namibia. Namibia gained its independence; South Africa deposed the old crocodile, President Botha, and readied itself for democracy; Mozambique achieved peace at the second attempt; but Angola continued to reap the whirlwind.
## 9
## Carnival at Luanda
_The Luanda carnival is the modern incarnation of street celebrations that have been a central part of the life of the city since the seventeenth century. In independent Angola the carnival has lost its association with the Catholic calendar, but it has retained political dimensions that date back to imperial days. A more important feature of the carnival than either politics or religion is its social dimension. The carnival represents a rivalry between the various communities of the city that takes the form of competitive dancing. The carnival becomes an exuberant display of wealth and ostentation as the fishing families dress their daughters in gorgeous costumes in which to dance to the rhythm of the drums. But this chapter also suggests that the carnival represents, and always has represented, the pride of the common folk as they vigorously deny their powerlessness before the western-suited brokers of city politics. An earlier version of this essay was included in a festschrift dedicated to Roland Oliver that appeared in 1988 in the_ Journal of African History _and is reproduced here with the kind permission of the publisher, Cambridge University Press. A Portuguese version of this account appeared in 1991 in the journal_ Análise Social.
On Friday, March 27, 1987, Luanda celebrated its carnival on the magnificent palm-fringed boulevard that sweeps along the bay past the pink Grecian dome of the Bank of Angola. The date was a political one, unrelated to the Lenten calendar, but the festival was deeply imbued with rich symbols of Angola's history. For four hundred years Luanda has been the premier city of Africa's Atlantic seaboard. The dynamic continuity of popular culture had been little ruffled by the coming and going of contrasted regimes—Spanish Hapsburgs in the 1580s, Protestant Netherlander in the 1640s, Brazilian planters in the 1660s, Portuguese mercantilists in the 1730s, black Creoles in the 1850s, army monarchists in the 1880s, white republicans in the 1910s, "fascist" authoritarians in the 1930s, industrializing capitalists in the 1960s, nationalist revolutionaries in the 1980s. The carnival and similar feast days have always represented a flexible response to the traumas of change, a tight hold on the values of the past, and an ironic portrayal of how to exorcise contemporary devils.
Popular acclaim for the Luanda carnival was as old as the city itself, though the occasion on which to celebrate had changed over the centuries. The choice of significant dates for public processions was as important to the political authorities of the past as it became to the party managers of 1987. In the seventeenth century the city fathers spent far more of their budget on feast days than on drainage or lighting. Attendance in full regalia at the great processions of state and church was compulsory. The choice of date for the 1987 carnival represented an attempt by the state, in its latest incarnation, to capture a popular base in Luanda. The Angolan state and its ruling MPLA party suffered, like most states and most ruling parties in Africa, from chronic weakness. The carnival therefore seemed to present an ideal forum in which to mobilize grassroots support. The constituency consisted of two hundred carnival groups located in all the numerous quarters, townships, parishes, shanties, fishing villages, asphalt suburbs, and high-rise slums of the metropolis. Each community had a sharply differentiated set of social attributes and reflected its identity in an ostentatious carnival image. The most successful groups appeared in the great procession, where thirty-three finalists competed for the prize and status of champion. The ruling political party decided in 1977 that the procession should henceforth be held on March 27, the date of the South African withdrawal from Angola after the "second war of liberation" in 1975-76. This date was chosen from among possible anniversaries with some care. May Day, in particular, was considered and rejected as being insufficiently linked to local political achievement. The expulsion of the South Africans was selected as the one unsullied patriotic achievement that might be grafted onto the growing charivari of a traditional carnival.
Breaking away from the Christian calendar and choosing a political date for the carnival was symbolically important. To revelers a four-day festival ending on what had once been Ash Wednesday was observed in full, and no government decree could restrict drinking time to a single bank holiday. For politicians the church was a potential threat to their state, and it was therefore important for the party to unravel even vestigial associations with organized religion and weave the threads of carnival into the fabric of civic loyalties. The Catholic Church, though weakened by proscription in the Portuguese revolutionary wars of the 1830s and by persecution under the imperial republic in the 1910s, remained the largest church in Angola and under a Marxist government was liable to become a focus of opposition as it had in Poland. In Angola the capture of a Catholic festival by the state was seen as a significant achievement, the creation of a civil religion, complete with borrowed ceremonials. Alongside the sponsorship of carnival, some MPLA party leaders maintained an uneasy relationship of adolescence with the Methodist Church in which they had been raised. More difficult was their relationship with the Ntoko Church, an independent black church influenced by the Kimbanguist Church of Congo. The growth of Ntokoism was seen as a disquieting challenge to the party's agenda for political education, and Ntokoists were treated as subversives, like Jehovah's Witnesses, rather than as good patriots. In the weeks before the 1987 carnival one Ntokoist congregation came into armed confrontation with nationalist security forces and blood was shed on the road to Catete, a symbolic place of colonial martyrdom where MPLA heroes had once fallen to Portuguese security forces.
The 1987 carnival was divided between witnesses and actors. The interaction between the two was complex. On the central stand, flanked by the ministerial fleet of black Mercedes-Benz cars, the party hierarchy were witnesses. For them the carnival was the tenth Carnival of Victory, and the odd placard was hoisted before them portraying Botha-the-enemy or Kaunda-the-comrade. But at another level the hierarchy was much more intimately linked to the historical reality of the carnival as experienced by the participants. In origin, the carnival is emphatically the carnival of Luanda, and some members of the ruling party's political bureau had their cultural roots firmly embedded in the traditions of the city. They belonged to the historic Creole families that survived the social Darwinism of the early twentieth century, survived the mass influx of settlers at midcentury, survived the wilderness at the ends of the earth during the guerrilla war. They had now recovered the military tradition and past influence of their forebears to govern the army-led regime of the popular front. Some branches of the great congeries of Creole families dated back to the seventeenth century and had a long familiarity with carnival-type celebrations, though as members of the government they were detached from the popular participants.
The second covered stand of witnesses constituted the haute bourgeoisie of middle-class Luanda. In contrast to the old black Creole elite, many among the twentieth-century upper class were of recent mixed ancestry. Marrying light had commonly been the racial ambition of social climbers in Luanda, and the women and children seated under the civilian awning at carnival were a testimony to the continuity of custom. The first great influx of white immigrants to Angola consisted of republican carpetbaggers flocking to Africa in search of petty government employment after the Portuguese revolution of 1910. On arrival they drove the old Creole functionaries out of their bureaucratic positions with loud racist cries of self-justification. Once settled, however, they were often forced by circumstance to marry black wives and thereafter give preferment to their own _mestiço_ children. A second wave of immigrants came with the coffee boom of the 1950s. Although this brought a significant influx of white females, some male immigrants continued to marry black and even to make obeisance to African custom by adopting circumcision for themselves and giving bridewealth to their in-laws. In the colonial generation having a white parent facilitated access to scarce education and hence to employment and status. The educated mestiço population temporarily enhanced its prominence when in 1975 90 percent of expatriate whites left Angola. The mestiço class was no match, however, for the Creole elite in the political struggles of the 1980s, although mestiço wives and children of the administrative bureaucracy were given a covered stand at the carnival.
A third witness enclosure at the carnival contained the diplomatic corps, hot, bored, and overdressed. The only symbolism that they recognized was the intrusion into the carnival of an alien troupe of Afro-Caribbean dancers, jesters, drummers, and harlequins sent from Cuba, Angola's Iberian Creole partner. The diplomats were constantly on the qui vive for relative rise and decline in the pecking order of international friendships. Angolans, by contrast, were not interested in mere foreigners. The Cuban troupe did not even feature on the printed order of ceremony. The populace had given the final cheers and begun to depart before the supposedly grand Cuban finale. After the intense symbolism of the home groups, the death masks and painted skeletons of the Cuban artistes were irrelevant to the huge crowds of plebeian witnesses who had lined the streets for six hours without shade. Historically, however, carnival in Catholic America exceeded the parallel developments in Latin Africa, and the Cuban link of 1987 mirrored the ties between Luanda's great festivals and Brazilian ones.
The feast that can be most readily compared to the modern-day carnival is the canonization feast of St. Francis Xavier in 1620, which is fully described in Ralph Delgado's history of Angola. The Jesuits of 1620, like the politicians of later times, wanted to harness customary celebrations to their own cause. They therefore contributed richly ornate floats to St. Francis's procession. They also included moral sketches such as the seven-headed monster that depicted pride as a lion, avarice as an ass, covetousness as a dog, libertinism as a sow, rage as a leopard, and gluttony as a wolf; sloth was somehow linked to Brazil. The governor, a conquistador from one of the great Jewish finance houses with slave-trading interests, ordered naval cannonades and night illuminations throughout the city. The army paraded and gave musket and arquebus salutes. The Luanda bards competed to write praise songs for the new saint. But for all the innovations and all the need of persons in authority to be seen in their most resplendent finery, the strength of the 1620 procession lay in its popular cultural roots.
Traditionally carnivals exorcise social ills and traumas through ridicule. The 1620 procession was led by three white giants (presumably puppets), far too large to have been comfortable in the cramped berth of a European sailing vessel. Dressed in formal wear, they were accompanied by their "father," a black dwarf captured in the Ndongo wars, who wore a velvet tunic of scarlet with white shoes and a rainbow beret. He railed at his huge white "children" with a thousand jokes and witticisms. To a country that had recently gone through the most devastating of all its slaving wars, the tableau must have had considerably more poignancy than trite ecclesiastical homilies about the seven deadly sins. But joking relationships were not enough to help a venerable but impotent old man overcome the striding might of the conquistadores.
The second group in the 1620 procession was a dance troupe sent by Creole society in São Tomé, the offshore island where slave-worked sugar plantations had been developed to a lucrative level of proficiency even before Columbus opened up the New World. The troupe was led, like its twentieth-century counterparts, by a puppet king whose prowess was acclaimed by accompanying praise singers. These were followed by sword dancers in the Portuguese style and by a ballet in which the daughters of the city worthies portrayed shepherdesses. The central float represented a Neptune-like god, the local saint of the sea folk of the Luanda shore, and the worker of miracles. The float, shaped realistically in the form of a whale, had an altar shrine on its back and was pulled by the king of the sea. Its tail was decorated with gold, shells, and silk braid. The lord of the waves, a deity lounging in a green, white, and red costume, was serenaded by an orchestra of four sirens. His dancing cortege sang ballads associated with fish, the mainstay of the shoreline economy.
Three further tableaux represented the condition of Africa in 1620. Angola, the colonial kingdom, was a figure dressed in green, with a blue turban quartered like a crown, a train of gem-encrusted gauze, and white boots covered in buttons and gold chain. Kongo, the Creole kingdom straddling the merchant world of the Atlantic and the terrestrial world of Africa, was similarly attired and brought gifts of rare price. Ethiopia, the land of the blacks, was dressed in the mode of the country, a simple loincloth around his waist, but he was the one who gave alms to the poor and scattered small silver coins, like Maundy groats, which the crowds caught rapturously like crowds catching beads at New Orleans carnival. The procession represented a syncretism of pagan rituals from the Mediterranean pantheon, of Christian rituals from the Iberian church, and of Mbundu and Kongo rituals designed to foster fertility and prosperity. Carnival was also an attempt to come to terms with the disasters of colonial conquest. The tradition carried on down the centuries, with constant modification, before being reenacted in the "Popular Republic" of independent Angola.
In the nineteenth century the colonial aspect of Luanda festivals was significantly modified. The Portuguese Revolution of 1820 to 1851, like the French Revolution that preceded it, was vigorously anticlerical. In 1834 the monastic houses were dissolved and their lands were sold to a new class of barons to ensure their support for the changes that revolution had wrought. In Angola one Luanda church, heralding a new scientific age, became a meteorological observatory. Meanwhile the monarchy passed into the hands of the ubiquitous house of Saxe-Coburg and the style of the Portuguese kings and queens came to resemble that of the bourgeois monarchies of Louis-Philippe and Victoria rather than the imperial grandeur of Louis Napoleon. The royal birthday became the official occasion for public ceremony and the competitive display of sartorial ostentation. For the 1846 birthday of King-Consort Ferdinand an effigy of the queen of Portugal was carried in a cortege through Luanda. Historical festivals continued to be celebrated under the bourgeois monarchy. Every August 15 the defeat of the Dutch by Salvador de Sà was a feast day, as indeed it would be until the fall of the last colonial government in 1975. Regattas also became part of the nineteenth-century festival scene: children paraded in striped uniforms while brass bands supplemented the traditional drum ensemble and the orchestras that used both African and Mediterranean stringed instruments. And despite all the innovations adopted at Luanda, religious festivals soon crept back.
In 1887 Héli Chatelain witnessed the Luanda carnival at a dinner attended by some of the stiffest members of Portugal's colonial bureaucracy. The gentlemen, preening themselves in cravats, went apoplectic when their wives suddenly abandoned all decorum, rose up from the feminine ranks of the oppressed, and put boot polish on the faces of their guests before showering them with bags of flour. The Presbyterian bachelor looked on with apparent good humor as the self-important world of the colonial elite was turned upside down. He also went out into the street to see how the black population of nineteenth-century Angola had made carnival its very own form of cultural expression and protest. The people of the street enthusiastically mocked the fearsome powers of their overlords.
The key scholarly document relating to the evolution of the Luanda carnival is the Parisian doctoral thesis of Rui Duarte de Carvalho, who, while studying the fisher folk of Luanda, witnessed the reincarnation of Luanda's traditional street ceremonial in the postcolonial victory carnival. The Luanda carnival is rooted in the customs of the fishing communities but linked to the various levels of modern society. In 1987 each carnival group was governed by a _comandante_ (commander), representing an office once feared and respected for both its military rank and its colonial authority. His headquarter was called the _quartel_ (cantonment), a term likewise advisedly military by choice. The comandante was chosen for his skill alone and could be an immigrant or a peasant from the land rather than a true coast dweller, so long as his managerial skills warranted his appointment. The group also elected a president, who had to be a Luanda citizen, but "one of them" rather than "one of us." He wore a suit and tie, went into the "paved city," and talked to bureaucrats, administrators, and even politicians if the need arose. The president played no role in the dance but was related to the prestigious carnival sponsors. For its dance master the carnival group chose a king, who dressed up in the finest clothing his group could afford and wore a crown as in the procession of 1620. The king's consort was a resplendent queen who enthused the crowd and caught the eye of the judges. The royal couple were attended by the _conde_ (count), who resembled the knave in a pack of cards, and by a princess. The king, the queen, and all the dancers were chosen for their talent rather than their social status. The lead singer, the standard bearer, and the bandmaster were normally married into an elite family. Charm carriers protected the drums from evil rivals, and straw torches were used to warm the drum skins. Gourd bearers supplied liquid refreshment to the musicians. The dancers were decked in vigorous colors, the wealthier teams being clothed in matching cotton prints. Each group symbolically demonstrated its profession, such as the fishermen by net throwing, while marshals kept the crowds back by vigorously mimicking police truncheon blows. The rear of the procession was brought up by jogging supporters, surging forward or dropping back as the dancers progressed.
The financing of the carnival groups was by public gift. The spectators put banknotes between the lips of the queen as she danced. The notes were then passed to the treasurer, who gave them to the nurse, who locked them in her bag. The military nursing sister as a key figure of dance societies is also found among the dance societies of German East Africa. Similar rituals of public fundraising can also be witnessed in the great parades of the Kimbanguist Church of Congo, where columns of dancing worshippers file past their leader depositing their contributions. Carnival revenue was used to regenerate the group and renew its costumes and musical instruments. It was also consumed, part of the income becoming the private revenue of the king, the queen, and the comandante, but some was put aside for mutual assistance and funeral funds.
Banners and emblems were carried in the carnival procession. One homemade placard bore that quintessential message of class warfare in Africa: _cuidade com o cão_ (beware of the dog), a message seen as _mbwa mkali_ on wrought iron gates in Nairobi, as _chien méchant_ in Libreville, or in its Amharic version on the aristocratic portals of Addis Ababa. Guard dogs and the protection of property and privilege live on in the proletarian subconscious long after colonialism has died. Protests at continued social stratification were bravely displayed to the well-fed witnesses on the carnival tribune. More historic tableaux involved slave whipping, litter bearing, and other forms of remembered subservience and indignity. Independence had not yet alleviated the great stress felt as Angola went through the anguish of social transformation and periodic bouts of civil war.
The dance societies of the Luanda carnival resemble the _beni_ dance societies of eastern Africa in ways that raise interesting questions. How far is the mimicking of ceremonial sartorial styles based on antagonism and mockery and how far is it based on admiration and the aspiration to achieve European status? In colonial Lubumbashi the Belgian authorities saw dance societies as mocking colonial authority through satire and threatened dance troupes with arrest. In Malawi, by contrast, the clean orderliness of the dance societies was appreciated by the colonial administration, which saw no hostility to itself. In Luanda the carnival seems to have been officially encouraged during the 1950s, and there was even talk of dances being performed by colonial police sepoys. After the 1961 uprisings, however, carnival was banned, in a belated acknowledgment by colonial administrators of its potential for mocking the establishment and encouraging national aspiration.
A second comparative question concerns the importance of colonial consciousness in the dance societies. In eastern Africa the dance societies resolved tensions between former slaves and old-style freedmen, between coast-born Swahili and up-country peasants, between the "posh" and the "vulgar." In postcolonial Luanda rivalry between competing groups was more important than outward political messages. Carnival was highly localized, and preferred rivals were those adopting similar dance traditions and belonging to the same interlocking territories and families. Well-known neighborhood personalities were challenged with provocative songs that revealed moral weaknesses, private foibles, and social lapses. Allegations of incest and witchcraft in particular provided grist for the song mills of the Luanda bards. After each carnival it became necessary to lower the social temperature until a normal joking level of badinage had been restored.
One dance group that expected to become champions in the 1987 Luanda carnival was the Union of 54, a fisherman's union like some of those in the 1620 parade. The community from which it arose had lost its fishing bank in 1944 when unusual tides swept away the middle section of Luanda island. The families settled under the baobabs on a mainland beach and rebuilt their canoe fleet, made up of dugouts carved from the light trunks of _mafumeira_ trees cut on the banks of the Bengo river. For centuries boatmen had earned a living by ferrying canoeloads of Bengo river water around the headland and into the bay to supply the city with salt-free domestic water. Real economic muscle, however, derived not from water carrying but from fishing. Seine nets, weighted with shells and buoyed with wooden floats, were drawn toward the beach, while larger prey had to be caught on lines equipped with expensive lead weights. Fishing has always been a man's task, and the heaviest work was done by migrant youths known as Bailundos, an ethnic label derived from the labor- exporting kingdom of Mbailundu, which lost control of its own highland destiny in the colonial war of 1902. The powerful master fishermen are Muxiluanda, and it is they who run the serious competitive business of commanding the _varina_ style of carnival dance group.
The women of the fishing communities are responsible for fish marketing and sometimes wield as much power and influence as men. In the late colonial period, however, fishwives lost control of the wholesale trade to white European fishmongers who elbowed their way in with trucks. Women retained only the retailing side of the business until after independence, when they regained the initiative. A hierarchy of marketing evolved, with control in the hands of six dominant female entrepreneurs. Influential fishermen recognized the power of women in the community and chose their place of residence carefully to gain best matrimonial advantage. Both men and women sponsored carnival groups, but the money contributed by women determined the success or failure of a dance troupe's reputation.
Fish catchers and fish traders did not invest much in producer goods other than nets and canoes. It was ostentation, even hedonistically conspicuous consumption, that enabled one to cut a good figure at the carnival. Clothing was expensive, and carnival officers wore uniforms that Louis XIV might have admired. Dancers were decked in the finest wax prints (batiks) that money or influence could obtain, and alcohol was important in winning friends and supporters. Beer had to be hoarded well in advance for carnival, and sugar that would be surreptitiously distilled into cane rum had to be bought at bootleg prices in the townships. The old sixteenth-century Jaga custom of topping palm trees for palm wine was revived, much to the grief of farming communities that grew palm trees on the coastal plain. A skilled peasant knew how to tap a tree prudently to gain a modest but long-term supply of wine sap, but in the 1980s the palm groves of the Luanda plain were decimated by foraging speculators with a ready market for carnival palm wine but no thought for the morrow.
Despite its wealth, the prestigious Union of 54 did not win the 1987 carnival. The gods were displeased. The sea became rough. The canoes could not put out. The omens needed to be read again and the water gods had to be propitiated. The whale remained the fisherman's friend and was expected to bring messages, while the sirens so vividly described in 1620 remained a vibrant feature of life in the beach villages. If household gods and water deities could not remedy social and commercial failure, more drastic measures had to be taken. Forty miles down the coast the shrine of the chief priest of the sea god was in a baobab tree. It was there that the priest accepted offerings of expensive whiskey, port wine, and sweetmeats, all laid out on a fine tablecloth. When even the gods could not remedy a carnival failure, the law was taken into the hands of the hard men and gangs were sent out to gain revenge from opponents.
A more prosaic explanation of why the Union of 54 failed to win the competition was that its closest rival, the Union of N'Zumba, whose presence had traditionally spurred the Union of 54 on to greater prowess, did not participate, and a new rival, Island World, had gained in standing. Long based on the same beach as the Union of 54, the Union of N'Zumba had recruited its followers among the same elite families of the fishing aristocracy. It lost many supporters in the 1950s, however, when white immigrants took over the coast and drove black Africans to settle a few hundred yards inland in the Prenda quarter of the city. In the 1980s the group was still in financial crisis and, fearing a loss of status, withdrew from competitive dancing. The Union of N'Zumba attempted to recover its influence by sponsoring a football team, but even football was no match for a dance troupe. Island World's members belonged to the seamen's union. On trips abroad they could buy costumes that far outshone the drab government-issue textiles. By origin also a fisherman's union like the Union of 54, Island World used the same varina style of dance. Of the six types of dance displayed at the carnival, the varina was the one that thrilled the shoreline crowds, and it was Island World that won the competition.
The economic influence of the fishing communities and the dominance of the varina dancers did not mean that other boroughs of the city were excluded from the carnival competition. Inland carnival groups had their own dance traditions and aspirations. _Kazukuta_ was the dance style of the old proletarian quarter of the upper city. It flourished in the Bairo Operário, built early in the twentieth century as the residential zone for a declining black elite that had been driven out of the city center by white immigrants. The new suburb gained a twilight reputation for managing the informal sector of the city economy with unusual finesse. The borough was supposed to be a residential showpiece of colonial town planning, but the roads were never paved and the poets of Angola's protest literature decried the lack of electric light. Beyond the Bairo Operário lay Sambizanga, a poorer quarter for working-class black people. Here too the carnival dance tradition was kazukuta, but the passion for carnival was equaled, if not surpassed, by the passion for football. It was in the hungry slums of Sambizanga, under the cloak of a local football management committee, that the revolution of the dispossessed broke out in 1977. Since then security had been tight, and young soldiers were everywhere.
The attitude of the carnival to the soldiers was ambiguous. The conscripts were the children of the slums, and it was to the slums that they returned for safety when desertion seemed to be the only escape from eternal national service. The soldiers were also the champions of freedom, the people's militia, the protection against the vividly remembered foreign invasions from north and south that had so nearly captured the city on November 10, 1975. But the soldiers were also the arm of authority, and they brought fear to the slums. It was soldiers who crushed the popular uprising of 1977, who threatened the lucrative trade in bootleg spirits, who spied on black-market contraband, who had the guns to commit daylight robbery, and who sometimes held up law-abiding citizens on dark nights. Carnival in 1987 was still, as it was in colonial times, about exorcising the fear of authority.
The nominal heroes of the Angolan revolution and liberation were the peasants, but their participation in the carnival was meager. They sent only one group to the final competition to dance the _kabetula_ dance of the farmers. To the big spenders of the fishing community, farming seemed a poor way of life. Even the great farms of the Bengo river, which sponsored _dizanda_ dancers, sent only two competitive groups. Yet coastal farming was a growth sector of the Luanda economy. When government failed to maintain the living standards of the provinces and lost control of rural Angola to a rebellious peasant movement, the capital city underwent a crisis. The popular response was to bypass the bureaucratic structures of production and distribution and create a dynamic if informal free-market economy. New peri-urban farmers turned wasteland into cassava fields and tomato gardens. Parallel markets sprang up at which everything from champagne and television sets to canvas shoes and transistor batteries could be bought or sold or exchanged. Carnival groups, like everyone else in Luanda, learnt to juggle with state controls, seize market opportunities, and function with a two-tier exchange rate with unprecedented fiscal distortion.
Carnival is at heart a celebration. Politicians would have liked it to be a celebration of their strength and their popularity, but it was not. It was a celebration of ingenuity and survival: survival in a war without end, a war that began as a colonial war in 1961 and became a civil war in 1975. It was a celebration of the identity in which people rejoiced—not a national identity, not even a city identity, but an identity with neighbors and kinfolk in the closest, safest community that they could preserve. It was a celebration of wealth, of ostentatious display, of purchasing power, of conspicuous consumption. It was a celebration of freedom and a challenge to the awesome figures of authority that trespassed across the historic stage and had to be brought low by allegorical displays and carefully ritualized mockery. It was a celebration of youth in which grandmothers paraded the offspring of their daughters with pride and finery. It was a celebration of defiance before the perplexed bourgeoisie in a city bursting with class conflict. Above all, carnival was a celebration of historical tenacity and endurance. Five centuries of fishermen absorbed and tamed peoples, cultures, religions, and rituals from all over the world and made them a part of their very own Luanda carnival.
## 10
## The Struggle for Power
_In addition to a colonial war (1961–74) and an international war (1975–91), the history of Angola was darkened by two civil wars that pitted two surviving liberation movements against one another. The MPLA and UNITA fought each other mercilessly in both 1992–94 and 1998–2002_. A History of Postcolonial Lusophone Africa _(2002), edited by Patrick Chabal, carried three chapters on Angola: one on the wars of the 1970s, one on the war of the 1980s, and a third on the wars of the 1990s. The third chapter is reprinted here by kind permission of the publishers, Christopher Hurst and Indiana University Press_.
When Angola emerged from the cold war in 1991, it was a different country from the one that had emerged from the colonial war in 1974. In 1974 the major export had been coffee, efficiently carried by truck on asphalted highways built for strategic purposes. In 1991 one of the exports that exceeded coffee was scrap metal, quarried from the half a million tons of nonferrous junk attached to thousands of military and civilian vehicles that had been blown up during the years of bitter conflict and left along Angola's ruined roads. The graveyard of military vehicles was matched by the graveyard of human victims. Those who had died—of hunger, wounds, disease, or gunshots—were buried and uncounted, but those who survived—maimed, crippled, displaced, and unemployed—were all too visible and had to be counted by the agencies that supplied them with basic meals and artificial limbs. The war of destabilization, begun on the southern plain in the early 1980s, had spread through the highlands and lowlands until conflict reached Angola's northern frontier by the later 1980s. Only the larger urban enclaves of the highlands and the cities strung along the Atlantic shore were spared day-to-day fighting. Even these cities, however, felt the rigorous consequences of war when hundreds of thousands of war victims emerged from the countryside to seek refuge from the trauma that had engulfed all of village Angola.
It would be too simplistic to say that the reason a southern army had moved through the countryside destroying everything in its path was in order to terrify rural Angola's peasants into accepting the rule of Savimbi and UNITA and rejecting the rule of dos Santos and the MPLA. Much of the success of the southern army had been due to widespread rural disaffection with the MPLA government, and a civil war had therefore gone in tandem with the cold war, the war of destabilization. The difficult initial struggle of the MPLA to gain and hold the capital city had meant that rural Angola had been neglected. One of the causes of the great uprising of 1977 had been the suggestion that unemployed urban youths should be dispatched to the country as work brigades to pick the unharvested coffee crop. The youths were alarmed at the prospect. Their dignity depended on being sophisticated urbanites with, despite their minimal amount of schooling, a taste for sharp dressing. To be sent to a countryside full of yokels and wild animals, not to mention magicians and poisonous snakes, would have been a terrifying experience. Their disdain for the countryside was shared by salaried workers in the city, who contemptuously described the people of the countryside as the "Bantu," vernacular-speaking rustics quite unlike themselves with their smooth Portuguese manners. The antagonism between the town and the countryside had paved the way for the war of the 1980s to spread like a bush fire from neglected province to neglected province. Regional distrust remained a dreadful burden as the nation sought a sustainable peace for the 1990s.
Burdensome though the legacies of war may have been, the eighteen months from May 1991 to September 1992 were the most spectacular months of optimism and freedom that Angola had ever witnessed. Savimbi and his entourage of generals moved down from the highlands and set up their opulent residential quarters in the Miramar district of Luanda, overlooking the palm-fringed bay. Thousands of highland refugees in the coastal cities loaded their meager possessions on their heads and set off for the interior to rediscover their villages and seek out their surviving relatives. International observers poured into the country to marvel at the peace process, at the new economic opportunities, at the adoption by Africa of a democratic procedure to settle differences. The political parties hired public relations firms to run sophisticated election campaigns on television stations, and the political leaders drew large crowds of cheering supporters to their rallies in the country's town squares. The representative of the United Nations, Margaret Anstee, flew everywhere in decrepit aircraft parsimoniously funded by the United States and courageously flown by intrepid Russians. She endeavored to harmonize the two partisan armies that were to be partly demobilized and partly integrated into a single national force. The euphoria of peace made supervised demobilization virtually impossible, however. The government conscripts vanished into civilian society, while the opposition ones were hidden away in provincial redoubts in case "the leader" should require their services later. The most obsolete of UNITA weapons were handed over to teams of international inspectors, but sophisticated military equipment was cached away in arms dumps strategically chosen around the provinces by Savimbi himself. On the government side a new security force, dressed in a sinister black costume, was armed and trained for action against civilians should circumstances lead to urban guerrilla warfare after the election. While people danced in the streets and vowed that war should never return to their land, the pragmatic power brokers of both sides made contingency plans.
When, after a year of blissful peace, Angola finally went to the polls to elect a parliament and a new president, the voters divided cleanly and clearly between the town and the countryside. The towns had more or less survived the war of destabilization on the basis of imported food paid for with oil revenue and supplemented by philanthropy from organizations across the world. The countryside had done much less well, having suffered a sharp loss of earning capacity following the collapse of the colonial infrastructure and the total failure of the Soviet-style economy to create any rural network that could purchase produce from the farmers or distribute to them essential commodities such as soap, salt, and cooking oil. In September 1992 the countryside voted for the opposition—for Savimbi and for change—while the towns voted for the government—for preferential economic treatment and for armed protection from the hungry raiders out in the rural areas. Some modification of the voting pattern was effected by historic or ethnic loyalties, but the UNITA leaders were greatly dismayed to find that some urban Ovimbundu, in both highland towns and coastal cities, had failed to support them in their election bid and had adopted the national townsmen's strategy of voting for the MPLA. Even more dismaying to Savimbi was the betrayal of the United States, which, to all intents and purposes, had promised him that if he stopped the war and went to the polls he would undoubtedly win the election. When Savimbi failed to win, by a clear margin of two to one in the parliamentary election and by a decisive if not absolute majority vote for dos Santos in the presidential election, he immediately went back to war. Western-style democracy had no consolation prizes for coming second in its first-past-the-post, winner-takes-all system of voting.
The civil war that broke out in Angola in November 1992 was quite different from the colonial war of 1961 and from the interventionist war of 1975 and its destabilizing aftermath in the 1980s. Those wars had been fought in the countryside and had only indirectly affected the towns. The defeated opposition in 1992 could do its electoral sums quite as effectively as any UN observer and recognized that it was in the urban heartlands that it had lost its bid for power through the ballot box. UNITA therefore set out to destroy those urban heartlands and to destroy a government that had proven itself totally unwilling to make any concessions to its opponents by offering a significant postwar redistribution of the economic spoils of the extractive economy. The civil war of 1992 first broke out in Luanda itself, triggered by UNITA's intransigent rejection of the election result but initially launched and pursued with vigor by the government. Within days the city had been violently cleansed of politicians unwilling to abandon Savimbi's cause. Worse still, the urban militias were given the license to settle old scores by attacking townsmen who were thought to have voted for UNITA. Savimbi refused any compromise solutions, recognizing that the presidential system—so attractive to him when he thought he could win—gave all power to the president rather than to the prime minister, the cabinet, or the elected parliament. Savimbi calculated that his only hope of gaining the power that he had craved almost pathologically since his student days in Switzerland was to seize it through the barrel of a gun.
The war of 1992 brought even heavier weapons to Angola than those used in previous wars, and in the new conflict the big towns of the interior—Huambo, Kuito, Malange—were severely damaged while their populations were almost starved. Savimbi no longer had support from South Africa, but he did have access to relatively cheap secondhand weapons bought, ironically enough, from the countries of the former Soviet empire. He discovered in particular that the huge republic of Ukraine, with fifty million people struggling to make a living, was willing to sell redundant military equipment and had an air cargo fleet with the capacity to fly weapons, ammunition, and fuel oil to makeshift airstrips hidden in the orchard savanna of eastern Angola. Payment for the new UNITA arsenal came from the wild digging of diamonds extracted from rivers of the interior and flown out through cloak-and-dagger channels to Antwerp, the Belgian capital of the diamond-cutting world. In the expensive business of modern war, fought with technologically sophisticated weapons that required imported ammunition, UNITA recognized that its diamond wealth could not compete with the ten-times-greater oil wealth of the Angolan government. In 1993 UNITA attacked the onshore oil installations at the mouth of the Congo River, either to deprive the MPLA of revenue or to capture an oil supply of its own. The oil port of Soyo temporarily fell into opposition hands, but ruptured storage tanks only caused massive pollution while the oil platforms on the ocean horizon were never at risk from military activity. By 1994 Savimbi was forced to recognize that his early military successes had exhausted his resources and could bring no immediate political victory. For long-term survival he needed to seek a truce on the best terms he could extract.
Ending the war proved a particularly intractable diplomatic challenge. Margaret Anstee, having orchestrated the election with aplomb, negotiated valiantly to win the peace as well, but it was not until late in 1994 that a new UN peacemaker, Alioune Beye, eventually secured a cease-fire in Lusaka. The accord generated none of the euphoria that had accompanied the peace signed at Bicesse in 1991. Savimbi showed his contempt for the unpalatable necessity of suspending hostilities by staying away from the signing ceremony. He had no desire to come face to face with Eduardo dos Santos, who had now outwitted him both in a patently free and fair election and in prolonged siege warfare, which had given him control of highland cities that Savimbi deemed his birthright. Savimbi retired to the small highland town of Bailundu to plot future political or military developments. Dos Santos returned from Lusaka to Luanda to consolidate his personal power by both political and financial means. Savimbi evaded all forms of peace monitoring by the United Nations and refused to demobilize under the terms of the Lusaka Accord. Dos Santos basked in his international acclaim as a peacemaker who now enjoyed almost unlimited western support for his government. But war remained the agenda on the horizon, and each side tried to provoke the other into being the first to break the Lusaka cease-fire and incur the international opprobrium of being the guilty party that returned Angola to civil war. In the highlands the cold hostility, neither war nor peace, lasted for four years. Meanwhile civil society had been changing down on the coast.
The end of the cold war and the signing of the 1991 peace at Bicesse in Portugal had brought important changes to the status and role of the churches in Angola. For fifteen years after independence, the state and its nominally Marxist-Leninist government had ignored the churches. Members of the Luanda political elite who had remained affiliated to religious congregations had used great discretion, almost secrecy, when attending church services. Even the Methodist Church, in which several eminent leaders, including Agostinho Neto himself, had been nurtured, received only minimal official toleration. In the 1990s the government moved away from its initial hostility, and an attitude of toleration gradually gave way to an actual wooing of the churches by the presidency. Although 90 percent of Angolans now belonged to a church, the political influence of the congregations remained weak and the churches proved incapable of preventing further outbreaks of war. No church actively advocated war, but none was openly willing to condemn the concept of "peace through victory," and church members were trapped by a loss of liberty and human rights that scarred Angolan society throughout the 1990s.
The largest and most united church was the Catholic Church, which, although it had historically been split between foreign missionaries who stood up for black colonial subjects and Portuguese bishops who stood up for the white colonial state, was nevertheless firmly structured around a single authoritative voice legitimized by Rome. The Protestant churches, although riven by contrasts of ideology, pastoral tradition, and ethnicity, might have expected to benefit from a folk memory of Catholics as supporters of empire and Protestants as anti-imperialists who gave succor to the liberation movements. Any such legacy of sympathy between nationalists and Protestants was eroded after independence, however, by growing government conservatism. The Luanda elite, attempting to rebuild the country's traditions of power and subordination, approved of the Catholic Church's authoritarian hierarchy, and the old colonial tradition of treating the Catholic hierarchy as the natural ally of government was revived at the end of the cold war. Traditional Protestants and the old independent churches of the Kimbanguists and the Tokoists were left fragmented on the margins of society, and it was a new Pentecostal religious tradition that provided a spiritual home for the victims of war who crowded once more into the coastal cities after the abortive peace of 1991.
Once the pannational Catholic Church had overcome the stigma of its reactionary legacy from the colonial era, its influence began to grow and it became an attractive symbol of power for those who wanted to be associated with the elite. But Catholics also began to match Protestants in dispensing charity to the dispossessed and listening to the voice of the voiceless, thus usurping the role of the Methodists, who had lost their heroic status as defenders of the oppressed and were now perceived by some as the traditional partners of an uncaring MPLA government. Their standing was further diminished when the government presumed on Methodist loyalty and gave Methodists fewer state resources with which to alleviate poverty than it gave to the Catholic Church, whose endorsement it solicited in the game of power politics. As the government wooing of the churches progressed, Christians were openly welcomed into membership in the once-atheist ruling party and dos Santos appointed church leaders to his privy council. But coopting church leaders into the establishment weakened rather than strengthened the congregations. In Angola no "peace and justice" commission was set up, no "truth and reconciliation" were attempted, no rehabilitation of social relations between former enemies took place, and the state remained in full control of everyday life. Church members who wanted their lives to be independent of the politics of clientship found that they could not subsist without compromise. Nonsmoking and teetotaling Protestants who had been morally outraged by the government issue of rations of tobacco and alcohol could not refuse to receive their allocation, since it was only by selling perquisites on the wild market for one hundred times their posted price that state sector employees could realize the value of their salary substitute and buy all the real necessities of life. The impotence of dependency reached down through the ranks of society and turned everyone into a vassal of the MPLA. While church members were becoming dependent on the party, the party began using the church as a symbol of its own power and prestige. Eduardo dos Santos, the Soviet-trained technocrat, chose to have his son baptized as a Catholic and invited the Pope himself to celebrate Mass for the millions in a Luanda football stadium. During the war-torn 1990s this cohabitation between church and state handicapped the efforts of the churches to find a means of satisfying the intense popular desire for peace. It was not until mid-2000 that the churches finally began to cooperate, regardless of the wrath of Dos Santos, and bring the people out into the streets of the capital to demonstrate for peace. An interchurch congress on the rights and wrongs of entering into dialogue with the enemy, rather than allowing the war to drag on, finally broke the silence of fear, and an open public debate about Angola's future was launched.
One political initiative designed to prevent a renewed outbreak of war had occurred in 1997. As part of the search for a policy that would defuse the anger of the opposition and minimize the danger of a return to war, the president created a "government of national unity." A limited number of junior posts were offered to members of the southern elite who were willing to leave the highlands and join the ruling circle in Luanda. Some seventy UNITA members who had been elected to parliament in September 1992 moved to the comforts of the city and took their seats in the legislative chamber; seven of their leaders became ministers and vice ministers in a cabinet whose padded payroll also included sixty MPLA members. This low-key concession to power sharing was silently undermined, however, by the continuing rise of presidential authority. One of the most potent effects of the failure of the UN election of 1992 and of the catastrophic war that followed was the decision by the president to concentrate more power in his own hands. From being a single-party state with a disaffected opposition thinly scattered in the provinces and abroad, Angola became a presidential state in which power emanated from the palace. Dos Santos, like Louis XIV, built his palace on the outskirts of the restless city, safely removed from the fickle mob, and it was there that political decisions began to bypass government ministries, party cells, and state bureaucracies. Angola was no longer a "people's republic," and the president's huge, well-fortified presidential complex, the Futungo, ostentatiously resembled the extravagant luxury of Mobutu in Congo-Zaire rather than the austere highland hideouts in which Savimbi dodged from night to night to avoid capture or assassination by his many personal foes and political enemies. But for all the gilding on his cage, dos Santos was almost as much a prisoner as Savimbi. After 1992 he virtually ceased to travel around the country, and even when he visited his own capital city he went with a heavily armed guard. The caged president orchestrated a personality cult, which led to an extravagant adulation that constantly emphasized his image as the man of peace, in shining contrast to Savimbi, who was always portrayed as the man of war. The presidential court even suggested that dos Santos, who had been at war with his own people for twenty years, be nominated for the Nobel Peace Prize.
In 1998 the presidential personality cult reached a climax during a week-long birthday party for dos Santos. He ceremonially visited the restoration work on the seventeenth-century chapel of Our Lady of Muxima, he launched a regatta and a parachuting competition, he awarded new costumes to paramount chiefs, he unveiled a commemorative postage stamp, and he opened an exhibition on "protecting the sea and its riches," thereby showing his ultramodern concern for ecology and the environment while many of his human subjects went on starving. The country's horrific medical plight gave the president an opportunity to visit favored hospitals bearing gifts and seeking loyalty, to visit a leper colony and a camp for displaced children, to express solidarity with those who campaigned against polio or sustained the victims of AIDS. An American-style fundraising dinner was devoted to the rehabilitation of the victims of land mines, which his government had probably done as much as Savimbi's opposition to scatter over the country. Amidst the hopeless despair that presaged an imminent return to war, the week ended with gymnastics, sporting competitions, the cutting of a birthday cake, and the awarding of a Brazilian honorary degree to the president. The bread-and-circus fantasies were an attempt to overcome rising popular disaffection and an increasing fear of police surveillance. As the president became all-powerful, even the Luanda elders of the MPLA found themselves marginalized, as was demonstrated when a prime minister from the prestigious Van Dunem Creole family was humiliatingly made to carry the blame for government unpopularity. But while the people on the street saw the junketing and partying as an extravagant display of scandal and corruption in high places, the establishment in the bureaucracies saw hero worship as the necessary gateway to power and status on the fringes of the court. As money poured into the presidency without let or hindrance, the politics of clientship became ever more pronounced, and success depended on largesse.
The development of the untamed market economy in Angola had serious consequences for the middle class. The purchasing power of state salaries dwindled with rampant inflation, and bureaucrats were driven, like the displaced poor of the shanties, to live by their wits. Economic insecurity led to corruption, violence, and crime, which touched the lives of all sectors of society. As in many other parts of Africa, public servants had no choice but to spend their time and energy working at second jobs in the private sector and retaining their formal jobs in the public sector to ensure for themselves structural positions and state privileges rather than monetary or material reward. As public services withered away, only those who offered bribe money, preferably in American dollars, could obtain the necessary medicines or documents to survive. Under these circumstances clientship became at least as necessary as it had been under the Soviet-style economic system, but the _nomenklatura_ , the privileged elite, had to find new ways of obtaining patrons. This need for a patron was most acute in Luanda, where the bourgeois cost of living was comparable to that in Tokyo and yet salaries ranged from an utterly derisory 200 to a merely inadequate 2,000 dollars per month. Members of parliament, police officers, senior civil servants, and army commanders all came to depend on the president in person, who had pockets deep enough to award those whom he favored an annual Christmas bonus that was sometimes as high as 25,000 dollars, the equivalent of ten years' salary for a junior government employee. The sweetening of those on whom the regime depended was matched by the crushing of those who might dissent. During 1998, as the expectation of a new civil war rose to a certainty, the presidential office increased the range of organizations that became dependent on its bounty and were therefore trapped into silent complicity. Benefactions were used both to minimize grassroots protest from the hungry slums, which profoundly feared a return to war, and to manipulate the factions that kept the traditional cadres of the MPLA in disputatious disarray.
One of the small institutionalized steps on the road to totalitarian presidentialism in Angola had been the creation in 1996 of the Eduardo dos Santos Foundation. The foundation was designed to implement a widespread policy of privatizing the assets of the state in order that they could be used to consolidate the power of the president rather than meet any of the more objectively assessed political needs of the nation. The idea was far from new; a similar transfer of state resources from democratically accountable local and central government to quasi-non-governmental agencies run by political favorites had already been undertaken on a large scale in Thatcher's Britain—"selling the family silver," as Harold Macmillan pithily called it. In Angola, however, the process was masked by rather more opaque layers of secrecy and cloaked in even more dubious forms of legality than any adopted in other countries. Privatization policies were politically motivated, designed to prevent the overthrow of the government either by democratic vote in the world's north or by mob restlessness in the countries of the less affluent south. In Angola the president's patrimonial foundation refined the politics of patronage by a further concentration of power in the Futungo palace. The funds of the foundation derived from a presidential "tax" that mirrored the state taxes levied on international trading firms, petroleum prospectors, construction companies, banking corporations, and small domestic businesses. Having creamed off a top slice of the nation's assets, the presidential foundation went into competition with the state to provide services that had ceased to be available through official channels but that now became privately accessible to the president's clients. A private presidential university was set up to compete with the underfunded downtown Agostinho Neto University, which subsequently saw both foreign finance and foreign personnel eroded. Even greater finesse was shown in the case of a home for abandoned children in the suburb of Cacuaco, for which the foundation gained public credit with a small subsidy while the core funding was siphoned out of the city of Luanda's own budget. Some of the largesse reached the provinces, but the presidential bounty was predominantly spent in the city, the political base with the greatest capacity to make or unmake presidents in the event of revolution.
Manipulating power by offering carrots and showing sticks to the elite was rather easier than winning support among the urban masses, who saw poverty as the consequence of widespread corruption at the highest levels. It became necessary to generate "spontaneous" outbursts of popular enthusiasm for the president, but only authorized spontaneity could be tolerated. The restless workers of Sambizanga—the black parish in which the president had been born, but also the parish in which Nito Alves had mounted his 1977 challenge to the government of Agostinho Neto—were persuaded to come down into the asphalt town and demonstrate their loyalty to the president. The spontaneity had been so well prepared that the chanting crowds wore specially prepared T-shirts bearing pictures of "their" president. The mobilization of the dispossessed rapidly soured, however, when the crowds were permitted to search out approved public enemies against whom to vent their rage over their shabby poverty. The first permitted target was an ethnic one—the demonstrators chanted anti-Ovimbundu slogans as they intimidated anyone who had come down from the highland and might have UNITA sympathies. In order to separate out the faithful from the faithless, it was suggested in parliament that identity cards should be issued naming the "tribe" of each bearer, but this calamitous recipe for urban warfare was not carried out. By 1996 the orchestrated politics of violence were extended to include xenophobia, and crowds were permitted to attack anyone who might be branded as foreign. A government campaign against aliens was given the chilling code name "Cancer Two," and the search for enemies was directed not only at Africans, particularly "Zaireans" from Congo, but also at the communities of Lebanese and other Asian businessmen, whom the population saw as exploiters but whom the president's men now wished to supplant in the lucrative import-export sector of wholesale trade.
While corruption was being orchestrated by politicians down in the city, the highlands were getting ready for war. By the end of 1996 it was estimated that Savimbi's war chest had grown to two billion U.S. dollars and that he had recently been able to buy another 450 tons of weapons flown in from Bulgaria to the airstrip that UNITA conscripts had built near Bailundu. At this time no less than 20,000 of Angola's government troops were being tied down in Cabinda, where three armed secessionist movements were threatening the security of the oil wells. Each movement had the potential to secure active support from Angola's northern neighbors, Congo-Brazzaville and Congo-Zaire, either of which would gladly have conquered Cabinda. In May 1997 this foreign situation suddenly altered when the thirty-year-old dictatorship of Mobutu collapsed in Congo-Zaire and a new military dictator, Laurent Kabila, who had a shadowy past in the Lumumba era, took control of Kinshasa and entered into an alliance with the dos Santos government in Luanda. As a result of the change, some 10,000 of Savimbi's troops who had been sheltered by Mobutu in preparation for the next Angolan civil war found themselves temporarily stranded, and some tried to seek refuge in Congo-Brazzaville. Within a month of the Kinshasa revolution a similar revolution broke out in Brazzaville, followed by four months of a peculiarly savage civil war in which 10,000 townsmen were killed. To ensure that the outcome did not threaten its own security, Angola sent an army into Congo-Brazzaville and occupied the oil port of Pointe Noire. The troops also enabled General Sasso—with the tacit connivance of oil interests in both France and the United States—to overthrow the elected Brazzaville government despite mercenary interventions from new international players such as Uzbekistan and Croatia. The turbulence in Kinshasa and Brazzaville disrupted UNITA's war preparation, but during 1998 Savimbi retrieved his scattered units and some highly trained members of Mobutu's fleeing presidential guard and mobilized a force of 15,000 trained men and 10,000 auxiliary conscripts. He also recruited some of the genocidal Rwanda militants who were hiding in Congo-Kinshasa, some orphaned military companies that had lost out in the civil war in Brazzaville, and some Serbian mercenaries; and he commissioned Morocco to train a new officer corps for UNITA to replace those generals who had been handsomely seduced into moving to Luanda to set up a renegade UNITA faction in the city.
The dos Santos government prepared for war as actively as did UNITA in the months that followed the apparently accidental death of the UN peacekeeper Alioune Beye in June 1998. Thirty battalions were deployed around the country, and an air force equipped with Brazilian jets was put on standby near Benguela, ready to strafe the highlands. Spanish counterinsurgency specialists retrained 25,000 commandos in special police units prepared to repress any civilian unrest caused by war. The city politicians hoped quickly and definitively to drive UNITA's forces out of the country and into Zambia. The city generals aspired to capture the Kwango valley, where the most plentiful alluvial diamonds were to be found. In the last weeks of 1998 a section of dos Santos's army persuaded the president that any further delay in dealing a death blow to Savimbi's forces would be strategically foolish. Savimbi had been arming so heavily, however, that it was already too late to strike a surprise winning blow. The government forces, inappropriately armed and inadequately trained, were fiercely repulsed when they tried to take the highlands. During the first half of 1999 UNITA held the military advantage, and even its reluctant recruits, kidnapped from nominally friendly Ovimbundu territory, fought for their lives, terrified that if they lost the war to the _mestiços_ of the city they would be packed off to the lowlands as despised farm labor. The civil war of 1998 became the cruellest yet seen in Angola. UNITA starved the cities, notably Malange and Kwito, by refusing to allow humanitarian food supplies to be flown in by the international agencies. It hoped that the Angolan government would be forced by world opinion to stop a war that was killing thousands of civilians. Savimbi may also have hoped that the coast would rise up in revolt as new waves of displaced persons descended from the shattered highland towns. But the world, fascinated by the wealth of Angola's oil wells, did not press the government to negotiate a peace, and the civilians did not risk mounting any public protest when their streets were patrolled by black-clad security police.
The depraved war between a government mesmerized by wealth and an opposition obsessed by power carried on throughout 1999 and into 2000. In some successful engagements UNITA captured government weapons, but a shortage of fuel caused it serious logistical difficulties. One solution was to buy diesel covertly from dealers in the enemy camp. Personal relations across the divide between the two warring elites were much closer than ethnic or ideological enmity would have led one to suppose; successive peace negotiations had accustomed rival delegations to do business with one another while drinking together in expensive nightclubs staffed by the seductive hostesses of Abidjan or Lusaka. For UNITA to buy fuel on a black market run by enemy officers required a large supply of fresh diamonds, and in the late 1990s it was estimated that 100,000 men and women were being forced to dig the cold alluvial mud of the Kwango River for some of the world's most sought-after gem diamonds. Although Angola's diamonds earned only about one-tenth of the seven billion dollars a year derived from oil, a significant proportion of them were marketed by UNITA, bypassing official cartel channels licensed by De Beers and enabling Savimbi to continue military operations after cold war funding had ceased. With diamond money UNITA leaders were able to win support from French client regimes in Burkina Fasso, Togo, and Ivory Coast, all of which provided them with travel documents. So much blood money became involved in the sale of Angola's diamonds, as in the sale of those from Sierra Leone, that the United Nations imposed penalties on nations that facilitated the diamonds-for-weapons trade. At the same time De Beers feared that if it did not stop the diamond cutters and polishers from buying bargain-price diamonds from war zones, the world might mount a humanitarian campaign against the wearing of diamonds similar to the one that animal rights activists had used to make the wearing of furs socially unacceptable in western society. Despite all the protests, guns were still flown into highland Angola under cover of darkness, carried by mercenary planes using unsupervised airstrips in countries that were rewarded for closing their eyes. The crisis in diamond sales from Angola became acute only when it was realized that the government supply of legitimate diamonds, dug from a deep-level kimberlite mine, was being enhanced by conflict diamonds that freewheeling generals were buying from their cash-strapped opponents and legitimizing with forged certificates of provenance. President dos Santos, whose daughter held a diamond-dealing license, had to act to make sure that Angolan diamond licenses were above any suspicion of forgery, which might have closed down the industry on both sides of the war zone. As oil prices fell temporarily to ten dollars a barrel, the government was almost as anxious as UNITA to protect diamond export revenues. Although the MPLA had written off four billion of its eleven billion dollars of old war debts, it had been forced, at a time when oil prices were dropping, to buy hugely expensive new weapons with which to conduct the war of 1998. When oil prices recovered, the military tide turned, UNITA lost its highland headquarters in Bailundu, and the fighting was once more concentrated in the dry, empty plains on the borders of Zambia, through which Savimbi moved in his mobile command caravan visiting his shifting guerrilla camps.
The international scramble to obtain a stake in the Angolan oil industry reached gold-rush proportions. The giant exploration companies, those of Britain and France to the fore, calculated that the North Sea and Alaskan fields would run out of viable new reserves in the new century and that it was in the ultradeep concessions off Angola's Atlantic coast that the best prospects were to be expected. Although drilling oil from a seabed two miles deep called for a technology that had not yet been perfected, with underwater stations serviced by automated submarines and flexible extraction pipes attached to surface platforms out in the ocean, the companies were willing to make down payments of 300 million dollars for the right to explore each concession block in Angola's deep waters. In the early months of the new millennium, Luanda's "jungle capitalism," to use Tony Hodges's felicitous phrase, was once more awash with money. The benefits, however, did not trickle down to the people. Schoolteachers continued to be outnumbered two-and-a-half to one by soldiers, while the elite spent a generous share of the national education budget on sending its children abroad to obtain a sound education. Voices of complaint, including that of the editor of the one significant independent news sheet in Luanda, were silenced, apparently by MPLA death squads, much as newspapermen had previously been murdered in Huambo by UNITA death squads. In the countryside the totalitarian savagery of UNITA continued unabated with the kidnapping of all available children for military duty and the burning of dissidents after accusations of witchcraft. While the slaughter went on in the highlands, members of Savimbi's own family sheltered in a haven of exile controversially afforded to them by the West African president of the Republic of Togo. In the city oil continued to be the fuel that inflamed civil war, as it had been since the dawn of independence on November 11, 1975.
By the year 2000 Angola had come full circle in the thirty years since the death of Salazar, Portugal's old dictator. The civil wars of the 1990s, like the colonial wars of the 1960s, had reached a stalemate. The lives of many people were disrupted, but no solution to the military confrontation between the central system of government and the guerrillas on the periphery seemed in sight. The economy had changed from a dependence on the fluctuating price of coffee to a dependence on the equally unpredictable price of petroleum. In neither case was the industrial sector of production able to cushion the country significantly against the uncertainties of the world market for raw commodities. Politics in 2000 was as unresponsive to public opinion as it had been in 1969, though the dictator who balanced the powers of the several factions of the property-owning class was now a member of the homegrown Luso-African elite of Luanda rather than the imperially oriented haute bourgeoisie of fascist Portugal. In each case the army kept an eye on political decision making and a finger in the economic pie. Senior officers in the colonial army of the 1960s built their wealth on a black market underpinned by coffee exports and currency speculation and invested it in real estate in Lisbon. In the national army of the 1990s officers dominated the now-privatized trade in diamonds and invested their wealth in the Luanda housing market, earning large fortunes as landlords to the foreign employees of oil companies, diplomatic missions, and philanthropic aid agencies. Wealth was as sharply polarized in 2000 as it had been in late colonial times, but the city slums had grown from half a million established members of the _musseque_ families to four million displaced transients camped on the Luanda coastal plain. The colonial class of three hundred thousand privileged and semiprivileged expatriates had been replaced by a similar number of black Portuguese-speaking Angolans who retained many of the old colonial attitudes of social and moral superiority and worshipped in the same Catholic churches that had sustained Salazar's brand of authoritarianism. On the streets the Angolan press of the 1990s was as circumscribed in its news and opinions as ever the censored press of the 1960s had been, and Angolan citizens who held political views were as wary of the political police as colonial subjects had been when trying to evade Salazar's secret agents. Freedom of opinion and of opportunity, which had been stifled in the days of empire, proved virtually incapable of resuscitation in the era of liberation.
## 11
## A Journey through Angola
_In May 2003 I was invited to accompany four members of a British parliamentary delegation to Angola. This personal account of some of the things we saw and heard represents a background to the official all-party report_. _It also draws on the short piece published in the house journal of the Royal Institute of International Affairs, whose staff, attached to the British Angola Forum, organized the visit. Our purpose was to witness the reconstruction and resettlement being attempted in both urban and rural Angola one year after the end of the series of wars chronicled in this book. I am very grateful to Hilton Dawson, Andrew Robathan, Tony Colman, and Francis Listowel for their companionship; their indefatigable professionalism quite restored my faith in politicians_.
The most unexpected aspect of postwar Angola in May 2003 is the vibrancy of the free press. Every Saturday the streets of the _cidade asfaltada_ (asphalt city) are alive with runners selling no less than five titles. The competition is fierce as editors struggle to devise ever more eye-catching stories of financial malfeasance and political infighting. The most noted, and most persecuted, of the weeklies so suffered from the strain that some of its editorial board members left and two rival papers are now on the street: the _Angolense_ , number 226, boasting its sixth year of survival, and the _Semanário Angolense_ , number 11, in its first year of publication.
Government attempts at curbing the explosion of pent-up frustrations that seethe among the cultured Angolan middle classes have been a mixture of the crude and the subtle. The attempt at direct prepublication censorship seems to have backfired. When political newspapers appeared with blank spaces, reader curiosity knew no bounds and samizdat photocopies of offending articles, which had often previously appeared in the Portuguese press, were passed from hand to hand through business and government offices. Editors seem not to have adopted the ingenious scam of the old _Central African Examiner_ , which, in the first days of Ian Smith's Rhodesian rebellion, offered prizes to readers who correctly filled in the censor's blank spaces. The government in Luanda backtracked on censorship, not having the skills of the old Salazar dictatorship, which filled blank spaces with government copy tailored to fit the gaps. More covert forms of pressure are therefore applied by latter-day minders of the public consciousness.
The big story, which has run and run, is "Who are the richest men in Angola?" The names and photographs of the fifty-nine top candidates are now very much in the public domain. Each is allegedly worth more than fifty million United States dollars, some of them more than one hundred million. The list may not be very accurate either in names or in magnitudes, but its publication has had very interesting consequences. One Angolan entrepreneur took the offending newspaper to court and charged it with destroying his credit rating on the international financial markets. He was, he protested, worth very much more than the alleged one hundred million dollars he was accused of siphoning off the common weal. The paper's editor also went to court and accused the government of libel. The official press, he said, had so intemperately denounced the millionaire story as a slur on the integrity of members of the ruling elite that it had undermined his reputation for journalistic probity. The government, whose supporters featured prominently on the list, referred the story to the attorney general, but he very publicly washed his hands of the matter and passed the case to the criminal police. The destination of Angola's three billion dollars' worth of missing oil earnings is one of the fascinating facets of a new stirring of public awareness.
The defining moment of Angola's loss of innocence came a generation ago, on May 27, 1977. When the younger folk in Luanda feel reasonably safe from the prying ears of the security services, they ask ever more insistently, "Daddy, where were you on May 27?" In a country where most women are politically marginalized, they might even ask mother where she was hiding while the blood flowed in the prisons. The pervasive fear of "preventive detention," which normally reduces freedom of speech to mere freedom of conversation, is based on folk memory of the extensive witch hunts that followed the 1977 attempt by young idealists, including some radical young women, to overthrow pragmatic male graybeards. That was the day when the Angolan dream began to unravel. That was when the old president's cancer began to take a terminal hold. That was when old scores between guerrilla factions were resolved. That was when freedom died. It is therefore astonishing to find the free press in Luanda carrying an old speech by Nito Alves, the martyred hero of the impetuous young who had planned the rebellion of 1977 under the cloak of managing the Sambizanga football club—doubly astonishing since the working-class suburb of Sambizanga, focus of subversion in the popular mind, has once again been without water for five days in May 2003 and people are thirsty, dirty, and cross.
The memory of 1977 remains vivid, tragically refreshed in the autumn of 1992, when Luanda witnessed another bloodbath. In that year the old cold war veteran Jonas Savimbi, the man of violent passions, of towering arrogance, of fiery oratory, had been assured by the United States that if only he would submit himself to a democratic election he would surely gain the power that had eluded him for twenty-six long years. But he and his American sponsors did not win. He was worsted at the polls by the gray José Eduardo dos Santos, whom the voters deemed the less fearsome of the presidential candidates. Savimbi cried foul, but even the most brilliant of psephologists could never have devised a rigged result as complex, surprising, sophisticated, and varied as the verdict given by the people of Angola when they turned out in their full millions to take part in the country's one and only general election. It was back to the gun for Savimbi, but before he could strike the government launched a virulent preemptive blow. Those too young to remember the killings of 1977 vividly remember the killings of 1992, and when conversation in public places turns to politics local people fall eerily silent. As the embassy's armored car, with its Gurkha backup, drives me through the night city to the airport, key points of that urban war are pointed out. At notorious intersections cars thought to contain opposition sympathizers had been stopped, the occupants shot, and the vehicles set ablaze. Ten years later, when peace was finally signed over Savimbi's remote grave at the end of the earth, the remnants of his UNITA movement donned three-piece suits and moved into the city. The leaders began to seek a coherent political purpose and to use the new columns of the independent weekly press.
Freedom of the press may irritate the Angolan government, but it does not really threaten it. The newspapers are far too expensive for most ordinary salaried workers, and the expatriate business and diplomatic corps is not sophisticated enough to read between the editorial lines. The real requirement for the building of an open society, a civil society, a democratic society, an informed society is the radio. It was radio that got the nationalist movements up and running in Angola in the 1960s; it was a broadcasting station that was the focus of confrontation between the coup leaders of 1977 and the Cuban expeditionary force, which silenced their microphones; it was UNITA's Voice of the Black Cockerel that kept Savimbi's name alive during the civil wars. In 2003 the capacity and willingness of the Catholic Church to finance and shelter a radio station that asked impertinent questions causes the government qualms, and occasionally bishops have to go to the Bunker to apologize to the president's men. In this power game embassies play an interesting role. At one level an embassy is there to ensure that business contracts fall to its home country's citizens, though in truth only Portugal and Brazil have the cultural and linguistic fluency to score well at this game. The other role of an embassy is a mildly subversive one as diplomats daily test the democratic temperature. In this game the powerless patronize the diplomatic cocktail circuit to discuss political openness and fiscal transparency with hosts who are excluded from the closed magic circle of Luanda's power brokers. The newfound voices of "civil society" provide a glimmer of hope that politics may change; if these voices could gain access to broadcasting, debate might really become a part of the political scene. But when the 2003 parliamentary delegation from Britain visited a church radio station, its members found a palace general waiting for them in the editor's office.
The parliamentary visitors of 2003 are met at midnight by a formally attired ambassador who begins their briefing in the VIP lounge at Luanda airport while efficient airport minions expedite luggage and paperwork. And so through lit but potholed streets to the Tropico Hotel, lavishly rebuilt home of visiting oil executives, diamond smugglers, Bretton Woods bankers, and opposition turncoats and the drinking hole for all the semiwealthy flotsam of the asphalt lower city who wish to keep their ears tuned to the groundswell of political and financial gossip. Next morning the pink dawn rises as the uniformed flunkeys change the guard at the portico and the street urchins urgently skip up and down the dual carriageway selling newspapers through car windows to commuters. Government policy seems to be a question of how to enlarge the begging bowl and get people around the table for a so-called donor's conference. This agenda is apparently in direct contradiction to the policies of the several hundred nongovernmental agencies that help keep Angola's nose above water level, and whose desire is to devise exit strategies that will leave government to staff and fund the social rehabilitation program that the world's voluntary sector has initiated. Aid fatigue is seen to be imminent. Advisers to government recognize that if fiscal transparency is not improved and the $740 million-a-year leakage of oil revenues to private offshore investments is not stemmed, the world will wash its hands of Angola, starvation or no starvation. The organization Transparency International ranks Angola number 124 in its Corruption Perceptions Index. The country may be rich, but the people are very poor.
Along the once-colonial streets of Luanda's lower town the divide between the haves and the have-nots is conspicuous as one passes the bored excombatants standing as armed guards by the iron grills of parlors where the beautiful ones are having their bouffant hairstyles refreshed. But lift your eyes up to the new De Beers skyscraper and you are liable to fall through a sewage manhole with a broken cover or be crowded by a chauffeur-driven six-cylinder saloon skirting months of accumulated street rubbish. The policeman pirouetting on his upturned tub still directs traffic with a deft flick of the wrist in front of the city hall, while elsewhere intermittent electric traffic lights control chaotic drivers and pedestrians at crowded intersections. The once-ubiquitous taxis with green fenders, symbols of lower-middle-class wellbeing, have vanished, and those who do not have cars line up for rusting privatized buses. Beyond the city the roads to and from the provinces are overburdened with worn-out trucks straining to carry oversized cargoes across makeshift bridges.
In between the tinted Mercedes owner and the matriarchal onion hawker it is difficult to find the voice of middle-class Angola. Especially if you are a mere Briton. One hundred years ago the British ruled Luanda, or at least the service sector and the import-export business. Now Victoria's Portuguese stepchildren have taken over, and self-respecting Angolans do business only with the best—the Portuguese, the former grandmasters, the imperialists whom the nationalists have tamed, the devils whom everyone knows and trusts. Any foreigner who is not Portuguese is mildly despised in Angola's profoundly neocolonial political culture. When an ambassador throws a dinner party, he cannot predict who will turn up: a hungry excombatant, a marginalized politician in a tight-fitting suit, a local functionary hoping to find a bursary. A few articulate agency voices call themselves the "civil society," but their unifying agenda was the call for peace, and now that the guns in all but the ultrarich oil enclave of Cabinda have ceased thundering, the civil society has lost some of its cohesion of purpose. The clearest vision of a new future may come from women, but in Angola the macho pecking order remains very Latinized. The bronze figure of the seventeenth-century Queen Nzinga may stand on the pedestal where Agostinho Neto once commanded the entrance to the Avenue of the Old Veterans, but this symbolism is not reflected in twenty-first-century gender politics.
The enclave of downtown Luanda, with its political intrigue, its sexual gossip, its irrelevant cocktail parties, its billion-dollar loan sharks, is not Angola. To visit the real Angola, removed from the suffocating obsession with power and wealth, you need to travel five or ten miles from the center, beyond the end of the old colonial tarmac. Suddenly you are immersed in a vibrant world of free-market dynamism that is probably as energetic and inventive as any economic system in the world. It has had to be if people and politicians alike were to survive the bizarre distortions that a soi-disant Marxist system of command economics imposed on the country in the 1980s. The survival strategies that women merchants and male artisans developed in those years now serve the country exceedingly well. The hundred-acre open-air workshops are hives of industry where there is virtually nothing that cannot be made or bartered by Angolans of incredible ingenuity. When local supplies of recycled scrap fail, raw materials come in on the oil budget, plywood from the plundered forests of Indonesia, calico from the cotton mills of Congo, roofing nails from the iron furnaces of Europe. A Heath Robinson (what Americans might call a Rube Goldberg) diesel generator has been hauled in by a shrewd speculator who sells power through a web of tangled cables to hundreds of open-air carpenters making every conceivable item of furniture or style of roofing timber, not to mention fleets of wooden wheelbarrows. The metals section of the market is filled with blacksmiths who can turn wrecked cars into workshop tools or domestic utensils. Portly businesswomen sit under huge colored umbrellas selling everything from small brass screws to heavy headloads of cotton prints displayed at a dollar a yard. Up on the cliff, beyond the great workshops, the thousand-acre market known as Tira Bikini will have the bikini off your back and resell it before you can bat an eyelid. There the market women reach far beyond the neighboring countries of Central Africa and wholesalers have tentacles that stretch from Brazil to Dubai. There is nothing that cannot be bartered, be it a window motor for your air-conditioned Mercedes or half a gross of plastic sandals to be hawked through the township alleyways.
There are difficulties to be overcome in this wild world of free enterprise. Death from disease stalks stallholders and customers alike when they work all day in beating sun with neither a water supply nor a latrine for miles around. The big social insurance associations are the ones concerned with funeral expenses, which might at any moment suddenly impose great financial burdens on any small family firm. But savings systems for community tragedies are not the only form of inventiveness that Angolans have had to develop. The currency system also requires skill and experience. At the height of the "Marxist" experiment prices were calculated in six-packs of lager and wealth was stored in commodities rather than currencies. Now the home for a market woman's cash is the American dollar. Money changers take their cut, but when inflation runs at about 100 percent, stallholders prefer to change their money back and forth, if necessary every week, to preserve value while at the same time having the liquidity for daily transactions in local bundles of banknotes. The system works—Angolans are shrewd and competent businesspersons—but of late the government has been trying to put its oar in and an intrusive agency, the "fiscal police," is making urban survival more difficult, more dangerous, more needful of ingenuity.
There are some surprises in this world of unbridled capitalism. In a country where more than 40 percent of the black population speaks Portuguese as its preferred language and where many people barely think of themselves as Africans at all, it turns out that adult education in the market suburb is being conducted in French. When a stranger visits a technical college classroom his words of greeting are translated not into the local Kimbundu but into the Lingala lingua franca of western Congo. Enterprise in Greater Luanda often remains in the hands of networks of people who gained their commercial experience either as emigrants during the colonial heyday or as refugees during the colonial and postcolonial wars. The children of returnees from Kinshasa, the Congo metropolis where French is the language of government and Lingala the language of the marketplace, have not yet been reabsorbed into the Lusophone culture of their ancestral homeland.
The four million provincial migrants who have settled around Greater Luanda during the four successive wars that tore the country apart with ever-increasing virulence are not all engaged in business and bustle. Some of them live in tented cities of unspeakable poverty and deprivation, and even the majority who are sheltered by cement-block walls and corrugated roofs have no access to standpipes for their water and no system of sanitation more sophisticated than squatting behind a bush—as the actor Joseph Fiennes discovered when making a publicity film for Christian Aid. This is where the white Land Rover brigade comes into its own. Building public latrines that market stallholders and their customers can use for a penny a visit really is a service that can immediately reduce levels of disease and infection and ought to open the way for a system of local government that actually provides services. One organization goes even further and has devised a prototype system of municipal refuse collection, using old oil drums on swivel sticks that can easily be emptied into a cart drawn by an agency tractor. In Kilamba Kiaxi, where the baobab stands under which the founding hero of the nation allegedly wrote up his plan for a liberation war, municipal seminars are held at which people publicly speak out on the subject of latrines and refuse collection. The memory of Agostinho Neto makes the place where this first forum was held into hallowed ground to which politicians, and even the old hero's widow herself, come to drink at the fonthead of political wisdom. The idea of holding local debates might—or might not—catch on in less privileged places and become the beginning of participatory democracy in a city accustomed to the old Portuguese practice of rule by fascist decree. Sooner or later elections will be held, and meeting the newly expressed demand for public service could be good for electoral advantage.
Water is the big political issue in Luanda. Historically the city sent canoes up the coast to the nearest river to bring down more or less drinkable water. Nowadays water is privately distributed by tanker truck to all except the privileged few who receive it, effectively cost free though sometimes only intermittently, by municipal pipe. The cost of water is normally about a penny a liter, but in one camp, to which beggars and their dependents had been banished to prettify the downtown area, the price jumped several-fold when a storm washed a bridge away and the tankers had to make a detour along unreliable mud paths. Even at the best of times water from the entrepreneur who owns the truck may cost a family half of its combined weekly income. Thus it is that another very welcome foreign initiative is the laying of suburban pipelines to the shanty towns, sometime with small decoy pipes laid above the main ones so that when water robbers, who belong to an industry that earns $35 million a year, try to tap into the supply to fill their trucks they will only siphon water from the minor pipe and not destroy the pressure in the heavy-duty mains.
Nightmares are not confined to that half of the population living in cities of displaced persons strung along the coast. The wheeling and dealing of black marketeers may have enabled the swollen coastal slums to survive, but no such options were available in the provinces. Both the third war (1992–94) and the fourth war (1998–2002) were designed to starve and destroy whole populations. The victims of the third war still live in the shells of cities that had been blasted by UNITA for failing to vote for Savimbi in the electoral beauty contest of 1992. Huambo, the highland city built between the 1920s and 1960s as the Benguela railway workshop and the hub of a planter economy, remains shattered, although a road up from the coast is now navigated by intrepid truckers and a road down from the interior has been cleared by the Halo Trust, an international mine-disposal agency. A year after the peace was signed, survivors still huddled in encampments over which the ruling party hoisted its partisan flag of victory rather than a national flag of reconciliation. Opposition regiments are being demobilized, though not with quite the care that had been promised in the form of resettlement kits, tools, seeds, and blankets, let alone any training for civilian jobs. Even the limited support given to former combatants was reserved for men, and little recognition was afforded to the women who had been enrolled into the armies to provide every variety of domestic, portering, and sexual service to the boys in the bush. As for children in army service, the government refused to recognize their existence or to grant them demobilization papers. Adolescents remain at risk, after being informally demobilized, of being formally mobilized all over again when they reach conscription age. They fear the city generals sent up from the coast to rule the wilderness.
Only four million of Angola's people live in Luanda. The other ten million do not, and they have not had a good war. In the colonial war (1961–74) many of their fathers and mothers were conscripted as ultracheap labor to work for a pittance on settler farms and in the mines. When city politicians, both in government and in opposition, dream about a golden past flowing with milk and honey and rich in coffee and diamonds, rural people become apprehensive. In the cold war (1975–91) matters got worse and conscripted peasants had to serve whichever army caught them first, to do the bidding of Moscow and Havana or of Washington and Pretoria. The first civil war (1992–94) was more murderous again, though this time it was towns that were destroyed, particularly the highland cities that had "betrayed" Savimbi by voting to join ranks with the wealthy urban elites on the coast. By the time of the second civil war (1998-2002) survival strategies were wearing thin and the eternal optimism for which Africa is noted was becoming threadbare. It was this last war that caused the greatest distress to farmers on the wide eastern plains. Along the upper Zambezi peasants had been compelled to feed UNITA irregulars, and so the Luanda government bluntly decided to starve half a million people across the whole region to make sure that there was no food that the "bandits" could steal. Such brutal policies caused some farming families to flee to Zambia in search of vacant common land on which to subsist. The lucky ones found employment on Zambian farms or were sheltered in refugee camps sustained with flour, beans, cooking oil, and salt by the World Food Program. The less fortunate offended their hosts by accepting menial ill-paid jobs that undercut the wage-bargaining power of local Zambian laborers.
Now that the war is over, people from the Zambian camps are moving back to the small towns inside the Angolan border and activity is beginning to hum in the villages. Two hundred children in a country-town orphanage, half of them resident in double bunks with mosquito nets and half coming in from local families where they have been socially placed, start the day by raucously singing a Portuguese adaptation of the old song "If you're happy and you know it clap your hands, if you're happy and you know it nod your head, if you're happy and you know it—and you _really_ want to show it—stamp your feet...." Each verse has vigorous gestures, and when it is led by a British earl and a Labour parliamentarian the shining faces of the children are marvelous to behold. If infant mortality in Angola is to be reduced, many more mosquito nets will be needed and government will have to change some of its spending priorities to supply the chemicals to treat them. If the half of Angola's children who get no schooling at all are to be enrolled in classes such as the orphanage provides, education budgets will have to be introduced into all eighteen provinces.
Although peace finally arrived in 2002, the peace dividend appears to be spent in town rather than spread across the provinces. Since half of all Angola's people are children, the biggest peacetime demand is for education. Those within reach of a school find conditions Spartan: blackboard and chalk, one teacher for each hundred new entrants, very few textbooks, no exercise books. One such school, with an enthusiastic enrolment of pint-sized tots, was reached by Britain's parliamentary visitors after they had crossed the swollen Zambezi in a very small dugout canoe. The school, together with a small leper colony, is being supported by an evangelical aid agency. Elsewhere Unicef is making up for forty lost years in a country where parents had sometimes gained a little colonial schooling but the children of the war are illiterate. In the two heartland provinces of Malange and Huambo, 250,000 children need schools and several thousand teachers need training. Now that the war is over, however, every adult who did not manage to get to the coastal cities as a refugee hopes to get there as a job seeker. No Angolan, whether a teacher or any other ambitious citizen, wants a posting to the countryside. Urban drift is universal, not simply a feature of war.
If education is one responsibility that government cannot indefinitely delegate to philanthropic agencies, health is another. As exiles test the viability of a return to their ancestral villages, the question of health provision will be much on their minds before they bring parents and children home to Angola. The task of providing even minimal care in communities of returnees will be made difficult by bureaucratic obstructionism. The qualifications gained by Angolans in Congo or in Zambia are sneered at by petty patriots who use bureaucratic chicanery to protect their jobs, their status, their self-perceived superiority. Already Angola suffers from a chronic brain drain as qualified Angolan teachers, nurses, and administrators leave to earn more reliable salaries in Johannesburg, Lausanne, Baltimore, or Lisbon. White foreigners in white Land Rovers can temporarily help to bridge the gap in health provision and initiate the necessary vaccination and inoculation campaigns that will protect one cohort of infants. In 2003 seven million children were vaccinated in an effort to reduce Angola's horrendous infant mortality rate of 25 percent. Suitable long-term health plans need to be adopted by government.
The bombshell that might hit Angola now that peace has facilitated the free movement of people is a potential epidemic of HIV infection. No figures on current infection levels are reliable, though the Luanda maternity hospital did report an 8 percent incidence in 2001. The difficult balancing act will be to make people aware of the risks, including the risks brought by long-distance truckers now coming across the borders from highly infected areas in Namibia and Zambia, while avoiding starting a witch hunt that will target returnees from asylum as potential carriers of AIDS. Witch hunting has suddenly become not only a metaphor in Angola, but also a widely present practice. As people try to understand their malaise, their poverty, their ill health one year after the guns fell silent, they look to witchcraft as the evil force that blights their lives. The risks of AIDS are carried not only by "witches" and by returnees from countries where labor migration has fostered focal points of sexually transmitted infection, but also by men and women coming out of armies where unprotected sexual activity was rife. It is among the demobilized and among the cohorts of juveniles who never quite reached the age of conscription that HIV-awareness education needs to be most intense. Luanda's prime-time television showed a demure young lady stopping men on the street and asking them if they knew how to roll on a condom. Replies varied from Blimpish outrage at the very thought to open willingness to discuss the issue on camera. In the villages street theater takes the place of television and demonstrators get through a considerable supply of wooden penises during their performances before issuing free demonstration supplies in crowded rustic discos.
Street theater is also used to raise awareness of land mines and other abandoned ordnance that litter the roadside verges of the Angolan battlefields. Whether the number of land mines the armies left behind is one million or ten million is not known, but in some areas the risks are very real. Highly professional and experienced Angolan staff are clearing the highways, making the delivery of heavy food cheap by truck rather than expensive by air. Opening up farmland closed by minefields in order to coerce starving farmers to become army camp followers will take longer. Where the risk of mines is low and soil fertility is adequate, peasant men as well as women are hard at work. A return to the best maize land will, however, be bedeviled by legal disputes over title. Displaced farmers, returning refugees, demobilized conscripts all fear that any rich, well-watered tracts may fall into the hands of the country's ambitious generals with their cohorts of slick city lawyers and their blueprints for a return to colonial-style plantations.
In Angola the layers of government—municipal, provincial, and central—are not yet noted as purveyors of well-being. The country languishes alongside Uzbekistan at the bottom of the Human Development Index for oil-producing countries. The situation in the churches is somewhat more promising than in government. The Catholic Church runs its semifree radio station, but has not risked broadcasting any whispers of "liberation theology" and some radical priests with egalitarian sympathies—notably those from the Basque provinces of Spain—have been compelled to take home leave. In the last phase of the war one archbishop did win the Sakharov Prize for speaking out in favor of peace, liberty, and human rights. Other churchmen, however, aspire to join the generals in the ranks of the establishment and enroll their dependents and supporters into the private, very expensive Catholic university. The Methodist Church, historically the spiritual home of a radical, at one time even Marxist, segment of the city's elite, continues to flourish, and pew space was at a premium on Mother's Day when the venerable old bishop, who survived both the colonial oppression and the Marxist austerity, was in attendance alongside his young successor. Empowering the poor has been the agenda some of the churches have proposed. The more dynamic churches are Pentecostal, and it is to them that people have flocked during the years of insecurity. Congregations have mushroomed in ethnically mixed communities that have no traditional institutions of conflict resolution and where the church replaces kinship as the focus for a reconstructed society. Evangelical churches also flourish in the provinces, and Britain's parliamentary visitors were welcomed in the tradition of the Plymouth Brethren by a local pastor in one of the remotest villages in all Angola. He spoke no Portuguese but gently recounted in Lovale how his community was beginning to come back from the forests and rethatch its clay-built houses, harvest its surviving groves of cooking bananas, and rebuild its cultivation mounds of rich Zambezi alluvium. On such quiet firmness is the future of Angola based. Problems there may be, failings there may be too, but the spirit of hope lives on.
## Notes
Chapter 1
For details of the creation of the Atlantic colonial system see David Birmingham, _Trade and Empire in the Atlantic, 1400–1600_ (London: Routledge, 2000).
Chapter 2
José Mendes Ribeiro Norton de Matos, _Memórias e trabalhos da minha vida_ , 4 vols. (Lisbon: Editora Marítimo-Colonial, 1944–45).
José Capela, _O vinho para o preto: notas e textos sobre a exportação do vinho para África_ (Oporto: Afrontamento, 1973).
Charles von Onselen, _Studies in the Social and Economic History of the Witwatersrand_ , 2 vols. (London: Longman, 1982). For a wider illumination of the role of alcohol in southern African society see Jonathan Crush and Charles Ambler, eds., _Liquor and Labor in Southern Africa_ (Athens: Ohio University Press, 1992).
William A. Cadbury, _Labour in Portuguese West Africa_ (London: G. Routledge, 1909).
An English edition of General Delgado's memoirs is available: Humberto da Silva Delgado, _The Memoirs of General Delgado_ (London: Cassell, 1964); as well as a revised Portuguese edition: _Memórias de Humberto Delgado,_ ed. Iva Delgado and Antonio de Figueiredo (Lisbon: Dom Quixote, 1991).
Chapter 3
David Birmingham, "Merchants and Missionaries in Angola," _Lusotopie_ (1998): 345–55.
The latest and perhaps most lively of Beatrix Heintze's studies is _Pioneiros Africanos: caravanas de carregadores na África Centro-Ocidental entre 1850 e 1890_ (Lisbon: Caminho, 2004).
David Livingstone, _Missionary Travels and Researches in South Africa_ (London: J. Murray, 1857). For a gloss on Livingstone's visit to Angola see David Birmingham, _Portugal and Africa_ (Athens: Ohio University Press, 2004).
Mary Kingsley, _West African Studies_ (London: Macmillan, 1899).
The family biography of Héli Chatelain was written by his sister: Alida Chatelain and Amy Roch, _Héli Chatelain, l'ami de l'Angola, 1859–1908, fondateur de la Mission philafricaine d'après sa correspondence_ (Lausanne: Mission Philafricaine, 1918). She makes extensive but selective use of his letter books. By very good fortune these letter books and other papers have survived in the archives of the Alliance Missionnaire Evangélique held by the Schweizer Allianz Mission in Winterthur. I am exceedingly grateful to the mission, and particularly to Albert Zimmerli, for the welcome hospitality that they have afforded me when working in Switzerland. Chapter 4 of the present book makes use of these papers to explore the second half of Chatelain's career and may, I hope, form the basis for a wider study of his life and times both as a member of the Luanda "Methodist" mission and as the founder of the Swiss mission at Kalukembe.
António Francisco Ferreira da Silva Porto, _Viagens e apontamentos de um Portuense em África,_ ed. Maria Emília Madeira Santos (Coimbra: Coimbra University, 1986) is the first—and so far only—edited volume of Silva Porto's extensive archive. Frederick Stanley Arnot, _Missionary Travels in Central Africa_ (London: Alfred Holness, 1914).
Chapter 4
In _A África e a instalacao do sistema colonial,_ ed. Maria Emília Madeira Santos, 418–29 (Lisbon: IICT, 2000).
For a recent study of the Basel mission see Sonia Abun-Nasr, _Afrikaner und Missionar: Die Lebensgeschichte von David Asante_ (Basel: Schlettwein, 2003).
Henry W. Nevinson, _A Modern Slavery_ (London: Harper, 1906; reprint, Essex: Background Books, 1963). Nevinson's essays were first published in _Harper's_ magazine in 1906.
Chapter 5
For further details and sources on Belgium and Portugal see David Birmingham, Muriel Chamberlain, and Chantal Metzger, _L'Europe et l'Afrique de 1914 à 1970_ (Paris: SEDES, 1994), and also volume 2 of David Birmingham and Phyllis M. Martin, _History of Central Africa_ (London: Longman, 1983).
See "The Coffee Barons of Cazengo," in Birmingham, _Portugal and Africa_.
Chapter 6
Christine Messiant, _L'Angola colonial: histoire et société_ (Basel: Schlettwein, forthcoming).
Basil Davidson, _The African Awakening_ (London: Cape, 1955).
Chapter 7
Marcelo Bittencourt, _"Estamos juntos": o MPLA e a luta anticolonial_ (Luanda: Angolan National Archives, forthcoming).
Drumond Jaime and Helder Barber, eds., _Angola: depoimentos para a história recente_ , vol. 1 (Lisbon: Drumond Jaime and Helder Barber, 1999).
Chapter 8
David Birmingham, _Frontline Nationalism in Angola and Mozambique_ (Trenton, NJ: Africa World Press; London: James Currey, 1992).
Fred Bridgland, _Jonas Savimbi: A Key to Africa_ (Edinburgh: Mainstream, 1986).
Chapter 9
David Birmingham, "O carnaval em Luanda," _Análise Social_ 26 (1991): 417–29; Birmingham, "Carnival at Luanda," _Journal of African History_ 29, no. 1 (1988): 93–103.
Ralph Delgado, _História de Angola,_ 4 vols. (Lobito: Ralph Delgado, 1948–53). Two promised later volumes appear not to have been completed, but Delgado did also write books on Benguela and its hinterland, both as history and in the form of a novel, _O Amor a 12 graus de latitude sul_ (Oporto: Emprêsa Industrial Gráfica, 1935).
Rui Duarte de Carvalho, "Ana Manda—les enfants du filet: identité collective, créativité sociale et production de la différence culturelle: un cas Muxiluanda" (PhD diss., L'École des Hautes Etudes en Sciences Sociales, Paris, 1986).
Terence Ranger, _Dance and Society in Eastern Africa, 1890–1970: The Beni Ngoma_ (Berkeley: University of California Press; London: Heinemann, 1975).
Chapter 10
Patrick Chabal, ed., _A History of Postcolonial Lusophone Africa_ (Bloomington: Indiana University Press; London: Christopher Hurst, 2002).
Christine Messiant, "The Eduardo dos Santos Foundation: Or, how Angola's Regime is Taking over Civil Society," _African Affairs_ 100 (2001): 287–309.
Tony Hodges, _Angola: From Afro-Stalinism to Petro-Diamond Capitalism_ (Bloomington: Indiana University Press; Oxford: James Currey, 2001).
Chapter 11
All-Party Parliamentary Group for Angola, _Impressions and Recommendations on a Visit to Angola, 3–10 May 2003_ (London: Royal Institute for International Affairs, 2003).
Transparency International Corruption Perceptions Index 2003, <http://www.transparency.org/cpi/2003/cpi2003.en.html>. In the 2004 index Angola was at number 133.
## Further Reading
Birmingham, David. _A Concise History of Portugal_. 2nd ed. Cambridge: Cambridge University Press, 2003. Translated into Portuguese by Ana Mafalda Tello as _História de Portugal: uma perspectiva mundial_ (Lisbon: Terramar, 1998).
—. _The Decolonization of Africa_. Athens: Ohio University Press; London: Routledge, 1995.
—. _Portugal and Africa_. Athens: Ohio University Press, 2004. Translated into Portuguese by Arlindo Barbeitos as _Portugal e Africa_ (Lisbon: Vega, 2003).
—. _Trade and Conflict in Angola: The Mbundu and Their Neighbours under the Influence of the Portuguese, 1483–1790_. Oxford: Clarendon Press, 1966. Translated into Portuguese by João B. Borges as _Alianças e conflitos: os primórdios da ocupação estrangeira em Angola, 1483–1790_ (Luanda: Arquivo Histórico de Angola, 2004).
—. _Trade and Empire in the Atlantic, 1400–1600_. London: Routledge, 2000.
Birmingham, David, and Phyllis M. Martin. _History of Central Africa_. 2 vols. London: Longman, 1983.
—. _History of Central Africa: The Contemporary Years; Since 1960_. London: Longman, 1998.
Bittencourt, Marcelo. _"Estamos juntos": o MPLA e a luta anticolonial_. Luanda: Angolan National Archives, forthcoming
Clarence-Smith, William Gervase. _Slaves, Peasants and Capitalists in Southern Angola, 1840–1926_. Cambridge: Cambridge University Press, 1979.
—. _The Third Portuguese Empire, 1825–1975: A Study in Economic Imperialism_. Manchester: Manchester University Press, 1985.
Coppé, Margrit, and Fergus Power, eds. _Stories for Trees: Stories and Images of Angola_. Luanda: Development Workshop, 2002.
Davidson, Basil. _The African Awakening_. London: Cape, 1955.
—. _The Black Man's Burden: Africa and the Curse of the Nation-State_. New York: Times Books, 1992.
—. _Black Mother: The Years of the African Slave Trade_. Boston: Little, Brown, 1961.
—. _In the Eye of the Storm: Angola's People_. Garden City, NY: Doubleday, 1972.
Dias, Jill. "Angola." In _Nova história da expansão Portuguesa,_ vol. 10, 319–556. Lisbon: Estampa, 1998.
Ferreira, Manuel Ennes. _A indústria em tempo de guerra: Angola 1975–91_. Lisbon: Cosmos, 1999.
Freudenthal, Aïda. "Angola." In _Nova história da expansão Portuguesa,_ vol. 11, 259–417. Lisbon: Estampa, 2001.
—. _Arimos e fazendas: a transição agrária em Angola_. Luanda: Caxinde, 2005.
Heintze, Beatrix. _Pioneiros Africanos: Caravanas de carregadores na África Centro-Ocidental entre 1850 e 1890_. Lisbon: Caminho, 2004.
Jaime, Drumond, and Helder Barber, eds. _Angola: depoimentos para a história recente_. Vol. 1. Lisbon: Drumond Jaime and Helder Barber, 1999.
Kapuscinski, Ryszard. _Another Day of Life_. Translated by William R. Brand and Katarzyna Mroczkowska-Brand. London: Pan, 1987.
Lara, Lúcio. _Um amplo movimento: itinerário do MPLA através de documentos e anotações de Lúcio Lara_. Vol. 1. Luanda: Ruth and Lúcio Lara, 1997.
Mabeko-Tali, Jean-Michel. _Barbares et citoyens: l'identité nationale à l'épreuve des transitions africaines: Congo-Brazzaville, Angola_. Paris: Harmattan, 2005.
—. _Dissidências e poder de estado: o MPLA perante si próprio, 1962–1977_. 2 vols. Translated from French to Portuguese by Manuel Ruas. Luanda: Editorial Nzila, 2001.
Maier, Karl. _Angola: Promises and Lies_. London: Serif, 1996.
Marcum, John A. _The Angolan Revolution_. 2 vols. Cambridge, MA: MIT Press, 1969–1978.
Marques, João Pedro. _The Sounds of Silence: Nineteenth-Century Portugal and the Abolition of the Slave Trade_. Translated by Richard Wall. Oxford: Berghahn, 2005.
Martin, Phyllis M. _Leisure and Society in Colonial Brazzaville_. Cambridge: Cambridge University Press, 1995.
Mendes, Pedro Rosa. _Baía dos tigres_. Lisbon: Dom Quixote, 1999. Translated by Clifford Landers as _Bay of Tigers: An African Odyssey_. Orlando: Harcourt, 2003.
Messiant, Christine. _L'Angola colonial: histoire et société_. Basel: Schlettwein, forthcoming.
—. "Luanda (1945–1961): colonisés, société coloniale et engagement nationaliste." In _Bourgs et villes en Afrique lusophone,_ edited by Michel Cahen, 125–200. Paris: Harmattan, 1989.
—. "Em Angola, até o passado é imprevisível." In _Construindo o passado angolano: as fontes e a sua interpretação,_ 803–59. Lisbon: Comissão Nacional para as Comemorações dos Descobrimentos Portugueses, 2000.
Miller, Joseph C. _Way of Death: Merchant Capitalism and the Angolan Slave Trade, 1730–1830_. Madison: University of Wisconsin Press, 1988. Nascimento, Augusto. _Desterro e contrato: Moçambicanos a caminho de S. Tomí e Principe, 1940–1960_. Maputo: Arquivo Histórico de Moçambique, 2002.
Nevinson, Henry W. _A Modern Slavery_. London: Harper, 1906. Reprint, Essex: Background Books, 1963.
Newitt, M. D. D. _A History of Mozambique_. London: Hurst, 1995.
—. _Portuguese Settlement on the Zambesi: Exploration, Land Tenure and Colonial Rule in East Africa_. Harlow: Longman, 1973.
Péclard, Didier. "Etat colonial, missions chrétiennes et nationalisme en Angola, 1920–1975: aux racines sociales de l'UNITA." PhD diss, IEP Paris, 2005.
Pepetela (Artur Carlos Maurício Pestana). _Mayombe_. Translated by Michael Wolfers. London: Heinemann, 1983.
—. _The Return of the Water Spirit_. Translated by Luís R. Mitras. Oxford: Heinemann, 2002.
—. _Yaka_. Translated by Marga Holness. Oxford: Heinemann, 1996.
Vos, Jelmer. "The Kingdom of Kongo and Its Borderlands." PhD diss., London University, 2005.
Walker, John Frederick. _A Certain Curve of Horn: The Hundred-Year Quest for the Giant Sable Antelope of Angola_. New York: Atlantic Monthly Press, 2002.
The novels of Artur Pestana, who writes under the pen name Pepetela, are some of the most valuable works in the canon of Angolan literature and illuminate the country's historical experience from the seventeenth to the twenty-first centuries as no other source can. Only _Mayombe_ (1983), _Yaka_ (1996), and _Desejo de Kianda_ (The Return of the Water Spirit) (2002) are available in English. The other half-dozen works urgently deserve to be made available to a wider audience, notably the legend of the old tortoise, which graphically displays the wartime experiences of rural Angolans, and the detective adventures of Jaime Bunda, which portray the aspirations and fears of city dwellers.
The essays of Christine Messiant, predominantly in French but a few in English and Portuguese, are the most important collection of academic works on the contemporary history of Angola, beginning with her essay on Luanda (1989) and culminating in her perceptively humorous Portuguese lecture describing how in Angola even the past is unpredictable (2000). The book version of Messiant's doctoral thesis is forthcoming from Schlettwein in Basel and an anthology of her writings is in preparation at Karthala in Paris.
Two of the most important historical studies of Angola to have appeared in Portuguese are in the _Nova história da expansão Portuguesa_. Jill Dias has written the most comprehensive survey so far attempted of nineteenth-century Angola's history. She has drawn on her thirty years of archival research and her long-running editorship of the _Revista Internacional de Estudos Africanos_. Aïda Freudenthal carries the story forward, also at near book length, with perceptive insights carried forward to 1930. More recently she has published _Arimos e fazendas: a transição agrária em Angola_ (2005). João Pedro Marques (2005) has written an important innovative study of the changing imperial attitudes of Portugal in the nineteenth century. The irony of Marques's title, _The Sounds of Silence_ , when referring to Portugal and the abolition of the slave trade, is highlighted by the work of Augusto Nascimento (2002), which documents the continuing transoceanic trade in convicts and exiles in the 1950s. A study of more recent history, focusing on Angola with a good economic emphasis, is by Manuel Ennes Ferreira (1999).
In Germany, Beatrix Heintze's extensive pioneering studies, in Portuguese as well as in German, cover many aspects of Angola's experience from the seventeenth century onward. Her latest book, _Pioneiros Africanos_ , contains brilliant vignettes of daily life on the colonial frontier.
In Britain, two scholars have made great contributions to the understanding of Portuguese imperial activities in Africa. Gervase Clarence-Smith followed up his _Slaves, Peasants and Capitalists in Southern Angola, 1840–1926_ (1979) with a pathbreaking survey, _The Third Portuguese Empire 1825–1975_ (1985). Malyn Newitt similarly followed up his focused study of Zambezi colonization, _Portuguese Settlement on the Zambesi_ (1973), with a comprehensively documented survey, _A History of Mozambique_ (1995). The latest London University doctorate on Angola is by Jelmer Vos, _The Kingdom of Kongo and its Borderlands 1885–1913_ (2005).
My own work on Angola includes a companion volume to the present one, _Portugal and Africa_ (2004), with fifteen essays on topics from Iron Age commercial history to the postcolonial urban rebellion of 1977. Aspects of the comparisons between Angola and its neighbors will be found in the three volumes of the Longman _History of Central Africa_ (1983, 1998), which I edited with Phyllis M. Martin, a historian who wrote the classic study _Leisure and Society in Colonial Brazzaville_ (1995). _Alianças e conflitos_ (2004) is the Portuguese edition of my _Trade and Conflict in Angola_ (1966). For the imperial context of empire see _A Concise History of Portugal_ (2nd ed., 2003), and for the context of decolonization see _The Decolonization of Africa_ (1995). See also my companion volume to _Decolonization_ , a work dealing with the origins of the Atlantic empires, _Trade and Empire in the Atlantic_ , _1400–1600_ (2000).
Scholars in the United States have produced extensive and excellent work on Angola. One of the most fundamental books ever written on the country is Joseph C. Miller's _Way of Death_ (1988). For the twentieth century the classic two-volume study is John A. Marcum's _Angolan Revolution_ (1969–1978). More recently, in a rather different vein, John Frederick Walker came to understand Angola very well when conducting his study of ecological survival in a war zone, a story written up in _A Certain Curve of Horn_ (2002).
Many travelers have recorded their impressions of Angola over many centuries. Henry W. Nevinson's _Modern Slavery_ (1906) is a travel diary whose observations on the nature of early twentieth-century slavery were published in _Harper's_ magazine and confirmed at the time by William A. Cadbury and Héli Chatelain. Basil Davidson's _African Awakening_ (1955) describes a journey into Angola and the Congo by a scholar who went on to observe the region closely for fifty years. Other relevant works by Davidson include _Black Mother_ (1961), _In the Eye of the Storm_ (1972), and _The Black Man's Burden_ (1992), which is perhaps the most ambitious attempt yet to survey the imperial legacy in Africa. A third traveler whose eye leaves a lasting legacy is Ryszard Kapuscinski ( _Another Day of Life,_ 1987) a Polish journalist who experienced the birth pangs of the Angolan republic. Another outstanding journalist came from America to experience the daily life of war-torn Angola and record his sympathetic insights: Karl Maier, _Angola: Promises and Lies_ (1996). A fifth distinguished observer, who saw more of Angola than most and wrote with unrivaled perception, is Pedro Rosa Mendes ( _Baía dos tigres_ , 1999).
Inside Angola, new works of great insight are beginning to be published. _Stories for Trees_ (2002), edited by Margrit Coppé and Fergus Power, contains not only charming vignettes of daily life under the shadow of war but also a fine photographic record of the late twentieth century. A much heavier documentary work, _Angola: depoimentos para a história recente,_ the first volume of which was edited by Drumond Jaime and Helder Barber in 1999, contains transcripts of interviews with many of Angola's leading citizens. One such leading citizen and intellectual was Lúcio Lara, and although he never wrote his own autobiography, his friends assembled a collection of his writings: _Um amplo movimento_ (1997). A two-volume history of the party to which Lara belonged has been published by Jean-Michel Mabeko-Tali, _Dissidências e poder de estado_. A French study by Mabeko-Tali, _Barbares et citoyens_ (2005), compares the search for national identity in Angola and in its northern neighbor, Congo-Brazzaville. A Brazilian dissertation by Marcelo Bittencourt, "Estamos juntos" (to be published by the Angolan National Archives), will provide new insights into the war of decolonization, which stretched Angola's peoples to the limit from 1961 to 1974.
A little fresh archival material, and some new travel experiences, have enlivened the essays in this book. The archives of the Swiss mission in Angola, safely preserved over the years in the mission headquarters at Winterthur, were a fund of inspiration. In Luanda the national archive contains unbelievable quantities of documentation sent to the capital from the provinces at various points in time. The collection that I most recently consulted concerns the highland district of Caconda, where the Swiss polymath, preacher, linguist, and wagon merchant Héli Chatelain built his Christian village. His insights were what inspired me to put this collection of essays together.
## Index
**The index that appeared in the print version of this title was intentionally removed from the eBook. Please use the search function on your eReading device to search for terms of interest. For your reference, the terms that appear in the print index are listed below.**
Active Revolt (Congo-Brazzaville)
Afonso (King)
African National Congress
Afrikaners
agriculture. _See specific crops_
alcohol
and Dutch colonizers
as labor inducement
and labor productivity
production of
and the Taylor mission
trade impacts
Alves, Nito
American League of Friends
Angola
attack on South African passenger jet
Boers in
capitalism, unbridled
churches, political and social roles of
civil war (1992–94)
civil war (1998–2002)
cold war
economic privatization
education
exports
foreign business interests in
free press
health
Human Development Index
inflation
living conditions
oil industry
Protestantism and regional patriotism
regional distrust
rural rebellion, postindependence
South African destabilization
tribal conflict in
in the 21st century
and U.S. policies
wars of destabilization
water, politics of
wealth and status discrepancies
_Angolense_
anticlericalism
Antsee, Margaret
Arab empire
Arnot, Frederick
Asia
assimilados
Axum
Balmer, Alfred
Baptist Church
British
Missionary Society
Barroso, Father
Belgian Congo.
_See also_ Congo Reform Movement
Belgium
Benguela
Benguela Railway Company
Beye, Alioune
Bittencourt, Marcelo
black bourgeoisie
black consciousness movement
Boer War
Boers
in Angola
Boma (Congo)
Botha, P.W
Bowskill, J. S.
Britain
Baptist church
and the Portuguese empire
and South Africa
and the transformation of Africa
British and Foreign Bible Society
Brussels Conference Act
Bush, George H.W.
Cabinda Gulf
Cadbury, William
Cadornega, Antóónio de Oliveira de
Caetano, Marcello
Canary Islands
"Cancer Two,"
Carlos of Saxe-Coburg
carnival
celebration date
and Héli Chatelain
churches' roles in
dances
dance societies
financing
social ills and traumas, exorcism of
social roots
witnesses and actors
Carter, Jimmy
Carthage
Carvalho, Rui Duarte de
Casement, Roger
Catholic Church
in Angola
and Héli Chatelain
in Congo
imperial agenda
and industrial workers
in Portugal
and South Africa
cattle ranching
_Central African Examiner_
Central African revolution
in Angola
in Congo
_See also under specific countries_
Chatelain, Héli
and alcohol trade
and carnival
and Catholic Church
commercial enterprises
mission work
ox wagons
religious beliefs
return to Switzerland
and slavery
China
Chipenda, Daniel
Christianity
introduction to Central Africa
missions
_See also specific churches_
clerical workers
cocoa growing
coffee growing
cold war
Colman, Tony
colonial abuses in Central Africa
colonial crisis of 1908
Congo-Brazzaville
Active Revolt
Congo Reform Movement
Congo state
Congo Reform Movement
and the humanitarian movement
and Leopold II of Saxe-Coburg
Congo-Zaire
Congregationalist Church
conquistadores
contratados
convicts
copper mining
Corruption Perceptions Index
cotton farming
Creoles
in Angola
in South Africa
Cruz, Viriato da
Cuba
Darbyites. _See_ Plymouth Brethren
Davidson, Basil
Dawson, Hilton
De Beers
decolonization
Delgado, Humberto
diamond mining
Dias, Bartholomeu
dizanda dance
dos Santos, (José) Eduardo
Eduardo dos Santos Foundation
Dowie, John Alexander
Dutch empire
and alcohol
in Transvaal
Eco, Umberto, _The Name of the Rose_
Eduardo dos Santos Foundation
education
Protestant schooling
évolués. _See_ clerical workers
fascism
First, Ruth
fishing industry
FNLA. _See_ Frente Nacional de Libertação de Angola (FNLA)
Forsythe, Frederick, _The Dogs of War_
freemasonry
Frelimo. _See_ Frente de Libertação de Moçambique (Frelimo)
Frente de Libertação de Moçambique (Frelimo)
Frente Nacional de Libertação de Angola (FNLA)
frères larges. _See_ Plymouth Brethren
Galvão, Henrique
Gandhi, Mohandas
Germany
gin. _See_ alcohol
"government of national unity,"
Guinea
gunpowder
Heintze, Beatrix
Hertzog, James B. M.
HIV/AIDS
Huambo
humanitarian movement, Central African
India
industry. _See_ specific industries
International Labour Organization
ivory
Jameson raid
Jews
Joaquina, Dona Ana
kabetula dance
Kabila, Laurent
kazukuta dance
Kimbangu, Simon
Kingsley, Mary
Kinshasa (Léopoldville)
Kongo kingdom
Kruger, Paul
labor
agricultural
conditions
recruitment
sources
trade in
wages
and women
working conditions
_See also_ serviçais; slavery
land ownership
Lara, Lúcio
_Leaves of Healing_
Leopold II of Saxe-Coburg
Leuba, Frederic
Liberia
Libreville
Liga Nacional Africana
Listowel, Francis
Livingstone, David
Lobito
Lonrho Corporation
"low-intensity conflicts,"
Luanda
Creole life in
mission (see _also_ Chatelain, Héli; Taylor, William)
Lusaka Accord
lusotropicalism
Machel, Samora
Macmillan, Harold
Marques Pires and Company
Messiant, Christine
Methodist Church
of Angola
mining industry
social impacts
urban work force
missionaries and merchants, cooperation between
Mission Philafricaine
Mondlane, Eduardo
money
Movimento Popular da Libertação de Angola (MPLA)
and churches
civil war
and rural dissaffection
and urban Ovimbundu
Mozambique
and Britain
education
independence
nation building
1960 revolt
and South Africa
South African policies toward
women in colonial society
Mozambique National Resistance. _See_ Resistência Nacional Mocambicana (Renamo)
MPLA. _See_ Movimento Popular da Libertação de Angola (MPLA)
Mueda
Namibia
and Germany
HIV/AIDS
independence struggles
and South Africa
uranium supplies
and U.S. policies
nation building
Nazi ideology
Ndongo kingdom
Neto, Agostinho
Nevinson, Henry
Newton Carnegie and Co.
Norton de Matos, José Mendes Ribeiro
Nova Oeiras
Ntoko Church
oil industry
Oliver, Roland
Onselen, Charles van
Ottoman empire
Ovimbundu
Pan African Congress
Persian Empire
PIDE. _See_ Polícia Internacional de Defesa do Estado (PIDE)
Pieren, Ali
Pinto Andrade, Joachim de
plantation system
Plymouth Brethren
and slavery
Polícia Internacional de Defesa do Estado (PIDE)
Popular Movement for the Liberation of Angola (MPLA)
population growth
and race
Portugal
Catholic Church in
interwar dictatorship
Protestants in
and the slave trade
Portuguese empire
agricultural policies
churches
hybrid culture
lusotropicalism
neocolonial transformation
and the slave trade
press, freedom of
Protestant churches
cooperation with Portuguese colonizers
Free Church of Vaud
imperial agendas
mission churches
State Church of Vaud
_See also specific churches_
Pungo Adongo mission
racism
and assimilation
and social classes
Reagan, Ronald
Real Companhia Vinicula
Renamo. _See_ Resistência Nacional Moçambicana (Renamo)
Resistência Nacional Moçambicana (Renamo)
revolution of 1961 (coffee rebellion)
and economic boom
and international politics
Rhodesia
Rhodesian war
Robathan, Andrew
Roman Empire
Royal Institute of International Affairs
Royal Transafrica Railway Company
rubber industry
rum. _See_ alcohol
Sàà, Salvador de
Saint Louis
Salazar, António de Oliveira
salt industry
São Tomé
Savimbi, Jonas
"scramble for Africa,"
_Semanário Angolense_
serviçais
mortality rate
recruitment and trade
women
Severus, Septimius
Sharpeville massacre
Silva Porto, António da
slavery
Smith, Ian
Smuts, Jan
social classes
and land alienation
and Protestant schooling
and racial attitudes
and social engineering
"social mestiços,"
South Africa
and Angolan destabilization
army
assimilados in
black consciousness movement
Boers
British interests
creolized population
labor controls
labor from Mozambique
mine workers
and Mozambique
and Namibia
Nazi ideology in
and occupation of Angola
and other African nations
passenger jet attacked
population
racism
and the Rhodesian war
women in colonial society
and Zambia
Soviet Union
Spain/Portugal
Stanley, Henry Morton
Studebaker company
sugar cane
Swahili
Taylor, William
and the Methodist Church of America
mission
Tippu Tip
tobacco
Transvaal
tribalism
Ukraine
União Nacional para a Independência Total de Angola (UNITA)
and civil war
and the "government of national unity,"
Unilever
Union Castle
UNITA. _See_ União Nacional para a Independência Total de Angola (UNITA)
United Nations
United States
and Angola
and Namibian uranium supplies
Valdez, José Travassos
vegeculture
Vorster, John
Welsh, C. E.
Wilhelm II (Kaiser)
wine. _See_ alcohol
women in colonial society
political power
and sexual violence
World Council of Churches
Zambia
Angolan refugees
Benguela railway
copper mines
HIV/AIDS
independence struggles
Savimbi in
and South Africa
Zimbabwe
Zion City, Illinois
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Le raïon de Tatarbounary est une subdivision administrative de l'Ukraine, située dans l'oblast d'Odessa et dans la région historique du Boudjak. Le centre administratif est situé à Tatarbounary.
Géographie physique
Le district de Tatarbounary est situé dans la partie sud-ouest de l'oblast d'Odessa distante de . Il est baigné au sud-est par la Mer Noire dont les côtes sont bordées par plusieurs limans : liman de Sassyk, liman de Chagany, liman de Burnas, liman d'Alibeï, liman de Khadjyder et liman de Karatchaous, tous séparés de la mer par un cordon littoral. Ce territoire est considéré comme une zone humide d'importance internationale (Convention de Ramsar)
Il est constitué d'une plaine côtière en très légère pente dans le sens nord-sud, sillonnée par le lit de rivières côtières, certaines intermittentes ou asséchées, aboutissant à un liman : Drakoulia (), Nérouchaï (), Kohylnyk, Kahatch (), Sarata (), Alkalia (), Khadjider ().
La végétation naturelle, poussant sur des terres noires, est de type steppique puis lagunaire (un tiers des plages de l'oblast d'Odessa s'y concentrent).
Les ressources naturelles les plus importantes de la région sont des sources d'eau sulfureuse et de boue curative ainsi que des carrières de pierre de construction (calcaire) et de sable de chantier.
Depuis 2010, le district abrite un parc naturel national, "les limans de Touzlov" ().
La région se caractérise par un climat continental tempéré avec des températures annuelles élevées (210 à sans gel) et de faibles précipitations (moyennes annuelles de 350 à ).
Communications
Le raïon ne comporte pas de réseau ferroviaire. Les gares les plus proches sont situées à Artsyz et à Sarata
L'axe routier international Odessa - Galați (Roumanie) le traverse sur sa longueur, soit 60 kimomètres.
De sa gare routière, par navette (marchroutka), Tararbounary est située sur le trajet de lignes internationales vers Chișinău (Moldavie) ; inter-régionales vers Kiev via Odessa, Kherson et Sébastopol via Odessa et Mykolaïv ; intra-régionales vers Odessa, Chernomorsk, Bilhorod-Dnistrovskyï, Bolhrad, Reni, Izmaïl, Kilia, Vylkové ou directement à certaines localités des raïons limitrophes ; locales à toutes les agglomérations du raïon. Le nombre de rotations varie de deux à trois par semaine à plus d'une dizaine quotidiennes.
Géographie humaine
La population est de , soit une densité de au km². Le raïon comprend 18 structures administratives (une ville et 17 communes rurales) à savoir (les données sont de 2001, sauf indication contraire) :
Nationalités représentées : Ukrainiens (71 %) majoritaires dans les communes citées, sauf indication contraire, Bulgares (12 %), Moldaves (9 %), Russes (6 %), Tsiganes, Gagaouzes, Biélorusses...
Les principaux secteurs d'emploi sont l'agriculture (55 %, céréales, tournesol, viticulture, productions animales), le commerce, les services et loisirs (14 %, stations balnéaires et thermales), l'industrie (1 %), la construction (1 %).
Le taux de chômage est de 3 %.</div> Les retraités représentent 28 % de la population.
</div>
Éducation, culture
Le district compte scolaires secondaires, 18 avec des sections de lycée dont 2 avec des sections technologiques/professionnelles (domaine maritime), un lycée avec des sections artistiques ainsi qu'un établissement médico-pédagogique. Le français est enseigné dans les deux établissements de Tatarbounary et dans celui de Hlyboké.
Il compte également un musée régional ethnologique, hébergeant une exposition permanente de peintres locaux et cinq petits musées locaux.
La presse écrite locale se compose de trois titres : Tatarbounarskyï Visnyk (bulletin du district), Raïtsentr (journal associatif) et Lyman (journal privé).
La vie spirituelle se déroule dans 19 églises orthodoxes, le couvent de la Sainte Transfiguration de Boryssivka et 23 communautés protestantes.
Développement des réseaux
Eau qu'il s'agisse de l'alimentation ou de l'évacuation, il n'existe pas de systèmes centralisés. L'eau provient de puits artésiens ou de forage regroupés sur 8 usines de traitement. Il n'existe plus de ressources nouvelles ; un chantier d'aqueduc venant de Kilia est en cours et devra être complété par la mise aux normes du réseau local, très dégradé sur un tiers du district.
Gaz de ville pas de réseau de gaz ou de chauffage urbain, d'où des solutions semi-collectives ou particulières, basées principalement sur le gaz liquide. Un gazoduc, venant d'Artsyz, est en projet.
Électricité le réseau dépend de la centrale d'Izmaïl ; il est de médiocre qualité.
Routes locales chantier en cours des axes très dégradés Tatarbounary — Rosseïka (Prymorské), Tatarbounary — Dyviziïa (T1610), Tatarbounary — Nérouchaï.
Galerie
Notes et références
Tatarbounary | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,878 |
\section{Introduction}
Over the last decade, Cloud computing has emerged as an important trend which assisted in moving the computing, control, and data storage resources across the geographically distributed data centers. Today, however, Cloud computing is encountering growing challenges such as-unpredictable high communication latency, privacy gaps and related traffic loads of networks connecting to end devices \cite{b1,b2,b3}. To address some of these limitations, Fog computing has emerged as a promising paradigm that brings the computation closer to the physical IoT devices deployed at the network edge; commonly referred to as 'Things' (sensors, mobile phones, edge switches/routers, and vehicles, etc.) \cite{b1,b2,b3,b4,b5}. For example, commercial edge routers having high processing speed and equipped with a large number of cores and communicating with the external or border layer of the network, have the potential to become new servers for Fog networks \cite{b20}.
Essentially, Fog computing is a highly virtualized platform that offers computing capabilities to allow various applications to run anywhere. To tackle the scalability issues of traditional centralized control architectures, Software-Defined Network (SDN) is the most viable network technology in the Fog environment \cite{b4,b5,b6,b7}. SDN-based Fog architectures provide a centralized controller with global knowledge of the network state which is capable of controlling Fog nodes services while Fog nodes simply accept policies from the controller without understanding various network protocols standards \cite{b22}. In \cite{b21}, the scenario showed that the Fog node includes an SDN controller which handles the programmability of the network edge devices by either a fully distributed or centralized management.
Although Fog networking is a promising technology to cope with the disadvantages of Cloud and the existing networks, but there are still challenges that remain to be assessed in the future. Most importantly, there is a need for a distributed intelligent platform at the edge that manages distributed computing, networking, and storage resources. In Fog networks, however, making an optimal distribution decision faces a lot of challenges due to uncertainties associated with task demands and resources available at the Fog nodes \cite{b11} and the wide range of computing power capacities of nodes. Furthermore, the distribution decision should also consider the communication delay between nodes, which can lead to prolonged processing time [8]–[10]. Therefore, the challenges being faced by Fog computing paradigm are varied and many; they include crucial decisions about i) whether Fog nodes should be offloaded or not, ii) an optimal number of tasks to be offloaded, and iii) mapping of incoming tasks to available Fog nodes under their respective resource capacities.
The existing schemes proposed in the literature have majorly concentrated on load balancing and cooperative offloading in Fog environment \cite{b8, b9, b10, b11, b12}. However, most of these approaches impose many restrictions and specific assumptions over the networks; which often do not relate with realistic Fog networks \cite{b15}. Additionally, these approaches require non-trivial mathematical equations when more and more restrictions are considered.
Under the scope of the above challenges, the proposed algorithm formulates the offloading problem as a Markov decision process (MDP) subject to the dynamics of the system in terms of Fog nodes behavior. This problem allows Fog nodes to offload their computation intensive tasks by selecting the most suitable neighboring Fog node in the presence of uncertainties on the task demands and resource availability at the Fog nodes. However, the system cannot precisely predict the transition probabilities and rewards due to dynamically changing incoming task demands and resource status. To solve this problem, this paper uses the classic model-free reinforcement learning algorithm, Q-learning, which can be used to solve MDPs with unknown reward and transition functions by making observations from experience.
Because of a reinforcement learning methods' advantages, it has been largely discussed in developing load balancing problems. For instance, the authors, in \cite{b17}, implemented reinforcement learning for distributed static load balancing of data-intensive applications in a heterogeneous environment. Dutreilh \textit{et al.} \cite{b18} proposed a learning process for automatic resource allocation in a cloud. Likewise, authors in \cite{b14} proposed a deep reinforcement learning based offloading scheme for a user in an Ad-hoc Mobile Cloud. However, the proposed Q-learning in Fog network differs from existing state-of-the-art in terms of three primary reasons. First, the environment is changed by multiple Fog nodes according to individual incoming requests from end devices. Second, each Fog node may have a different computing capacity which spells out the differences in resource availability between Fog nodes. Third, the distance between Fog nodes affects decision-making for an action which is chosen in a way that minimizes the communication time.
\subsection{Contributions}
The main objective of the proposed algorithm is to choose optimal offloading decisions while minimizing tasks processing delay and node overload probability. In addition, to provide more distributed and scalability for Fog network model. The novelty of the proposed algorithm is the fact that it accommodates variable incoming task rates and different computing capacities of each Fog node as well as the distance between Fog nodes in its formulation. Moreover, it simplifies the algorithmic framework without any specific assumption on the considered network model, whereas other related works generally impose restrictions on the network so as to simplify non-trivial mathematical equations. Towards this end, the proposed model merges with SDN-based Fog computing where SDN Fog controller directly controls, programs, orchestrates and manages network resources. Further, SDN Fog nodes serve the end users' request and deliver information based on collected traffic information to the controller \cite{b15}. Besides, the proposed reward function is defined with the aim to minimize the processing time and the overall overloading probability. Since a controller has the global view of the network, it can observe the reward and next state for current state and current action. This advantage can make the proposed Q-learning with guarantees both performance and convergence rates \cite{b15}. As a result, SDN Fog controller can find optimal actions which help nodes to optimally select a neighboring node to which it can offload its requested tasks. Then, nodes also determine how many tasks to offload to a neighboring node chosen based on the size of its task demand and the number of tasks currently remaining in its queue. The key contributions of this paper are listed as follows.
\begin{itemize}
\item The proposed Q-learning based offloading decision lets the controller decide the optimal actions according to the reward function. This is an attractive feature because the controller has the ability to define their reward function based on the required performance.
\item The proposed Q-learning based offloading decision can expect how good the present offloading is over the future, which achieves the exceptional overall system performance.
\item The decision-making process using reinforcement learning is on-demand and pay-as-you-go, which makes the proposed method compatible with SDN architecture \cite{b15}. Hence, it can be regarded as a load balancing application running on an SDN Fog controller. These attractive features provide an important opportunity to apply our load balancing problem within SDN Fog networks.
\end{itemize}
\subsection{Organization}
In section 2, this paper discusses load balancing in Fog networks. Section 3 is dedicated to both the system description for the proposed algorithm and the problem formulation. Details on the proposed reinforcement learning-based offloading algorithm is described in Section 4. The simulation results and our analysis of these results are presented in Section 5. Finally, Section 6 concludes this paper and gives some insight on possible future work.
\section{Load balancing in Fog networks}
Load balancing in Fog networks refers to efficiently distributing incoming workload across a group of processing Fog nodes so that the capacity of concurrent users and the reliability of requested tasks increase. It can be categorized under two different methods; static and dynamic load balancing methods \cite{b16}. Static load balancing distributes the workload using prior knowledge of task requests, which is determined at the beginning of the execution. The main drawback of static methods is that the allocation of tasks cannot be changed during the process execution to reflect changes in traffic load. In contrast to the static method, dynamic load balancing allocates tasks dynamically when one of the nodes becomes under-loaded. In other words, it can update the allocation of tasks continuously depending upon the newest knowledge of traffic loads. Hence, the accurate real-time load prediction is necessary for effective load balancing. In virtual machine (VM) environments, multiple VMs share the resources on the same physical machine (PM). Therefore, a load balancing function would be much more complicated in this environment \cite{b7,b12}.
Comparing to traditional servers, Fog node is essentially designated based on the characteristics and features of end devices. Fog nodes are formed by at least one or more physical devices with high processing capabilities \cite{b6}. For a better understanding, a Fog node would be a logical concept, with a heterogeneous type of devices as its physical infrastructure. In this way, the Fog node encompasses end devices together while the processing capacity in these end devices should be presented in terms of virtual computing units. Hence, all physical devices of a Fog node are aggregated as one single logical entity able to seamlessly execute distributed services as if those were on a single device \cite{b6}. In this paper, the Fog node is described as a processing server like a mobile base station but also including the integrated computing capacity of end devices. For example, vehicles and desktop computers can also share their computational power to provide task requests. The Fog node of these end devices can be responsible for managing their resources and also communicate between all Fog nodes.
\section{system model}
This section fisrtly define the proposed Fog architecture design and the system model. Then, the MDP-based offloading problem formulation are presented.
\subsection{System description}
In virtualized Fog network, the SDN Fog controller and the SDN Fog nodes are essential infrastructures in the load balancing (Hereinafter, SDN Fog controller and SDN Fog nodes are called controller and nodes, respectively). End users are directly connected to the nodes where they can submit application requests locally. The nodes are connected according to certain network topology, and are logically connected to the controller. In Fig. 1, the paper show a Fog network architecture of the proposed system. This paper considers a dynamic load balancing technique using distributed Fog computing mechanism in which nodes can offload their computation tasks to a neighboring node with available queue spaces in terms of computing capabilities and task demands distributions. Load balancing algorithms are typically based on a load index, which provides a measure of the workload at a node relative to some global average. The load index is used to detect a load imbalance state, in which the load index at one node is much higher or much lower than the load index on the other nodes. The length of the CPU queue has been shown to be remaining resource information that can be used as a good load index on time-shared workstations when the performance measure of interest is the average response time \cite{b14,b18}.
The controller is responsible for information aggregation and decision-making, whereas the nodes work directly to serve the end users and deliver information based on collected traffic information and queue status to controllers. The system works as follows. First, the controller builds a network map based on the queue information delivered from the nodes. Next, the controller runs algorithms to determine whether the present node should offload its requested tasks or not and if so, how many tasks to offload to a neighboring node chosen based on the size of its task demand and the number of tasks currently remaining in its queue. The main objective is to choose optimal offloading actions while minimizing tasks processing delay and node overload probability.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{Fig1_new.png}
\caption{The system model for the SDN-based Fog architecture.}
\label{figure1}
\end{figure}
\subsection{Problem formulation}
To achieve the required performance, the proposed load balancing problem is formulated as an MDP for which this paper proposes an algorithm with performance guarantees. MDP involves a decision agent that repeatedly observes the current state $s$ of the controlled system, takes a decision $a$ among the ones allowed in that state (${a}\in{A(s)}$) and then observes a transition to a new state $s'$ and a reward $r$ that will affect its following decisions. MDP is a stochastic rule by which the agent selects actions as a function of states. Hence, the new state and the reward observed from the transition probability distributions can fully characterize the behavior of the underlying controlled system. In this system, the controller selects actions as a function of present states, taking into account the following states and the rewards observed from all nodes.
In particular, our MDP is characterized by a 4-tuple $\big \langle S,A,P,R \big \rangle $, detailed as below:
\begin{itemize}
\item $S=\{s=$($n^l,w,Q$)$\}$ is the state space
\begin{itemize}
\item ${n^l}\in{\mathbb{N}}$ ($1 \le n^l \le N$) is the node which has the requested tasks to be allocated from end users
\item ${w}\in{\mathbb{N}}$ ($1 \le w \le {W_{max}}$) is the number of tasks to be allocated per unit time
\item $Q=\{$($Q_1,...,Q_N$)$|Q_i\in$\{$0,1,...,Q_{i,max}$\}$\}$ is the number of tasks currently remaining in the nodes' queue
\end{itemize}
\item $A=\{a=$($n^o,w^o$)$\}$ is the action space
\begin{itemize}
\item ${n^o}\in{\mathbb{N}}$ ($1 \le n^o \le N,n^o\neq n^l$) is defined as a neighboring node within the considered Fog network and that is being offloaded by the node $n^l$.
\item ${w^o}\in{\mathbb{N}}$ ($1 \le w^o \le {W_{max}}$) is the number of tasks to be offloaded to a neighboring fog node $n^o$.
Let $A(s) \subseteq A$ be the set of actions that can be taken at the state $s$.
$A(s)$ is determined such that the node $n^l$ can only offload the tasks to another node with equal or less number of tasks currently requested.
Depending on an action $a$, the number of tasks to be locally processed ($w^l$) is decided with regard to the available queue space of the node $n^l$.
\end{itemize}
\item $P : S \times A \times S \rightarrow [0,1] $ is the transition probability distribution $P(s'|s,a)$ of a new state $s'$ given that the system is in state $s$ and action $a$ is chosen.
\item $R : S \times A \rightarrow \mathbb{R}$ is the reward when the system is in state $s$ and action $a$ is taken. The main goal of the system is to make an optimal offloading action at each system with the objective of maximizing the utility while minimizing the processing delay and overload probability. Hence, the proposed system defines the immediate reward function $R(s,a)$ given an action $a$ at state $s$ as follows.
\begin{equation} \label{one}
R(s,a) = U(s,a) - (D(s,a) + O(s,a)),
\end{equation}
where $U(s,a)$, $D(s,a)$ and $O(s,a)$ represent the immediate utility, immediate delay and overload probability function, respectively.
\begin{itemize}
\item The immediate utility $U(s,a)$ is calculated as
\begin{equation} \label{two}
U(s,a) = r_u\log (1+w^l+w^o),
\end{equation}
where, $r_u$ is a utility reward.
\item The immediate delay $D(s,a)$ is calculated as
\begin{equation} \label{three}
D(s,a) = \chi_d\cdot\frac{t^w+t^c+t^e}{(w^l+w^o)},
\end{equation}
where, $\chi_d$ is a delay weight,
\begin{enumerate}
\item The average waiting time $t^w$ at the queue of the fog node $n^l$ and the offloading fog node $n^o$ is
\begin{equation} \label{four}
t^w = \frac{Q^l}{\overline{\mu ^l}}\mathbb{1}(w^l\neq 0)+\Big(\frac{Q^l}{\overline{\mu ^l}}+\frac{Q^o}{\overline{\mu ^o}}\Big)\mathbb{1}(w^o\neq 0),
\end{equation}
where, $\mu ^i$ is the computing service rate of node $n^i$.
\item The communication time of task offloading $t^c$ is
\begin{equation} \label{five}
t^c = \frac{2\cdot T\cdot w^o}{r_{l,o}},
\end{equation}
where, $T$ is the data size of a task, $r_{l,o}$ is the fog transmission service rate from $n^l$ to $n^o$ given by:
\begin{equation} \label{six}
r_{i,j} = B\cdot\log \Big(1+\frac{g_{i,j}\cdot P_{t_{x},i}}{B\cdot N_0}\Big),
\end{equation}
where, $B$ is the bandwidth per a node, $g_{i,j}\triangleq \beta_1 {d_{i,j}}^{-\beta_2}$ is the channel gain between nodes $n^i$ and $n^j$ with $d_{i,j}$, $\beta_1$, and $\beta_2$ being the distance between two nodes, the path loss constant and path loss exponent, respectively. The variable $P_{t_{x},i}$ denotes the transmission power of node $n^i$ and $N_0$ is the noise power spectral density, which is defined at normalized thermal noise power -174 dBm/Hz.
\item The execution time $t^e$ by the fog node $n^l$ and the offloaded fog node $n^o$ is
\begin{equation} \label{seven}
t^e = \frac{I\cdot CPI \cdot w^l}{f^l}+\frac{I\cdot CPI \cdot w^o}{f^o},
\end{equation}
where, $I$ is the number of instructions per task, $CPI$ is CPU cycles per instruction and $f^i$ is the CPU speeds of a node $n^i$.
\end{enumerate}
\item The overloaded probability $O(s,a)$ is calculated as
\begin{equation} \label{eight}
O(s,a)=\chi_o\cdot\frac{w^l\cdot P_{overload,l}+w^o\cdot P_{overload,o}}{w^l+w^o},
\end{equation}
\begin{equation}
P_{overload,i} =\frac{\max(0,\lambda_i-(Q_{i,max}-Q'_i))}{\lambda_i},
\end{equation}
\begin{equation}
Q'_i = \min(\max(0,Q_i-\overline{\mu^i})+w^i,Q_{i,max}),
\end{equation}
where, $\chi_o$ is an overload weight and $\lambda_i$ is the task arrival rate arriving at node $n^i$ which can be modeled by a Poisson process. In (10), $Q'_i$ represents the next estimated queue state of node $n^i$ in state $s$ and action $a$ is taken.
\end{itemize}
\end{itemize}
With the transition probability $P$ and reward $R$ being determined prior to the execution of the controlled system, the MDP can thus be solved through traditional dynamic programming (DP) algorithms. The key idea of DP is the use of value functions to organize and structure the search for good decisions. The optimal action at each state is defined as the action that gives the maximum long-term reward, which is the discounted sum of the expected immediate rewards of all future decisions about state-action starting from the current state. An immediate reward received $k$ time steps further in the future is worth only $\gamma^{k-1}$ times what it would be worth if it were received immediately where $\gamma$ is called a \textit{discount factor}(0$<$$\gamma$$<$1) \cite{b13}. The optimal value function is defined, which satisfy the Bellman optimality equations:
\begin{equation}
\begin{split}
v^*(s) &= \max_{a} \mathbb{E}(R_{t+1}+\gamma v^*(S_{t+1})|S_t=s,A_t=a) \\
& = \max_{a}\sum_{s',r}p(s',r|s,a)[r+\gamma v^*(s')].
\end{split}
\end{equation}
\section{The proposed Reinforcement learning-based offloading algorithm}
In this Section, the reinforcement learning-based offloading algorithm is proposed to address the limitations of the traditional approaches.
As defined in the previous section, in the case where the system can have transition probability functions $P$ and reward $R$ for any state-action pair, the MDP can be solved through DP methods. However, for most cases, the system cannot precisely predict $P$, and $R$; and the system may change the transition probability distributions or rewards. To address these limitations, reinforcement learning is proposed. In reinforcement learning, the lack of information is solved by making observations from experience. Classical DP algorithms are of limited features in reinforcement learning both because of their assumption of a perfect model and because of their great computational cost \cite{b13}. Among the different reinforcement learning techniques, Q-learning is the classic model-free algorithm. It is usually used to find the optimal state-action policy for any MDP without an underlying policy. Given the controlled system, the learning controller repeatedly observes the current state $s$, takes action $a$, and then a transition occurs, and it observes the new state $s'$ and the reward $r$. From these observations, it can update its estimation of the Q-function for state $s$ and action $a$ as follows:
\begin{equation}
Q(s,a) \leftarrow (1-\alpha)Q(s,a)+\alpha \Big[R(s,a)+\gamma\max_{a'\in A_{s'}}Q(s',a')\Big],
\end{equation}
where, $\alpha$ is the \textit{learning rate} (0$<$$\alpha$$<$1), balancing the weight of what has already been learned with the weight of the new observation.
The simplest action selection rule is to select one of the actions with the highest estimated value, i.e., Greedy selection ($a_t\doteq\arg\max_{a}Q_t(a)$). Thus, the greedy action selection always exploits current knowledge to maximize immediate reward. An important element of Q-learning is the $\epsilon$-greedy algorithm. This latter behaves greedily most of the time, but with small probability, $\epsilon$, select randomly from all the available actions with equal probabilities, independently. Reinforcement learning call this greedy selection and the $\epsilon$ probability of random selection as exploitation and exploration policies, respectively. Exploitation is the right action to do to maximize the expected reward on the one step, whereas exploration may produce the greater total reward in the long term. An advantage of $\epsilon$-greedy algorithm is that, as the number of steps increases, every action will be visited an infinite number of times, thus ensuring that $Q(s,a)$ converges to the optimal value.
The appropriate reward function that the proposed system have chosen is calculated using Eq.(1). The approximate next state $s'$ is obtained after defining its three components mentioned in the Section \RomanNumeralCaps{3}-B; While the next queue state is a deterministic entity, the next fog node which has task to be allocated and the size of tasks following each node tasks arrival are of stochastic nature and therefore the proposed system models them using Poisson random variables.
The procedures of the proposed Q-learning algorithm is presented in Algorithm 1.
\begin{algorithm}
\DontPrintSemicolon
\LinesNumbered
\textbf{Input} learning rate ($\alpha$), discount factor ($\gamma$), exploration policy ($\epsilon$), service rate ($\mu$), task arrival rate ($\lambda$), and distance vector ($\mathcal{D}$)
\textbf{Output} Optimized offloading table ($\mathcal{Q}$)
\textbf{Set} $Q(s,a)$ := 0 ($\forall$$s \in S$) ($\forall$$a \in A(a)$), \textit{iter} := 0, and $s$ := (1,1,$\{$($Q_1,...,Q_N$)$|Q_i=0\}$)
\While{(iter $\le$ maximum iteration)}{
Choose $a \in A(a)$ using $\epsilon$-greedy algorithm\;
Offload the tasks according to action $a$ and observe next state $s'$ and reward $r$\;
$Q(s,a)\leftarrow (1-\alpha)Q(s,a)+\alpha\Big[R(s,a)+\gamma\max_{a'\in A_{s'}}Q(s',a')\Big]$\;
$s \leftarrow s'$
\textit{iter}$\leftarrow$\textit{iter}$+1$
}
\caption{The proposed Q-learning algorithm\label{QR}}
\end{algorithm}
\section{Performance evaluation}
\subsection{Simulation settings}
This Section analyzes the performance of the proposed load balancing algorithm in simulations in which the proposed system considers a fog network consisting of $N$ nodes. The system sets a network area of $100\times 100$ $m^{2}$ where nodes were randomly allocated. At the beginning of the Q-learning, since the first Q-value is zero, the algorithm encourages exploration more. That is why the optimal action selection worked for $\epsilon$ = 0.9 and 0.7 between the initial iteration and the last iteration. In addition, utility reward $r_u$ is set to 10.
The performance of the proposed Q-learning algorithm was compared to some of the existing offloading methods. Specifically, the simulation used least-queue, nearest, and random node selection offloading methods as benchmarks. Least-queue is a classic offloading method, which is implemented by offloading tasks always to the node with minimum queue status. Nearest node selection is an offloading algorithm widely used in IoT, device-to-device communications since the node selects the most adjacent neighboring node aiming to minimize communication delay and energy consumption.
For different evaluation scenarios, the simulation was set to most parameters to default values and vary the exclusively following parameters. 1) task arrival process of nodes; 2) computing service rate of nodes. Over the variation of the task arrival rate, the simulation was considered the average computing service rate of fog node as 1.8. Likewise, over the variation of the computing service rate, the average task arrival rate was kept to 5.2. The detailed simulation parameters are given in Table \ref{table1}.
\begin{table}[h]
\caption{Parameter values in simulations.}
\label{table1}
\centering
\begin{tabular}{c|c|c}
\hline\noalign{\smallskip}
\multicolumn{2}{c}{\textbf{Parameter}} & \textbf{Value}\\
\hline
\hline
\multicolumn{2}{c}{Number of fog nodes ($N$)} & 5\\
\hline
\multicolumn{2}{c}{Maximum CPU queue size ($Q_{max}$)} & 10\\
\hline
\multicolumn{2}{c}{Learning rate ($\alpha$)} & 0.5\\
\hline
\multicolumn{2}{c}{Discount factor ($\gamma$)} & 0.5\\
\hline
\multicolumn{2}{c}{Data size per a task ($T$)} & 500 Mbytes\\
\hline
\multicolumn{2}{c}{Number of instructions per a task ($I$)} & 200 $\times 10^6$\\
\hline
\multicolumn{2}{c}{Number of cycles per a an instruction ($CPI$)} & 5\\
\hline
\multicolumn{2}{c}{System bandwidth per node ($B$)} & 2 MHz\\
\hline
\multicolumn{2}{c}{Path loss parameter ($\beta_1$,$\beta_2$)} & $(10^{-3},4)$\\
\hline
\multicolumn{2}{c}{Transmission power of fog node ($P_{tx,i}$)} & 20 dBm\\
\hline
\multicolumn{2}{c}{Weights ($\chi_d$,$\chi_o$)} & $(1, 150)$\\
\hline
\end{tabular}
\end{table}
\subsection{Performance analysis}
Simulation results are presented in Fig. 2, 3, and 4 under different system configurations. Fig. 2(a) and 2(b) show the average reward over the variation of the task arrival rate and the computing service rate, respectively. Throughout the different configurations, the proposed load balancing algorithm using reinforcement learning shows higher average rewards than the three other classic load balancing methods. The main reason is that the proposed offloading decision is to offload one of the actions with the highest Q-value. This algorithm implies that the Q-learning based offloading decision minimize processing time and overload probability according to the proposed reward function. Furthermore, the Q-learning can expect how good the present offloading is over the future, which achieves the exceptional overall system performance. The average reward greatly increases as the computing service rate is increased since more number of tasks are executed. Meanwhile, the average reward consistently decreases as the arrival rate is increased because the fog nodes have a relatively high number of tasks in their queues; therefore a lower number of tasks can be allocated by them.
Fig. 3(a) and 3(b) show the average delay over the variation of the task arrival rate and the computing service rate, respectively. The proposed algorithm achieves the minimum average delay. The main reason is that the proposed offloading algorithm accommodates both the node's queue status and the distance between fog nodes in its formulation. On the other hand, when the computing service rate increases from 3 to 7, the average delay hardly decreases. This result indicates that once a certain level of computing service rate is achieved, for a fixed arrival rate, the delay that is occurring is caused by the communication latency which is defined as the transmission service rate from one fog node to another one offloaded.
In addition, the average overload probability was observed as shown in Fig. 4(a) and 4(b). The results show that the proposed algorithm greatly reduces the overload probability. Specifically, when the task arrival rate is 9, the proposed algorithm offers 5.77$\%$, 5.17$\%$, and 6.23$\%$ lower overload probability than random, least-queue, and nearest offloading methods. These results highlight the fact that when the node makes an offloading decision with the proposed algorithm, it considers not only nodes' queue states but also their task arrival distribution. Therefore, the proposed algorithm can minimize the failed allocation and with that the risk that a task can be lost if it arrives when the nodes' queue is already full. These improvements can be attributed to the Q-learning based offloading decision in accordance with different service rate and task arrival rate of nodes.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{ramda_mu_R2.png}
\caption{Average reward.}
\label{figure2}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{ramda_mu_D2.png}
\caption{Average delay.}
\label{figure3}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{ramda_mu_O2.png}
\caption{Average overload probability.}
\label{figure4}
\end{figure}
\section{Conclusion}
In this paper, the Fog dynamic load balancing algorithm has been proposed using a reinforcement learning technique. Load balancing in Fog networks faces a lot of challenges due to uncertainties related to workloads and the wide range of computing power capacities of nodes. The proposed process formulates a Markov decision process to find optimal actions which help nodes to select optimally a neighboring node to which it can offload its requested tasks. Then, nodes also determine how many tasks to offload to a neighboring node chosen based on the size of its task demand and the number of tasks currently remaining in its queue. Actions considered to be optimal if they allow minimal processing time and a minimum overall overload probability. To guarantee the optimal action-selection, the proposed algorithm applied Q-learning with $\epsilon$-greedy algorithm. The performance of the proposed load balancing algorithm is evaluated in different configurations in which nodes offload to their neighboring nodes within a realistic network. The simulation results show that the proposed algorithm can achieve lower average processing delay and lower failed allocation probability due to overloading as compared to existing methods. As the future works, the proposed model-free learning merged with the model-based learning will be considered to experiment, which may give less dependency to exploration policy and bias-free results.
\section*{Acknowledgement}
This research was supported by the NSERC B-CITI CRDPJ 501617-16 grant
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,944 |
/*
* decaffeinate suggestions:
* DS102: Remove unnecessary code created because of implicit returns
* Full docs: https://github.com/decaffeinate/decaffeinate/blob/master/docs/suggestions.md
*/
const _ = require("underscore")
const sd = require("sharify").data
const benv = require("benv")
const Backbone = require("backbone")
const sinon = require("sinon")
const moment = require("moment")
const { resolve } = require("path")
const { fabricate } = require("@artsy/antigravity")
const Fair = require("../../../../models/fair.coffee")
xdescribe("ForYouView", function () {
before(function (done) {
return benv.setup(() => {
sd.API_URL = "localhost:3003"
sd.CURRENT_PATH = ""
sd.CURRENT_USER = "hello"
sd.NODE_ENV = "test"
benv.expose({ $: benv.require("jquery") })
sinon.stub(Backbone, "sync")
this.OverviewView = require("../../client/overview.coffee")
Backbone.$ = $
this.fair = new Fair(fabricate("fair"))
return done()
})
})
after(function () {
benv.teardown()
Backbone.sync.restore()
return (sd.CURRENT_USER = undefined)
})
return describe("#initialize", () =>
it("renders personalized artist list", function () {
const view = new this.OverviewView({
el: $(`<div>
<div class='clock'></div>
<div class='container-left'><div class='large-section-subheading'></div></div>
</div>`),
fair: this.fair,
model: this.model,
})
Backbone.sync.args[0][2].success({
time: moment().subtract("minutes", 2).format("YYYY-MM-DD HH:mm:ss ZZ"),
})
Backbone.sync.args[1][2].success([{ artist: fabricate("artist") }])
Backbone.sync.args[2][2].success([])
view.$el.html().should.not.containEql("undefined")
view.$el.html().should.not.containEql("#{")
view.$el.html().should.not.containEql("NaN")
view.$(".container-left .large-section-subheading").length.should.equal(1)
return view
.$(".container-left .large-section-subheading")
.text()
.should.containEql("Pablo Picasso")
}))
})
| {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"http:\/\/mathhelpforum.com\/algebra\/86629-4-5-litre.html","text":"1. 4.5 Litre\n\nNot sure how to approach this question. If i filled a 4.5 litre bottle with pennys how much would be in there.\n\nAny help appreciated?\n\n2. Originally Posted by craig_r\nNot sure how to approach this question. If i filled a 4.5 litre bottle with pennys how much would be in there.\n\nAny help appreciated?\nYou'd need to find the volume of a penny which is $V = \\frac{\\pi D^2h}{4}$\n\nwhere\n\u2022 V = Volume\n\u2022 D = Diameter\n\u2022 h = thickness\n\nOnce you know the volume of a penny you can do\n\n$n = \\frac{4.5 \\times 10^6}{V}$\n\nto get the number of pennies (n). That formula only works for measuring the penny's thickness and diameter\/radius in mm\n\nBear in mind this will be an overestimate because it does not account for gaps between the pennies.","date":"2016-10-25 12:08:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 2, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6302305459976196, \"perplexity\": 1036.0454499948298}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-44\/segments\/1476988720062.52\/warc\/CC-MAIN-20161020183840-00044-ip-10-171-6-4.ec2.internal.warc.gz\"}"} | null | null |
St Swithun's Church is the smallest ancient Church of England parish church in the English county of Hampshire. Newnham and Nately Scures are part of the Anglican United Parish which includes: Greywell, Mapledurwell and Up Nately, which in turn are covenanted with a further seven churches in the area.
History
The Church was built of flint and rubble around 1175. It is considered to be the best largely unspoilt example of a Norman single-cell apsidal church in England. There are only four examples remaining in the UK. A gallery was installed in 1591 and rebuilt together with the roof in 1786. Binstead stone forms the door and window dressings.
Services
Services normally take place in each of the churches within the United Parish including St Swithun's twice per month. The church is never locked by day.
Burials
General The 1st Baron Dorchester
Colonel Thomas Carleton
Lieutenant-General Sir Christopher Wallace
Violet Edith Potter
External links
St Swithun's Church Nately Scures England - Showing Memorial for Guy Carleton, Governor of Quebec, and Thomas Carleton, Governor of New Brunswick
Buildings and structures completed in 1175
12th-century church buildings in England
Church of England church buildings in Hampshire
Churches with Norman architecture
Basingstoke and Deane | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,898 |
import torch
import torch.optim as opt
from torch.autograd import Variable
from torch import FloatTensor as FT
import agent
from replay_buffer import ExperienceReplay
import numpy as np
import dill
from torch.utils.serialization import load_lua
import model_defs.ddpg_models.mountain_cart.critic as critic
import model_defs.ddpg_models.mountain_cart.actor as actor
import random
#Default hyperparameter values
REPLAY_BUFFER_SIZE = 1000000
DISCOUNT_FACTOR = 1
LEARNING_RATE_CRITIC = 0.01
LEARNING_RATE_ACTOR = 0.01
ACTOR_ITER_COUNT = 1000
CRITIC_ITER_COUNT = 1000
BATCH_SIZE = 100
EPSILON = 0.01
class DQNController:
"""
Attributes:
replay_buffer: The DDPGAgent replay buffer
"""
"""
@property
def actor(self):
return self.actor
@property
def critic(self):
return self.critic
@property
def replay_buffer(self):
return self.replay_buffer
"""
def __init__(self, config, controllers, env):
"""Constructor for the DDPG_agent
Args:
actor_path: location of the actor_t7
critic_path: location of the critic_t7
buffer_size: size of the replay buffer
alpha: The learning rate
gamma: The discount factor
Returns:
A DDPGAgent object
"""
super(DDPGAgent, self).__init__(auxiliary_losses)
#state_size + 1 because we append reward to the input
state_size = state_size + 1
#Initialize experience replay buffer
self.replay_buffer = ExperienceReplay(state_size, action_size, buffer_size)
#TODO
self._caller = caller
subcontroller_idxs = config[""]
for idx in config
self.subcontrollers = []
#initialize parameters
self.epsilon = 0.35
self._alpha = alpha
self.iter_count = iter_count
self._gamma = gamma
self._batch_size = batch_size
self._state_size = state_size
self._action_size = action_size
#Specify model locations
self._critic_path = critic_path
self._Q = create_model(config)
#initialize models
#self.load_models()
#Initialize optimizers
self._critic_optimizer = opt.Adam(self.critic.parameters(), lr=self._critic_alpha)
def train(self, q_caller = None):
"""Trains the agent for a bit.
Args:
Returns:
None
"""
self.epsilon = self.epsilon * 0.99992
#get argmax_a' Q(s,a')
s_t,a_t,r_t,s_t1,done,act_dur= self.replay_buffer.batch_sample(self._batch_size)
loss_total = 0
argmax_t = np.empty((self._batch_size,1))
for j in range(self._batch_size):
argmax = 0
for i in range(self._action_size):
if (self._Q.forward(s_next, i)> self._Q.forward(s_next, argmax)):
argmax_t[j,1] = i
#CONCAT STATE ACTIONS
#DQN loss
loss = torch.nn.loss(self._Q.forward(s_t, a_t), r_t + self._Q(s_t1,argmax_t)) #ADD OPTIMIZER
self._optimizer.zero_grad()
loss.backward(self._Q.parameters())
self._optimizer.step()
#update_critic
def act(self,
env,
caller_id,
is_test=False):
# Keep track of total reward and duration
tot_r = 0
tot_dur = 0
while True:
# Forward step
cur_state = env.cur_obs
cur_action = self.choose_action(cur_state, is_test)
r, dur, done, ret = self.execute_action(env, cur_action)
# Store the tuple in replay
self.replay_buffer.put(cur_state, cur_action, r, done, dur, caller_id)
# Update totals
tot_r += r
tot_dur += dur
# End if episode end or return action called
if done or ret:
break
# Return control to caller
return tot_r, dur, done
def choose_action(self, cur_state, is_test=False):
"""Get the next action from the agent.
Takes a state and returns the next action from the agent.
The agent may cache the reward and state
Args:
cur_state: The current state of the enviroment
is_test: Check to see if the agent is done
Returns:
The next action
"""
cur_action = None
if is_test:
a, _ = self.actor.forward(upcast(np.expand_dims(cur_state,axis=0)),[])
cur_action = a.data.cpu().numpy()
elif random.random() < self.epsilon:
cur_action = np.expand_dims(np.random.randn(self._action_size),axis=0)
else:
a, _ = self.actor.forward(upcast(np.expand_dims(cur_state,axis=0)),[])
cur_action = a.data.cpu().numpy()
return cur_action
def execute_action(self, env, a, is_test):
ret = False
# Return control to caller
if a == 0:
r = 0
dur = 0
done = False
ret = True
# Perform real actual action
elif a < action_size + 1:
_, r, done = env.next_obs(a)
dur = 1
# Make a subroutine call
else:
sub_idx = a - (action_size + 1)
r, dur, done = subcontrollers[sub_idx].act(env, self._id, self.)
return r, dur, done, ret
def save_models(self, locations=None):
"""Save the model to a given locations
Args:
Locations: where to save the model
Returns:
None
"""
#Return all weights and buffers to the cpu
self.actor.cpu()
self.critic.cpu()
#Save both models
actor_file=open(self._actor_path,"wb")
dill.dump(self.actor,actor_file)
critic_file=open(self._critic_path,"wb")
dill.dump(self.critic,critic_file)
def load_models(self, locations=None):
# TODO: Make it actually do what it says
#TODO: Remove hard coding of data
"""Loads the models from given locations
Args:
Locations: from where to load the model
Returns:
None
"""
actor_file=open(self._actor_path,"rb")
self.actor = actor.Actor(self._state_size,self._action_size) #dill.load(actor_file)
critic_file=open(self._critic_path,"rb")
self.critic = critic.Critic(self._state_size + self._action_size, 1)#dill.load(critic_file)
self._target_critic = critic.Critic(self._state_size + self._action_size,1)#dill.load(critic_file)
#Move weights and bufffers to the gpu if possible
if torch.cuda.is_available():
self.actor.cuda()
self.critic.cuda()
self._target_critic()
def upcast(x):
''' Upcasts x to a torch Variable.
'''
#TODO: Where does this go?
return Variable(FT(x.astype(np.float32)))
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,107 |
Sony Pictures Networks India and WWE® Expand Partnership in India
MUMBAI, India & STAMFORD, Conn.–(BUSINESS WIRE)–Sony Pictures Networks India Private Limited, (SPN) and WWE (NYSE: WWE) today announced a five-year extension with an expanded broadcast agreement that will deliver WWE's weekly flagship programming and localized content in India and the Indian subcontinent, making SPN the exclusive home to WWE in-ring content across its television and digital platforms.
As part of the agreement, SPN will have the rights to air Raw®, SmackDown®, NXT® as well as WWE pay-per-view specials live on its sports and digital platforms in English, Hindi and regional languages.
The expanded partnership also provides SPN the rights to WWE Network which hosts thousands of hours of content to be made available to audiences in India exclusively through SonyLIV, SPN's OTT platform. In addition and for the first time, SPN will curate content from WWE's extensive video library which includes live events, iconic matches, interviews with Legends, reality shows, documentaries and more, on its own platforms.
WWE will also return to India with even bigger live events and Superstars will visit the region for promotional tours to engage with fans.
Rajesh Kaul, Head Sports and Chief Revenue Officer Distribution, Sony Pictures Networks India
"WWE has been consistently ranking in the top three properties across all sports networks in India, dominating the sports entertainment space. We are delighted to extend our partnership with WWE in a first of its kind deal that will give us an increased exposure to valuable content and reach out to a broader viewer base across India on multiple platforms."
James Rosenstock, WWE Executive Vice-President, International
"India is a strategically important market for WWE and Sony Pictures Networks has been an extraordinary partner in helping to grow our fanbase in the region making it one of the most-watched sports properties. The expanded partnership gives our passionate fans new opportunities to engage with the WWE brand."
The agreement comes at a time when WWE's popularity in India continues to grow. WWE is one of the most-watched sports properties in India, and India ranks #1 for WWE YouTube consumption and #1 in WWE Facebook followers. In addition, WWE has been actively recruiting elite athletes and performers from the country. Last year, WWE hosted its largest tryout in history, where more than 70 top athletes from India showcased their abilities in Mumbai. Four Indian recruits were selected to begin training at the WWE Performance Center in Orlando, Florida, U.S.
About Sony Pictures Networks India (SPN)
Sony Pictures Networks India (SPN), is an indirect wholly owned subsidiary of Sony Corporation, Japan.
SPN has several channels including Sony Entertainment Television (SET and SET HD), one of India's leading Hindi general entertainment television channels; MAX, India's premium Hindi movies and special events channel; MAX 2, another Hindi movie channel showcasing great India Cinema; MAX HD, a high definition Hindi movie channel airing premium quality films; WAH, the FTA channel for Hindi movies; SAB and SAB HD the family-oriented Hindi comedy entertainment channels; PAL, a genre leader in rural Hindi speaking markets (HSM) showcasing the best of Hindi general entertainment and Hindi movies from SPN's content library; PIX and PIX HD, the English movie channels; AXN and AXN HD, the channels showcasing the best in Reality, Entertainment and Drama; Sony BBC Earth and Sony BBC Earth HD, the premium factual entertainment channels, Sony AATH, the Bangla entertainment channel; MIX a refreshing Hindi music channel; YAY!, the kids entertainment channel; sports entertainment channels – SONY SIX, SONY SIX HD, SONY ESPN, SONY ESPN HD, SONY TEN 1, SONY TEN 1 HD, SONY TEN 2, SONY TEN 2 HD, SONY TEN 3, SONY TEN 3 HD; Sony Marathi, the Marathi general entertainment channel; SonyLIV – the digital entertainment VOD platform; SPN Productions, the networks' film production arm and Studio NEXT the independent production venture for original content and IPs for TV and digital media. SPN reaches out to over 700 million viewers in India and is available in 167 countries.
The network is recognized as an employer of choice within and outside the media industry. SPN is a recipient of several awards, including the 'Aon Best Employers India' Award in recognition of SPN's unique workplace culture and exceptional people practices, consistently ranking amongst India's Top 10 Companies with Best Health & Wellness Practices by SHRM & CGP Partners, listed by Working Mother & AVTAR as one of the 100 Best Companies for Women in India and adjudged one of India's Great Workplaces by the Great Place to Work® Institute. Sony Pictures Networks India Private Limited is in its 24th year of operations in India. It has a subsidiary, MSM-Worldwide Factual Media Private Limited and an affiliate, Bangla Entertainment Private Limited in India.
For more information, log onto www.sonypicturesnetworks.com
About WWE
WWE, a publicly traded company (NYSE: WWE), is an integrated media organization and recognized leader in global entertainment. The company consists of a portfolio of businesses that create and deliver original content 52 weeks a year to a global audience. WWE is committed to family friendly entertainment on its television programming, pay-per-view, digital media and publishing platforms. WWE's TV-PG, family-friendly programming can be seen in more than 800 million homes worldwide in 28 languages. WWE Network, the first-ever 24/7 over-the-top premium network that includes all live pay-per-views, scheduled programming and a massive video-on-demand library, is currently available in more than 180 countries. The company is headquartered in Stamford, Conn., with offices in New York, Los Angeles, London, Mexico City, Mumbai, Shanghai, Singapore, Dubai, Munich and Tokyo.
Additional information on WWE (NYSE: WWE) can be found at wwe.com and corporate.wwe.com. For information on our global activities, go to http://www.wwe.com/worldwide/.
Trademarks: All WWE programming, talent names, images, likenesses, slogans, wrestling moves, trademarks, logos and copyrights are the exclusive property of WWE and its subsidiaries. All other trademarks, logos and copyrights are the property of their respective owners.
Forward-Looking Statements: This press release contains forward-looking statements pursuant to the safe harbor provisions of the Securities Litigation Reform Act of 1995, which are subject to various risks and uncertainties. These risks and uncertainties include, without limitation, risks relating to: the impact of the COVID-19 outbreak on our business, results of operations and financial condition; entering, maintaining and renewing major distribution agreements; a rapidly evolving media landscape; WWE Network (including the risk that we are unable to attract, retain and renew subscribers); our need to continue to develop creative and entertaining programs and events; the possibility of a decline in the popularity of our brand of sports entertainment; the continued importance of key performers and the services of Vincent K. McMahon; possible adverse changes in the regulatory atmosphere and related private sector initiatives; the highly competitive, rapidly changing and increasingly fragmented nature of the markets in which we operate and greater financial resources or marketplace presence of many of our competitors; uncertainties associated with international markets including possible disruptions and reputational risks; our difficulty or inability to promote and conduct our live events and/or other businesses if we do not comply with applicable regulations; our dependence on our intellectual property rights, our need to protect those rights, and the risks of our infringement of others' intellectual property rights; the complexity of our rights agreements across distribution mechanisms and geographical areas; potential substantial liability in the event of accidents or injuries occurring during our physically demanding events including without limitation, claims alleging traumatic brain injury; large public events as well as travel to and from such events; our feature film business; our expansion into new or complementary businesses and/or strategic investments; our computer systems and online operations; privacy norms and regulations; a possible decline in general economic conditions and disruption in financial markets; our accounts receivable; our indebtedness including our convertible notes; litigation; our potential failure to meet market expectations for our financial performance, which could adversely affect our stock; Vincent K. McMahon exercises control over our affairs, and his interests may conflict with the holders of our Class A common stock; a substantial number of shares are eligible for sale by the McMahons and the sale, or the perception of possible sales, of those shares could lower our stock price; and the volatility of our Class A common stock. In addition, our dividend is dependent on a number of factors, including, among other things, our liquidity and historical and projected cash flow, strategic plan (including alternative uses of capital), our financial results and condition, contractual and legal restrictions on the payment of dividends (including under our revolving credit facility), general economic and competitive conditions and such other factors as our Board of Directors may consider relevant. Forward-looking statements made by the Company speak only as of the date made and are subject to change without any obligation on the part of the Company to update or revise them. Undue reliance should not be placed on these statements. For more information about risks and uncertainties associated with the Company's business, please refer to the "Management's Discussion and Analysis of Financial Condition and Results of Operations" and "Risk Factors" sections of the Company's SEC filings, including, but not limited to, our annual report on Form 10-K and quarterly reports on Form 10-Q.
Kaumudi Naithani +91 9833931953 | kaumudi.naithani@setindia.com
Matthew Altman 203-352-1177 | Matthew.Altman@wwecorp.com
Investor:
Michael Weitz 203-352-8642 | Michael.Weitz@wwecorp.com
Categories Select Category 2D 3D 4K 5G Accessories AI AIS Animation AR Article articles Audio Augmented Reality AV Batteries Black Market Blockchain Brand Protection Broadcast Business Call Center Camera Cameras Career Case Study Cell Phone Cinema Cinematographer Cloud Color Correction Color Grading Communications Compositing Concert Consumer Electronics Content Content Creation Corporate Covid Cryptocurrency Customers DaIlies Dance Design Digital Cinema Distribution DIT Documentary Drone Earphones Education Effects Electronics Entertainment eSports Facial Recognition Feature Feature news Film FilmMaker Finishing Gaming Grading HDR HR IBC Image Editing Indie Film Industrial Interview IP Video IQS Laptop Lead Story LED Lenses Live Event Live Production Live Streaming M&E Manufacturing Marketing Mastering Media Production Microphone Mixing Mobile Mobile Video Monitor Music Music Video NAB NAPTE Netflix Networked Video News People Photography Plugins PoE Post Post Production Presentation Production Project Projectors ProRes RAW Radio Remote Shooting Retail Review Robotics Security Shaders Short Film Slow Motion Capture Small Business SMPTE Speakers Speech Recognition SSD Stock media Storage Storytelling Streaming Streaming Services Switcher Technology Transcription Transmitter Tutorial VFX Video Video Editing Video Games Video Processor Video Wall Visual Effects VOIP Volumetric Video VR Web Design Wireless Workflow
It's Time for the Next Generation of Streaming Service
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Producer/Artist Thomas Statnick Invests in a Neve® 8424 Console
Evolve Partners with Theatrixx to Distribute Theatrixx-branded Products in the US
"The Bathtub," Starring Sonic Youth Drummer, Streams Free Online
H.I.G. Realty Invests in Production Studios & Content Hub in Madrid
Whistleblower and Data Rights Activist Brittany Kaiser to be featured in Collective[i] Forecast speaker series
K-Motion and Perfect Game Join Forces to Create PG TECH
Apple Launches Major New Racial Equity and Justice Initiative Projects to Challenge Systemic Racism, Advance Racial Equity Nationwide
Perfect Corp. Wins 2021 BIG Innovation Award for Its AI Beauty Tech Solutions
AlefEdge Partners With Netris to Enable Automated Networking for Enterprise Edge Services
GK Software Launches Omnibasket, Allowing Solution Providers to Easily Integrate Their Applications With GK's cloud4retail
Octo Awarded Spot on Army's RS3 92 Situational Awareness and Augmented Reality Technologies Contract
Step Inside the Home of the Future – Virtually – at P&G's CES 2021 LifeLab Experience
Hasbro to Webcast February 2021 Investor Events
Pandemic Pushes Digital Transformation from Pilots to Production as Enterprises Contend with Disruptions
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Mavenir Cloudifies its Portfolio
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Assimilate Apps FREE For The Next 6 Months!
Copyright © Digital Producer Magazine All rights reserved. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,104 |
I Aim to Misbehave
By Dustin Rowles | Industry | October 15, 2009 |
It's not exactly what you think, but Nathan Fillion — who is finally on a show that's lasted until its second season — will be reprising his Captain Mal role from "Firefly," in a matter of speaking. For the Halloween episode of "Castle," Fillion's Richard Castle will be dressed as, well, his former "Firefly" self.
And if you're not watching "Castle," you really should be. I knocked it after the pilot episode, but I re-reviewed it over the summer after watching the entire first season. It's not exactly "The Wire," or anything, but it may very well be the best procedural on television now. There's not a lot of novelist turned murderers who exclaim, "Best. Murder. Ever!" The cases themselves are mediocre, but the banter is gold. And if Nathan Fillion is not your idea of eye candy, Stana Katic is, hands down, the hottest female cop in the history of television.
Anyway, here's a grainy image (via Fireflyfans) of what you can expect on the October 26th episode (big H/T to bistro):
Never Go With a Hippie to a Second Location
F**k Me, Running. Ryan Reynolds: Dude in Drag
Dustin is the founder and co-owner of Pajiba. You may email him here, follow him on Twitter, or listen to his weekly TV podcast, Podjiba. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,259 |
function openNav() {
document.getElementById("mySidenav").style.width = "350px";
}
function closeNav() {
document.getElementById("mySidenav").style.width = "0";
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Ashok Binayak () is a Hindu temple located in Kathmandu district at eastern side of Kathmandu Durbar Square in Maru Tole. This temple is of Lord Ganesha also known as Binayak. The god is worshipped as the god of luck by Hindus. The temple hosts one of the four original Ganesh shrines of Kathmandu valley. It is a popular place of worship for Hindus.It is worshipped both by Hindus and Buddhists. The temple is visited during Tuesdays every week as it is the day which is considered the day of Ganesh.
History
Ashok Vinayak is one of the four original Ganesh of Kathmandu valley. The other three Ganesh shrines being Chandra Binayak, Surya Binayak and Jal Binayak. Additionally, there are other Ganesh temples like Kamal Binayak and Karya Binayak which are also popular Ganesh temples situated inside the valley.
Structure
The temple is a single-storeyed structure. The design is also noted for not having a gajur or a pointed design on the center-top of the roof. This design choice is also responsible for naming the locality "Maru" which literally in newari language means "does not have". The entire exterior of temple is plated in gold and is surrounded by bars at the present moment.
The temple hosts the holy shrine of Ganesh or Ashok Vinayak. On special days like Tuesdays and during special occasions like Dashain and Indra Jatra, the statue is decorated with a metallic cover made of silver or other metals. Right across the narrow road where the temple is, you will find a mouse which is known as a loyal bahan of the god Ganesh. You will also find structures where people can light and place their oil candles. The temple is also decorated with bells.
See also
List of Hindu temples in Nepal
References
External links
About Ashok Binayak Temple
Ashok Binayak Temple Kathmandu's Bagmati Attractions.
Information about Ashok Binayak Temple.
Hindu temples in Kathmandu District
Ganesha temples | {
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} | 6,593 |
\section{Introduction}
\IEEEPARstart{T}{opic} modeling offers a suite of useful tools that automatically learn the latent semantic structure of a large collection of documents or images, with
latent Dirichlet allocation (LDA)~\cite{lda} as one of the most popular examples.
The vanilla LDA is an unsupervised model built on input contents of documents or images. In many applications side information is often available apart from raw contents, e.g., user-provided rating scores of an online review text or user-generated tags for an image. Such side signal usually provides additional information to reveal the underlying structures of the data in study. There have been extensive studies on developing topic models that incorporate various side information, e.g., by treating it as supervision. Some representative models are supervised LDA (sLDA) \cite{slda} that captures a real-valued regression response for each document, multiclass sLDA \cite{multislda} that learns with discrete classification responses, discriminative LDA (DiscLDA) \cite{disclda} that incorporates classification response via discriminative linear transformations on topic mixing vectors, and MedLDA \cite{medlda,gibbsmedlda} that employs a max-margin criterion to learn discriminative latent topic representations for accurate prediction.
Topic models are typically learned by finding maximum likelihood estimates (MLE) through local search or sampling methods \cite{variationallda,gibbslda,emlda},
which may get trapped in local optima. Much recent progress has been made on developing spectral decomposition~\cite{a:twosvd,a:tensordecomp,spechmm} and nonnegative matrix factorization (NMF) \cite{practicalalgo,nmf1,beyondsvd,nmf2} methods to estimate the topic-word distributions. Instead of finding MLE estimates, which is a known NP-hard problem \cite{beyondsvd}, these methods assume that the documents are i.i.d. sampled from a topic model, and attempt to recover the underlying model parameters. Compared to local search and sampling algorithms, these methods enjoy the advantage of being provably effective. In fact, sample complexity bounds have been proved to show that given a sufficiently large collection of documents, these algorithms can recover the model parameters accurately with a high probability.
Recently, some attention has been paid on supervised topic models with NMF methods. For example, Nguyen et al.~\cite{a:sAnchor} present an extension of the anchor-word methods~\cite{practicalalgo} for LDA to capture categorical information in a supervised LDA for sentiment classification.
However, for spectral methods, previous work has mainly focused on unsupervised latent variable models, leaving the broad family of supervised models (e.g., sLDA) largely unexplored.
The only exception
is~\cite{specregression} which presents a spectral method for mixtures of regression models, quite different from sLDA.
Such ignorance is not a coincidence as supervised models impose new technical challenges. For instance, a direct application of previous techniques~\cite{a:twosvd,a:tensordecomp} on sLDA
cannot handle regression models with duplicate entries.
In addition, the sample complexity bound gets much worse if we try to match entries in regression models
with their corresponding topic vectors.
In this paper, we extend the applicability of spectral learning methods by presenting novel spectral decomposition algorithms to recover the parameters of sLDA models from low-order empirical moments estimated from the data. We present two variants of spectral methods.
The first algorithm is an extension of the spectral methods for LDA,
with an extra power update step of recovering the regression model in sLDA, including the variance parameter.
The power-update step uses a newly designed empirical moment to recover regression model entries directly from the data and reconstructed topic distributions.
It is free from making any constraints on the underlying regression model.
We provide a sample complexity bound and analyze the identifiability conditions.
In fact, the two-stage method does not increase the sample complexity much compared to that of the vanilla LDA.
However, the two-stage algorithm could have some disadvantages because of its separation that the topic distribution matrix is recovered in an unsupervised manner without considering
supervision and the regression parameters are recovered by assuming a fixed topic matrix.
Such an unwarranted separation often leads to inferior performance compared to Gibbs sampling methods in practice (See Section~\ref{sec:exp}).
To address this problem, we further present a novel single-phase spectral method for supervised topic models,
which jointly recovers all the model parameters (except the noise variance) by doing a single-step of robust tensor decomposition of a newly designed
empirical moment that takes both input data and supervision signal into consideration.
Therefore, the joint method can use supervision information in recovering both the topic distribution matrix and
regression parameters. The joint method is also provably correct and we provide a sample complexity bound to achieve the $\epsilon$
error rate in a high probability.
Finally, we provide thorough experiments on both synthetic and real datasets to demonstrate the practical effectiveness of our spectral methods.
For the two-stage method, by combining with a Gibbs sampling procedure, we show superior performance in terms of language modeling, prediction accuracy and running time
compared to traditional inference algorithms.
Furthermore, we demonstrate that on a large-scale review rating dataset
our single-phase method alone can achieve comparable or even better results than the state-of-the-art methods (e.g., sLDA with Gibbs sampling and MedLDA).
Such promising results are significant to the literature of spectral methods, which were often observed to
be inferior to the MLE-based method
; and a common heuristic was to use the outputs
of a spectral method to initialize an EM algorithm, which sometimes improves the performance~\cite{a:meetem}.
\iffalse
Generally speaking, spectral methods are not better than MLE-based methods \cite{a:tensordecomp}
although a combination (e.g. use the output of spectral method as initialization) of them sometimes improves
the performance significantly \cite{a:yiningwang}, \cite{a:meetem}. However, empirical studies show that
even using our method alone, we can achieve comparable or superior performance in large-scale dataset, which demonstrate the effectiveness of our method.
\fi
The rest of the paper is organized as follows. Section $2$ reviews basics of supervised topic models. Section $3$ introduces the background knowledge and notations for sLDA and high-order tensor decomposition.
Section $4$ presents the two-stage spectral method, with a rigorous theoretical analysis, and Section $5$ presents the joint spectral method together with a sample complexity bound.
Section $6$ presents some implementation details to scale up the computation.
Section $7$ presents experimental results on both synthetic and real datasets.
Finally, we conclude in Section $8$.
\section{Related Work}
Based on different principles, there are various methods to learn supervised topic models. The most natural one is maximum-likelihood estimation (MLE). However, the
highly non-convex property of the learning objective makes the optimization problem very hard. In the original paper \cite{slda}, where the MLE is used, the authors choose variational
approximation to handle intractable posterior expectations. Such a method tries to maximize a lower bound which is built on some variational distributions and a mean-field assumption is usually imposed for tractability. Although the method works well in practice, we do not have any guarantee that the distribution we learned is close to the true one. Under Bayesian framework, Gibbs sampling is an attractive method which enjoys the property that the stationary distribution of the chain is the target posterior distribution. However, this does not mean that we can really get accurate samples from posterior distribution in practice. The slow mixing rate often makes the sampler trapped in a local minimum which is far from the
true distribution if we only run a finite number of iterations.
Max-margin learning is another principle on learning supervised topic models, with maximum entropy discrimination LDA (MedLDA)~\cite{medlda} as a popular example. MedLDA explores the max-margin principle to learn sparse and discriminative topic representations. The learning problem can be defined under the regularized Bayesian inference (RegBayes)~\cite{regBayes} framework, where the max-margin posterior regularization is introduced to ensure that the topic representations are good at predicting response variables.
Though both carefully designed variational inference~\cite{medlda} and Gibbs sampling methods~\cite{gibbsmedlda} are given, we still cannot guarantee the quality of the learnt model in general.
Recently, increasing efforts have been made to recover the parameters directly with provable correctness for topic models, with the main focus on unsupervised models such as LDA.
Such methods adopt either NMF or spectral decomposition approaches.
For NMF, the basic idea is that even NMF is an NP-hard problem in general, the topic distribution matrix can be recovered under
some separable condition, e.g., each topic has at least one anchor word.
Precisely, for each topic, the method first finds an anchor word that has non-zero probability only in that topic.
Then a recovery step reconstructs the topic distribution given such
anchor words and a second-order moment matrix of word-word co-occurrence~\cite{beyondsvd,practicalalgo}.
The original reconstruction step only needs a part of the matrix which is not robust in practice.
Thus in \cite{practicalalgo}, the author recovers the topic distribution based on a probabilistic framework.
The NMF methods produce good empirical results on real-world data.
Recently, the work \cite{a:sAnchor} extends the anchor word methods to handle supervised topic models. The method
augments the word co-occurrence matrix with additional dimensions for metadata such as sentiment, and
shows better performance in sentiment classification.
Spectral methods start from computing some low-order moments based on the samples and then relate them with the model parameters.
For LDA, tensors up to three order are sufficient to recover its parameters~\cite{a:twosvd}.
After centralization, the moments can be
expressed as a mixture of the parameters we are interested in.
After that whitening and robust tensor decomposition steps are adopted
to recover model parameters. The whitening step makes that the third-order tensor can be decomposed as a set of
orthogonal eigenvectors and their corresponding eigenvalues after some operations based on its output
and the robust tensor decomposition step then finds them.
Previous work only focuses on unsupervised LDA models and
we aim to extend the ability for spectral methods to handle response variables. Finally,
some preliminary results of the two-stage recovery algorithm have been reported in~\cite{a:yiningwang}.
This paper presents a systematical analysis with a novel one-stage spectral method,
which yields promising results on a large-scale dataset.
\vspace{-.15cm}
\section{Preliminaries}
We first overview the basics of sLDA, orthogonal tensor decomposition and the notations to be used.
\vspace{-.15cm}
\subsection{Supervised LDA}\label{sec:sLDA-model}
Latent Dirichlet allocation (LDA)~\cite{lda} is a hierarchical generative model for topic modeling of text documents or images represented in a bag-of-visual-words format~\cite{ldaimage}. It assumes $k$ different \emph{topics} with topic-word distributions $\bm\mu_1,\cdots,\bm\mu_k\in\Delta^{V-1}$, where $V$ is the vocabulary size and $\Delta^{V-1}$ denotes the probability simplex of a $V$-dimensional random vector.
For a document, LDA models a
topic mixing vector $\bm h\in\Delta^{k-1}$
as a probability distribution over the $k$ topics. A conjugate Dirichlet prior with parameter $\bm \alpha$ is imposed on the topic mixing vectors. A bag-of-words model is then adopted, which first generates a topic indicator for each word $z_j \sim \textrm{Multi}(\bm h)$ and then generates the word itself as $w_j \sim \textrm{Multi}(\bm \mu_{z_j})$.
Supervised latent Dirichlet allocation (sLDA)~\cite{slda} incorporates an extra response variable $y\in\mathbb R$ for each document.
The response variable is modeled by a linear regression model $\bm\eta\in\mathbb R^k$
on either the topic mixing vector $\bm h$ or the averaging topic assignment vector $\bar{\bm z}$,
where $\bar z_i = \frac{1}{M}\sum_{j=1}^M {1_{[z_j=i]}}$ with $M$ being the number of words in the document and $1_{[\cdot]}$ being the indicator function (i.e., equals to 1 if the predicate holds; otherwise 0).
The noise is assumed to be Gaussian with zero mean and $\sigma^2$ variance.
Fig.~\ref{slda} shows the graph structure of the sLDA model using $\bm h$ for regression.
Although previous work has mainly focused on the model using averaging topic assignment vector $\bar{\bm z}$, which is convenient for collapsed Gibbs sampling and variational inference with $\bm h$ integrated out due to the conjugacy between a Dirichlet prior and a multinomial likelihood,
we consider using the topic mixing vector $\bm h$ as the features for regression because it will considerably simplify our spectral algorithm and analysis.
One may assume that whenever a document is not too short, the empirical distribution of its word topic assignments should be close to the document's topic mixing vector.
Such a scheme was adopted to learn sparse topic coding models~\cite{Zhu:stc11}, and has demonstrated promising results in practice.
Our results also prove that this is an effective strategy.
\iftrue
\begin{figure}[t]\vspace{-.1cm}
\centering
\begin{tikzpicture}
\tikzstyle{main}=[circle, minimum size = 5mm, thick, draw =black!80, node distance = 6mm]
\tikzstyle{connect}=[-latex, thick]
\tikzstyle{box}=[rectangle, draw=black!100]
\node[main, fill = white!100] (alpha) [label=center:$\alpha$] { };
\node[main] (h) [right=of alpha,label=center:h] { };
\node[main] (z) [right=of h,label=center:z] {};
\node[main] (mu) [above=of z,label=center:$\bm{\mu}$] { };
\node[main, fill = black!10] (x) [right=of z,label=center:x] { };
\node[main, fill = black!10] (y) [below= of h, label=center:y]{ };
\node[main] (eta) [below=of alpha, label=center:$\bm{\eta}$]{};
\node (exp) [below= of z]{$y = \bm{\eta}^{\top}\bm{h} + \epsilon $};
\path (alpha) edge [connect] (h)
(h) edge [connect] (z)
(z) edge [connect] (x)
(mu) edge [connect] (x)
(h) edge [connect] (y)
(eta) edge [connect] (y);
\node[rectangle, inner sep=-0.9mm, fit= (z) (x),label=below right:M, xshift=5mm] {};
\node[rectangle, inner sep=3.6mm,draw=black!100, fit= (z) (x)] {};
\node[rectangle, inner sep=1mm, fit= (z) (x) (y) ,label=below right:N, xshift=5mm] {};
\node[rectangle, inner sep=5mm, draw=black!100, fit = (h) (z) (x) (y) ] {};
\end{tikzpicture}\vspace{-.3cm}
\caption{A graphical illustration of supervised LDA.
See text for details.}
\label{slda} \vspace{-.4cm
\end{figure}
\fi
\vspace{-.15cm}
\subsection{High-order tensor product and orthogonal tensor decomposition}
Here we briefly introduce something about tensors,
which mostly follow the same as in \cite{a:tensordecomp}.
A real \emph{$p$-th order tensor} $A\in\bigotimes_{i=1}^p{\mathbb R^{n_i}}$ belongs to the tensor product of Euclidean spaces $\mathbb R^{n_i}$.
Without loss of generality, we assume $n_1=n_2=\cdots=n_p=n$. We can identify each coordinate of $A$ by a $p$-tuple
$(i_1,\cdots,i_p)$, where $i_1,\cdots,i_p\in [n]$.
For instance, a $p$-th order tensor is a vector when $p=1$ and a matrix when $p=2$. We can also consider a $p$-th order tensor $A$ as a multilinear mapping.
For $A\in\bigotimes^p\mathbb R^n$ and matrices $X_1,\cdots,X_p\in\mathbb R^{n\times m}$,
the mapping $A(X_1,\cdots,X_p)$ is a $p$-th order tensor in $\bigotimes^p\mathbb R^m$, with
$[A(X_1,\cdots,X_p)]_{i_1,\cdots,i_p}
\triangleq \sum_{j_1,\cdots,j_p\in [n]}{A_{j_1,\cdots,j_p}[X_1]_{j_1,i_1}[X_2]_{j_2,i_2}\cdots [X_p]_{j_p,i_p}}$.
Consider some concrete examples of such a multilinear mapping. When $A$, $X_1$ and $X_2$ are matrices, we have $A(X_1,X_2) = X_1^\top AX_2$. Similarly, when $A$ is a matrix and $x$ is a vector, we have $A(I, x) = Ax$.
An orthogonal tensor decomposition of a tensor $A\in\bigotimes^p\mathbb R^n$ is a collection of orthonormal vectors $\{\bm v_i\}_{i=1}^k$
and scalars $\{\lambda_i\}_{i=1}^k$ such that
$A = \sum_{i=1}^k{\lambda_i\bm v_i^{\otimes p}}$, where we use $\bm v^{\otimes p} \triangleq \bm v\otimes \bm v\otimes\cdots\otimes \bm v$ to denote the $p$-th order tensor
generated by a vector $\bm v$.
Without loss of generality, we assume $\lambda_i$ are nonnegative when $p$ is odd since we can change the sign of $\bm{v_i}$ otherwise.
Although orthogonal tensor decomposition in the matrix case
can be done efficiently by singular value decomposition (SVD),
it has several delicate issues in higher order tensor spaces~\cite{a:tensordecomp}.
For instance, tensors may not have unique decompositions,
and an orthogonal decomposition may not exist for every symmetric tensor~\cite{a:tensordecomp}.
Such issues are further complicated when only noisy estimates of the desired tensors are available.
For these reasons, we need more advanced techniques to handle high-order tensors. In this paper, we will apply \emph{robust tensor power} methods~\cite{a:tensordecomp} to recover robust eigenvalues and eigenvectors of an (estimated) third-order tensor. The algorithm recovers eigenvalues and eigenvectors up to an absolute error $\varepsilon$,
while running in polynomial time w.r.t the tensor dimension and $\log(1/\varepsilon)$.
Further details and analysis of the robust tensor power method are in Appendix~A.2 and \cite{a:tensordecomp}.
\iffalse
\begin{figure}[t]\vspace{-.3cm}
\centering
\subfigure[$y_d=\bm\eta_d^\top\bm h_d + \varepsilon_d$]{
\scalebox{0.8}{
\begin{tikzpicture}
\node[obs] (x) {$x$} ; %
\node[latent, left=of x] (z) {$z$} ; %
\node[latent, left=of z] (h) {$\bm h$}; %
\node[const, left=of h] (alpha) {$\bm\alpha$};
\node[latent, right=of x] (mu) {$\bm\mu$};
\node[const, below=of alpha] (eta) {$\bm\eta,\sigma$};
\node[obs, below=of h, right=of eta] (y) {$y$};
\node[const, below=of mu] (beta) {$\bm\beta$};
\edge {alpha} {h};
\edge {h} {z};
\edge {z} {x};
\edge {mu} {x};
\edge {h} {y};
\edge {eta} {y};
\edge {beta} {mu};
\plate {plate1} { (z) (x) } {$M$};
\plate {plate2} { (h) (y) (plate1) } {$N$};
\plate {plate3} { (mu) } {$k$};
\end{tikzpicture}
}
\label{fig_slda_h}
}
\subfigure[$y_d=\bm\eta_d^\top\bar{\bm z}_d + \varepsilon_d$]{
\scalebox{0.8}{
\begin{tikzpicture}
\node[obs] (x) {$x$} ; %
\node[latent, left=of x] (z) {$z$} ; %
\node[latent, left=of z] (h) {$\bm h$}; %
\node[const, left=of h] (alpha) {$\bm\alpha$};
\node[latent, right=of x] (mu) {$\bm\mu$};
\node[const, below=of alpha] (eta) {$\bm\eta,\sigma$};
\node[obs, below=of h, right=of eta] (y) {$y$};
\node[const, below=of mu] (beta) {$\bm\beta$};
\edge {alpha} {h};
\edge {h} {z};
\edge {z} {x};
\edge {mu} {x};
\edge {z} {y};
\edge {eta} {y};
\edge {beta} {mu};
\plate {plate1} { (z) (x) } {$M$};
\plate {plate2} { (h) (y) (plate1) } {$N$};
\plate {plate3} { (mu) } {$k$};
\end{tikzpicture}
}
\label{fig_slda_z}
}
\vspace{-.2cm}
\caption{Plate notations for two variants of sLDA}\label{fig_slda}\vspace{-.4cm}
\end{figure}
\fi
\iffalse
\begin{figure}
\centering
\includegraphics[width = 15cm]{images/slda.png}
\caption{Plate notations for two variants of sLDA}
\label{slda}
\end{figure}
\fi
\vspace{-.15cm}
\subsection{Notations}
We use $\|\bm v\| = \sqrt{\sum_i{v_i^2}}$ to denote the Euclidean norm of vector $\bm v$,
$\|M\|$ to denote the spectral norm of matrix $M$, $\|T\|$ to denote the operator norm of a high-order tensor, and
$\|M\|_F = \sqrt{\sum_{i,j}{M_{ij}^2}}$ to denote the Frobenious norm of $M$. We use an one-hot vector $\bm x\in\mathbb R^V$ to represent a word in a document (i.e., for the $i$-th word in a vocabulary, only $x_i = 1$ all other elements are $0$).
In the two-stage spectral method, we use $O \triangleq (\bm\mu_1,\bm\mu_2,\cdots,\bm\mu_k)\in\mathbb R^{V\times k}$ to denote
the topic distribution matrix, and $\widetilde O\triangleq (\widetilde{\bm\mu}_1,\widetilde{\bm\mu}_2,\cdots,\widetilde{\bm\mu}_k)$
to denote the canonical version of $O$, where $\widetilde{\bm\mu}_i = \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}}\bm\mu$ with
$\alpha_0\triangleq \sum_{i=1}^k{\alpha_i}$. For the joint spectral method, we combine the topic distribution $\bm{\mu}_i$ with its
regression parameter to form a joint topic distribution vector $\bm{v}_i \triangleq [\bm{\mu}_i',\eta_i]'$. We use notation $O^* \triangleq (\bm{v}_1, \bm{v}_2, ... ,
\bm{v}_k) \in \mathbb{R}^{(V+1) \times k} $ to denote the joint topic distribution matrix and $\widetilde{O}^* \triangleq [\widetilde{\bm{v}}_1, \widetilde{\bm{v}}_2, ...,
\widetilde{\bm{v}}_k]$ to denote its canonical version where $\widetilde{\bm{v}}_i \triangleq \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0 + 1)}} \bm{v}_i$.
\vspace{-.15cm}
\section{A Two-stage Spectral Method}
We first present a two-stage spectral method to recover the parameters of sLDA.
The algorithm consists of two key components---an orthogonal tensor decomposition of observable moments to recover the topic distribution matrix $O$ and a power update method to recover the regression model $\bm\eta$. We present these techniques and a rigorous theoretical analysis below.
\vspace{-.15cm}
\subsection{Moments of observable variables}
Our spectral decomposition methods recover the topic distribution matrix $O$ and the linear regression model $\bm\eta$ by manipulating moments of observable variables. In Definition~\ref{def_moment}, we define a list of moments on random variables from the underlying sLDA model.
\begin{deff} We define the following moments of observable variables:
{\small \setlength\arraycolsep{1pt} \begin{eqnarray}
M_1 &=& \mathbb E[\bm x_1],\quad
M_2 = \mathbb E[\bm x_1\otimes\bm x_2] - \frac{\alpha_0}{\alpha_0+1}M_1\otimes M_1,\\
M_3 &=& \mathbb E[\bm x_1\otimes\bm x_2\otimes\bm x_3] - \frac{\alpha_0}{\alpha_0+2}(\mathbb E[\bm x_1\otimes\bm x_2\otimes M_1] \nonumber \\
& &+ \mathbb E[\bm x_1\otimes M_1\otimes \bm x_2] + \mathbb E[M_1\otimes \bm x_1\otimes \bm x_2])\nonumber\\
& &+ \frac{2\alpha_0^2}{(\alpha_0+1)(\alpha_0+2)}M_1\otimes M_1\otimes M_1,\\
M_y &=& \mathbb E[y\bm x_1\otimes \bm x_2] - \frac{\alpha_0}{\alpha_0+2}(\mathbb E[y] \mathbb E[\bm x_1\otimes\bm x_2]
+ \mathbb E[\bm x_1]\otimes\mathbb E[y\bm x_2] \nonumber\\
& & + \mathbb E[y\bm x_1]\otimes\mathbb E[\bm x_2]) + \frac{2\alpha_0^2}{(\alpha_0+1)(\alpha_0+2)}\mathbb E[y]M_1\otimes M_1.
\end{eqnarray}}\label{def_moment}
\end{deff}\vspace{-.25cm}
Note that the moments $M_1$, $M_2$ and $M_3$ were also defined in~\cite{a:twosvd,a:tensordecomp} for recovering the parameters of LDA models. For sLDA, we need to define a new moment $M_y$ in order to recover the linear regression model $\bm\eta$.
The moments are based on observable variables in the sense that they can be estimated from i.i.d. sampled documents.
For instance, $M_1$ can be estimated by computing the empirical distribution of all words,
and $M_2$ can be estimated using $M_1$ and word co-occurrence frequencies. Though the moments in the above forms look
complicated, we can apply elementary calculations based on the conditional independence structure of sLDA to significantly simplify them and more importantly to get them connected with the model parameters to be recovered, as summarized in Proposition~\ref{prop_moments}, whose proof is elementary and deferred to Appendix C for clarity.
\begin{prop}The moments can be expressed using the model parameters as:\\[-.6cm]
{\small \setlength\arraycolsep{1pt} \begin{eqnarray}
M_2 &=& \frac{1}{\alpha_0(\alpha_0+1)}\sum_{i=1}^k{\alpha_i\bm\mu_i\otimes\bm\mu_i}, \\
M_3 &=& \frac{2}{\alpha_0(\alpha_0+1)(\alpha_0+2)}\sum_{i=1}^k{\alpha_i\bm\mu_i\otimes\bm\mu_i\otimes\bm\mu_i},\\
M_y &=& \frac{2}{\alpha_0(\alpha_0+1)(\alpha_0+2)}\sum_{i=1}^k{\alpha_i\eta_i\bm\mu_i\otimes\bm\mu_i}.
\end{eqnarray}}\label{prop_moments}
\end{prop
\iffalse
\begin{prof*}
All the three equations can be proved in a similar mannar. We only give the proof to the equation on $M_y$.
In sLDA the topic mixing vector $\bm{h}$ follows a Dirichlet prior distribution with parameter $\bm{\alpha}$.
Therefore, we have:
\begin{equation*}
\begin{aligned}
& \mathbb{E}[h_i] = \frac{\alpha_i}{\alpha_0}, \mathbb{E}[h_i,hj] = \left \{ \begin{array}{ll}
\frac{(\alpha_i + 1)\alpha_i}{\alpha_0(\alpha_0 + 1)} & if\ i = j \\
\frac{\alpha_i\alpha_j}{\alpha_0(\alpha_0 + 1)} & if \ i \not = j \end{array}\right . \\
&\mathbb{E}[h_ih_jh_k] = \left \{ \begin{array}{ll}
\frac{\alpha_i(\alpha_i + 1)(\alpha_i + 2)}{\alpha_0(\alpha_0 + 1)(\alpha_0 + 2)} & if \ i= j = k\\
\frac{\alpha_j\alpha_i(\alpha_i+1)}{\alpha_0(\alpha_0 + 1)(\alpha_0 + 2)} & if \ i = j \not = k\\
\frac{\alpha_i\alpha_j\alpha_k}{\alpha_0(\alpha_0 + 1)(\alpha_0 + 2)} & if \ i \not = j \not = k\\
\end{array}\right .
\end{aligned}
\end{equation*}
Next, note that:
\begin{equation*}
\begin{aligned}
&\mathbb{E}[y|\bm{h}] = \bm{\eta}'\bm{h}, \mathbb{E}[\bm{x}_1|\bm{h}] = \sum\limits_{i=1}^{k}h_i\bm{\mu}_i \\
& \mathbb{E}[\bm{x}_1 \otimes \bm{x}_2 | \bm{h}] = \sum\limits_{i,j =1}^{k}h_ih_j\bm{\mu}_i\bm{\mu}_j\\
& \mathbb{E}[y\bm{x}_1\otimes \bm{x}_2 | \bm{h}] = \sum\limits_{i,j,l=1}^{k} h_ih_jh_l\cdot \eta_i\bm{\mu}_j \otimes
\bm{\mu}_l
\end{aligned}
\end{equation*}
the proposition can then be easily proved by taking expectation over the topic mixing vector $\bm{h}$. \
\hfill\QED
\end{prof*}
\fi
\begin{algorithm}[t]
\caption{spectral parameter recovery algorithm for sLDA. Input parameters: $\alpha_0, L, T$.}
\centering
\begin{algorithmic}[1]
\State Compute empirical moments and obtain $\widehat M_2, \widehat M_3$ and $\widehat M_y$.
\State Find $\widehat W\in\mathbb R^{n\times k}$ such that $\widehat M_2(\widehat W, \widehat W) = I_k$.
\State Find robust eigenvalues and eigenvectors $(\widehat\lambda_i,\widehat{\bm v}_i)$ of $\widehat M_3(\widehat W,\widehat W,\widehat W)$
using the robust tensor power method \cite{a:tensordecomp} with parameters $L$ and $T$.
\State Recover prior parameters: $\widehat\alpha_i \gets \frac{4\alpha_0(\alpha_0+1)}{(\alpha_0+2)^2\widehat\lambda_i^2}$.
\State Recover topic distributions: \vspace{-.3cm}
$$\widehat{\bm\mu}_i\gets \frac{\alpha_0+2}{2}\widehat\lambda_i(\widehat W^+)^\top\widehat{\bm v}_i.$$\vspace{-.55cm}
\State Recover the linear regression model: \vspace{-.2cm}
$$\widehat\eta_i \gets \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i.$$\vspace{-.55cm}
\State \textbf{Output:} $\widehat{\bm\eta}$, $\widehat{\bm\alpha}$ and $\{\widehat{\bm\mu}_i\}_{i=1}^k$.
\end{algorithmic}
\label{alg1}
\end{algorithm}
\vspace{-.7cm}
\subsection{Simultaneous diagonalization}
Proposition~\ref{prop_moments} shows that the moments in Definition \ref{def_moment} are all the weighted sums of tensor products of $\{\bm\mu_i\}_{i=1}^k$ from the underlying sLDA model.
One idea to reconstruct $\{\bm\mu_i\}_{i=1}^k$ is to perform \emph{simultaneous diagonalization} on tensors of different orders.
The idea has been used in a number of recent developments of spectral methods for latent variable models~\cite{a:twosvd,a:tensordecomp,specregression}.
Specifically, we first whiten the second-order tensor $M_2$ by finding a matrix $W\in\mathbb R^{V\times k}$ such that
$W^\top M_2 W = I_k$.
This whitening procedure is possible whenever the topic distribuction vectors $\{\bm\mu_i\}_{i=1}^k$ are linearly independent
(and hence $M_2$ has rank $k$).
This is not always correct since in the ``overcomplete" case~\cite{a:overcomplete},
it is possible that the topic number $k$ is larger than vocabulary size $V$. However, the
linear independent assumption gives us a more compact representation for the topic model and
works well in practice. Hence we simply assume that the whitening procedure is possible.
The whitening procedure and the linear independence assumption also
imply that $\{W\bm\mu_i\}_{i=1}^k$ are orthogonal vectors (see Appendix A.2 for details),
and can be subsequently recovered by performing an orthogonal
tensor decomposition on the simultaneously whitened third-order tensor $M_3(W,W,W)$.
Finally, by multiplying the pseudo-inverse of the whitening matrix $W^+$ we obtain the topic distribution vectors $\{\bm\mu_i\}_{i=1}^k$.
It should be noted that Jennrich's algorithm \cite{jennrich1,jennrich2,ankur-review} could recover $\{\bm\mu_i\}_{i=1}^k$ directly from the 3-rd order tensor $M_3$ alone when $\{\bm\mu_i\}_{i=1}^k$ is linearly independent.
However, we still adopt the above simultaneous diagonalization framework because the intermediate vectors $\{W\bm\mu_i\}_{i=1}^k$
play a vital role in the recovery procedure of the linear regression model $\bm\eta$.
\vspace{-.1cm}
\subsection{The power update method}
Although the linear regression model $\bm\eta$ can be recovered in a similar manner by performing simultaneous diagonalization on $M_2$ and $M_y$, such a method has several disadvantages, thereby calling for novel solutions. First, after obtaining entry values $\{\eta_i\}_{i=1}^k$ we need to match them to the topic distributions $\{\bm\mu_i\}_{i=1}^k$ previously recovered.
This can be easily done when we have access to the true moments,
but becomes difficult when only estimates of observable tensors are available
because the estimated moments may not share the same singular vectors due to sampling noise.
A more serious problem is that when $\bm\eta$ has duplicate entries the orthogonal decomposition of $M_y$ is no longer unique.
Though a randomized strategy similar to the one used in~\cite{a:twosvd} might solve the problem,
it could substantially increase the sample complexity~\cite{a:tensordecomp} and render the algorithm impractical.
In \cite{spechmm}, the authors provide a method for the matching problem by reusing eigenvectors.
We here develop a power update method to resolve the above difficulties with a similar spirit. Specifically, after obtaining the whitened (orthonormal) vectors $\{\bm v_i\} \triangleq c_i\cdot W^{\top}\bm\mu_i$ \footnote{$c_i \triangleq \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}} $ is a scalar coefficient that depends on $\alpha_0$ and $\alpha_i$. See Appendix A.2 for details.} we recover the entry $\eta_i$ of the linear regression model directly by computing a power update $\bm v_i^\top M_y(W,W)\bm v_i$. In this way, the matching problem is automatically solved because we know what topic distribution vector $\bm\mu_i$ is used when recovering $\eta_i$. Furthermore, the singular values (corresponding to the entries of $\bm \eta$) do not need to be distinct because we are not using any unique SVD properties of $M_y(W,W)$. As a result, our proposed algorithm works for any linear model $\bm\eta$.
\vspace{-.1cm}
\subsection{Parameter recovery algorithm}
Alg.~\ref{alg1} outlines our parameter recovery algorithm for sLDA (Spectral-sLDA). First, empirical estimations of the observable moments in Definition \ref{def_moment} are computed from the given documents.
The simultaneous diagonalization method is then used
to reconstruct the topic distribution matrix $O$ and its prior parameter $\bm\alpha$. After obtaining $O=(\bm\mu_1,\cdots,\bm\mu_k)$,
we use the power update method introduced in the previous section to recover the linear regression model $\bm\eta$. We can also recover the noise level parameter $\sigma^2$ with the other
parameters in hand by estimating $\mathbb{E}[y]$ and $\mathbb{E}[y^2]$ since
$\mathbb{E}[y] = \mathbb{E}[\mathbb{E}[y|\bm{h}]] = \bm{\alpha}^{\top}\bm{\eta}$ and
$\mathbb{E}[y^2] = \mathbb{E}[\mathbb{E}[y^2|\bm{h}]] =
\bm{\eta}^{\top}\mathbb{E}[\bm{h}\otimes \bm{h}]\bm{\eta} + \sigma^2$, where the term $\mathbb{E}[\bm{h}\otimes \bm{h}]$ can be computed in an analytical form using the model parameters, as detailed in Appendix C.1.
Alg.~1 admits three hyper-parameters $\alpha_0$, $L$ and $T$. $\alpha_0$ is defined as the sum of all entries in the prior parameter $\bm\alpha$. Following the conventions in~\cite{a:twosvd,a:tensordecomp}, we assume that $\alpha_0$ is known a priori and use this value to perform parameter estimation. It should be noted that this is a mild assumption, as in practice usually a homogeneous vector $\bm\alpha$ is assumed and the entire vector is known~\cite{lsa}. The $L$ and $T$ parameters are used to control the number of iterations in the robust tensor power method. In general, the robust tensor power method runs in $O(k^3LT)$ time.
To ensure sufficient recovery accuracy, $L$ should be at least a linear function of $k$ and
$T$ should be set as $T = \Omega(\log(k) + \log\log(\lambda_{\max}/\varepsilon))$,
where $\lambda_{\max} = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$
and $\varepsilon$ is an error tolerance parameter.
Appendix~A.2 and~\cite{a:tensordecomp} provide a deeper analysis into the choice of $L$ and $T$ parameters.
\vspace{-.1cm}
\subsection{Sample Complexity Analysis}
\vspace{-.1cm}
We now analyze the sample complexity of Alg.~\ref{alg1} in order to achieve $\varepsilon$-error with a high probability. For clarity, we focus on presenting the main results, while deferring the proof details to Appendix~A, including the proofs of important lemmas that are needed for the main theorem.
\begin{thm
Let $\sigma_1(\widetilde O)$ and $\sigma_k(\widetilde O)$ be the largest and the smallest singular values of the canonical topic distribution matrix $\widetilde O$.
Define $\lambda_{\min} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\max}}}$
and $\lambda_{\max} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$'∫?
with $\alpha_{\max}$ and $\alpha_{\min}$ the largest and the smallest entries of $\bm\alpha$.
Suppose $\widehat{\bm\mu}$, $\widehat{\bm\alpha}$ and $\widehat{\bm\eta}$ are the outputs of Algorithm 1,
and $L$ is at least a linear function of $k$.
Fix $\delta\in(0,1)$. For any small error-tolerance parameter $\varepsilon > 0$,
if Algorithm \ref{alg1} is run with parameter $T = \Omega(\log(k) + \log\log(\lambda_{\max}/\varepsilon))$
on $N$ i.i.d. sampled documents (each containing at least 3 words) with
$N\geq \max(n_1, n_2, n_3)$, where\vspace{-.2cm}
{\small
\begin{equation*}
\begin{aligned}
n_1 &= C_1\cdot \left(1+\sqrt{\log(6/\delta)}\right)^2\cdot \frac{\alpha_0^2(\alpha_0+1)^2}{\alpha_{\min}}, \nonumber \\
n_2 &= C_2\cdot \frac{(1+\sqrt{\log(15/\delta)})^2}{\varepsilon^2\sigma_k(\widetilde O)^4} \\
&\cdot\max\left(\left( \|\bm\eta\| - \sigma \Phi^{-1}\left( \frac{\delta}{60}\right)\right)^2, \alpha_{\max}^2\sigma_1(\widetilde O)^2\right), \nonumber \\
n_3 &= C_3\cdot \frac{(1+\sqrt{\log(9/\delta)})^2}{\sigma_k(\widetilde O)^{10}}\cdot\max\left(\frac{1}{\varepsilon^2},
\frac{k^2}{\lambda_{\min}^2}\right), \nonumber
\end{aligned}
\end{equation*} }
and $C_1, C_2$ and $C_3$ are universal constants,
then with probability at least $1-\delta$, there exists a permutation $\pi:[k]\to [k]$ such that for every topic $i$, the following holds:\vspace{-.2cm}
\setlength\arraycolsep{-1pt} {\small \begin{eqnarray}
&&|\alpha_i-\widehat{\alpha}_{\pi(i)}| \!\leq\! \frac{4\alpha_0(\alpha_0 \!+\! 1)(\lambda_{\max} \!+\! 5\varepsilon)}{(\alpha_0 \!+\! 2)^2\lambda_{\min}^2(\lambda_{\min} \!-\! 5\varepsilon)^2}\cdot 5\varepsilon,~if~\lambda_{\min} > 5\varepsilon \nonumber \\
&&\|\bm\mu_i - \widehat{\bm\mu}_{\pi(i)}\| \!\leq\! \left(3\sigma_1(\widetilde O)\left(\frac{8\alpha_{\max}}{\lambda_{\min}} + \frac{5(\alpha_0+2)}{2}\right) + 1\right)\varepsilon \nonumber \\
&&|\eta_i - \widehat\eta_{\pi(i)}| \!\leq\! \left(\frac{\|\bm\eta\|}{\lambda_{\min}} + (\alpha_0+2)\right) \varepsilon . \nonumber
\end{eqnarray}}\vspace{-.4cm}
\label{thm_main}
\end{thm}
In brevity, the proof is based on matrix perturbation lemmas (see Appendix~A.1) and analysis to the orthogonal tensor decomposition methods (including SVD and robust tensor power method) performed on inaccurate tensor estimations (see Appendix~A.2).
The sample complexity lower bound consists of three terms, from $n_1$ to $n_3$.
The $n_3$ term comes from the sample complexity bound for the robust tensor power method~\cite{a:tensordecomp};
the $(\|\bm\eta\|- \sigma \Phi^{-1}(\delta/60))^2$ term in $n_2$ characterizes the recovery accuracy for the linear regression model $\bm\eta$,
and the $\alpha_{\max}^2\sigma_1(\widetilde O)^2$ term arises when we try to recover the topic distribution vectors $\bm\mu$;
finally, the term $n_1$ is required so that some technical conditions are met.
The $n_1$ term does not depend on either $k$ or $\sigma_k(\widetilde O)$,
and could be largely neglected in practice.
\begin{rem}
An important implication of Theorem~\ref{thm_main} is that it provides a sufficient condition
for a supervised LDA model to be identifiable, as shown in Remark~\ref{rem_identifiability}.
To some extent, Remark~\ref{rem_identifiability} is the best identifiability result possible under our inference framework,
because it makes no restriction on the linear regression model $\bm\eta$, and the linear independence assumption is unavoidable
without making further assumptions on the topic distribution matrix $O$.
\end{rem}
\begin{rem} Given a sufficiently large number of i.i.d. sampled documents with at least 3 words per document, a supervised LDA model $\mathcal M=(\bm\alpha,\bm\mu,\bm\eta)$ is identifiable if $\alpha_0 = \sum_{i=1}^k{\alpha_i}$ is known and $\{\bm\mu_i\}_{i=1}^k$ are linearly independent.
\label{rem_identifiability}
\end{rem}
We now take a close look at the sample complexity bound in Theorem~\ref{thm_main}.
It is evident that $n_2$ can be neglected when the number of topics $k$ gets large,
because in practice the norm of the linear regression model $\bm\eta$ is usually assumed to be small in order to avoid overfitting.
Moreover, as mentioned before, the prior parameter $\bm\alpha$ is often assumed to be homogeneous with $\alpha_i = 1/k$ \cite{lsa}.
With these observations, the sample complexity bound in Theorem~\ref{thm_main}
can be greatly simplified.
\begin{rem}
Assume $\|\bm\eta\|$ and $\sigma$ are small and $\bm\alpha = (1/k, \cdots, 1/k)$.
As the number of topics $k$ gets large, the sample complexity bound in Theorem~\ref{thm_main} can be simplified as\vspace{-.3cm}
\setlength\arraycolsep{1pt}\begin{equation}
N \geq \Omega\left( \frac{\log(1/\delta)}{\sigma_k(\widetilde O)^{10}}\cdot\max(\varepsilon^{-2}, k^3)\right).
\end{equation}\label{rem_simplebound1}
\end{rem}\vspace{-.6cm}
The sample complexity bound in Remark~\ref{rem_simplebound1} may look formidable
as it depends on $\sigma_k(\widetilde O)^{10}$.
However, such dependency is somewhat necessary because we are using third-order tensors to recover the underlying model parameters.
\section{Joint Parameter Recovery}
The above two-stage procedure has one possible disadvantages, that is, the recovery of the topic distribution matrix does not
use any supervision signal, and thus the recovered topics are often not good enough for prediction tasks, as shown in experiments.
The disadvantage motivates us to develop a joint spectral method with theoretical guarantees.
We now describe our single-phase algorithm.
\subsection{Moments of Observable Variables}
We first define some moments based on the observable variables including the information that we need to recover the model parameters.
Since we aim to recover the joint topic distribution matrix $O^*$, we combine the word vector $\bm{x}$ with the response variable $y$ to form a joint vector $ \bm{z} = [ \bm{x}', y]'$ and define the following moments:
\iffalse
\begin{figure*}[t]
\centering
\begin{tikzpicture}[scale=0.3]
\tikzstyle{ann} = [fill=white, font=\footnotesize, inner sep =1pt]
\foreach [count=\i]\x in {1,...,5}
\foreach [count=\j]\y in {6,...,2}
\foreach [count=\k]\z in {2,...,6}{
\drawbox{\x}{\y}{\z}{blue!25};
}
\foreach [count=\i]\y in {6,...,2}{
\drawbox{6}{\y}{1}{yellow!25};
}
\foreach [count=\i]\x in {1,...,5}{
\drawbox{\x}{1}{1}{yellow!25};
}
\foreach [count=\i]\y in {6,...,2}
\foreach [count=\j]\z in {2,...,6}{
\drawbox{6}{\y}{\z}{green!25};
}
\foreach [count=\i]\x in {1,...,5}
\foreach [count=\j]\z in {2,...,6}{
\drawbox{\x}{1}{\z}{green!25};
}
\drawbox{6}{1}{1}{red!25}
\foreach [count=\i]\z in {2,...,6}{
\drawbox{6}{1}{\z}{yellow!25};
}
\foreach [count= \i]\x in {16,...,21}
\foreach [count =\j]\y in {7,...,3} {
\drawbox{\x}{\y}{1}{green!25};
}
\foreach [count=\i]\x in {16,...,21}
\foreach [count=\j]\y in {7,...,3}
\foreach [count=\k]\z in {3,...,7}{
\drawbox{\x}{\y}{\z}{blue!25};
}
\foreach [count=\i]\y in {7,...,3}{
\drawbox{23}{\y}{1}{yellow!25};
}
\foreach [count=\i]\x in {16,...,21}{
\drawbox{\x}{1}{1}{yellow!25};
}
\foreach [count=\i]\y in {7,...,3}
\foreach [count=\j]\z in {3,...,7}{
\drawbox{23}{\y}{\z}{green!25};
}
\foreach [count=\i]\x in {16,...,21}
\foreach [count=\j]\z in {3,...,7}{
\drawbox{\x}{1}{\z}{green!25};
}
\drawbox{23}{1}{1}{red!25}
\foreach [count=\i]\z in {3,...,7}{
\drawbox{23}{1}{\z}{yellow!25};
}
\draw[arrows=<-,line width=1pt](15.8,8)--(13.5,8.5);
\draw[arrows=<-,line width=1pt](27.2,8)--(29.0,8.5);
\draw[arrows=<-,line width=1pt](27.2,3)--(29.0,3.5);
\draw[arrows=<-,line width=1pt](24.2,0.7)--(26.0,0.7);
\node[ann] at (13,9.2) {$\mathbb{E}[\bm{x}_1 \otimes \bm{x}_2 \otimes \bm{x}_3]$};
\node[ann] at (32,8.7) {$\mathbb{E}[y\bm{x}_1 \otimes \bm{x}_2]$};
\node[ann] at (32,3.7) {$\mathbb{E}[y^2\bm{x}_1]$};
\node[ann] at (28,0.7) {$\mathbb{E}[y^3]$};
\end{tikzpicture}
\caption{Interaction between response variable $y$ and word vector $\bm{x}$ in the 3rd-order tensor $M_3$.}
\label{fig10}
\end{figure*}
\fi
\begin{deff} (Centerized Moments
{\small \begin{equation}
\begin{aligned}
& N_1 = \mathbb{E}[\bm{z}_1] \\
& N_2 = \mathbb{E}[ \bm{z}_1 \otimes \bm{z}_2] - \dfrac{\alpha_0}{\alpha_0 + 1 } N_1 \otimes N_1 - \sigma^2\bm{e} \otimes \bm{e} \\
& N_3 = \mathbb{E}[ \bm{z}_1 \otimes \bm{z}_2 \otimes \bm{z}_3] - \dfrac{\alpha_0}{\alpha_0 + 1}( \mathbb{E}[\bm{z}_1 \otimes \bm{z}_2 \otimes N_1] \\
& \ \ \ \ + \mathbb{E}[\bm{z}_1 \otimes N_1 \otimes \bm{z}_2] + \mathbb{E}[N_1 \otimes \bm{z}_1 \otimes \bm{z}_2] ) \\
& \ \ \ \ + \dfrac{2\alpha_0^2}{(\alpha_0 + 1)(\alpha_0 +2)}N_1 \otimes N_1 \otimes N_1 \\
& \ \ \ \ + 3\sigma^2N_{1_{V+1}}\bm{e} \otimes \bm{e} \otimes \bm{e}
+ \dfrac{1}{\alpha_0 + 1} \sigma^2 (\bm{e} \otimes \bm{e} \otimes N_1 \\
& \ \ \ \ + \bm{e} \otimes N_1 \otimes \bm{e} + N_1 \otimes \bm{e} \otimes \bm{e} ),
\end{aligned}
\end{equation}}
where $\bm{e}$ is the $(V$+$1)$-dimensional vector with the last element equaling to $1$ and all others zero, $N_{1_{V+1}}$ is the $V+1$-th element of $N_1$.
\end{deff}
The intuition for such definitions is derived from
an important observation that once the latent variable $\bm{h}$ is given, the mean value of $y$ is a weighted combination of the regression parameters and $\bm{h}$
(i.e., $\mathbb{E}[y|\bm{h}] = \sum_{i = 1}^{k}\eta_i h_i$), which has the same form as for $\bm{x}$
(i.e., $\mathbb{E}[\bm{x}|\bm{h}] = \sum_{i=1}^{k} {\bm \mu}_i h_i$). Therefore, it is natural to regard $y$ as an additional dimension of the word vector $\bm{x}$, which gives the new vector $\bm{z}$.
This combination leads to some other terms involving the high-order moments of $y$, which introduce the variance parameter $\sigma$ when we centerize the moments.
Although we can recover $\sigma$ in the two-stage method,
recovering it jointly with the other parameters seems to be hard. Thus we treat $\sigma$ as a hyper-parameter.
One can determine it via a cross-validation procedure.
\begin{figure}[t
\centering
\includegraphics[width = .5\textwidth]{images/tensor2.png}\vspace{-.2cm}
\caption{Interaction between response variable $y$ and word vector $\bm{x}$ in the 3rd-order tensor $M_3$.}
\label{fig10}\vspace{-.3cm}
\end{figure}
As illustrated in Fig.~\ref{fig10}, our 3rd-order moment can be viewed as a centerized version of the combination of $N_3' \triangleq \mathbb{E}[\bm{x}_1 \otimes \bm{x}_2 \otimes \bm{x}_3] $,
$N_y' \triangleq \mathbb{E}[y\bm{x}_2\otimes \bm{x}_2]$ and some high-order statistics of the response variables.
Note that this combination has already aligned the regression parameters with the corresponding topics. Hence, we do not need an extra matching step.
In practice, we cannot get the exact values of those moments. Instead, we estimate them from the i.i.d. sampled documents.
Note that we only need the moments up to the third order, which means any document consisting of at least three words can be used in this estimation.
Furthermore, although these moments seem to be complex, they can be expressed via the model parameters in a graceful manner, as
summarized in Proposition~\ref{p1} which can be proved by expanding the terms by definition, similar as in the proof of Proposition 1.
\begin{prop} \label{p1}
The moments in Definition 2 can be expressed by using the model parameters as follows:\vspace{-.2cm}
\begin{equation}
\begin{aligned}
& N_2 = \dfrac{1}{\alpha_0(\alpha_0 + 1)} \sum\limits_{i =1}^{k} \alpha_i \bm{v}_i \otimes \bm{v}_i \\
& N_3 = \dfrac{2}{\alpha_0(\alpha_0 +1)(\alpha_0 +2)} \sum\limits_{i=1}^{k} \alpha_i \bm{v}_i \otimes \bm{v}_i \otimes \bm{v}_i,
\end{aligned}\vspace{-.2cm}
\end{equation}
where $\bm{v}_i$ is the concatenation of the $i$-th word-topic distribution $\bm{\mu}_i$ and
regression parameter $\eta_i$.
\end{prop}
\iffalse
\begin{prof*}
Similar to proposition $1$, the equations can be proved via taking the expectation over the topic
mixing vector $\bm{h}$ and noting that:
\begin{equation*}
\begin{aligned}
&\mathbb{E}[\bm{z}_1|\bm{h}] = \sum\limits_{i=1}^{k} h_i\bm{v}_i \\
&\mathbb{E}[\bm{z}_1 \otimes \bm{v}_2|\bm{h}] = \sum\limits_{i,j = 1}^{k} h_ih_j\bm{v}_i \otimes \bm{v}_j
+ \sigma^2 \bm{e}_{V+1} \otimes \bm{e}_{V+1}\\
&\mathbb{E}[\bm{z}_1 \otimes \bm{z}_2 \otimes \bm{z}_3 | \bm{h}] = \sum\limits_{i,j,l=1}^{k} h_ih_jh_l
\bm{v}_i \otimes \bm{v}_j \otimes \bm{v}_l \\
&\ \ \ \ \ \ \ + 3\sigma^2 \bm{\eta}'\bm{h}\cdot \bm{e}\otimes \bm{e} \otimes \bm{e} +
\sigma^2 \sum\limits_{i=1}^{k} h_i(\bm{e} \otimes \bm{e} \otimes \bm{v}_i \\
& \ \ \ \ \ \ \ + \bm{e}\otimes \bm{v}_i \otimes \bm{e} +
\bm{v}_i \otimes \bm{e} \otimes \bm{e})\\
\end{aligned}
\end{equation*}
\hfill\QED
\end{prof*}
\fi
\subsection{Robust Tensor Decomposition}
Proposition~\ref{p1} shows that the centerized tensors are weighted sums of the tensor products
of the parameters $\{\bm{v}_i\}_{i=1}^k$ to be recovered.
A similar procedure as in the two-stage method can be followed in order to develop our joint spectral method, which consists
of {\it whitening } and {\it robust tensor decomposition} steps.
First, we whiten the 2nd-order tensor $N_2$ by finding a matrix $W \in \mathbb{R}^ {(V +1) \times k}$
such that $W^{\top}N_2W = I_k$.
This whitening procedure is possible whenever the joint topic distribution vectors $\{\bm{v}_i\}^k_{i=1}$
are linearly independent, that is, the matrix has rank $k$. The whitening procedure and the linear independence
assumption also imply that $\{W^{\top}\bm{v}_i\}^k_{i=1}$
are orthogonal vectors and
can be subsequently recovered by performing an orthogonal tensor decomposition on the simultaneously
whitened third-order tensor $N_3(W, W, W)$ as summarized in the following proposition.
\begin{prop}
Define $\bm{\omega}_i = \sqrt{\dfrac{\alpha_i}{\alpha_0(\alpha_0 +1)}}W^{\top}\bm{v}_i$. Then:
\begin{itemize}
\item
$\{\bm{\omega}\}_{i=1}^{k}$ is an orthonormal basis.
\item
$N_3(W,W,W)$ has pairs of robust eigenvalue and eigenvector $(\lambda_i, \bm{v}_i)$
with $\lambda = \dfrac{2}{\alpha_0 + 2} \sqrt{\dfrac{\alpha_0(\alpha_0+1)}{\alpha_i}} $
\end{itemize}
\end{prop}
Finally, by multiplying the pseudo-inverse of the whitening matrix $W^+$ we obtain the joint topic distribution vectors $\{\bm{v}_i\}^k_{i=1}$.
\begin{algorithm}[t]
\caption{a joint spectral method to recover sLDA parameters. Input parameters: $\alpha_0$, $L$, $T$, $\sigma$ }\label{a1}
\begin{algorithmic}[1]
\State Compute empirical moments and obtain $\widehat{N}_2, \widehat{N_3}$.
\State Find $\widehat{W} \in \mathbb{R}^{V + 1\times k}$ such that $\widehat{N}_2(\widehat{W},\widehat{W}) = I_k $.
\State Find robust eigenvalues and eigenvectors $(\widehat{\lambda}_i, \widehat{\bm{v}}_i)$
of $\widehat{N}_3(\widehat{W},\widehat{W},\widehat{W})$ using the robust tensor
power method with parameters $L$ and $T$.
\State Recover prior parameters: $\widehat{\alpha}_i \gets \frac{4\alpha_0(\alpha_0 + 1)}{(\alpha_0+2)^2\widehat{\lambda}_i^2}.$
\State Recover topic distribution: $$\widehat{\bm{v}}_i \gets \frac{\alpha_0 + 2}{2}\widehat{\lambda}_i(W^+)^{\top}\widehat{\bm{\omega}}_i.$$
\State {\bf Output: } model parameters $\widehat{\alpha}_i$, $\widehat{\bm{v}}_i$ $i = 1,...,k$
\end{algorithmic}
\label{alg2}
\end{algorithm}
We outline our single-phase spectral method in Alg.~\ref{a1}. Here we assume that the noise variance $\sigma$ is given. Note that in the two-stage spectral method, it does
not need the parameter $\sigma$ because it does not use the information of the variance of prediction error.
Although there is a disadvantage that we need to tune it, the introduction of $\sigma$ sometimes increases
the flexibility of our methods on incorporating some prior knowledge (if exists)
We additionally need three hyper-parameters $\alpha_0, L $ and $T$, similar as in the two-stage method. The parameter $\alpha_0$ is defined as the summation of all the entries of the prior parameter $\bm{\alpha}$.
$L$ and $T$ are used to control the number of iterations in robust tensor decomposition. To ensure a sufficiently high recovery accuracy, $L$ should
be at least a linear function of $k$, and $T$ should be set as $T = \Omega(\log(k) + \log\log(\lambda_{max}/\epsilon))$,
where $\lambda_{max} = \frac{2}{\alpha_0 + 2}\sqrt{\frac{\alpha_0(\alpha_0 + 1)}{\alpha_{min}}}$ and $\epsilon$ is the error
rate.
\vspace{-.1cm}
\subsection{Sample Complexity Analysis}
\vspace{-.1cm}
We now analyze the sample complexity in order to achieve $\epsilon$-error with a high probability.
For clarity, we defer proof details to Appendix B.
\begin{thm}
Let $ \sigma_1(\widetilde{O}^*)$ and $\sigma_k(\widetilde{O}^*)$ be the largest and smallest singular
values of the joint canonical topic distribution matrix $\widetilde{O}^*$. Let
$\lambda_{max} \triangleq \frac{2}{\alpha_0 + 2}\sqrt{\frac{\alpha_0(\alpha_0 +1)}{\alpha_{min}}}$ where
$\alpha_{min}$ is the smallest element of $\bm{\alpha}$;
$\lambda_{min} \triangleq \frac{2}{\alpha_0 + 2}\sqrt{\frac{\alpha_0(\alpha_0 +1)}{\alpha_{max}}}$ where
$\alpha_{max}$ is the largest element of $\bm{\alpha}$.
For any error-tolerance parameter $\epsilon > 0$, if Algorithm $2$ runs at least
$T \!=\! \Omega(\log(k) \!+\! \log\log(\lambda_{max}/\epsilon))$
iterations on $N$ i.i.d. sampled documents with $N \geq (n_1^\prime, n_2^\prime, n_3^\prime)$, where:\vspace{-.3cm}
{\small \begin{equation*}
\begin{aligned}
& n_1^\prime = K_1 \cdot \dfrac{\alpha_0^2(\alpha_0+1)^2C^2(\delta/36N)\cdot(2 + \sqrt{2\log(18/\delta)})^2 }{\alpha_{min}^2} \\
& n_2^\prime = K_2 \cdot C^2(\delta/144N)(2 + \sqrt{2 \log(72/\sigma)})^2 \\
& n_3^\prime = K_3 \cdot \dfrac{C^2(\delta/36N)(2+\sqrt{2\log(18/\sigma)})^2}
{\sigma_k(\widetilde{O}^*)^{10}}\cdot\max(\dfrac{1}{\epsilon^2}, \dfrac{k^2}{\lambda_{min}^2}),\\
\end{aligned}
\end{equation*}}\vspace{-.3cm}
\noindent $C(x)$ is a polynomial of inverse CDF of normal distribution and the norm of regression parameters $\| {\bm \eta}\|$; $K_1,K_2,K_3$ are some universal constants.
Then with probability at least $1 - \delta$, there exist a permutation $\pi: [k] \to [k]$
such that the following holds for every $i \in [k]$:\vspace{-.4cm}
\begin{equation*}
\begin{aligned}
& |\alpha_i - \widehat{\alpha}_{\pi(i)} |
\leq \dfrac{4\alpha_0(\alpha_0 + 1) }{(\alpha_0 + 2)^2\lambda_{min}^2(\lambda_{min} - 5 \epsilon)^2} \cdot 5\epsilon \\
& \|\bm{v}_i - \widehat{\bm{v}}_{\pi(i)} \| \leq \left ( \sigma_1(\widetilde{O}^*)(\alpha_0 + 2)(\frac{7}{2} +
\frac{8}{\lambda_{min}})\right )\cdot \epsilon.
\end{aligned}
\end{equation*}\vspace{-.1cm}
\label{thm_main2}
\end{thm}
\vspace{-.5cm}
Similar to Theorem 1, the sample complexity bound consists of three terms.
The first and second terms do not
depend on the error rate $\epsilon$, which are required so that some technical conditions
are met. Thus they could be largely neglected in practice.
The third $n_3^\prime$ term comes from the sample complexity bound for the robust tensor power method~\cite{a:tensordecomp}.
\begin{rem}
Note the RHS of the requirements of $N$ includes a
function of $N$~(i.e., $C(1/N)$). As mentioned above, $C(x)$ is polynomial of inverse CDF of normal
distribution with low degree. Since the inverse CDF grows very slowly~(i.e., $|\Phi^{-1}(1/N)| = o(\log(N))$).
We can omit it safely.
\end{rem}
\begin{rem}
Following the above remark and assume that $\|\bm{\eta}\|$ and $\sigma$ are small and $\bm{\alpha}$ are homogeneous, the
sample complexity can be simplified as (a function of $k$):\vspace{-.3cm}
$$
N = O\left( \dfrac{\log(1/\sigma)}{\sigma_k(\widetilde{O}^*)^{10}} \cdot \max(\dfrac{1}{\epsilon^2}, k^3)\right).
$$\label{rem_simplebound2} \vspace{-.45cm}
\end{rem}\vspace{-.1cm}
The factor $1/\sigma_k(\widetilde{O}^*)^{10}$ is large, however, such a factor is necessary since we use the third order tensors. This factor roots in the tensor
decomposition methods and one can expect to improve it if we have other better methods to decompose $\widehat{N}_3$.
\subsection{Sample Complexity Comparison}
\label{compp}
As mentioned in Remark \ref{rem_simplebound1} and Remark \ref{rem_simplebound2}, the joint spectral method shares the
same sample complexity as the two-stage algorithm in order to achieve $\epsilon$ accuracy, except two minor differences.
First, the sample complexity depends on the smallest singular value of (joint) topic distribution $\sigma_k(\widetilde{O})$. For the joint method, the joint topic distribution
matrix consists of the original topic distribution matrix and one extra row of the regression parameters. Thus from Weyl's inequality~\cite{a:weyl}, the smallest singular value
of the joint topic distribution matrix is larger than that of the original topic distribution matrix,
and then the sample complexity of the joint method is a bit lower than that of the two-stage method, as empirically justified in experiments.
Second, different from the two-stage method, the errors of topic distribution $\bm{\mu}$ and regression parameters $\bm{\eta}$ are estimated together in the joint method
(i.e., $\bm{v}$), which can potentially give more accurate estimation of regression parameters considering that the number of
regression parameters is much less than the topic distribution.
\section{Speeding up moment computation}
We now analyze the computational complexity and present some implementation details to make the algorithms more efficient.
\subsection{Two-Stage Method}
In Alg.~1, a straightforward computation of the third-order tensor $\widehat M_3$ requires
$O(NM^3)$ time and $O(V^3)$ storage, where $N$ is corpus size, $M$ is the number of words per document and $V$ is the vocabulary size.
Such time and space complexities are clearly prohibitive for real applications, where the vocabulary usually contains tens of thousands of terms.
However, we can employ a trick similar as in~\cite{speedup} to speed up the moment computation.
We first note that only the whitened tensor $\widehat M_3(\widehat W,\widehat W,\widehat W)$ is needed in our algorithm, which only takes $O(k^3)$ storage.
Another observation is that the most difficult term in $\widehat M_3$ can be written as
$\sum_{i=1}^r{c_i\bm u_{i,1}\otimes \bm u_{i,2}\otimes\bm u_{i,3}}$,
where $r$ is proportional to $N$ and $\bm u_{i,\cdot}$ contains at most $M$ non-zero entries.
This allows us to compute $\widehat M_3(\widehat W,\widehat W,\widehat W)$ in $O(NMk)$ time by computing
$\sum_{i=1}^r{c_i\bm (W^\top\bm u_{i,1})\otimes (W^\top\bm u_{i,2})\otimes (W^\top\bm u_{i,3})}$.
Appendix~C.2 provides more details about this speed-up trick.
The overall time complexity is $O(NM(M+k^2)+V^2+k^3LT)$
and the space complexity is $O(V^2+k^3)$.
\subsection{Joint Method}
For the single-phase algorithm, a straightforward computation of the third-order tensor $\widehat N_3$ has the same complexity of $O(NM^3)$ as in the two-stage method.
And a much higher time complexity is needed for computing $\widehat{N}_3(\widehat{W},\widehat{W},\widehat{W})
, which is prohibitive.
Similar as in the two-stage method, since we only need $\widehat{N}_3(\widehat{W},\widehat{W},\widehat{W})$ in Alg.~\ref{a1}, we turn to compute this term directly.
We can then use the trick mentioned above to do this.
The key idea is to decompose the third-order tensor into different parts based on the occurrence of words and compute them respectively.
The same time comlexity and space complexity is needed for the single-phase method.
Sometimes $N_2$ and $N_3$ are not ``balanced" (i.e., the value of some elements are much larger than the others).
This situation happens when either
the vocabulary size is too large or the range of $\bm{\eta}$ is too large. One can image that
if we have a vocabulary consisting of one million words while $\min{\eta}_i = 1$, then the energy
of the matrix $N_2$ concentrates on $(N_2)_{V+1,V+1}$.
As a consequence, the SVD performs badly when the matrix is ill-conditioned.
A practical solution
to this problem is that we scale the word vector $\bm{x}$ by a constant, that is, for the $i$-th word in the dictionary, we set $x_i = C , x_j = 0, \forall i \not = j$, where $C$ is a constant.
The main effect is that we can make the matrix more stable after this manipulation. Note that when we fix $C$,
this makes no effect on the recovery accuracy. Such a trick is primarily for computational stability. In our experiments, $C$ is set to be $100$.
\subsection{Dealing with large vocabulary size $V$}
One key step in the whitening procedure of both methods is to perform SVD on the second order moment $M_2 \in \mathbb{R}^{V \times V}$ (or $N_2 \in \mathbb{R}^{(V+1) \times (V+1)}$).
A straightforward implementation of SVD has complexity $O(k V^2)$,\footnote{It is not $O(V^3)$ as we only need top-$k$ truncated SVD.} which is unbearable when the vocabulary size $V$ is large.
We follow the method in \cite{a:nystrom} and perform random projection to reduce dimensionality. More precisely, let $S \in \mathbb{R}^{\widetilde{k}}$ where $\widetilde{k} < V$ be
a random matrix and then define $C = M_2 S$ and $\Omega = S^{\top}M_2S$. Then a low rank approximation of $M_2$ is given by $ \widetilde{M_2} = C\Omega^{+}C^{\top} $.
Now we can obtain the whitening matrix without directly performing an SVD on $M_2$ by appoximating $C^{-1}$ and $\Omega$ separately.
The overall algorithm is provided in Alg.~\ref{alg3}. In practice, we set $\widetilde{k} = 10 k$ to get a sufficiently accurate approximation.
\begin{algorithm}[t]
\caption{Randomized whitening procedure. Input parameters: second order moment $M_2$ (or $N_2$).}
\centering
\begin{algorithmic}[1]
\State Generate a random projection matrix $S \in \mathbb{R}^{V\times \widetilde{k}}$.
\State Compute the matrices $C$ and $\Omega$:\vspace{-.3cm}
$$C = M_2S \in \mathbb{R}^{V \times \widetilde{k}},~\textrm{and}~ \Omega = S^{\top}M_2S \in \mathbb{R}^{ \widetilde{k}\times \widetilde{k}}.$$\vspace{-.7cm}
\State Do SVD for both $C$ and $\Omega$:\vspace{-.3cm}
$$C = U_C \Sigma_C D_C^\top,~\textrm{and}~ \Omega = U_\Omega \Sigma_\Omega D^\top_\Omega$$\vspace{-.7cm}
\State Take the rank-$k$ approximation: $U_C \leftarrow U_C(\colon ,1\colon k)$ \vspace{-.3cm}
$$\Sigma_C \leftarrow \Sigma_C(1\colon k,1\colon k), D_C \leftarrow D_C(\colon,1\colon k)$$ \vspace{-.6cm}
$$ D_{\Omega} \leftarrow D_{\Omega}(1\colon k,1\colon k),
\Sigma_{\Omega} \leftarrow \Sigma_{\Omega}(1\colon k,1 \colon k)$$\vspace{-.7cm}
\State Whiten the approximated matrix: \vspace{-.3cm}
$$W = U_C\Sigma_C^{-1}D_C^{\top}D_{\Omega}\Sigma_{\Omega}^{1/2}. $$\vspace{-.7cm}
\State \textbf{Output:} Whitening matrix $W$.
\end{algorithmic}
\label{alg3}
\end{algorithm}
\section{Experiments}\label{sec:exp}
We now present experimental results on both synthetic and two real-world datasets.
For our spectral methods,
the hyper-parameters $L$ and $T$ are set to be $100$, which is sufficiently large for our experiment settings.
Since spectral methods can only recover the underlying parameters, we first run them to
recover those parameters in training and then use Gibbs sampling to infer the topic mixing vectors $\bm{h}$ and topic assignments for each word $t_i$ for testing.
Our main competitor is sLDA with a Gibbs sampler (Gibbs-sLDA), which is asymptotically accurate and often outperforms variational methods.
We implement an uncollapsed Gibbs sampler, which alternately draws samples from the local
conditionals of $\bm{\eta}$, $\bm z$, $\mathbf{h}$, or $\bm{\mu}$,
when the rest variables are given. We monitor the behavior of the
Gibbs sampler by observing the relative
change of the training data log-likelihood, and terminate when the average change
is less than a given threshold (e.g., $1e^{-3}$) in the last $10$ iterations.
The hyper-parameters of the Gibbs sampler are set to be the same as our methods, including topic numbers and $\bm{\alpha}$.
We evaluate a hybrid method that uses the parameters $\bm{\mu},\bm{\eta}, \bm{\alpha}$ recovered by our joint spectral method
as initialization for a Gibbs sampler.
This strategy is similar to that in~\cite{a:meetem}, where
the estimation of a spectral method is used to initialize an EM method for further refining. In our hybrid method, the Gibbs sampler plays the similar role of refining.
We also compare with MedLDA~\cite{medlda}, a state-of-the-art topic model for classification and regression, on real datasets. We use the Gibbs sampler with data augmentation~\cite{gibbsmedlda}, which is more accurate than the original variational methods, and adopts the same stopping condition as above.
On the synthetic data, we first use $L_1$-norm
to measure the difference between the reconstructed parameters
and the underlying true parameters.
Then we compare the prediction accuracy and per-word likelihood on both synthetic and real-world datasets.
The quality of the prediction on the synthetic dataset is measured by mean squared error (MSE) while
the quality on the real-word dataset is assessed by predictive $R^2$ ($pR^2$), a normalized version of MSE, which is defined as
$pR^2 = 1 - \frac{ \sum_i(y_i - \widehat{y}_i)^2 }{ \sum_i(y_i - \bar{y})^2 },
where $\bar{y}$ is the mean of
testing data and $\widehat{y}_i$ is the estimation of $y_i$. The per-word log-likelihood
is defined as $\log p(\omega|\hm{h},O) = \log \sum_{j=1}^{k}p(\omega|t =j, O)p(t = j|\bm{h})$.
\begin{figure*}[t]\vspace{-.1cm
\centering
\includegraphics[width=4.9cm]{images/250_alpha.png}
\includegraphics[width=4.9cm]{images/250_eta.png}
\includegraphics[width=4.9cm]{images/250_mu.png}\vspace{-.2cm}
\caption{Reconstruction errors of two spectral methods when each document contains $250$ words.
$X$ axis denotes the training size $n$ in log domain with base $2$ (i.e., $n = 2^k, k \in \{8,...,15\}$). Error bars denote the standard deviations measured on 3 independent trials under each setting.}
\label{fig_convergence1}\vspace{-.1cm}
\end{figure*}
\begin{figure*}[t
\centering
\includegraphics[width=4.9cm]{images/500_alpha.png}
\includegraphics[width=4.9cm]{images/500_eta.png}
\includegraphics[width=4.9cm]{images/500_mu.png}\vspace{-.3cm}
\caption{Reconstruction errors of two spectral methods when each document contains $500$ words.
$X$ axis denotes the training size $n$ in log domain with base $2$ (i.e., $n = 2^k, k \in \{8,...,15\}$). Error bars denote the standard deviations measured on 3 independent trials under each setting.}
\label{fig_convergence2}\vspace{-.45cm}
\end{figure*}
\vspace{-.15cm}
\subsection{Synthetic Dataset}\label{sec:synthetic}
We generate our synthetic dataset following the generative process of sLDA, with a vocabulary of
size $V = 500$ and topic number $k = 20$.
We generate the topic distribution matrix $O$ by first sampling each entry from a uniform distribution and then normalizing
every column of it. The linear regression model $\bm{\eta}$ is sampled from a standard Gaussian distribution.
The prior parameter $\bm{\alpha}$ is assumed to be homogeneous, i.e., $\bm{\alpha} = (1/k,... , 1/k)$.
Documents and response variables are then generated from the sLDA model specified in Section~\ref{sec:sLDA-model}.
We consider two cases where the length of each document is set to be $250$ and $500$ repectively.
The hyper-parameters are set to be the same as the ones that used to
generate the dataset
\footnote{The methods are insensitive to
the hyper-parameters in a wide range. e.g., we still get high accuracy even we set the hyper-parameter $\alpha_0$ to be twice as large as the true value.}.
\vspace{-.15cm}
\subsubsection{Convergence of estimated model parameters}
Fig.~\ref{fig_convergence1} and Fig.~\ref{fig_convergence2} show the $L_1$-norm reconstruction errors of $\bm{\alpha}$, $\bm{\eta}$ and $\bm{\mu}$ when each document contains different number of words.
Note that due to the unidentifiability of topic models, we only get a permutated estimation of the underlying parameters. Thus we run a bipartite
graph matching to find a permutation that minimizes the reconstruction error. We can find that as the sample size increases, the reconstruction errors for all parameters decrease consistently to zero in both methods, which verifies the correctness of our theory.
Taking a closer look at the figures, we can see that the empirical convergence rates for $\bm{\alpha}$ and $\bm{\mu}$ are
almost the same for the two spectral methods. However, the convergence rate for regression parameters $\bm{\eta}$ in the joint
method is much higher than the one in the two-stage method, as mentioned in the comparison
of the sample complexity in Section~\ref{compp} , due to the fact that the joint method can bound the
estimation error of $\bm{\eta}$ and $\bm{\mu}$ together.
Furthermore, though Theorem~\ref{thm_main} and Theorem~\ref{thm_main2} do not involve the number of words per document,
the simulation results demonstrate a significant improvement when more words are observed in each document,
which is a nice complement for the theoretical analysis.
\vspace{-.15cm}
\subsubsection{Prediction accuracy and per-word likelihood}
Fig.~\ref{fig_prediction1} shows that both spectral methods consistently outperform Gibbs-sLDA.
Our methods also enjoy the advantage of being less variable,
as indicated by the curve and error bars.
Moreover, when the number of training documents is sufficiently large, the performance of the reconstructed model is very close to the true model\footnote{Due to the randomness in the data generating process, the true model has a non-zero prediction error.}, which implies that our spectral methods
can correctly identify an sLDA model from its observations, therefore supporting our theory.
The performances of the two-stage spectral method and the joint one are comparable this time, which is largely because of the
fact the when giving enough training data, the recovered model is accurate enough. The Gibbs method is easily caught in a local
minimum so we can find as the sample size increases, the prediction errors do not decrease monotonously.
\begin{figure}[t]\vspace{-.1cm}
\centering
\includegraphics[width=4cm]{images/250_mse.png}
\includegraphics[width=4cm]{images/250_likeli.png}
\includegraphics[width=4cm]{images/500_mse.png}
\includegraphics[width=4cm]{images/500_likeli.png}\vspace{-.2cm}
\caption{
Mean square errors and negative per-word log-likelihood of Alg.~\ref{alg1} and Gibbs sLDA.
Each document contains $M=500$ words. The $X$ axis denotes the training size ($\times 10^3$).
The ''ref. model" denotes the one with the underlying true parameters.
}
\label{fig_prediction1}\vspace{-.2cm}
\end{figure}
\iffalse
\begin{figure}[ht!
\centering
\includegraphics[width=4.2cm]{images/hotel_pr2_alpha_0_1_small.png}
\includegraphics[width=4.2cm]{images/hotel_likeli_alpha_0_1_med_out_small.png}\vspace{-.2cm}
\caption{$pR^2$ scores and negative per-word log-likelihood on the Hotel Review dataset. The $X$ axis indicates the number of topics. Error bars
indicate the standard deviation of $5$-fold cross-validation. Vocabulary size $V = 5,000$}
\label{hotel}\vspace{-.3cm}
\end{figure}
\begin{figure}[ht!
\centering
\includegraphics[width=4.1cm]{images/hotel_pr2_alpha_0_1_tall.png}
\includegraphics[width=4.1cm]{images/hotel_likeli_alpha_0_1_med_out_tall.png}\vspace{-.2cm}
\caption{$pR^2$ scores and negative per-word log-likelihood on the Hotel Review dataset. The $X$ axis indicates the number of topics. Error bars
indicate the standard deviation of $5$-fold cross-validation. Vocabulary size $V = 5,000$}
\label{hotel}\vspace{-.3cm}
\end{figure}
\fi
\begin{figure}[ht!
\centering
\includegraphics[width=7cm]{images/hotel_pr2_alpha_0_1_fat.png}
\includegraphics[width=7cm]{images/hotel_likeli_alpha_0_1_med_out_fat.png}\vspace{-.2cm}
\caption{$pR^2$ scores and negative per-word log-likelihood on the Hotel Review dataset. The $X$ axis indicates the number of topics. Error bars
indicate the standard deviation of $5$-fold cross-validation.
\label{hotel}\vspace{-.4cm}
\end{figure}
\subsection{Hotel Reviews Dataset}
For real-world datasets, we first test on a relatively small Hotel Review dataset, which consists of $15,000$ documents for training and $3,000$ documents for
testing that are randomly sampled from TripAdvisor website. Each document is associated with
a rating score from $1$ to $5$ and our task is to predict it.
We pre-process the dataset by shifting the review scores so that they have zero mean and unit variance as in~\cite{gibbsmedlda}.
Fig.~\ref{hotel} shows the prediction accuracy and per-word likelihood when the vocabulary size
is $5,000$ and the mean level of $\bm{\alpha}$ is $\hat{\alpha} = 0.1$. As MedLDA adopts a quite different objective from sLDA, we only compare on the prediction accuracy. Comparing with
traditional Gibbs-sLDA and MedLDA, the two-stage spectral method is much worse, while
the joint spectral method is comparable at its optimal value. This result is not surprising
since the convergence rate of regression parameters for the joint method is faster than
that of the two-stage one.
The hybrid method (i.e., Gibbs sampling initialized with the joint spectral method) performs as
well as the state-of-the-art MedLDA. These results show that spectral methods are
good ways to avoid stuck in relatively bad local optimal solution.
\vspace{-.1cm}
\subsection{Amazon Movie Reviews Dataset}
Finally, we report the results on a large-scale real dataset, which is built on Amazon movie reviews~\cite{a:amazon}, to demonstrate
the effectiveness of our spectral methods on improving the prediction accuracy as well as finding discriminative topics.
The dataset consists of $7,911,684$ movie reviews written by $889,176$ users from Aug $1997$ to Oct $2012$. Each review is accompanied
with a rating score from $1$ to $5$ indicating how a user likes a particular movie. The median number of words per
review is $101$. We consider two cases where a vocabulary with $V = 5,000$ terms or $V = 10,000$ is built by selecting high frequency words and deleting
the most common words and some names of characters in movies.
When the vocabulary size $V$ is small (i.e., $5,000$), we run exact SVD for the
whitening step; when $V$ is large (i.e., $10,000$), we run the randomized SVD to approximate the result.
As before, we also pre-process the dataset by shifting the review scores so that they have zero mean and unit variance.
\begin{figure*}[ht!]\vspace{-.1cm
\centering
\iffalse
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_1_2.png}
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_01_2.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_1_2.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_01_2.png}
\fi
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_1_med.png}
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_01_med.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_1_3.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_01_2.png}\vspace{-.2cm}
\caption{$pR^2$ scores and negative per-word log-likelihood on Amazon dataset. The $X$ axis indicates the number of topics. Error bars
indicate the standard deviation of $5$-fold cross-validation. Vocabulary size $V = 5,000$}
\label{fig1}
\end{figure*}
\begin{figure*}[ht!]\vspace{-.1cm
\centering
\iffalse
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_1_1w_2.png}
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_01_V_1w_2.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_1_1w_2.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_01_V1w_2.png}
\fi
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_1_1w_med.png}
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_01_V_1w_med.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_1_1w_med_out.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_01_V_1w_med_out.png}\vspace{-.2cm}
\caption{$pR^2$ scores and negative per-word log-likelihood on Amazon dataset. The $X$ axis indicates the number of topics. Error bars
indicate the standard deviation of $5$-fold cross-validation. Vocabulary size $V = 10,000$}
\label{fig2}\vspace{-.4cm}
\end{figure*}
\vspace{-.15cm}
\subsubsection{Prediction Performance}
Fig.~\ref{fig1} shows the prediction accuracy and per-word log-likelihood when $\bar{\alpha}$ takes different values and the vocabulary size $V = 5,000$, where
$\bar{\alpha} = \alpha_0/k$ denotes the mean level for $\bm{\alpha}$.
We can see that comparing to the classical Gibbs sampling method, our spectral method is
a bit more sensitive to the hyper-parameter $\bm{\alpha}$.
But in both cases, our joint method alone outperforms the Gibbs sampler and the two-stage spectral method.
MedLDA is also sensitive to the hyper-parameter $\bm{\alpha}$.
When $\bm{\alpha}$ is set properly, MedLDA achieves the best result comparing with the other
methods, however, the gap between our joint method and MedLDA is small.
This result is significant for spectral methods, whose practical performance was often much inferior to likelihood-based estimators.
We also note that if $\bar{\alpha}$ is not set properly (e.g., $\bar{\alpha} = 0.01$), a hybrid method that initializes a Gibbs sampler by the results of our spectral methods
can lead to high accuracy, outperforming the Gibbs sampler and MedLDA
with a random initialization.
We use the results of the joint method for initialization because this
gives better performance compared with the two-stage method.
Fig.~\ref{fig2}
shows the results when the vocabulary size $V = 10,000$.
This time the joint spectral method gets the best result while the two-stage method
is comparable with Gibbs sampling but worse than MedLDA.
The hybrid method is comparable with the joint method, demonstrating that this strategy works
well in practice again.
An interesting phenomenon is that the spectral method
gets good results when the topic number is only $3$ or $4$, which means the spectral method can fit the data using fewer topics.
Although there is a rising trend on prediction accuracy
for the hybrid method, we cannot verify this because we cannot get the results of spectral methods when $k$ is large.
The reason is that when $k > 9$, the spectral method fails in the robust
tensor decomposition step, as we get some negative eigenvalues. This phenomenon can be explained by the nature of our methods --- one crucial step in Alg.~\ref{alg1} and Alg.~\ref{a1}
is to whiten $\widehat{M}_2$ which can be done when the underlying topic matrix $O$ ( or joint topic matrix $\widehat{O}^*$) is of full rank. For the Amazon
review dataset, it is impossible to whiten it with more than $9$ topics. This fact can be used for model selection to avoid using too many extra topics.
There is also a rising trend in the Gibbs sampling when $V = 10,000$ as we measure the $pR^2$ indicator,
it reaches peak when topic size $k = 40$ which is about $0.22$ no matter $\alpha_0$ is $0.1$ or $0.01$.
The results may indicate that with a good initialization, the Gibbs sampling method could get much better performance.
Finally, note that Gibbs sampling and the hybrid Gibbs sampling methods get better log-likelihood values. This result is not
surprising because Gibbs sampling is based on MLE while spectral methods do not. Fig.~\ref{fig1} shows that the hybrid Gibbs
sampling achieves the best per-word likelihood. Thus if one's main goal is to maximize likelihood, a hybrid technique is
desirable.
\vspace{-.15cm}
\subsubsection{Parameter Recovery}
We now take a closer investigation of the recovered parameters for our spectral methods.
Table $1$ shows the estimated regression parameters of both methods, with $k=8$ and $V=5,000$.
We can see that the two methods have different ranges of the possible predictions ---
due to the normalization of ${\bm h}$, the range of the predictions by a model with estimate $\widehat{ {\bm \eta} }$ is $[\min( \widehat{ {\bm \eta}}), \max(\widehat{{\bm \eta}})]$.
Therefore, compared with the range provided by the two-stage method (i.e., $[-0.75,0.83]$), the
joint method gives a larger one (i.e. $[-2.00,1.12]$) which better matches the range of the true labels (i.e., $[-3, 1]$) and therefore
leads to more accurate predictions as shown in Fig.~\ref{fig1}.
We also examine the estimated topics by both methods.
For the topics with the large value of $\eta$, positive words (e.g., ``great")
dominate the topics in both spectral methods because the frequencies for them are much higher than negative ones (e.g., ``bad"). Thus we mainly focus on the ``negative" topics where the
difference can be found more expressly.
Table 2 shows the topics correspond to the smallest value of ${\bm \eta}$ by each method.
To save space, for the topic in each method we show the non-neutral words from the top $200$ with highest probabilities.
For each word, we show its probability as well as the rank (i.e., the number in bracket)
in the topic distribution vector.
\begin{wraptable}{r}{.23\textwidth} \vspace{-.5cm}
\caption{Estimated $\boldsymbol \eta$ by the two spectral methods (sorted for ease of comparison).}\label{table:param} \vspace{-.2cm}
\centering
\begin{tabular}{|c|c|}
\hline\hline
Two-stage & Joint \\
\hline
-0.754 & -1.998 \\
-0.385 & -0.762 \\
-0.178 & -0.212 \\
-0.022 & -0.098 \\
0.321 & 0.437 \\
0.522 & 0.946 \\
0.712 & 1.143 \\
0.833 & 1.122 \\
\hline
\end{tabular}\vspace{-.3cm}
\end{wraptable}
We can see that the negative words (e.g., ``bad", ``boring") have a higher rank (on average) in the topic by the joint spectral method than in the topic by the two-stage method,
while the positive words (e.g., ``good", ``great") have a lower rank (on average) in the topic by the joint method than in the topic by the two-stage method.
This result suggests that this topic in the joint method is more strongly associated with the negative reviews, therefore yielding a
better fit of the negative review scores when combined with the estimated ${\bm \eta}$.
Therefore, considering the supervision information can lead to improved topics.
Finally, we also observe that in both topics some positive words (e.g., ``good") have a rather high rank.
This is because the occurrences of such positive words are much frequent than the negative ones.
\begin{table}[t]\vspace{-.15cm}
\caption{Probabilities and ranks (in brackets) of some non-neutral words. N/A means that the word does not appear in the top $200$ ones with highest probabilities.}\vspace{-.2cm}
\centering
\begin{tabular}{|c|l|l|}
\hline\hline
words & Two-stage spec-slda & Joint spec-slda \\
\hline
bad & $ 0.006905 ~(14) $ & $ 0.009864 ~(8) $ \\
boring & $ 0.002163 ~(101) $ & $ 0.002433 ~(63) $ \\
stupid & $ 0.001513 ~(114) $ & $ 0.001841 ~(93) $ \\
horrible & $ 0.001255 ~(121) $ & $ 0.001868 ~(89) $ \\
terrible & $0.001184 ~(136) $ & $0.001868 ~(88) $ \\
waste & $0.001157 ~(140) $ & $ 0.001896 ~(84) $ \\
disappointed & $ 0.000926 ~(171)$ & $0.001282 ~(127) $ \\
\hline
good & $0.012549 ~(7) $ & $0.012033 ~(5) $ \\
great & $0.012549 ~(11) $ & $0.003568 ~(37) $ \\
love & $0.007513 ~(12) $ & $0.003137 ~(46)$ \\
funny & $ 0.004334 ~(26) $ & $ 0.003346 ~(39) $ \\
enjoy & $ 0.002163 ~(77) $ & $ 0.001317 ~(123) $ \\
awesome & $0.001208 ~(132) $ & N/A \\
amazing & $0.001199 ~(133) $ & N/A \\
\hline
\end{tabular}\vspace{-.4cm}
\end{table}
\vspace{-.15cm}
\subsection{Time efficiency}
Finally, we compare the time efficiency with Gibbs sampling.
All algorithms are implemented in C++.
Our methods are very time efficient
because they avoid the time-consuming iterative steps in traditional variational inference and Gibbs sampling methods.
Furthermore, the empirical moment computation, which is the most time-consuming part in Alg.~\ref{alg1} and Alg.~\ref{alg2} when dealing with large-scale datasets,
consists of only elementary operations and can be easily optimized.
Table~\ref{table:time} shows the running time on the synthetic dataset with various sizes in the setting where the topic number is $k=10$, vocabulary size is $V=500$ and document length is $100$.
We can see that both spectral methods are much faster than Gibbs sampling, especially when the data size is large.
Another advantage of our spectral methods is that we can easily parallelize the
computation of the low-order moments over multiple compute nodes, followed by a single step of synchronizing the local moments.
Therefore, the communication cost will be very low, as compared to the distributed algorithms for topic models~\cite{a:yahoolda}
which often involve intensive communications in order to synchronize the messages for (approximately) accurate inference.
\begin{table}[t]\vspace{-.1cm}
\caption{Running time (seconds) of our spectral learning methods and Gibbs sampling.}
\label{table:time}\vspace{-.2cm}
\centering
\begin{tabular}{lllllll}
\hline
$n(\times 5 \times 10^3)$ & 1 & 2 & 4 & 8 & 16 & 32 \\
\hline
Gibbs sampling& 47 & 92 & 167 & 340 & 671 & 1313\\
Joint spec-slda& 11 & 15 & 17 & 28 & 45 & 90\\
Two-stage spec-slda& 10& 13& 15& 22 & 39 & 81\\
\hline
\end{tabular}\vspace{-.3cm}
\end{table}
As a small $k$ is sufficient for the Amazon review dataset, we report the results with different $k$ values on a synthetic dataset where the vocabulary size $V=500$, the document length $m=100$ and the document size $n=1,000,000$. As shown in Fig.~\ref{fig:distribute}, the distributed implementation of our spectral methods (both two-stage and joint) has almost ideal (i.e., linear) speedup
with respect to the number of threads for moments computing.
The computational complexity of the tensor decomposition step is $O(k^{5+\delta})$ for a third-order tensor $T\in \mathbb{R}^{k \times k \times k}$, where $\delta$ is small~\cite{a:tensordecomp}.
When the topic number $k$ is large (e.g., as may be needed in applications with much larger datasets), one can follow the
recent developed stochastic tensor gradient descent (STGD) method to compute the
eigenvalues and eigenvectors~\cite{a:sgdtd}, which
can significantly reduce the running time in the tensor decomposition stage.
\begin{figure}[t
\centering
\includegraphics[width=0.36\textwidth,height=0.25\textwidth]{images/time_new.png}\vspace{-.3cm}
\caption{Running time of our method w.r.t the number of threads. Both $x$ and $y$ axes are plotted in log scale with base $e$.}
\label{fig:distribute}\vspace{-.5cm}
\end{figure}
\vspace{-.15cm}
\section{Conclusions and Discussions}
We propose two novel spectral decomposition methods to recover the parameters of supervised LDA models from labeled documents.
The proposed methods enjoy a provable guarantee of model reconstruction accuracy and are highly efficient
and effective.
Experimental results on real datasets demonstrate that
the proposed methods, especially the joint one, are superior to existing methods.
This result is significant for spectral methods, which were often inferior to MLE-based methods in practice.
For further work, it is interesting to recover parameters when the regression model is non-linear.
\section*{Acknowledgements}
This work is supported by the National 973 Basic Research Program of China (Nos. 2013CB329403, 2012CB316301), National NSF of China (Nos. 61322308, 61332007), and Tsinghua Initiative Scientific Research Program (No. 20141080934).
\bibliographystyle{plain}
\section{Proof to Theorem 1}
In this section, we prove the sample complexity bound given in Theorem 1. The proof consists of three main parts.
In Appendix A.1, we prove perturbation lemmas that bound the estimation error of the whitened tensors
$M_2(W,W), M_y(W,W)$ and $M_3(W,W,W)$ in terms of the estimation error of the tensors themselves.
In Appendix A.2, we cite results on the accuracy of SVD and robust tensor power method when performed on estimated tensors,
and prove the effectiveness of the power update method used in recovering the linear regression model $\bm\eta$.
Finally, we give tail bounds for the estimation error of $M_2,M_y$ and $M_3$ in Appendix A.3
and complete the proof in Appendix A.4.
We also make some remarks on the indirect quantities (e.g. $\sigma_k(\widetilde O)$) used in Theorem 1
and simplified bounds for some special cases in Appendix A.4.
All norms in the following analysis, if not explicitly specified, are 2 norms in the vector and matrix cases and the operator norm in the high-order tensor case.
\subsection{Perturbation lemmas}
We first define the canonical topic distribution vectors $\widetilde{\bm\mu}$ and estimation error of observable tensors,
which simplify the notations that arise in subsequent analysis.
\begin{deff}[canonical topic distribution]
Define the canonical version of topic distribution vector $\bm\mu_i$, $\widetilde{\bm\mu}_i$, as follows:
\begin{equation}
\widetilde{\bm\mu}_i \triangleq \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}}\bm\mu_i.
\label{def_can_mu}
\end{equation}
We also define $O,\widetilde O\in\mathbb R^{n\times k}$ by $O = [\bm\mu_1,\cdots,\bm\mu_k]$ and $\widetilde{O} = [\widetilde{\bm\mu_1},\cdots,\widetilde{\bm\mu_k}]$.
\end{deff}
\begin{deff}[estimation error]
Assume
\begin{eqnarray}
\|M_2-\widehat M_2\| &\leq& E_P,\\
\|M_y-\widehat M_y\| &\leq& E_y,\\
\|M_3-\widehat M_3\| &\leq& E_T.
\end{eqnarray}
for some real values $E_P,E_y$ and $E_T$, which we will set later.
\end{deff}
The following lemma analyzes the whitening matrix $W$ of $M_2$.
Many conclusions are directly from \cite{a:twosvd}.
\begin{lem}[Lemma C.1, \cite{speclda}]
Let $W, \widehat W\in\mathbb R^{n\times k}$ be the whitening matrices such that $M_2(W,W) = \widehat M_2(\widehat W,\widehat W) = I_k$. Let $A = W^\top\widetilde O$ and $\widehat A = \widehat W^\top\widetilde O$. Suppose $E_P \leq \sigma_k(M_2)/2$. We have
\begin{eqnarray}
\|W\| &=& \frac{1}{\sigma_k(\widetilde O)},\\
\|\widehat W\| &\leq& \frac{2}{\sigma_k(\widetilde O)},\label{eq6_pert}\\
\|W-\widehat W\| &\leq& \frac{4E_P}{\sigma_k(\widetilde O)^3},\label{eq_wwhat}\\
\|W^+\| &\leq& 3\sigma_1(\widetilde O),\\
\|\widehat W^+\| &\leq& 2\sigma_1(\widetilde O),\\
\|W^+ - \widehat W^+\| &\leq& \frac{6\sigma_1(\widetilde O)}{\sigma_k(\widetilde O)^2}E_P,\\
\|A\| &=& 1,\\
\|\widehat A\| &\leq & 2,\\
\|A - \widehat A\| &\leq& \frac{4E_P}{\sigma_k(\widetilde O)^2},\\
\|AA^\top - \widehat A\widehat A^\top\| &\leq& \frac{12E_P}{\sigma_k(\widetilde O)^2}.\label{eq4_pert}
\end{eqnarray}
\label{lem_whiten}
\end{lem}
\begin{proof}
Proof to Eq. (\ref{eq_wwhat}):
Let $\widehat W^\top\widehat M_2\widehat W = I$ and $\widehat W^\top M_2\widehat W = BDB^\top$, where $B$ is orthogonal and $D$ is a positive definite diagonal matrix.
We then see that $W = \widehat WBD^{-1/2}B^\top$ satisfies the condition $W M_2 W^\top = I$.
Subsequently, $\widehat W = WBD^{1/2}B^\top$.
We then can bound $\|W-\widehat W\|$ as follows
$$ \|W-\widehat W\| \leq \|W\|\cdot \|I-D^{1/2}\| \leq \|W\|\cdot \|I-D\| \leq \frac{4E_P}{\sigma_k(\widetilde O)^3},$$
where the inequality $\|I-D\|\leq \frac{4E_P}{\sigma_k(\widetilde O)^2}$ was proved in \cite{speclda}.
Proof to Eq. (\ref{eq4_pert}):
$\|AA^\top-\widehat A\widehat A^\top\| \leq \|AA^\top-A\widehat A^\top\| + \|A\widehat A^\top - \widehat A\widehat A^\top\|
\leq \|A-\widehat A\|\cdot (\|A\|+\|\widehat A\|) \leq \frac{12E_P}{\sigma_k(\widetilde O)^2}$.
All the other inequalities come from Lemma C.1, \cite{speclda}.
\end{proof}
We are now able to provide perturbation bounds for estimation error of whitened moments.
\begin{deff}[estimation error of whitened moments]
Define
\begin{eqnarray}
\varepsilon_{p,w} &\triangleq& \|M_2(W, W) - \widehat M_2(\widehat W,\widehat W)\|,\\
\varepsilon_{y,w} &\triangleq& \|M_y(W, W) - \widehat M_y(\widehat W,\widehat W)\|,\\
\varepsilon_{t,w} &\triangleq& \|M_3(W,W,W) - \widehat M_3(\widehat W,\widehat W,\widehat W)\|.
\end{eqnarray}
\label{def_epsilon}
\end{deff}
\begin{lem}[Perturbation lemma of whitened moments]
Suppose $E_P\leq \sigma_k(M_2)/2$. We have
\begin{eqnarray}
\varepsilon_{p,w} &\leq& \frac{16E_P}{\sigma_k(\widetilde O)^2},\\
\varepsilon_{y,w} &\leq& \frac{24\|\bm\eta\|E_P}{(\alpha_0+2)\sigma_k(\widetilde O)^2} + \frac{4E_y}{\sigma_k(\widetilde O)^2},\\
\varepsilon_{t,w} &\leq& \frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5} + \frac{8E_T}{\sigma_k(\widetilde O)^3}.
\end{eqnarray}
\end{lem}
\begin{proof}
Using the idea in the proof of Lemma C.2 in \cite{speclda}, we can split $\varepsilon_{p,w}$ as
\begin{equation*}
\begin{aligned}
\varepsilon_{p,w}
=& \|M_2(W,W) - M_2(\widehat W,\widehat W) + M_2(\widehat W,\widehat W) - \widehat M_2(\widehat W,\widehat W)\|\\
\leq& \|M_2(W,W) - M_2(\widehat W,\widehat W)\| + \|M_2(\widehat W,\widehat W) - \widehat M_2(\widehat W,\widehat W)\|.
\end{aligned}
\end{equation*}
We can the bound the two terms seperately, as follows.
For the first term, we have
\setlength\arraycolsep{1pt}\begin{eqnarray*}
\|M_2(W,W) - M_2(\widehat W,\widehat W)\|
&=& \|W^\top M_2 W-\widehat W^\top\widehat M_2\widehat W\|\\
&=& \|AA^\top - \widehat A\widehat A^\top\|\\
&\leq& \frac{12E_P}{\sigma_k(\widetilde O)^2}.
\end{eqnarray*}
where the last inequality comes from Eq. (\ref{eq4_pert}).
For the second term, we have
$$\|M_2(\widehat W,\widehat W)-\widehat M_2(\widehat W,\widehat W)\| \leq \|\widehat W\|^2\cdot \|M_2-\widehat M_2\| \leq \frac{4E_P}{\sigma_k(\widetilde O)^2},$$
where the last inequality comes from Eq. (\ref{eq6_pert}).
Similarly, $\varepsilon_{y,w}$ can be splitted as $\|M_y(W,W)-M_y(\widehat W,\widehat W)\|$ and $\|M_y(\widehat W,\widehat W)-\widehat M_y(\widehat W,\widehat W)\|$, which can be bounded separately. For the first term, we have
\begin{equation*}
\begin{aligned}
\|M_y(W,W) - M_y(\widehat W,\widehat W)\|
=& \|W^\top M_y W - \widehat W^\top M_y\widehat W\|\\
=& \frac{2}{\alpha_0+2}\|A\mathrm{diag}(\bm\eta)A^\top - \widehat A\mathrm{diag}(\bm\eta)\widehat A^\top\|\\
\leq& \frac{2\|\bm\eta\|}{\alpha_0+2}\cdot \|AA^\top - \widehat A\widehat A^\top\|\\
\leq& \frac{24\|\bm\eta\|}{(\alpha_0+2)\sigma_k(\widetilde O)^2}\cdot E_P.
\end{aligned}
\end{equation*}
For the second term, we have
$$\|M_y(\widehat W,\widehat W)-\widehat M_y(\widehat W,\widehat W)\| \leq \|\widehat W\|^2\cdot\|M_y-\widehat M_y\| \leq \frac{4E_y}{\sigma_k(\widetilde O)^2}.$$
Finally, we bound $\varepsilon_{t,w}$ as below, following the work \cite{specregression}.
\begin{eqnarray*}
\varepsilon_{t,w} &=& \|M_3(W,W,W) - \widehat M_3(\widehat W,\widehat W,\widehat W)\|\\
&\leq& \|M_3\|\cdot \|W-\widehat W\|\cdot (\|W\|^2 + \|W\|\cdot \|\widehat W\|
+ \|\widehat W\|^2) + \|\widehat W\|^3\cdot \|M_3-\widehat M_3\|\\
&\leq& \frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5} + \frac{8E_T}{\sigma_k(\widetilde O)^3},
\end{eqnarray*}
where we have used the fact that $$\|M_3\| \leq \sum_{i=1}^k{\frac{2\alpha_i}{\alpha_0(\alpha_0+1)(\alpha_0+2)}} = \frac{2}{(\alpha_0+1)(\alpha_0+2)}.$$
\end{proof}
\subsection{SVD accuracy}
The key idea for spectral recovery of LDA topic modeling is the \emph{simultaneous diagonalization} trick,
which asserts that we can recover LDA model parameters by performing orthogonal tensor decomposition
on a pair of simultaneously whitened moments, for example, $(M_2, M_3)$ and $(M_2, M_y)$.
The following proposition details this insight, as we derive orthogonal tensor decompositions for the whitened tensor product
$M_y(W,W)$ and $M_3(W,W,W)$.
\begin{prop}
Define $\bm v_i\triangleq W^\top \widetilde{\bm\mu}_i = \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}}W^\top\bm\mu_i$. Then
\begin{enumerate}
\item $\{\bm v_i\}_{i=1}^k$ is an orthonormal basis.
\item $M_y$ has a pair of singular value and singular vector $(\sigma_i^y, \bm v_i)$ with $\sigma_i^y = \frac{2}{\alpha_0+2}\eta_j$ for some $j \in[k]$.
\item $M_3$ has a pair of robust eigenvalue and eigenvector \cite{a:tensordecomp} $(\lambda_i, \bm v_i)$ with $\lambda_i = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{j'}}}$ for some $j'\in[k]$.
\end{enumerate}
\end{prop}
\begin{proof}
The orthonormality of $\{\bm v_i\}_{i=1}^k$ follows from the fact that $W^\top M_2W = \sum_{i=1}^k{\bm v_i\bm v_i^\top} = I_k$.
Subsequently, we have
\setlength\arraycolsep{1pt}\begin{eqnarray*}
M_y(W, W) &=& \frac{2}{\alpha_0+2}\sum_{i=1}^k{\eta_i \bm v_i\bm v_i^\top},\\
M_3(W,W,W) &=& \frac{2}{\alpha_0+2}\sum_{i=1}^k{\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_i}}\bm v_i\otimes\bm v_i\otimes\bm v_i}.
\end{eqnarray*}
\end{proof}
The following lemmas (Lemma \ref{lem_eta} and Lemma \ref{lem_mu})
give upper bounds on the estimation error of $\bm\eta$ and $\bm\mu$
in terms of $|\widehat\lambda_i-\lambda_i|$, $|\widehat{\bm v}_i-\bm v_i|$
and the estimation errors of whitened moments defined in Definition \ref{def_epsilon}.
\begin{lem}[$\eta_i$ estimation error bound]
Define $\widehat\eta_i \triangleq \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i$,
where $\widehat{\bm v}_i$ is some estimation of $\bm v_i$.
We then have
\begin{equation}
|\eta_i - \widehat\eta_i| \leq 2\|\bm\eta\|\|\widehat{\bm v}_i-\bm v_i\| + \frac{\alpha_0+2}{2}(1+2\|\widehat{\bm v}_i-\bm v_i\|)\cdot\varepsilon_{y,w}.
\end{equation}
\label{lem_eta}
\end{lem}
\begin{proof}
First, note that $\bm v_i^\top M_y(W,W)\bm v_i = \frac{2}{\alpha_0+2}\eta_i$ because $\{\bm v_i\}_{i=1}^k$ are orthonormal.
Subsequently, we have
\begin{equation*}
\begin{aligned}
\frac{2}{\alpha_0+2}|\eta_i - \widehat\eta_i|
=& \Big|\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - \bm v_i^\top M_y(W,W)\bm v_i\Big| \\
\leq& \Big| (\widehat{\bm v}_i-\bm v_i)^\top \widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i\Big| + \Big|\bm v_i^\top\left(\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\right)\Big|\\
\leq& \|\widehat{\bm v}_i-\bm v_i\|\|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i\| + \|\bm v_i\|\|\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\|.\\
\end{aligned}
\end{equation*}
Note that both $\bm v_i$ and $\widehat{\bm v}_i$ are unit vectors. Therefore,
\begin{equation*}
\begin{aligned}
\frac{2}{\alpha_0+2}|\eta_i-\widehat\eta_i|
\leq& \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i-\bm v_i\| + \|\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\|\\
\leq& \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i-\bm v_i\| + \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i - \bm v_i\| + \|\widehat M_y(\widehat W,\widehat W)-M_y(W,W)\|\|\bm v_i\|\\
\leq& 2\|\widehat{\bm v}_i-\bm v_i\|\left(\frac{2}{\alpha_0+2}\|\bm\eta\| + \varepsilon_{y,w}\right) + \varepsilon_{y,w}.
\end{aligned}
\end{equation*}
The last inequality is due to the fact that $\|M_y(W,W)\| = \frac{2}{\alpha_0+2}\|\bm\eta\|$.
\end{proof}
\begin{lem}[$\bm\mu_i$ estimation error bound]
Define $\widehat{\bm\mu}_i \triangleq \frac{\alpha_0+2}{2}\widehat\lambda_i(\widehat W^+)^\top\widehat{\bm v}_i$,
where $\widehat\lambda_i, \widehat{\bm v}_i$ are some estimates of singular value pairs $(\lambda_i, \bm v_i)$ of $M_3(W,W,W)$.
We then have
\begin{equation}
\begin{aligned}
\|\widehat{\bm\mu}_i - \bm\mu_i\| \leq& \frac{3(\alpha_0+2)}{2}\sigma_1(\widetilde O)|\widehat\lambda_i-\lambda_i|
+ 3\alpha_{\max}\sigma_1(\widetilde O)\|\widehat{\bm v}_i-\bm v_i\|
+ \frac{6\alpha_{\max}\sigma_1(\widetilde O)E_P}{\sigma_k(\widetilde O)^2}.
\end{aligned}
\end{equation}
\label{lem_mu}
\end{lem}
\begin{proof}
First note that $\bm\mu_i = \frac{\alpha_0+2}{2}\lambda_i (W^+)^\top\bm v_i$. Subsequently,
\begin{equation*}
\begin{aligned}
\frac{2}{\alpha_0+2}\|\bm\mu_i - \widehat{\bm\mu}_i\|
=& \|\widehat\lambda_i(\widehat W^+)^\top\widehat{\bm v}_i - \lambda_i(W^+)^\top\bm v_i\|\\
\leq& \|\widehat\lambda_i\widehat W^+ - \lambda_i W^+\|\|\widehat{\bm v}_i\| + \|\lambda_i W^+\|\|\widehat{\bm v}_i-\bm v_i\|\\
\leq& |\widehat\lambda_i-\lambda_i|\|\widehat W^+\| + |\lambda_i|\|\widehat W^+ - W^+\| + |\lambda_i|\|W^+\|\|\widehat{\bm v}_i-\bm v_i\|\\
\leq& 3\sigma_1(\widetilde O)|\widehat\lambda_i-\lambda_i| + \frac{2\alpha_{\max}}{\alpha_0+2}\cdot \frac{6\sigma_1(\widetilde O) E_P}{\sigma_k(\widetilde O)^2}
+ \frac{2\alpha_{\max}}{\alpha_0+2}\cdot 3\sigma_1(\widetilde O)\cdot \|\widehat{\bm v_i} - \bm v_i\|.
\end{aligned}
\end{equation*}
\end{proof}
To bound the error of orthogonal tensor decomposition performed on the estimated tensors $\widehat M_3(\widehat W,\widehat W,\widehat W)$,
we cite Theorem 5.1 \cite{a:tensordecomp}, a sample complexity analysis on the robust tensor power method we used
for recovering $\widehat\lambda_i$ and $\widehat{\bm v}_i$.
\begin{lem}[Theorem 5.1, \cite{a:tensordecomp}]
Let $\lambda_{\max} = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$,
$\lambda_{\min} = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\max}}}$,
where $\alpha_{\min} = \min{\alpha_i}$ and $\alpha_{\max} = \max{\alpha_i}$.
Then there exist universal constants $C_1, C_2 > 0$ such that the following holds:
Fix $\delta'\in(0,1)$.
Suppose $\varepsilon_{t,w}\leq \varepsilon$ and
\begin{eqnarray}
\varepsilon_{t, w} &\leq& C_1\cdot \frac{\lambda_{\min}}{k},\label{eq1_tensorpower}
\end{eqnarray}
Suppose $\{(\widehat\lambda_i, \widehat{\bm v}_i)\}_{i=1}^k$ are eigenvalue and eigenvector pairs
returned by running Algorithm 1 in \cite{a:tensordecomp} with input $\widehat M_3(\widehat W,\widehat W,\widehat W)$ for $L = \text{poly}(k)\log(1/\delta')$ and $N \geq C_2\cdot(\log(k) + \log\log(\frac{\lambda_{\max}}{\varepsilon}))$ iterations.
With probability greater than $1-\delta'$, there exists a permutation $\pi':[k]\to[k]$ such that
for all $i$,
$$\|\widehat{\bm v}_i - \bm v_{\pi'(i)}\| \leq 8\varepsilon/\lambda_{\min},\quad |\widehat\lambda_i - \lambda_{\pi'(i)}| \leq 5\varepsilon.$$
\label{lem_tensorpower}
\end{lem}
\subsection{Tail Inequalities}
\begin{lem}[Lemma 5, \cite{specregression}]
Let $\bm x_1,\cdots,\bm x_N\in\mathbb R^d$ be i.i.d. samples from some distribution with bounded support (i.e., $\|\bm x\|_2\leq B$ with probability 1 for some constant $B$).
Then with probability at least $1-\delta$,
$$\left\|\frac{1}{N}\sum_{i=1}^N{\bm x_i} - \mathbb E[\bm x]\right\|_2 \leq \frac{2B}{\sqrt{N}}\left(1+\sqrt{\frac{\log(1/\delta)}{2}}\right).$$
\label{lem_chernoff}
\end{lem}
\begin{cor}
Let $\bm x_1,\cdots,\bm x_N\in\mathbb R^d$ be i.i.d. samples from some distributions with $\Pr[\|\bm x\|_2 \leq B] \geq 1-\delta'$.
Then with probability at least $1-N\delta'-\delta$,
$$\left\|\frac{1}{N}\sum_{i=1}^N{\bm x_i} - \mathbb E[\bm x]\right\|_2 \leq \frac{2B}{\sqrt{N}}\left(1+\sqrt{\frac{\log(1/\delta)}{2}}\right).$$
\label{cor_chernoff}
\end{cor}
\begin{proof}
Use union bound.
\end{proof}
\begin{lem}[concentration of moment norms]
Suppose we obtain $N$ i.i.d. samples (i.e., documents with at least three words each and their regression variables in sLDA models).
Define $R(\delta) \triangleq \|\bm\eta\| - \sigma\Phi^{-1}(\delta)$,
where $\Phi^{-1}(\cdot)$ is the inverse function of the CDF of a standard Gaussian distribution.
Let $\mathbb E[\cdot]$ denote the mean of the true underlying distribution
and $\widehat{\mathbb E}[\cdot]$ denote the empirical mean.
Then
\iffalse
\begin{enumerate}
\item $\Pr\left[\|\mathbb E[\bm x_1] - \widehat{\mathbb E}[\bm x_1]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[\bm x_1\otimes\bm x_2\otimes\bm x_3] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y] - \widehat{\mathbb E}[y]\| < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}} \right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y\bm x_1] - \widehat{\mathbb E}[y\bm x_1]\|_F < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[y\bm x_1\otimes\bm x_2]\|_F < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}}\right] \geq 1 - \delta$.
\end{enumerate}
\fi
\begin{equation}
\begin{aligned}
& \Pr \left [\|\mathbb E[\bm x_1] - \widehat{\mathbb E}[\bm x_1]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}} \right ] \geq 1-\delta,\\
&\Pr \left [\|\mathbb E[\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}} \right ] \geq 1-\delta,\\
&\Pr \left [\|\mathbb E[\bm x_1\otimes\bm x_2\otimes\bm x_3] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]\|_F< \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}} \right ] \geq 1-\delta,\\
&\Pr \left [\|\mathbb E[y] - \widehat{\mathbb E}[y]\| < R(\delta/4 N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}} \right ] \geq 1-\delta,\\
&\Pr \left [\|\mathbb E[y\bm x_1] - \widehat{\mathbb E}[y\bm x_1]\|_F < R(\delta/4 N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}} \right ] \geq 1-\delta,\\
&\Pr \left [\|\mathbb E[y\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[y\bm x_1\otimes\bm x_2]\|_F < R(\delta/4 N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}} \right ] \geq 1 - \delta.
\end{aligned}
\end{equation}
\label{lem_tail_ineq}
\end{lem}
\begin{proof}
Use Lemma \ref{lem_chernoff} and Corrolary \ref{cor_chernoff} for concentration bounds involving the regression variable $y$.
\end{proof}
\begin{cor}
With probability $1-\delta$ the following holds:
\begin{enumerate}
\item $E_P = \|M_2-\widehat M_2\| \leq 3\cdot \frac{2+\sqrt{2\log(6/\delta)}}{\sqrt{N}}$.
\item $E_y = \|M_y-\widehat M_y\| \leq 10R(\delta/60 N)\cdot \frac{2+\sqrt{2\log(15/\delta)}}{\sqrt{N}}$.
\item $E_T = \|M_3-\widehat M_3\| \leq 10\cdot \frac{2+\sqrt{2\log(9/\delta)}}{\sqrt{N}}$.
\end{enumerate}
\label{cor_ep}
\end{cor}
\begin{proof}
Corrolary \ref{cor_ep} can be proved by expanding the terms by definition and then using tail inequality in Lemma \ref{lem_tail_ineq} and union bound.
Also note that $\|\cdot\| \leq \|\cdot\|_F$ for all matrices.
\end{proof}
\subsection{Completing the proof}
We are now ready to give a complete proof to Theorem \ref{thm_main}
\iffalse
\jun{the following restatement of the theorem is not necessary. Refer to the theorem in the main text. If restating, make them consistent.}
\begin{thm}[Sample complexity bound]
Let $\sigma_1(\widetilde O)$ and $\sigma_k(\widetilde O)$ be the largest and the smallest singular values of the canonical topic distribution matrix $\widetilde O$.
Define $\lambda_{\min} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\max}}}$
and $\lambda_{\max} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$,
where $\alpha_{\max}$ and $\alpha_{\min}$ are the largest and the smallest entries in $\bm\alpha$.
Suppose $\widehat{\bm\mu}$, $\widehat{\bm\alpha}$ and $\widehat{\bm\eta}$ are the outputs of Algorithm 1,
and $L$ is at least a linear function in $k$.
Fix $\delta\in(0,1)$.
For any small error-tolerance parameter $\varepsilon > 0$,
if Algorithm 1 is run with parameter $T = \Omega(\log(k) + \log\log(\lambda_{\max}/\varepsilon))$
on $N$ i.i.d. sampled documents with
$N\geq \max(n_1, n_2, n_3)$, where
\setlength\arraycolsep{1pt} \begin{eqnarray}
n_1 &=& C_1\cdot \left(1+\sqrt{\log(6/\delta)}\right)^2\cdot \frac{\alpha_0^2(\alpha_0+1)^2}{\alpha_{\min}}, \label{eq_n1} \\
n_2 &=& C_2\cdot \frac{(1+\sqrt{\log(15/\delta)})^2}{\varepsilon^2\sigma_k(\widetilde O)^4}
\\
&\ &\cdot\max\left((\|\bm\eta\| + \Phi^{-1}(\delta/60\sigma))^2, \alpha_{\max}^2\sigma_1(\widetilde O)^2\right),
\label{eq_n2} \\
n_3 &=& C_3\cdot \frac{(1+\sqrt{\log(9/\delta)})^2}{\sigma_k(\widetilde O)^{10}}\cdot\max\left(\frac{1}{\varepsilon^2}, \frac{k^2}{\lambda_{\min}^2}\right),
\label{eq_n3}
\end{eqnarray}
and $C_1, C_2$ and $C_3$ are universal constants,
then with probability at least $1-\delta$, there exists a permutation $\pi:[k]\to [k]$ such that for every topic $i$, the following holds:\\
\begin{enumerate}
\item $|\alpha_i-\widehat{\alpha}_{\pi(i)}| \leq \frac{4\alpha_0(\alpha_0+1)(\lambda_{\max}+5\varepsilon)}{(\alpha_0+2)^2\lambda_{\min}^2(\lambda_{\min}-5\varepsilon)^2}\cdot 5\varepsilon$, if $\lambda_{\min} > 5\varepsilon$;\\
\item $\|\bm\mu_i - \widehat{\bm\mu}_{\pi(i)}\| \leq \left(3\sigma_1(\widetilde O)\left(\frac{8\alpha_{\max}}{\lambda_{\min}} + \frac{5(\alpha_0+2)}{2}\right) + 1\right)\varepsilon$;\\
\item $|\eta_i - \widehat\eta_{\pi(i)}| \leq \left(\frac{\|\bm\eta\|}{\lambda_{\min}} + (\alpha_0+2)\right) \varepsilon$.
\end{enumerate}
\label{thm_main}
\end{thm}
\fi
\begin{proof}{(Proof of Theorem ~\ref{thm_main})}
First, the assumption $E_P\leq\sigma_k(M_2)$ is required for error bounds on $\varepsilon_{p,w}, \varepsilon_{y,w}$ and $\varepsilon_{t,w}$.
Noting Corrolary \ref{cor_ep} and the fact that $\sigma_k(M_2) = \frac{\alpha_{\min}}{\alpha_0(\alpha_0+1)}$, we have
$$N = \Omega\left(\frac{\alpha_0^2(\alpha_0+1)^2(1+\sqrt{\log(6/\delta)})^2}{\alpha_{\min}^2}\right).$$
Note that this lower bound does not depend on $k$, $\varepsilon$ and $\sigma_k(\widetilde O)$.
For Lemma \ref{lem_tensorpower} to hold, we need the assumptions that
$\varepsilon_{t,w} \leq \min(\varepsilon, O(\frac{\lambda_{\min}}{k}))$.
These imply $n_3$,
as we expand $\varepsilon_{t,w}$ according to Definition \ref{def_epsilon}
and note the fact that the first term $\frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5}$ dominates the second one.
The $\alpha_0$ is missing in the third requirment $n_3$ because $\alpha_0+1 \geq 1$, $\alpha_0+2 \geq 2$ and we discard them both.
The $|\alpha_i-\widehat{\alpha}_{\pi(i)}|$ bound follows immediately by Lemma \ref{lem_tensorpower}
and the recovery rule $\widehat{\alpha}_i = \frac{\alpha_0+2}{2}\widehat\lambda_i$.
To bound the estimation error for the linear classifier $\bm\eta$, we need to further bound $\varepsilon_{y,w}$.
We assume $\varepsilon_{y,w}\leq\varepsilon$.
By expanding $\varepsilon_{y,w}$ according to Definition \ref{def_epsilon} in a similar manner
we obtain the $(\|\bm\eta\|+\Phi^{-1}(\delta/60\sigma))^2$ term in the requirment of $n_2$.
The bound on $|\eta_i-\widehat\eta_{\pi(i)}|$ follows immediately by Lemma \ref{lem_eta}.
Finally, we bound $\|\bm\mu_i-\widehat{\bm\mu}_{\pi(i)}\|$ using Lemma \ref{lem_mu}.
We need to assume that $\frac{6\alpha_{\max}\sigma_1(\widetilde O)E_P}{\sigma_k(\widetilde O)^2} \leq \varepsilon$,
which gives the $\alpha_{\max}^2\sigma_1(\widetilde O)^2$ term.
The $\|\bm\mu_i-\widehat{\bm\mu}_{\pi(i)}\|$ bound then follows by Lemma \ref{lem_mu} and Lemma \ref{lem_tensorpower}.
\end{proof}
We make some remarks for the main theorem. In Remark~\ref{rem_connection}, we establish links between indirect quantities appeared in Theorem~\ref{thm_main} (e.g., $\sigma_k(\widetilde O)$) and the functions of original model parameters (e.g., $\sigma_k(O)$).
These connections are straightforward following their definitions.
\begin{rem}
The indirect quantities $\sigma_1(\widetilde O)$ and $\sigma_k(\widetilde O)$ can be related to $\sigma_1(O)$, $\sigma_k(O)$ and $\bm\alpha$ in the following way:
{\small \begin{equation*}
\sqrt{\frac{\alpha_{\min}}{\alpha_0(\alpha_0+1)}}\sigma_k(O) \leq \sigma_k(\widetilde O)\leq \sqrt{\frac{\alpha_{\max}}{\alpha_0(\alpha_0+1)}}\sigma_k(O);~~~~
\end{equation*}
\begin{equation*}
\sigma_1(\widetilde O) \leq \sqrt{\frac{\alpha_{\max}}{\alpha_0(\alpha_0+1)}}\sigma_1(O) \leq \frac{1}{\sqrt{\alpha_0+1}}.
\end{equation*}}
\label{rem_connection}
\end{rem}
We now take a close look at the sample complexity bound in Theorem~\ref{thm_main}.
It is evident that $n_2$ can be neglected when the number of topics $k$ gets large,
because in practice the norm of the linear regression model $\bm\eta$ is usually assumed to be small in order to avoid overfitting.
Moreover, as mentioned before, the prior parameter $\bm\alpha$ is often assumed to be homogeneous with $\alpha_i = 1/k$ \cite{lsa}.
With these observations, the sample complexity bound in Theorem~\ref{thm_main}
can be greatly simplified.
\begin{rem}
Assume $\|\bm\eta\|$ and $\sigma$ are small and $\bm\alpha = (1/k, \cdots, 1/k)$.
As the number of topics $k$ gets large, the sample complexity bound in Theorem~\ref{thm_main} can be simplified as
\begin{equation}
N = \Omega\left( \frac{\log(1/\delta)}{\sigma_k(\widetilde O)^{10}}\cdot\max(\varepsilon^{-2}, k^3)\right).
\end{equation}\label{rem_simpbound}
\end{rem}\vspace{-.5cm}
The sample complexity bound in Remark~\ref{rem_simpbound} may look formidable
as it depends on $\sigma_k(\widetilde O)^{10}$.
However, such dependency is somewhat necessary because we are using third-order tensors to recover the underlying model parameters.
Furthermore, the dependence on $\sigma_k(\widetilde O)^{10}$ is introduced by the robust tensor power method to recover LDA parameters,
and the reconstruction accuracy of $\bm\eta$ only depends on $\sigma_k(\widetilde O)^4$ and $(\|\bm\eta\|+\Phi^{-1}(\delta/60\sigma))^2$.
As a consequence, if we can combine our power update method for $\bm \eta$ with LDA inference algorithms that have milder dependence on the singular value $\sigma_k(\widetilde O)$,
we might be able to get an algorithm with a better sample complexity.
\iffalse
\section{Discussions}
\subsection{Appendix C.1. Extension to other topic recovery algorithms}
One important advantage of our proposed inference algorithm is its flexibility---the algorithm can be combined with many other LDA inference algorithms to infer supervised LDA model parameters.
More specifically, given access to any algorithm that recovers the topic distribution matrix $O$
and the prior parameter $\bm\alpha$ from i.i.d. sampled documents,
an inference algorithm for a supervised LDA model can be immediately obtained,
as shown in Algorithm \ref{alg2}.
\begin{algorithm}
\caption{sLDA parameter recovery based on an existing LDA inference algorithm $\mathcal A$. Input parameter: $\alpha_0$.}
\centering
\begin{algorithmic}[1]
\State Compute empirical moments and obtain $\widehat M_2, \widehat M_y$. Set $\widehat{\bm\eta} = 0$.
\State Find $\widehat W\in\mathbb R^{n\times k}$ such that $\widehat M_2(\widehat W, \widehat W) = I_k$.
\State Run algorithm $\mathcal A$ with observed documents and parameter $\alpha_0$. Obtain the estimated
topic distribution matrix $\widehat O=(\widehat{\bm\mu}_1,\cdots,\widehat{\bm\mu}_k)$ and prior parameter $\widehat{\bm\alpha}$.
\State Compute $\widehat{\bm v}_i = \sqrt{\frac{\widehat{\alpha}_i}{\alpha_0(\alpha_0+1)}}\widehat W^\top\widehat{\bm\mu}_i$ for each topic $i$.
\State Recover linear classifier: $\widehat\eta_i \gets \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i$.
\State \textbf{Output:} $\widehat{\bm\eta}$, $\bm\alpha$ and $\{\widehat{\bm\mu}_i\}_{i=1}^k$.
\end{algorithmic}
\label{alg2}
\end{algorithm}
The sample complexity of Algorithm \ref{alg2} depends on the sample complexity of the LDA inference algorithm $\mathcal A$.
Although the LDA inference algorithm $\mathcal A$ is free to make any assumptions on the topic distribution matrix $O$,
we comment that the linear independence of topic distribution vectors $\bm\mu$ is still required,
because the power update trick (step 5 in Algorithm 1) is valid only $W\bm\mu_i$ are orthogonal vectors.
\subsection*{Appendix C.2. Going beyond SVD}
The proposed methods are based on spectral decomposition of observable moments
and are provably correct.
However, a major drawback of these methods is their assumption that the topic distributions $\bm\mu$ are linearly independent.
Although standard LDA models assume this topic independence \cite{lda},
in practice there is strong evidence that different topics might be related \cite{correlatelda}.
Therefore, it is important to consider possible extensions.
Recently, there has been much work on provable LDA inference algorithms that do not require the topic independence assumption \cite{practicalalgo,beyondsvd}.
Instead of making the independence assumption, these methods assume a $p$-separability condition on the topic distribution matrix,
that is, there exists an \emph{anchor word} for each topic such that the anchor word only appears in this specific topic,
and its probability (given that topic) is at least $p$.
Here we briefly describe an idea that might lead to an sLDA inference algorithm without assuming topic independence.
First, let $O\in\mathbb R^{V\times k}$ be the topic distribution matrix defined previously and $H = (\bm h_1,\cdots,\bm h_N)\in\mathbb R^{k\times N}$
be the matrix of topic mixing vectors for each document.
Suppose $Q = \mathbb E[\bm x_1\bm x_2^\top]$ is the word co-occurrence frequency matrix
and $Q_y = \mathbb E[y\bm x_1]$ is the co-occurrence frequency matrix between word and regression variable observations.
It is clear that both $Q$ and $Q_y$ can be estimated from the training data, and we have
\begin{equation}
Q = ORO^\top, Q_y = OR\bm\eta,
\end{equation}
where $R \triangleq \mathbb E[HH^\top]$.
Assuming the $p$-separability condition holds and using methods developed in \cite{beyondsvd,practicalalgo},
we can consistently recover the $O$ and $R$.
Now note that $Q_y = (OR)\cdot \bm\eta$.
As a result, knowing both $OR$ and $Q_y$ we reduce the $\bm\eta$ inference problem to solving a linear equation systems.
Future research could be done on (1) determining when the matrix $OR$ has full rank $r$ so that the linear equations can be solved, and
(2) investigating the sample complexity problem for such an inference algorithm.
\subsection*{Appendix C.3. From regression to classification}
A natural extension to our result is to consider supervised LDA models for classification purposes.
The simplest model is a logistic regression classification model, where we assume the response variable $y_d$
for each document $d$ is in $\{+1,-1\}$, and
$$\Pr[y_d = 1|\bm h_d] = \frac{\exp(\bm\eta^\top\bm h_d)}{1+\exp(\bm\eta^\top\bm h_d)},$$
where $\bm\eta$ is a linear classifier.
Though appears simple, such an extension incurs many fundamental problems.
A major obstacle is the fact that the conditional expectation $\mathbb E[y|\bm h]$ is no longer linear
in the topic mixing vector $\bm h$.
As a result, we cannot even evaluate $\mathbb E[y]$ (or higher order tensors like $\mathbb E[y\bm x_1\otimes\bm x_2]$)
in closed forms.
Further more, even if we have accurate estimation to the above moments,
it is rather difficult to infer $\bm\eta$ back due to its non-linearality.
To our knowledge, the classification problem remains unsolved in terms of spectral decomposition methods,
and there is doubt whether a provably correct algorithm exists in such scenarios.
\fi
\section{Proof of Theorem 2}
In this section we give the proof of Theorem 2, following the similar line of the proof of Theorem
1. We bound the estimation errors and the errors introduced by the tensor decomposition step
respectively
\subsection{Definitions}
We first recall the definition of joint canconical topic distribution.
\begin{deff}
(Joint Canconical topic distribution) Define the canconical version of joint topic distribution vector
$\bm{v}_i,\widetilde{\bm{v}}_i$ as follows:
$$
\widetilde{\bm{v}}_i = \sqrt{\dfrac{\alpha_i}{\alpha_0(\alpha_0 +1)}}\bm{v}_i,
$$
where $\bm{v}_i = [ \bm{\mu}_i' , \eta_i]'$ is the topic dictribution vector extended by its regression parameter.
We also define $O^*, \widetilde{O^*} \in \mathbb{R}^{(V+1) \times k}$ by $ O^* = [ \bm{v}_1, ... ,\bm{v}_k] $
and $ \widetilde{O^*} = [ \widetilde{\bm{v}}_1 , ..., \widetilde{\bm{v}}_k]$.
\end{deff}
\subsection{Estimation Errors}
The tail inequatlities in last section \ref{lem_chernoff},\ref{cor_chernoff} gives the following estimation:
\begin{lem}
\label{l1}
Suppose we obtain $N$ i.i.d. samples. Let $\mathbb{E}[\cdot]$ denote the mean of the true underlying distribution and $\hat{\mathbb{E}}[\cdot]$ denote
the empirical mean. Define
\begin{equation}
\begin{aligned}
& R_1(\delta) = \|\bm{\eta}\| - \sigma\Phi^{-1}(\delta),\\
& R_2(\delta) = 2\|\bm{\eta}\|^2 + 2\sigma^2[\Phi^{-1}(\delta)]^2, \\
& R_3(\delta) = 4\|\bm{\eta}\|^3 - 4\sigma^3[\Phi^{-1}(\delta)]^3, \\
\end{aligned}
\end{equation}
where $\Phi^{-1}(\cdot)$ is the inverse function of the CDF of a
standard Gaussian distribution. Then:
\begin{equation}
\begin{aligned}
& \Pr \left [ \| \mathbb{E}[ \mathbf{x_1} \otimes \mathbf{x_2} \otimes \mathbf{x_3}] - \hat{\mathbb{E}}[\mathbf{x_1} \otimes
\mathbf{x_2} \otimes \mathbf{x_3} ] \|_F < \dfrac{2 + \sqrt{2\log(1/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta; \\
& \Pr \left [ \| \mathbb{E}[ y^i ] - \hat{\mathbb{E}}[ y^i ] \|_F < R_i(\delta/4 N )\cdot
\dfrac{2 + \sqrt{2\log(2/\delta)} }{ \sqrt{N} } \right ] \geq 1 - \delta \ \ \ \ i = 1, 2, 3;\\
& \Pr \left [ \| \mathbb{E}[ y^i \mathbf{x_1} ] - \hat{\mathbb{E}}[ y^i \mathbf{x_1} ] \|_F
< R_i(\delta / 4 N) \cdot \dfrac{2 + \sqrt{2\log(2/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta \ \ \ \ i = 1,2;\\
& \Pr \left [ \| \mathbb{E}[ y^i \mathbf{x_1} \otimes \mathbf{x_2} ] - \hat{\mathbb{E}}[ y^i \mathbf{x_1} \otimes \mathbf{x_2} ] \|_F
< R_i(\delta / 4 N) \cdot \dfrac{2 + \sqrt{2\log(2/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta \ \ \ \ i = 1 , 2.\\
\end{aligned}
\end{equation}
\end{lem}
\begin{proof}
This lemma is a direct application of lemma \ref{lem_chernoff} and corollary \ref{cor_chernoff}.
\end{proof}
\begin{cor}
\label{c1}
With probability $1 -\delta$, the following holds:
\begin{equation}
\begin{aligned}
& Pr \left [ \| \mathbb{E}[ \mathbf{z_1}] - \hat{\mathbb{E}}[ \mathbf{z_1}] \| < C_1(\delta / 8 N)
\dfrac{2 + \sqrt{2\log(4/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta, \\
& Pr \left [ \| \mathbb{E}[ \mathbf{z_1} \otimes \mathbf{z_2}] - \hat{\mathbb{E}}[ \mathbf{z_1} \otimes \mathbf{z_2}] \| < C_2(\delta / 12 N)
\dfrac{2 + \sqrt{2\log(6/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta, \\
& Pr \left [ \| \mathbb{E}[ \mathbf{z_1} \otimes \mathbf{z_2} \otimes \mathbf{z_3}] - \hat{\mathbb{E}}[\mathbf{z_1} \otimes
\mathbf{z_2} \otimes \mathbf{z_3} ] \|_F < C_3(\delta/16N) \dfrac{2 + \sqrt{2\log(8/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta, \\
\end{aligned}
\end{equation}
where
\begin{equation}
\begin{aligned}
& C_1(\delta) = R_1(\delta) + 1, \\
& C_2(\delta) = R_1(\delta) + R_2(\delta) + 1, \\
& C_3(\delta) = R_1(\delta) + R_2(\delta) + R_3(\delta) + 1. \\
\end{aligned}
\end{equation}
\end{cor}
\begin{proof}
Use lemma \ref{l1} as well as the fact that $\sqrt{a + b} \leq \sqrt{a} + \sqrt{b}, \forall a,b \geq 0$ and $ Pr( X \leq t_1 , Y \leq t_2) \leq Pr( X + Y \leq t_1 + t_2)$.
\end{proof}
\begin{lem}
\label{l3}
(concentration of moment norms) Using notations in lemma \ref{l1} and suppose
$C_3(\delta/144N)\dfrac{2 +\sqrt{2\log(72/\sigma)}}{\sqrt{N}} \leq 1 $
we have:
\begin{equation}
\begin{aligned}
& E_P = \|N_2 - \hat{N}_2\| \leq 3C_2(\delta/36N)) \cdot \dfrac{2 + \sqrt{2 \log(18/\sigma)}}{\sqrt{N}},\\
& E_T = \|N_3 - \hat{N}_3\| \leq 10C_3(\delta/144N)) \cdot \dfrac{2 + \sqrt{2 \log(72/\sigma)}}{\sqrt{N}}.\\
\end{aligned}
\end{equation}
\end{lem}
\begin{proof}
This lemma can be proved by expanding the terms by definition and use lemma \ref{l1} , corollary \ref{c1} as well as union bound.
Also note that $\| \cdot\| \leq \|\cdot \|_F$.
\end{proof}
\begin{lem}
(estimation error of whitened moments)
Define
\begin{equation}
\begin{aligned}
& \epsilon_{p,w} = \| N_2(W,W) - \hat{N}_2(\hat{W},\hat{W})\|, \\
& \epsilon_{t,w} = \| N_3(W,W,W) - \hat{N}_3(\hat{W},\hat{W},\hat{W})\|. \\
\end{aligned}
\end{equation}
Then we have:
\begin{equation}
\begin{aligned}
& \epsilon_{p,w} \leq \dfrac{16E_p}{\sigma_k(\widetilde{O^*})^2}, \\
& \epsilon_{t,w} \leq \dfrac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde{O^*})^5} +
\dfrac{8E_T}{\sigma_k(\widetilde{O^*})^3}.
\end{aligned}
\end{equation}
\end{lem}
\subsection{SVD accuracy}
We rewrite a part of Lemma ~\ref{lem_whiten} which we will use in the following.
\begin{lem}
\label{l5}
Let $W,\hat{W} \in R^{(V+1)\times k}$ be the whitening matrices such that
$N_2(W,W) = \hat{W}_2(\hat{W},\hat{W}) = I_k$. Further more, we suppose
$ E_P \leq \sigma_k(N_2)/2$. Then we have:
\begin{equation}
\begin{aligned}
& \|W - \hat{W}\| \leq \dfrac{4E_P}{\sigma_k(\widetilde{O^*})^3},\\
& \|W^+\| \leq 3\sigma_1(\widetilde{O^*}), \\
& \|\widetilde{W}^+\| \leq 2\sigma_1(\widetilde{O^*}), \\
& \|W^+ - \hat{W}^+\| \leq \dfrac{6\sigma_1(\widetilde{O^*})E_P}{\sigma_k(\widetilde{O^*})^2}.\\
\end{aligned}
\end{equation}
\end{lem}
Using the above lemma, we now can estimate the error introduced by SVD.
\begin{lem}($\bm{v}_i$ estimation error)
\label{l6}
Define $\hat{\bm{v}}_i = \frac{\alpha_0 + 2}
{2}\hat{\lambda}_i(\hat{W}^+)^{\top}\hat{\bm{\omega}}_i$ where
$(\hat{\lambda}_i,\hat{\omega}_i)$ are some estimations of svd pairs
$(\lambda_i, \omega_i)$ of $ N_3(W,W,W)$. We then have:
\begin{equation}
\begin{aligned}
\|\hat{\bm{v}}_i - \bm{v}_i\| \leq& \dfrac{(\alpha_0 + 1)6\sigma_1(\widetilde{O^*})E_P}{\sigma_k(\widetilde{O^*})^2}
+ \dfrac{(\alpha_0 +2)\sigma_1(\widetilde{O^*})|\hat{\lambda}_i - \lambda_i|}{2}
+3(\alpha_0 +1)\sigma_1(\widetilde{O^*})\|\hat{\bm{\omega}}_i - \bm{\omega}_i\|.
\end{aligned}
\end{equation}
\end{lem}
\begin{proof}
Note that $\bm{v}_i = \frac{}{}\lambda_i(W^+)^{\top}\bm{\omega}_i$. Thus we have:
\begin{equation}
\begin{aligned}
\frac{2}{\alpha_0 + 2 } \|\hat{\bm{v}}_i - \bm{v}_i\|
= &\|\hat{\lambda}_i(\hat{W}^+)^{\top}\hat{\bm{\omega}_i}
- \lambda_i(W^+)^{\top}\bm{\omega}_i \| \\
=& \|\hat{\lambda}_i(\hat{W}^+)^{\top}\hat{\bm{\omega}_i}
- \lambda_i(W^+)^{\top}\hat{\bm{\omega}}_i
+ \lambda_i(W^+)^{\top}\hat{\bm{\omega}}_i
- \lambda_i(W^+)^{\top}\bm{\omega}_i \| \\
\leq& \|\hat{\lambda}_i(\hat{W}^+)^{\top} - \lambda_i(W^+)^{\top}\|
\|\hat{\bm{\omega}}_i\|
+ \|\lambda_i(W^+)^{\top}\|\|\hat{\bm{\omega}}_i - \bm{\omega}_i\|\\
\leq& |\lambda_i|\|\hat{W}^+ - W^+\|
\|\hat{\bm{\omega}}_i\| +
|\hat{\lambda}_i - \lambda_i|\|\hat{W}^+\|
\|\hat{\bm{\omega}}_i\|
+ |\lambda_i|\|W^+\|\|\hat{\bm{\omega}}_i - \bm{\omega}_i\|\\
\leq& \dfrac{2(\alpha_0 +1)}{\alpha_0 + 2}\dfrac{6E_P\sigma_1(\widetilde{O^*})}{\sigma_k(\widetilde{O^*})^2}
+ 3\sigma_1(\widetilde{O^*})|\hat{\lambda}_i - \lambda_i| +
\dfrac{2(\alpha_0 + 1)}{\alpha_0 + 2}\cdot
3\sigma_1(\widetilde{O^*})\cdot \|\hat{\bm{\omega}}_i - \bm{\omega}_i \|.
\end{aligned}
\end{equation}
\end{proof}
\iffalse
We now cite an lemma from \cite{a:tensor} that provides a bound for the estimation error of $\bm{\omega}_i$
and $\lambda_i$.
\begin{lem}
\label{l7}
Let $\lambda_{max} =
\frac{2}{\alpha_0 + 2}\sqrt{\dfrac{\alpha_0(\alpha_0 + 1)}{\alpha_{min}}}$ and
$\lambda_{min} =
\frac{2}{\alpha_0 + 2}\sqrt{\dfrac{\alpha_0(\alpha_0 + 1)}{\alpha_{max}}}$, where
$\alpha_{max} = \max \alpha_i ,\alpha_{min} = \min \alpha_i$. Then there exist universal
constants $C_1,C_2 > 0$ such that the following hods: Fix $\delta' \in ( 0,1)$. Suppose
$\epsilon_{t,w} \leq \epsilon$ and
$$
\epsilon_{t,w} \leq C_1 \cdot \dfrac{\lambda_{min}}{k}
$$
Suppose $\{(\hat{\lambda}_i, \hat{\bm{v}}_i\}_{i=1}^{k}\}$ are eigenvalue and eigenvector pairs
returned by running Algorithm $1$ in \cite{a:tensor} with input $\hat{W}_3(\hat{W},\hat{W},\hat{W}) $ for
$L = poly(k)\log(1/\delta')$ and $N \geq C_2\cdot (\log(k) +
\log\log(\frac{\lambda_{max}}{\epsilon}) $ iterations. With probability at least $1 - \delta'$,
there exists a permutation $\pi':[k]\to[k]$ such that for all i:
$$
\|\hat{\bm{\omega}}_i - \bm{\omega}_i\| \leq 8\epsilon / \lambda_{min}, |\hat{\lambda}_i - \lambda_i | \leq 5 \epsilon.
$$
\end{lem}
\fi
\subsection{Completeing the proof}
We are now ready to complete the proof of Theorem ~\ref{thm_main2}.
\iffalse
\begin{theorem}
(Sample complexity bound)
Let $ \sigma_1(\widetilde{O^*})$ and $\sigma_k(\widetilde{O^*})$ be the largest and smallest singular
values of the joint canonical topic dictribution matrix $\widetilde{O^*}$.
For any error-tolerance parameter $\epsilon > 0$, if Algorithm $1$ runs at least
$T = \Omega(\log(k)) + \log\log(\dfrac{\lambda_{max}}{\epsilon})$ iterations on $N$ i.i.d. sampled documents with $N$
satisying:
\begin{equation}
\begin{aligned}
& N \geq K_1 \cdot \dfrac{\alpha_0^2(\alpha_0+1)^2C_2^2(\delta/36N)\cdot(2 + \sqrt{2\log(18/\delta)})^2 }{\alpha_{min}^2} \\
& N \geq K_2 \cdot C_3^2(\delta/144N)(2 + \sqrt{2 \log(72/\sigma)})^2 \\
&N \geq K_3 \cdot \dfrac{C_3^2(\delta/36N)(2+\sqrt{2\log(18/\sigma)})^2}
{\sigma_k(\widetilde{O^*})^{10}}\cdot\max(\dfrac{1}{\epsilon^2}, \dfrac{k^2}{\lambda_{min}^2})\\
\end{aligned}
\end{equation}
then with probability at least $1 - \delta$, there exist a permutation $\pi: [k] \to [k]$
such that the following holds for every $i \in [k]$:
\begin{equation}
\begin{aligned}
& |\alpha_i - \hat{\alpha}_{\pi(i)} |
\leq \dfrac{4\alpha_0(\alpha_0 + 1) }{(\alpha_0 + 2)^2\lambda_{min}^2(\lambda_{min} - 5 \epsilon)^2} \cdot 5\epsilon \\
& \|\bm{v}_i - \hat{\bm{v}}_{\pi(i)} \| \leq \left ( \sigma_1(\widetilde{O^*})(\alpha_0 + 2)(\frac{7}{2} +
\frac{8}{\lambda_{min}})\right )\cdot \epsilon \\
\end{aligned}
\end{equation}
\end{theorem}
\fi
\begin{proof}{(Proof of Theorem~\ref{thm_main2})}
We check out the conditions that must be satisfied for $\epsilon$ accuracy.
First $E_P \leq \sigma_k(N_2)/2$ is required in lemma \ref{l5}. Noting that $\sigma_k(N_2) = \dfrac{\alpha_{min}}{\alpha_0(\alpha_0 + 1)} $, we have:
$$
N \geq O\left ( \dfrac{\alpha_0^2(\alpha_0+1)^2C_2^2(\delta/36N)\cdot(2 + \sqrt{2\log(18/\delta)})^2 }{\alpha_{min}^2}\right ).
$$
In lemma \ref{l3}, we need that $C_3(\delta/144N)\dfrac{2\log(72/\sigma)}{\sqrt{N}} \leq 1$, which means:
$$
N \geq O \left ( C_3^2(\delta/144N)(2 + \sqrt{2 \log(72/\sigma)})^2 \right ).
$$
For lemma \ref{lem_tensorpower} to hold, we need the assumption that $\epsilon_{t,w} \leq C_1 \cdot \frac{\lambda_{min}}{k}$, which implies:
$$
N \geq O \left ( \dfrac{C_3^2(\delta/36N)(2+\sqrt{2\log(18/\sigma)})^2}{\sigma_k(\widetilde{O^*})^{10}}\cdot\max(\dfrac{1}{\epsilon^2},
\dfrac{k^2}{\lambda_{min}^2}) \right ).
$$
From the recovery rule $\hat{\alpha_i} = \dfrac{4\alpha_0(\alpha_0 + 1)}{(\alpha_0 + 2)^2 \hat{\lambda_i}^2}$, we have:
\begin{equation}
\begin{aligned}
|\hat{\alpha}_i - \alpha_i |
\leq \dfrac{4\alpha_0(\alpha_0 + 1) }{(\alpha_0 + 2)^2}
\left |\dfrac{1}{\hat{\lambda}_i^2} - \dfrac{1}{\lambda_i^2} \right |
\leq \dfrac{4\alpha_0(\alpha_0 + 1) }{(\alpha_0 + 2)^2\lambda_{min}^2(\lambda_{min} - 5 \epsilon)^2} \cdot 5\epsilon. \\
\end{aligned}
\end{equation}
Finally, we bound $\|\hat{\bm{v}}_i - \bm{v}_i\|$, which is a direct consequence of lemma \ref{l6} and lemma \ref{lem_tensorpower}.
We additional assume the following condition holds: $\dfrac{6E_P}{\sigma_k(\widetilde{O^*})^2} \leq \epsilon $. In fact, this condition is already satisfied
if the above requirements for $N$ hold.
Under this condition, we have:
$$
\|\hat{\bm{v}}_i - \bm{v}_i \| \leq \left ( \sigma_1(\widetilde{O^*})(\alpha_0 + 2)(\frac{7}{2} +
\frac{8}{\lambda_{min}})\right )\cdot \epsilon.\\
$$
\end{proof}
\section{Moments of Observable Variables}
\subsection{Proof to Proposition 1}
The equations on $M_2$ and $M_3$ have already been proved in \cite{speclda} and \cite{a:tensordecomp}.
Here we only give the proof to the equation on $M_y$.
In fact, all the three equations can be proved in a similar manner.
In sLDA the topic mixing vector $\bm h$ follows a Dirichlet prior distribution with parameter $\bm\alpha$.
Therefore, we have
\begin{equation}
\begin{aligned}
&\mathbb E[h_i] = \frac{\alpha_i}{\alpha_0},\\
&\mathbb E[h_ih_j] = \left\{\begin{array}{ll} \frac{\alpha_i^2}{\alpha_0(\alpha_0+1)}, &\text{if }i=j,\\
\frac{\alpha_i\alpha_j}{\alpha_0^2}, &\text{if }i\neq j\end{array}\right.,\\
&\mathbb E[h_ih_jh_k] = \left\{\begin{array}{ll} \frac{\alpha_i^3}{\alpha_0(\alpha_0+1)(\alpha_0+2)}, & \text{if }i=j=k,\\
\frac{\alpha_i^2\alpha_k}{\alpha_0^2(\alpha_0+1)}, & \text{if }i=j\neq k,\\
\frac{\alpha_i\alpha_j\alpha_k}{\alpha_0^3}, &\text{if }i\neq j, j\neq k, i\neq k\end{array}\right. .
\end{aligned}
\end{equation}
Next, note that
\begin{equation}
\begin{aligned}
&\mathbb E[y|\bm h] = \bm\eta^\top\bm h,\\
&\mathbb E[\bm x_1|\bm h] = \sum_{i=1}^k{h_i\bm\mu_i},\\
&\mathbb E[\bm x_1\otimes\bm x_2|\bm h] = \sum_{i, j=1}^k{h_ih_j\bm\mu_i\otimes\bm\mu_j},\\
&\mathbb E[y\bm x_1\otimes\bm x_2|\bm h] = \sum_{i, j, k=1}^k{h_ih_jh_k\cdot \eta_k\bm\mu_j\otimes\bm\mu_k}.
\end{aligned}
\end{equation}
Proposition 1 can then be proved easily by taking expectation over the topic mixing vector $\bm h$.
\subsection{Details of the speeding-up trick}
In this section we provide details of the trick mentioned in the main paper to speed up empirical moments computations.
First, note that the computation of $\widehat M_1$, $\widehat M_2$ and $\widehat M_y$ only requires $O(NM^2)$ time and $O(V^2)$ space.
They do not need to be accelerated in most practical applications.
This time and space complexity also applies to all terms in $\widehat M_3$ except the $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]$ term,
which requires $O(NM^3)$ time and $O(V^3)$ space if using naive implementations.
Therefore, this section is devoted to speed-up the computation of $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]$.
More precisely, as mentioned in the main paper,
what we want to compute is the whitened empirical moment $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3](\widehat W,\widehat W,\widehat W) \in
\mathbb R^{k\times k\times k}$.
Fix a document $D$ with $m$ words.
Let $T \triangleq \widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3|D]$ be the empirical tensor demanded.
By definition, we have
\begin{equation}
T_{i,j,k} = \frac{1}{m(m-1)(m-2)}\left\{\begin{array}{ll}
n_i(n_j-1)(n_k-2),& i=j=k;\\
n_i(n_i-1)n_k,& i=j,j\neq k;\\
n_in_j(n_j-1),& j=k,i\neq j;\\
n_in_j(n_i-1),& i=k,i\neq j;\\
n_in_jn_k,& \text{otherwise};\end{array}\right.
\label{eq_T}
\end{equation}
where $n_i$ is the number of occurrences of the $i$-th word in document $D$.
If $T_{i,j,k} = \frac{n_in_jn_k}{m(m-1)(m-2)}$ for all indices $i,j$ and $k$,
then we only need to compute
$$T(W,W,W) = \frac{1}{m(m-1)(m-2)}\cdot (W\bm n)^{\otimes 3},$$
where $\bm n \triangleq (n_1,n_2,\cdots,n_V)$.
This takes $O(Mk + k^3)$ computational time because $\bm n$ contains at most $M$ non-zero entries,
and the total time complexity is reduced from $O(NM^3)$ to $O(N(Mk+k^3))$.
We now consider the remaining values, where at least two indices are identical.
We first consider those values with two indices the same, for example, $i=j$.
For these indices, we need to subtract an $n_in_k$ term, as shown in Eq. (\ref{eq_T}).
That is, we need to compute the whitened tensor $\Delta(W,W,W)$, where $\Delta\in\mathbb R^{V\times V\times V}$ and
\begin{equation}
\Delta_{i,j,k} = \frac{1}{m(m-1)(m-2)}\cdot \left\{\begin{array}{ll}
n_in_k,& i=j;\\
0,& \text{otherwise}.\end{array}\right.
\end{equation}
Note that $\Delta$ can be written as $\frac{1}{m(m-1)(m-2)}\cdot A\otimes\bm n$,
where $A = \text{diag}(n_1,n_2,\cdots,n_V)$ is a $V\times V$ matrix and $\bm n = (n_1,n_2,\cdots,n_V)$ is defined previously.
As a result, $\Delta(W,W,W) = \frac{1}{m(m-1)(m-2)}\cdot (W^\top AW)\otimes \bm n$.
So the computational complexity of $\Delta(W,W,W)$ depends on how we compute $W^\top AW$.
Since $A$ is a diagonal matrix with at most $M$ non-zero entries, $W^\top AW$ can be computed in $O(Mk^2)$ operations.
Therefore, the time complexity of computing $\Delta(W,W,W)$ is $O(Mk^2)$ per document.
Finally we handle those values with three indices the same, that is, $i=j=k$.
As indicated by Eq. (\ref{eq_T}), we need to add a $\frac{2n_i}{m(m-1)(m-2)}$ term for compensation.
This can be done efficiently by first computing $\widehat{\mathbb E}(\frac{2n_i}{m(m-1)(m-2)})$
for all the documents (requiring $O(NV)$ time),
and then add them up, which takes $O(Vk^3)$ operations.
\iffalse
\section{Discussions}
\subsection{Appendix C.1. Extension to other topic recovery algorithms}
One important advantage of our proposed inference algorithm is its flexibility---the algorithm can be combined with many other LDA inference algorithms to infer supervised LDA model parameters.
More specifically, given access to any algorithm that recovers the topic distribution matrix $O$
and the prior parameter $\bm\alpha$ from i.i.d. sampled documents,
an inference algorithm for a supervised LDA model can be immediately obtained,
as shown in Algorithm \ref{alg2}.
\begin{algorithm}
\caption{sLDA parameter recovery based on an existing LDA inference algorithm $\mathcal A$. Input parameter: $\alpha_0$.}
\centering
\begin{algorithmic}[1]
\State Compute empirical moments and obtain $\widehat M_2, \widehat M_y$. Set $\widehat{\bm\eta} = 0$.
\State Find $\widehat W\in\mathbb R^{n\times k}$ such that $\widehat M_2(\widehat W, \widehat W) = I_k$.
\State Run algorithm $\mathcal A$ with observed documents and parameter $\alpha_0$. Obtain the estimated
topic distribution matrix $\widehat O=(\widehat{\bm\mu}_1,\cdots,\widehat{\bm\mu}_k)$ and prior parameter $\widehat{\bm\alpha}$.
\State Compute $\widehat{\bm v}_i = \sqrt{\frac{\widehat{\alpha}_i}{\alpha_0(\alpha_0+1)}}\widehat W^\top\widehat{\bm\mu}_i$ for each topic $i$.
\State Recover linear classifier: $\widehat\eta_i \gets \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i$.
\State \textbf{Output:} $\widehat{\bm\eta}$, $\bm\alpha$ and $\{\widehat{\bm\mu}_i\}_{i=1}^k$.
\end{algorithmic}
\label{alg2}
\end{algorithm}
The sample complexity of Algorithm \ref{alg2} depends on the sample complexity of the LDA inference algorithm $\mathcal A$.
Although the LDA inference algorithm $\mathcal A$ is free to make any assumptions on the topic distribution matrix $O$,
we comment that the linear independence of topic distribution vectors $\bm\mu$ is still required,
because the power update trick (step 5 in Algorithm 1) is valid only $W\bm\mu_i$ are orthogonal vectors.
\subsection*{Appendix C.2. Going beyond SVD}
The proposed methods are based on spectral decomposition of observable moments
and are provably correct.
However, a major drawback of these methods is their assumption that the topic distributions $\bm\mu$ are linearly independent.
Although standard LDA models assume this topic independence \cite{lda},
in practice there is strong evidence that different topics might be related \cite{correlatelda}.
Therefore, it is important to consider possible extensions.
Recently, there has been much work on provable LDA inference algorithms that do not require the topic independence assumption \cite{practicalalgo,beyondsvd}.
Instead of making the independence assumption, these methods assume a $p$-separability condition on the topic distribution matrix,
that is, there exists an \emph{anchor word} for each topic such that the anchor word only appears in this specific topic,
and its probability (given that topic) is at least $p$.
Here we briefly describe an idea that might lead to an sLDA inference algorithm without assuming topic independence.
First, let $O\in\mathbb R^{V\times k}$ be the topic distribution matrix defined previously and $H = (\bm h_1,\cdots,\bm h_N)\in\mathbb R^{k\times N}$
be the matrix of topic mixing vectors for each document.
Suppose $Q = \mathbb E[\bm x_1\bm x_2^\top]$ is the word co-occurrence frequency matrix
and $Q_y = \mathbb E[y\bm x_1]$ is the co-occurrence frequency matrix between word and regression variable observations.
It is clear that both $Q$ and $Q_y$ can be estimated from the training data, and we have
\begin{equation}
Q = ORO^\top, Q_y = OR\bm\eta,
\end{equation}
where $R \triangleq \mathbb E[HH^\top]$.
Assuming the $p$-separability condition holds and using methods developed in \cite{beyondsvd,practicalalgo},
we can consistently recover the $O$ and $R$.
Now note that $Q_y = (OR)\cdot \bm\eta$.
As a result, knowing both $OR$ and $Q_y$ we reduce the $\bm\eta$ inference problem to solving a linear equation systems.
Future research could be done on (1) determining when the matrix $OR$ has full rank $r$ so that the linear equations can be solved, and
(2) investigating the sample complexity problem for such an inference algorithm.
\subsection*{Appendix C.3. From regression to classification}
A natural extension to our result is to consider supervised LDA models for classification purposes.
The simplest model is a logistic regression classification model, where we assume the response variable $y_d$
for each document $d$ is in $\{+1,-1\}$, and
$$\Pr[y_d = 1|\bm h_d] = \frac{\exp(\bm\eta^\top\bm h_d)}{1+\exp(\bm\eta^\top\bm h_d)},$$
where $\bm\eta$ is a linear classifier.
Though appears simple, such an extension incurs many fundamental problems.
A major obstacle is the fact that the conditional expectation $\mathbb E[y|\bm h]$ is no longer linear
in the topic mixing vector $\bm h$.
As a result, we cannot even evaluate $\mathbb E[y]$ (or higher order tensors like $\mathbb E[y\bm x_1\otimes\bm x_2]$)
in closed forms.
Further more, even if we have accurate estimation to the above moments,
it is rather difficult to infer $\bm\eta$ back due to its non-linearality.
To our knowledge, the classification problem remains unsolved in terms of spectral decomposition methods,
and there is doubt whether a provably correct algorithm exists in such scenarios.
\fi
\section{Proof to Theorem 1}
In this section, we prove the sample complexity bound given in Theorem 1. The proof consists of three main parts.
In Appendix A.1, we prove perturbation lemmas that bound the estimation error of the whitened tensors
$M_2(W,W), M_y(W,W)$ and $M_3(W,W,W)$ in terms of the estimation error of the tensors themselves.
In Appendix A.2, we cite results on the accuracy of SVD and robust tensor power method when performed on estimated tensors,
and prove the effectiveness of the power update method used in recovering the linear regression model $\bm\eta$.
Finally, we give tail bounds for the estimation error of $M_2,M_y$ and $M_3$ in Appendix A.3
and complete the proof in Appendix A.4.
We also make some remarks on the indirect quantities (e.g. $\sigma_k(\widetilde O)$) used in Theorem 1
and simplified bounds for some special cases in Appendix A.4.
All norms in the following analysis, if not explicitly specified, are 2 norms in the vector and matrix cases and the operator norm in the high-order tensor case.
\subsection{Perturbation lemmas}
We first define the canonical topic distribution vectors $\widetilde{\bm\mu}$ and estimation error of observable tensors,
which simplify the notations that arise in subsequent analysis.
\begin{deff}[canonical topic distribution]
Define the canonical version of topic distribution vector $\bm\mu_i$, $\widetilde{\bm\mu}_i$, as follows:
\begin{equation}
\widetilde{\bm\mu}_i \triangleq \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}}\bm\mu_i.
\label{def_can_mu}
\end{equation}
We also define $O,\widetilde O\in\mathbb R^{n\times k}$ by $O = [\bm\mu_1,\cdots,\bm\mu_k]$ and $\widetilde{O} = [\widetilde{\bm\mu_1},\cdots,\widetilde{\bm\mu_k}]$.
\end{deff}
\begin{deff}[estimation error]
Assume
\begin{eqnarray}
\|M_2-\widehat M_2\| &\leq& E_P,\\
\|M_y-\widehat M_y\| &\leq& E_y,\\
\|M_3-\widehat M_3\| &\leq& E_T.
\end{eqnarray}
for some real values $E_P,E_y$ and $E_T$, which we will set later.
\end{deff}
The following lemma analyzes the whitening matrix $W$ of $M_2$.
Many conclusions are directly from \cite{speclda}.
\begin{lem}[Lemma C.1, \cite{speclda}]
Let $W, \widehat W\in\mathbb R^{n\times k}$ be the whitening matrices such that $M_2(W,W) = \widehat M_2(\widehat W,\widehat W) = I_k$. Let $A = W^\top\widetilde O$ and $\widehat A = \widehat W^\top\widetilde O$. Suppose $E_P \leq \sigma_k(M_2)/2$. We have
\begin{eqnarray}
\|W\| &=& \frac{1}{\sigma_k(\widetilde O)},\\
\|\widehat W\| &\leq& \frac{2}{\sigma_k(\widetilde O)},\label{eq6_pert}\\
\|W-\widehat W\| &\leq& \frac{4E_P}{\sigma_k(\widetilde O)^3},\label{eq_wwhat}\\
\|W^+\| &\leq& 3\sigma_1(\widetilde O),\\
\|\widehat W^+\| &\leq& 2\sigma_1(\widetilde O),\\
\|W^+ - \widehat W^+\| &\leq& \frac{6\sigma_1(\widetilde O)}{\sigma_k(\widetilde O)^2}E_P,\\
\|A\| &=& 1,\\
\|\widehat A\| &\leq & 2,\\
\|A - \widehat A\| &\leq& \frac{4E_P}{\sigma_k(\widetilde O)^2},\\
\|AA^\top - \widehat A\widehat A^\top\| &\leq& \frac{12E_P}{\sigma_k(\widetilde O)^2}.\label{eq4_pert}
\end{eqnarray}
\end{lem}
\begin{proof}
Proof to Eq. (\ref{eq_wwhat}):
Let $\widehat W^\top\widehat M_2\widehat W = I$ and $\widehat W^\top M_2\widehat W = BDB^\top$, where $B$ is orthogonal and $D$ is a positive definite diagonal matrix.
We then see that $W = \widehat WBD^{-1/2}B^\top$ satisfies the condition $W M_2 W^\top = I$.
Subsequently, $\widehat W = WBD^{1/2}B^\top$.
We then can bound $\|W-\widehat W\|$ as follows
$$ \|W-\widehat W\| \leq \|W\|\cdot \|I-D^{1/2}\| \leq \|W\|\cdot \|I-D\| \leq \frac{4E_P}{\sigma_k(\widetilde O)^3},$$
where the inequality $\|I-D\|\leq \frac{4E_P}{\sigma_k(\widetilde O)^2}$ was proved in \cite{speclda}.
Proof to Eq. (\ref{eq4_pert}):
$\|AA^\top-\widehat A\widehat A^\top\| \leq \|AA^\top-A\widehat A^\top\| + \|A\widehat A^\top - \widehat A\widehat A^\top\|
\leq \|A-\widehat A\|\cdot (\|A\|+\|\widehat A\|) \leq \frac{12E_P}{\sigma_k(\widetilde O)^2}$.
All the other inequalities come from Lemma C.1, \cite{speclda}.
\end{proof}
We are now able to provide perturbation bounds for estimation error of whitened moments.
\begin{deff}[estimation error of whitened moments]
Define
\begin{eqnarray}
\varepsilon_{p,w} &\triangleq& \|M_2(W, W) - \widehat M_2(\widehat W,\widehat W)\|,\\
\varepsilon_{y,w} &\triangleq& \|M_y(W, W) - \widehat M_y(\widehat W,\widehat W)\|,\\
\varepsilon_{t,w} &\triangleq& \|M_3(W,W,W) - \widehat M_3(\widehat W,\widehat W,\widehat W)\|.
\end{eqnarray}
\label{def_epsilon}
\end{deff}
\begin{lem}[Perturbation lemma of whitened moments]
Suppose $E_P\leq \sigma_k(M_2)/2$. We have
\begin{eqnarray}
\varepsilon_{p,w} &\leq& \frac{16E_P}{\sigma_k(\widetilde O)^2},\\
\varepsilon_{y,w} &\leq& \frac{24\|\bm\eta\|E_P}{(\alpha_0+2)\sigma_k(\widetilde O)^2} + \frac{4E_y}{\sigma_k(\widetilde O)^2},\\
\varepsilon_{t,w} &\leq& \frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5} + \frac{8E_T}{\sigma_k(\widetilde O)^3}.
\end{eqnarray}
\end{lem}
\begin{proof}
Using the idea in the proof of Lemma C.2 in \cite{speclda}, we can split $\varepsilon_{p,w}$ as
\begin{equation*}
\begin{aligned}
&\varepsilon_{p,w} \\
=& \|M_2(W,W) - M_2(\widehat W,\widehat W) + M_2(\widehat W,\widehat W) - \widehat M_2(\widehat W,\widehat W)\|\\
\leq& \|M_2(W,W) - M_2(\widehat W,\widehat W)\| + \|M_2(\widehat W,\widehat W) - \widehat M_2(\widehat W,\widehat W)\|.
\end{aligned}
\end{equation*}
We can the bound the two terms seperately, as follows.
For the first term, we have
\begin{eqnarray*}
\|M_2(W,W) - M_2(\widehat W,\widehat W)\|
&=& \|W^\top M_2 W-\widehat W^\top\widehat M_2\widehat W\|\\
&=& \|AA^\top - \widehat A\widehat A^\top\|\\
&\leq& \frac{12E_P}{\sigma_k(\widetilde O)^2}.
\end{eqnarray*}
where the last inequality comes from Eq. (\ref{eq4_pert})
For the second term, we have
$$\|M_2(\widehat W,\widehat W)-\widehat M_2(\widehat W,\widehat W)\| \leq \|\widehat W\|^2\cdot \|M_2-\widehat M_2\| \leq \frac{4E_P}{\sigma_k(\widetilde O)^2},$$
where the last inequality comes from Eq. (\ref{eq6_pert}).
Similarly, $\varepsilon_{y,w}$ can be splitted as $\|M_y(W,W)-M_y(\widehat W,\widehat W)\|$ and $\|M_y(\widehat W,\widehat W)-\widehat M_y(\widehat W,\widehat W)\|$, which can be bounded separately.
For the first term, we have
\begin{equation*}
\begin{aligned}
&\|M_y(W,W) - M_y(\widehat W,\widehat W)\| \\
=& \|W^\top M_y W - \widehat W^\top M_y\widehat W\|\\
=& \frac{2}{\alpha_0+2}\|A\mathrm{diag}(\bm\eta)A^\top - \widehat A\mathrm{diag}(\bm\eta)\widehat A^\top\|\\
\leq& \frac{2\|\bm\eta\|}{\alpha_0+2}\cdot \|AA^\top - \widehat A\widehat A^\top\|\\
\leq& \frac{24\|\bm\eta\|}{(\alpha_0+2)\sigma_k(\widetilde O)^2}\cdot E_P.
\end{aligned}
\end{equation*}
For the second term, we have
$$\|M_y(\widehat W,\widehat W)-\widehat M_y(\widehat W,\widehat W)\| \leq \|\widehat W\|^2\cdot\|M_y-\widehat M_y\| \leq \frac{4E_y}{\sigma_k(\widetilde O)^2}.$$
Finally, we bound $\varepsilon_{t,w}$ as below, following the work \cite{specregression}.
\begin{eqnarray*}
\varepsilon_{t,w} &=& \|M_3(W,W,W) - \widehat M_3(\widehat W,\widehat W,\widehat W)\|\\
&\leq& \|M_3\|\cdot \|W-\widehat W\|\cdot (\|W\|^2 + \|W\|\cdot \|\widehat W\|\\
&\ \ +& \|\widehat W\|^2) + \|\widehat W\|^3\cdot \|M_3-\widehat M_3\|\\
&\leq& \frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5} + \frac{8E_T}{\sigma_k(\widetilde O)^3},
\end{eqnarray*}
where we have used the fact that $$\|M_3\| \leq \sum_{i=1}^k{\frac{2\alpha_i}{\alpha_0(\alpha_0+1)(\alpha_0+2)}} = \frac{2}{(\alpha_0+1)(\alpha_0+2)}$$.
\end{proof}
\subsection{SVD accuracy}
The key idea for spectral recovery of LDA topic modeling is the \emph{simultaneous diagonalization} trick,
which asserts that we can recover LDA model parameters by performing orthogonal tensor decomposition
on a pair of simultaneously whitened moments, for example, $(M_2, M_3)$ and $(M_2, M_y)$.
The following proposition details this insight, as we derive orthogonal tensor decompositions for the whitened tensor product
$M_y(W,W)$ and $M_3(W,W,W)$.
\begin{prop}
Define $\bm v_i\triangleq W^\top \widetilde{\bm\mu}_i = \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}}W^\top\bm\mu_i$. Then
\begin{enumerate}
\item $\{\bm v_i\}_{i=1}^k$ is an orthonormal basis.
\item $M_y$ has a pair of singular value and singular vector $(\sigma_i^y, \bm v_i)$ with $\sigma_i^y = \frac{2}{\alpha_0+2}\eta_j$ for some $j \in[k]$.
\item $M_3$ has a pair of robust eigenvalue and eigenvector \cite{tensorpower} $(\lambda_i, \bm v_i)$ with $\lambda_i = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{j'}}}$ for some $j'\in[k]$.
\end{enumerate}
\end{prop}
\begin{proof}
The orthonormality of $\{\bm v_i\}_{i=1}^k$ follows from the fact that $W^\top M_2W = \sum_{i=1}^k{\bm v_i\bm v_i^\top} = I_k$.
Subsequently, we have
\begin{eqnarray*}
M_y(W, W) &=& \frac{2}{\alpha_0+2}\sum_{i=1}^k{\eta_i \bm v_i\bm v_i^\top},\\
M_3(W,W,W) &=& \frac{2}{\alpha_0+2}\sum_{i=1}^k{\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_i}}\bm v_i\otimes\bm v_i\otimes\bm v_i}.
\end{eqnarray*}
\end{proof}
The following lemmas (Lemma \ref{lem_eta} and Lemma \ref{lem_mu})
give upper bounds on the estimation error of $\bm\eta$ and $\bm\mu$
in terms of $|\widehat\lambda_i-\lambda_i|$, $|\widehat{\bm v}_i-\bm v_i|$
and the estimation errors of whitened moments defined in Definition \ref{def_epsilon}.
\begin{lem}[$\eta_i$ estimation error bound]
Define $\widehat\eta_i \triangleq \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i$,
where $\widehat{\bm v}_i$ is some estimation of $\bm v_i$.
We then have
\begin{equation}
|\eta_i - \widehat\eta_i| \leq 2\|\bm\eta\|\|\widehat{\bm v}_i-\bm v_i\| + \frac{\alpha_0+2}{2}(1+2\|\widehat{\bm v}_i-\bm v_i\|)\cdot\varepsilon_{y,w}.
\end{equation}
\label{lem_eta}
\end{lem}
\begin{proof}
First, note that $\bm v_i^\top M_y(W,W)\bm v_i = \frac{2}{\alpha_0+2}\eta_i$ because $\{\bm v_i\}_{i=1}^k$ are orthonormal.
Subsequently, we have
\begin{equation*}
\begin{aligned}
&\frac{2}{\alpha_0+2}|\eta_i - \widehat\eta_i|\\
=& \Big|\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - \bm v_i^\top M_y(W,W)\bm v_i\Big| \\
\leq& \Big| (\widehat{\bm v}_i-\bm v_i)^\top \widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i\Big| + \Big|\bm v_i^\top\left(\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\right)\Big|\\
\leq& \|\widehat{\bm v}_i-\bm v_i\|\|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i\| + \|\bm v_i\|\|\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\|.\\
\end{aligned}
\end{equation*}
Note that both $\bm v_i$ and $\widehat{\bm v}_i$ are unit vectors. Therefore,
\begin{equation*}
\begin{aligned}
&\frac{2}{\alpha_0+2}|\eta_i-\widehat\eta_i|\\
\leq& \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i-\bm v_i\| + \|\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\|\\
\leq& \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i-\bm v_i\| + \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i - \bm v_i\|\\& \ \ \ \ + \|\widehat M_y(\widehat W,\widehat W)-M_y(W,W)\|\|\bm v_i\|\\
\leq& 2\|\widehat{\bm v}_i-\bm v_i\|\left(\frac{2}{\alpha_0+2}\|\bm\eta\| + \varepsilon_{y,w}\right) + \varepsilon_{y,w}.
\end{aligned}
\end{equation*}
The last inequality is due to the fact that $\|M_y(W,W)\| = \frac{2}{\alpha_0+2}\|\bm\eta\|$.
\end{proof}
\begin{lem}[$\bm\mu_i$ estimation error bound]
Define $\widehat{\bm\mu}_i \triangleq \frac{\alpha_0+2}{2}\widehat\lambda_i(\widehat W^+)^\top\widehat{\bm v}_i$,
where $\widehat\lambda_i, \widehat{\bm v}_i$ are some estimates of singular value pairs $(\lambda_i, \bm v_i)$ of $M_3(W,W,W)$.
We then have
\begin{equation}
\begin{aligned}
\|\widehat{\bm\mu}_i - \bm\mu_i\| \leq& \frac{3(\alpha_0+2)}{2}\sigma_1(\widetilde O)|\widehat\lambda_i-\lambda_i| \\
&+ 3\alpha_{\max}\sigma_1(\widetilde O)\|\widehat{\bm v}_i-\bm v_i\|
+ \frac{6\alpha_{\max}\sigma_1(\widetilde O)E_P}{\sigma_k(\widetilde O)^2}.
\end{aligned}
\end{equation}
\label{lem_mu}
\end{lem}
\begin{proof}
First note that $\bm\mu_i = \frac{\alpha_0+2}{2}\lambda_i (W^+)^\top\bm v_i$. Subsequently,
\begin{equation*}
\begin{aligned}
&\frac{2}{\alpha_0+2}\|\bm\mu_i - \widehat{\bm\mu}_i\|\\
=& \|\widehat\lambda_i(\widehat W^+)^\top\widehat{\bm v}_i - \lambda_i(W^+)^\top\bm v_i\|\\
\leq& \|\widehat\lambda_i\widehat W^+ - \lambda_i W^+\|\|\widehat{\bm v}_i\| + \|\lambda_i W^+\|\|\widehat{\bm v}_i-\bm v_i\|\\
\leq& |\widehat\lambda_i-\lambda_i|\|\widehat W^+\| + |\lambda_i|\|\widehat W^+ - W^+\| + |\lambda_i|\|W^+\|\|\widehat{\bm v}_i-\bm v_i\|\\
\leq& 3\sigma_1(\widetilde O)|\widehat\lambda_i-\lambda_i| + \frac{2\alpha_{\max}}{\alpha_0+2}\cdot \frac{6\sigma_1(\widetilde O) E_P}{\sigma_k(\widetilde O)^2}\\
&+ \frac{2\alpha_{\max}}{\alpha_0+2}\cdot 3\sigma_1(\widetilde O)\cdot \|\widehat{\bm v_i} - \bm v_i\|.
\end{aligned}
\end{equation*}
\end{proof}
To bound the error of orthogonal tensor decomposition performed on the estimated tensors $\widehat M_3(\widehat W,\widehat W,\widehat W)$,
we cite Theorem 5.1 \cite{tensorpower}, a sample complexity analysis on the robust tensor power method we used
for recovering $\widehat\lambda_i$ and $\widehat{\bm v}_i$.
\begin{lem}[Theorem 5.1, \cite{tensorpower}]
Let $\lambda_{\max} = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$,
$\lambda_{\min} = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\max}}}$,
where $\alpha_{\min} = \min{\alpha_i}$ and $\alpha_{\max} = \max{\alpha_i}$.
Then there exist universal constants $C_1, C_2 > 0$ such that the following holds:
Fix $\delta'\in(0,1)$.
Suppose $\varepsilon_{t,w}\leq \varepsilon$ and
\begin{eqnarray}
\varepsilon_{t, w} &\leq& C_1\cdot \frac{\lambda_{\min}}{k},\label{eq1_tensorpower}
\end{eqnarray}
Suppose $\{(\widehat\lambda_i, \widehat{\bm v}_i)\}_{i=1}^k$ are eigenvalue and eigenvector pairs
returned by running Algorithm 1 in \cite{tensorpower} with input $\widehat M_3(\widehat W,\widehat W,\widehat W)$ for $L = \text{poly}(k)\log(1/\delta')$ and $N \geq C_2\cdot(\log(k) + \log\log(\frac{\lambda_{\max}}{\varepsilon}))$ iterations.
With probability greater than $1-\delta'$, there exists a permutation $\pi':[k]\to[k]$ such that
for all $i$,
$$\|\widehat{\bm v}_i - \bm v_{\pi'(i)}\| \leq 8\varepsilon/\lambda_{\min},\quad |\widehat\lambda_i - \lambda_{\pi'(i)}| \leq 5\varepsilon.$$
\label{lem_tensorpower}
\end{lem}
\subsection{Tail Inequalities}
\begin{lem}[Lemma 5, \cite{specregression}]
Let $\bm x_1,\cdots,\bm x_N\in\mathbb R^d$ be i.i.d. samples from some distribution with bounded support (i.e., $\|\bm x\|_2\leq B$ with probability 1 for some constant $B$).
Then with probability at least $1-\delta$,
$$\left\|\frac{1}{N}\sum_{i=1}^N{\bm x_i} - \mathbb E[\bm x]\right\|_2 \leq \frac{2B}{\sqrt{N}}\left(1+\sqrt{\frac{\log(1/\delta)}{2}}\right).$$
\label{lem_chernoff}
\end{lem}
\begin{cor}
Let $\bm x_1,\cdots,\bm x_N\in\mathbb R^d$ be i.i.d. samples from some distributions with $\Pr[\|\bm x\|_2 \leq B] \geq 1-\delta'$.
Then with probability at least $1-N\delta'-\delta$,
$$\left\|\frac{1}{N}\sum_{i=1}^N{\bm x_i} - \mathbb E[\bm x]\right\|_2 \leq \frac{2B}{\sqrt{N}}\left(1+\sqrt{\frac{\log(1/\delta)}{2}}\right).$$
\label{cor_chernoff}
\end{cor}
\begin{proof}
Use union bound.
\end{proof}
\begin{lem}[concentraion of moment norms]
Suppose we obtain $N$ i.i.d. samples (i.e., documents with at least three words each and their regression variables in sLDA models).
Define $R(\delta) \triangleq \|\bm\eta\|+ \Phi^{-1}(\delta)$,
where $\Phi^{-1}(\cdot)$ is the inverse function of the CDF of a standard Gaussian distribution.
Let $\mathbb E[\cdot]$ denote the mean of the true underlying distribution
and $\widehat{\mathbb E}[\cdot]$ denote the empirical mean.
Then
\begin{enumerate}
\item $\Pr\left[\|\mathbb E[\bm x_1] - \widehat{\mathbb E}[\bm x_1]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[\bm x_1\otimes\bm x_2\otimes\bm x_3] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y] - \widehat{\mathbb E}[y]\| < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}} \right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y\bm x_1] - \widehat{\mathbb E}[y\bm x_1]\|_F < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[y\bm x_1\otimes\bm x_2]\|_F < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}}\right] \geq 1 - \delta$.
\end{enumerate}
\label{lem_tail_ineq}
\end{lem}
\begin{proof}
Use Lemma \ref{lem_chernoff} and Corrolary \ref{cor_chernoff} for concentration bounds involving the regression variable $y$.
\end{proof}
\begin{cor}
With probability $1-\delta$ the following holds:
\begin{enumerate}
\item $E_P = \|M_2-\widehat M_2\| \leq 3\cdot \frac{2+\sqrt{2\log(6/\delta)}}{\sqrt{N}}$.
\item $E_y = \|M_y-\widehat M_y\| \leq 10R(\delta/60\sigma N)\cdot \frac{2+\sqrt{2\log(15/\delta)}}{\sqrt{N}}$.
\item $E_T = \|M_3-\widehat M_3\| \leq 10\cdot \frac{2+\sqrt{2\log(9/\delta)}}{\sqrt{N}}$.
\end{enumerate}
\label{cor_ep}
\end{cor}
\begin{proof}
Corrolary \ref{cor_ep} can be proved by expanding the terms by definition and then using tail inequality in Lemma \ref{lem_tail_ineq} and union bound.
Also note that $\|\cdot\| \leq \|\cdot\|_F$ for all matrices.
\end{proof}
\subsection{Completing the proof}
We are now ready to give a complete proof to Theorem \ref{thm_main}.
\begin{thm}[Sample complexity bound]
Let $\sigma_1(\widetilde O)$ and $\sigma_k(\widetilde O)$ be the largest and the smallest singular values of the canonical topic distribution matrix $\widetilde O$.
Define $\lambda_{\min} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\max}}}$
and $\lambda_{\max} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$,
where $\alpha_{\max}$ and $\alpha_{\min}$ are the largest and the smallest entries in $\bm\alpha$.
Suppose $\widehat{\bm\mu}$, $\widehat{\bm\alpha}$ and $\widehat{\bm\eta}$ are the outputs of Algorithm 1,
and $L$ is at least a linear function in $k$.
Fix $\delta\in(0,1)$.
For any small error-tolerance parameter $\varepsilon > 0$,
if Algorithm 1 is run with parameter $T = \Omega(\log(k) + \log\log(\lambda_{\max}/\varepsilon))$
on $N$ i.i.d. sampled documents with
$N\geq \max(n_1, n_2, n_3)$, where
\begin{eqnarray}
n_1 &=& C_1\cdot \left(1+\sqrt{\log(6/\delta)}\right)^2\cdot \frac{\alpha_0^2(\alpha_0+1)^2}{\alpha_{\min}}, \label{eq_n1} \\
n_2 &=& C_2\cdot \frac{(1+\sqrt{\log(15/\delta)})^2}{\varepsilon^2\sigma_k(\widetilde O)^4}
\\
&\ &\cdot\max\left((\|\bm\eta\| + \Phi^{-1}(\delta/60\sigma))^2, \alpha_{\max}^2\sigma_1(\widetilde O)^2\right),
\label{eq_n2} \\
n_3 &=& C_3\cdot \frac{(1+\sqrt{\log(9/\delta)})^2}{\sigma_k(\widetilde O)^{10}}\cdot\max\left(\frac{1}{\varepsilon^2}, \frac{k^2}{\lambda_{\min}^2}\right),
\label{eq_n3}
\end{eqnarray}
and $C_1, C_2$ and $C_3$ are universal constants,
then with probability at least $1-\delta$, there exists a permutation $\pi:[k]\to [k]$ such that for every topic $i$, the following holds:\\
\begin{enumerate}
\item $|\alpha_i-\widehat{\alpha}_{\pi(i)}| \leq \frac{4\alpha_0(\alpha_0+1)(\lambda_{\max}+5\varepsilon)}{(\alpha_0+2)^2\lambda_{\min}^2(\lambda_{\min}-5\varepsilon)^2}\cdot 5\varepsilon$, if $\lambda_{\min} > 5\varepsilon$\\;
\item $\|\bm\mu_i - \widehat{\bm\mu}_{\pi(i)}\| \leq \left(3\sigma_1(\widetilde O)\left(\frac{8\alpha_{\max}}{\lambda_{\min}} + \frac{5(\alpha_0+2)}{2}\right) + 1\right)\varepsilon$\\;
\item $|\eta_i - \widehat\eta_{\pi(i)}| \leq \left(\frac{\|\bm\eta\|}{\lambda_{\min}} + (\alpha_0+2)\right) \varepsilon$.
\end{enumerate}
\label{thm_main}
\end{thm}
\begin{proof}
First, the assumption $E_P\leq\sigma_k(M_2)$ is required for error bounds on $\varepsilon_{p,w}, \varepsilon_{y,w}$ and $\varepsilon_{t,w}$.
Noting Corrolary \ref{cor_ep} and the fact that $\sigma_k(M_2) = \frac{\alpha_{\min}}{\alpha_0(\alpha_0+1)}$, we have
$$N = \Omega\left(\frac{\alpha_0^2(\alpha_0+1)^2(1+\sqrt{\log(6/\delta)})^2}{\alpha_{\min}^2}\right).$$
Note that this lower bound does not depend on $k$, $\varepsilon$ and $\sigma_k(\widetilde O)$.
For Lemma \ref{lem_tensorpower} to hold, we need the assumptions that
$\varepsilon_{t,w} \leq \min(\varepsilon, O(\frac{\lambda_{\min}}{k}))$.
These imply Eq. (\ref{eq_n3}),
as we expand $\varepsilon_{t,w}$ according to Definition \ref{def_epsilon}
and note the fact that the first term $\frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5}$ dominates the second one.
The $\alpha_0$ is missing in Eq. (\ref{eq_n3}) because $\alpha_0+1 \geq 1$, $\alpha_0+2 \geq 2$ and we discard them both.
The $|\alpha_i-\widehat{\alpha}_{\pi(i)}|$ bound follows immediately by Lemma \ref{lem_tensorpower}
and the recovery rule $\widehat{\alpha}_i = \frac{\alpha_0+2}{2}\widehat\lambda_i$.
To bound the estimation error for the linear classifier $\bm\eta$, we need to further bound $\varepsilon_{y,w}$.
We assume $\varepsilon_{y,w}\leq\varepsilon$.
By expanding $\varepsilon_{y,w}$ according to Definition \ref{def_epsilon} in a similar manner
we obtain the $(\|\bm\eta\|+\Phi^{-1}(\delta/60\sigma))^2$ term in Eq. (\ref{eq_n2}).
The bound on $|\eta_i-\widehat\eta_{\pi(i)}|$ follows immediately by Lemma \ref{lem_eta}.
Finally, we bound $\|\bm\mu_i-\widehat{\bm\mu}_{\pi(i)}\|$ using Lemma \ref{lem_mu}.
We need to assume that $\frac{6\alpha_{\max}\sigma_1(\widetilde O)E_P}{\sigma_k(\widetilde O)^2} \leq \varepsilon$,
which gives the $\alpha_{\max}^2\sigma_1(\widetilde O)^2$ term in Eq. (\ref{eq_n2}).
The $\|\bm\mu_i-\widehat{\bm\mu}_{\pi(i)}\|$ bound then follows by Lemma \ref{lem_mu} and Lemma \ref{lem_tensorpower}.
\end{proof}
We make some remarks for the main theorem. In Remark~\ref{rem_connection}, we establish links between indirect quantities appeared in Theorem~\ref{thm_main} (e.g., $\sigma_k(\widetilde O)$) and the functions of original model parameters (e.g., $\sigma_k(O)$).
These connections are straightforward following their definitions.
\begin{rem}
The indirect quantities $\sigma_1(\widetilde O)$ and $\sigma_k(\widetilde O)$ can be related to $\sigma_1(O)$, $\sigma_k(O)$ and $\bm\alpha$ in the following way:
{\small \begin{equation*}
\sqrt{\frac{\alpha_{\min}}{\alpha_0(\alpha_0+1)}}\sigma_k(O) \leq \sigma_k(\widetilde O)\leq \sqrt{\frac{\alpha_{\max}}{\alpha_0(\alpha_0+1)}}\sigma_k(O);~~~~
\end{equation*}
\begin{equation*}
\sigma_1(\widetilde O) \leq \sqrt{\frac{\alpha_{\max}}{\alpha_0(\alpha_0+1)}}\sigma_1(O) \leq \frac{1}{\sqrt{\alpha_0+1}}.
\end{equation*}}
\label{rem_connection}
\end{rem}
We now take a close look at the sample complexity bound in Theorem~\ref{thm_main}.
It is evident that $n_2$ can be neglected when the number of topics $k$ gets large,
because in practice the norm of the linear regression model $\bm\eta$ is usually assumed to be small in order to avoid overfitting.
Moreover, as mentioned before, the prior parameter $\bm\alpha$ is often assumed to be homogeneous with $\alpha_i = 1/k$ \cite{lsa}.
With these observations, the sample complexity bound in Theorem~\ref{thm_main}
can be greatly simplified.
\begin{rem}
Assume $\|\bm\eta\|$ and $\sigma$ are small and $\bm\alpha = (1/k, \cdots, 1/k)$.
As the number of topics $k$ gets large, the sample complexity bound in Theorem~\ref{thm_main} can be simplified as
\begin{equation}
N = \Omega\left( \frac{\log(1/\delta)}{\sigma_k(\widetilde O)^{10}}\cdot\max(\varepsilon^{-2}, k^3)\right).
\end{equation}\label{rem_simpbound}
\end{rem}\vspace{-.5cm}
The sample complexity bound in Remark~\ref{rem_simpbound} may look formidable
as it depends on $\sigma_k(\widetilde O)^{10}$.
However, such dependency is somewhat necessary because we are using third-order tensors to recover the underlying model parameters.
Furthermore, the dependence on $\sigma_k(\widetilde O)^{10}$ is introduced by the robust tensor power method to recover LDA parameters,
and the reconstruction accuracy of $\bm\eta$ only depends on $\sigma_k(\widetilde O)^4$ and $(\|\bm\eta\|+\Phi^{-1}(\delta/60\sigma))^2$.
As a consequence, if we can combine our power update method for $\bm \eta$ with LDA inference algorithms that have milder dependence on the singular value $\sigma_k(\widetilde O)$,
we might be able to get an algorithm with a better sample complexity.
Such an extension is discussed in Appendix~C.1.
\section{Moments of Observable Variables}
\subsection{Proof to Proposition 1}
The equations on $M_2$ and $M_3$ have already been proved in \cite{speclda} and \cite{tensorpower}.
Here we only give the proof to the equation on $M_y$.
In fact, all the three equations can be proved in a similar manner.
In sLDA the topic mixing vector $\bm h$ follows a Dirichlet prior distribution with parameter $\bm\alpha$.
Therefore, we have
\begin{equation}
\begin{aligned}
&\mathbb E[h_i] = \frac{\alpha_i}{\alpha_0},\\
&\mathbb E[h_ih_j] = \left\{\begin{array}{ll} \frac{\alpha_i^2}{\alpha_0(\alpha_0+1)}, &\text{if }i=j,\\
\frac{\alpha_i\alpha_j}{\alpha_0^2}, &\text{if }i\neq j\end{array}\right.,\\
&\mathbb E[h_ih_jh_k] = \left\{\begin{array}{ll} \frac{\alpha_i^3}{\alpha_0(\alpha_0+1)(\alpha_0+2)}, & \text{if }i=j=k,\\
\frac{\alpha_i^2\alpha_k}{\alpha_0^2(\alpha_0+1)}, & \text{if }i=j\neq k,\\
\frac{\alpha_i\alpha_j\alpha_k}{\alpha_0^3}, &\text{if }i\neq j, j\neq k, i\neq k\end{array}\right.
\end{aligned}
\end{equation}
Next, note that
\begin{equation}
\begin{aligned}
&\mathbb E[y|\bm h] = \bm\eta^\top\bm h,\\
&\mathbb E[\bm x_1|\bm h] = \sum_{i=1}^k{h_i\bm\mu_i},\\
&\mathbb E[\bm x_1\otimes\bm x_2|\bm h] = \sum_{i, j=1}^k{h_ih_j\bm\mu_i\otimes\bm\mu_j},\\
&\mathbb E[y\bm x_1\otimes\bm x_2|\bm h] = \sum_{i, j, k=1}^k{h_ih_jh_k\cdot \eta_k\bm\mu_j\otimes\bm\mu_k}.
\end{aligned}
\end{equation}
Proposition 1 can then be proved easily by taking expectation over the topic mixing vector $\bm h$.
\subsection{Details of the speeding-up trick}
In this section we provide details of the trick mentioned in the main paper to speed up empirical moments computations.
First, note that the computation of $\widehat M_1$, $\widehat M_2$ and $\widehat M_y$ only requires $O(NM^2)$ time and $O(V^2)$ space.
They do not need to be accelerated in most practical applications.
This time and space complexity also applies to all terms in $\widehat M_3$ except the $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]$ term,
which requires $O(NM^3)$ time and $O(V^3)$ space if using naive implementations.
Therefore, this section is devoted to speed-up the computation of $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]$.
More precisely, as mentioned in the main paper,
what we want to compute is the whitened empirical moment $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3](\widehat W,\widehat W,\widehat W) \in
\mathbb R^{k\times k\times k}$.
Fix a document $D$ with $m$ words.
Let $T \triangleq \widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3|D]$ be the empirical tensor demanded.
By definition, we have
\begin{equation}
T_{i,j,k} = \frac{1}{m(m-1)(m-2)}\left\{\begin{array}{ll}
n_i(n_j-1)(n_k-2),& i=j=k;\\
n_i(n_i-1)n_k,& i=j,j\neq k;\\
n_in_j(n_j-1),& j=k,i\neq j;\\
n_in_j(n_i-1),& i=k,i\neq j;\\
n_in_jn_k,& \text{otherwise};\end{array}\right.
\label{eq_T}
\end{equation}
where $n_i$ is the number of occurrences of the $i$-th word in document $D$.
If $T_{i,j,k} = \frac{n_in_jn_k}{m(m-1)(m-2)}$ for all indices $i,j$ and $k$,
then we only need to compute
$$T(W,W,W) = \frac{1}{m(m-1)(m-2)}\cdot (W\bm n)^{\otimes 3},$$
where $\bm n \triangleq (n_1,n_2,\cdots,n_V)$.
This takes $O(Mk + k^3)$ computational time because $\bm n$ contains at most $M$ non-zero entries,
and the total time complexity is reduced from $O(NM^3)$ to $O(N(Mk+k^3))$.
We now consider the remaining values, where at least two indices are identical.
We first consider those values with two indices the same, for example, $i=j$.
For these indices, we need to subtract an $n_in_k$ term, as shown in Eq. (\ref{eq_T}).
That is, we need to compute the whitened tensor $\Delta(W,W,W)$, where $\Delta\in\mathbb R^{V\times V\times V}$ and
\begin{equation}
\Delta_{i,j,k} = \frac{1}{m(m-1)(m-2)}\cdot \left\{\begin{array}{ll}
n_in_k,& i=j;\\
0,& \text{otherwise}.\end{array}\right.
\end{equation}
Note that $\Delta$ can be written as $\frac{1}{m(m-1)(m-2)}\cdot A\otimes\bm n$,
where $A = \text{diag}(n_1,n_2,\cdots,n_V)$ is a $V\times V$ matrix and $\bm n = (n_1,n_2,\cdots,n_V)$ is defined previously.
As a result, $\Delta(W,W,W) = \frac{1}{m(m-1)(m-2)}\cdot (W^\top AW)\otimes \bm n$.
So the computational complexity of $\Delta(W,W,W)$ depends on how we compute $W^\top AW$.
Since $A$ is a diagonal matrix with at most $M$ non-zero entries, $W^\top AW$ can be computed in $O(Mk^2)$ operations.
Therefore, the time complexity of computing $\Delta(W,W,W)$ is $O(Mk^2)$ per document.
Finally we handle those values with three indices the same, that is, $i=j=k$.
As indicated by Eq. (\ref{eq_T}), we need to add a $\frac{2n_i}{m(m-1)(m-2)}$ term for compensation.
This can be done efficiently by first computing $\widehat{\mathbb E}(\frac{2n_i}{m(m-1)(m-2)})$
for all the documents (requiring $O(NV)$ time),
and then add them up, which takes $O(Vk^3)$ operations.
\iffalse
\section{Discussions}
\subsection{Appendix C.1. Extension to other topic recovery algorithms}
One important advantage of our proposed inference algorithm is its flexibility---the algorithm can be combined with many other LDA inference algorithms to infer supervised LDA model parameters.
More specifically, given access to any algorithm that recovers the topic distribution matrix $O$
and the prior parameter $\bm\alpha$ from i.i.d. sampled documents,
an inference algorithm for a supervised LDA model can be immediately obtained,
as shown in Algorithm \ref{alg2}.
\begin{algorithm}
\caption{sLDA parameter recovery based on an existing LDA inference algorithm $\mathcal A$. Input parameter: $\alpha_0$.}
\centering
\begin{algorithmic}[1]
\State Compute empirical moments and obtain $\widehat M_2, \widehat M_y$. Set $\widehat{\bm\eta} = 0$.
\State Find $\widehat W\in\mathbb R^{n\times k}$ such that $\widehat M_2(\widehat W, \widehat W) = I_k$.
\State Run algorithm $\mathcal A$ with observed documents and parameter $\alpha_0$. Obtain the estimated
topic distribution matrix $\widehat O=(\widehat{\bm\mu}_1,\cdots,\widehat{\bm\mu}_k)$ and prior parameter $\widehat{\bm\alpha}$.
\State Compute $\widehat{\bm v}_i = \sqrt{\frac{\widehat{\alpha}_i}{\alpha_0(\alpha_0+1)}}\widehat W^\top\widehat{\bm\mu}_i$ for each topic $i$.
\State Recover linear classifier: $\widehat\eta_i \gets \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i$.
\State \textbf{Output:} $\widehat{\bm\eta}$, $\bm\alpha$ and $\{\widehat{\bm\mu}_i\}_{i=1}^k$.
\end{algorithmic}
\label{alg2}
\end{algorithm}
The sample complexity of Algorithm \ref{alg2} depends on the sample complexity of the LDA inference algorithm $\mathcal A$.
Although the LDA inference algorithm $\mathcal A$ is free to make any assumptions on the topic distribution matrix $O$,
we comment that the linear independence of topic distribution vectors $\bm\mu$ is still required,
because the power update trick (step 5 in Algorithm 1) is valid only $W\bm\mu_i$ are orthogonal vectors.
\subsection*{Appendix C.2. Going beyond SVD}
The proposed methods are based on spectral decomposition of observable moments
and are provably correct.
However, a major drawback of these methods is their assumption that the topic distributions $\bm\mu$ are linearly independent.
Although standard LDA models assume this topic independence \cite{lda},
in practice there is strong evidence that different topics might be related \cite{correlatelda}.
Therefore, it is important to consider possible extensions.
Recently, there has been much work on provable LDA inference algorithms that do not require the topic independence assumption \cite{practicalalgo,beyondsvd}.
Instead of making the independence assumption, these methods assume a $p$-separability condition on the topic distribution matrix,
that is, there exists an \emph{anchor word} for each topic such that the anchor word only appears in this specific topic,
and its probability (given that topic) is at least $p$.
Here we briefly describe an idea that might lead to an sLDA inference algorithm without assuming topic independence.
First, let $O\in\mathbb R^{V\times k}$ be the topic distribution matrix defined previously and $H = (\bm h_1,\cdots,\bm h_N)\in\mathbb R^{k\times N}$
be the matrix of topic mixing vectors for each document.
Suppose $Q = \mathbb E[\bm x_1\bm x_2^\top]$ is the word co-occurrence frequency matrix
and $Q_y = \mathbb E[y\bm x_1]$ is the co-occurrence frequency matrix between word and regression variable observations.
It is clear that both $Q$ and $Q_y$ can be estimated from the training data, and we have
\begin{equation}
Q = ORO^\top, Q_y = OR\bm\eta,
\end{equation}
where $R \triangleq \mathbb E[HH^\top]$.
Assuming the $p$-separability condition holds and using methods developed in \cite{beyondsvd,practicalalgo},
we can consistently recover the $O$ and $R$.
Now note that $Q_y = (OR)\cdot \bm\eta$.
As a result, knowing both $OR$ and $Q_y$ we reduce the $\bm\eta$ inference problem to solving a linear equation systems.
Future research could be done on (1) determining when the matrix $OR$ has full rank $r$ so that the linear equations can be solved, and
(2) investigating the sample complexity problem for such an inference algorithm.
\subsection*{Appendix C.3. From regression to classification}
A natural extension to our result is to consider supervised LDA models for classification purposes.
The simplest model is a logistic regression classification model, where we assume the response variable $y_d$
for each document $d$ is in $\{+1,-1\}$, and
$$\Pr[y_d = 1|\bm h_d] = \frac{\exp(\bm\eta^\top\bm h_d)}{1+\exp(\bm\eta^\top\bm h_d)},$$
where $\bm\eta$ is a linear classifier.
Though appears simple, such an extension incurs many fundamental problems.
A major obstacle is the fact that the conditional expectation $\mathbb E[y|\bm h]$ is no longer linear
in the topic mixing vector $\bm h$.
As a result, we cannot even evaluate $\mathbb E[y]$ (or higher order tensors like $\mathbb E[y\bm x_1\otimes\bm x_2]$)
in closed forms.
Further more, even if we have accurate estimation to the above moments,
it is rather difficult to infer $\bm\eta$ back due to its non-linearality.
To our knowledge, the classification problem remains unsolved in terms of spectral decomposition methods,
and there is doubt whether a provably correct algorithm exists in such scenarios.
\fi
\section{Introduction}
\IEEEPARstart{T}{opic} modeling offers a suite of useful tools that automatically learn the latent semantic structure of a large collection of documents or images, with
latent Dirichlet allocation (LDA)~\cite{lda} as one of the most popular examples.
The vanilla LDA is an unsupervised model built on input contents of documents or images. In many applications side information is often available apart from raw contents, e.g., user-provided rating scores of an online review text or user-generated tags for an image. Such side signal usually provides additional information to reveal the underlying structures of the data in study. There have been extensive studies on developing topic models that incorporate various side information, e.g., by treating it as supervision. Some representative models are supervised LDA (sLDA) \cite{slda} that captures a real-valued regression response for each document, multiclass sLDA \cite{multislda} that learns with discrete classification responses, discriminative LDA (DiscLDA) \cite{disclda} that incorporates classification response via discriminative linear transformations on topic mixing vectors, and MedLDA \cite{medlda,gibbsmedlda} that employs a max-margin criterion to learn discriminative latent topic representations for accurate prediction.
Topic models are typically learned by finding maximum likelihood estimates (MLE) through local search or sampling methods \cite{variationallda,gibbslda,emlda},
which may get trapped in local optima. Much recent progress has been made on developing spectral decomposition~\cite{a:twosvd,a:tensordecomp,spechmm} and nonnegative matrix factorization (NMF) \cite{practicalalgo,nmf1,beyondsvd,nmf2} methods to estimate the topic-word distributions. Instead of finding MLE estimates, which is a known NP-hard problem \cite{beyondsvd}, these methods assume that the documents are i.i.d. sampled from a topic model, and attempt to recover the underlying model parameters. Compared to local search and sampling algorithms, these methods enjoy the advantage of being provably effective. In fact, sample complexity bounds have been proved to show that given a sufficiently large collection of documents, these algorithms can recover the model parameters accurately with a high probability.
Recently, some attention has been paid on supervised topic models with NMF methods. For example, Nguyen et al.~\cite{a:sAnchor} present an extension of the anchor-word methods~\cite{practicalalgo} for LDA to capture categorical information in a supervised LDA for sentiment classification.
However, for spectral methods, previous work has mainly focused on unsupervised latent variable models, leaving the broad family of supervised models (e.g., sLDA) largely unexplored.
The only exception
is~\cite{specregression} which presents a spectral method for mixtures of regression models, quite different from sLDA.
Such ignorance is not a coincidence as supervised models impose new technical challenges. For instance, a direct application of previous techniques~\cite{a:twosvd,a:tensordecomp} on sLDA
cannot handle regression models with duplicate entries.
In addition, the sample complexity bound gets much worse if we try to match entries in regression models
with their corresponding topic vectors.
In this paper, we extend the applicability of spectral learning methods by presenting novel spectral decomposition algorithms to recover the parameters of sLDA models from low-order empirical moments estimated from the data. We present two variants of spectral methods.
The first algorithm is an extension of the spectral methods for LDA,
with an extra power update step of recovering the regression model in sLDA, including the variance parameter.
The power-update step uses a newly designed empirical moment to recover regression model entries directly from the data and reconstructed topic distributions.
It is free from making any constraints on the underlying regression model.
We provide a sample complexity bound and analyze the identifiability conditions.
In fact, the two-stage method does not increase the sample complexity much compared to that of the vanilla LDA.
However, the two-stage algorithm could have some disadvantages because of its separation that the topic distribution matrix is recovered in an unsupervised manner without considering
supervision and the regression parameters are recovered by assuming a fixed topic matrix.
Such an unwarranted separation often leads to inferior performance compared to Gibbs sampling methods in practice (See Section~\ref{sec:exp}).
To address this problem, we further present a novel single-phase spectral method for supervised topic models,
which jointly recovers all the model parameters (except the noise variance) by doing a single-step of robust tensor decomposition of a newly designed
empirical moment that takes both input data and supervision signal into consideration.
Therefore, the joint method can use supervision information in recovering both the topic distribution matrix and
regression parameters. The joint method is also provably correct and we provide a sample complexity bound to achieve the $\epsilon$
error rate in a high probability.
Finally, we provide thorough experiments on both synthetic and real datasets to demonstrate the practical effectiveness of our spectral methods.
For the two-stage method, by combining with a Gibbs sampling procedure, we show superior performance in terms of language modeling, prediction accuracy and running time
compared to traditional inference algorithms.
Furthermore, we demonstrate that on a large-scale review rating dataset
our single-phase method alone can achieve comparable or even better results than the state-of-the-art methods (e.g., sLDA with Gibbs sampling and MedLDA).
Such promising results are significant to the literature of spectral methods, which were often observed to
be inferior to the MLE-based method
; and a common heuristic was to use the outputs
of a spectral method to initialize an EM algorithm, which sometimes improves the performance~\cite{a:meetem}.
\iffalse
Generally speaking, spectral methods are not better than MLE-based methods \cite{a:tensordecomp}
although a combination (e.g. use the output of spectral method as initialization) of them sometimes improves
the performance significantly \cite{a:yiningwang}, \cite{a:meetem}. However, empirical studies show that
even using our method alone, we can achieve comparable or superior performance in large-scale dataset, which demonstrate the effectiveness of our method.
\fi
The rest of the paper is organized as follows. Section $2$ reviews basics of supervised topic models. Section $3$ introduces the background knowledge and notations for sLDA and high-order tensor decomposition.
Section $4$ presents the two-stage spectral method, with a rigorous theoretical analysis, and Section $5$ presents the joint spectral method together with a sample complexity bound.
Section $6$ presents some implementation details to scale up the computation.
Section $7$ presents experimental results on both synthetic and real datasets.
Finally, we conclude in Section $8$.
\section{Related Work}
Based on different principles, there are various methods to learn supervised topic models. The most natural one is maximum-likelihood estimation (MLE). However, the
highly non-convex property of the learning objective makes the optimization problem very hard. In the original paper \cite{slda}, where the MLE is used, the authors choose variational
approximation to handle intractable posterior expectations. Such a method tries to maximize a lower bound which is built on some variational distributions and a mean-field assumption is usually imposed for tractability. Although the method works well in practice, we do not have any guarantee that the distribution we learned is close to the true one. Under Bayesian framework, Gibbs sampling is an attractive method which enjoys the property that the stationary distribution of the chain is the target posterior distribution. However, this does not mean that we can really get accurate samples from posterior distribution in practice. The slow mixing rate often makes the sampler trapped in a local minimum which is far from the
true distribution if we only run a finite number of iterations.
Max-margin learning is another principle on learning supervised topic models, with maximum entropy discrimination LDA (MedLDA)~\cite{medlda} as a popular example. MedLDA explores the max-margin principle to learn sparse and discriminative topic representations. The learning problem can be defined under the regularized Bayesian inference (RegBayes)~\cite{regBayes} framework, where the max-margin posterior regularization is introduced to ensure that the topic representations are good at predicting response variables.
Though both carefully designed variational inference~\cite{medlda} and Gibbs sampling methods~\cite{gibbsmedlda} are given, we still cannot guarantee the quality of the learnt model in general.
Recently, increasing efforts have been made to recover the parameters directly with provable correctness for topic models, with the main focus on unsupervised models such as LDA.
Such methods adopt either NMF or spectral decomposition approaches.
For NMF, the basic idea is that even NMF is an NP-hard problem in general, the topic distribution matrix can be recovered under
some separable condition, e.g., each topic has at least one anchor word.
Precisely, for each topic, the method first finds an anchor word that has non-zero probability only in that topic.
Then a recovery step reconstructs the topic distribution given such
anchor words and a second-order moment matrix of word-word co-occurrence~\cite{beyondsvd,practicalalgo}.
The original reconstruction step only needs a part of the matrix which is not robust in practice.
Thus in \cite{practicalalgo}, the author recovers the topic distribution based on a probabilistic framework.
The NMF methods produce good empirical results on real-world data.
Recently, the work \cite{a:sAnchor} extends the anchor word methods to handle supervised topic models. The method
augments the word co-occurrence matrix with additional dimensions for metadata such as sentiment, and
shows better performance in sentiment classification.
Spectral methods start from computing some low-order moments based on the samples and then relate them with the model parameters.
For LDA, tensors up to three order are sufficient to recover its parameters~\cite{a:twosvd}.
After centralization, the moments can be
expressed as a mixture of the parameters we are interested in.
After that whitening and robust tensor decomposition steps are adopted
to recover model parameters. The whitening step makes that the third-order tensor can be decomposed as a set of
orthogonal eigenvectors and their corresponding eigenvalues after some operations based on its output
and the robust tensor decomposition step then finds them.
Previous work only focuses on unsupervised LDA models and
we aim to extend the ability for spectral methods to handle response variables. Finally,
some preliminary results of the two-stage recovery algorithm have been reported in~\cite{a:yiningwang}.
This paper presents a systematical analysis with a novel one-stage spectral method,
which yields promising results on a large-scale dataset.
\vspace{-.15cm}
\section{Preliminaries}
We first overview the basics of sLDA, orthogonal tensor decomposition and the notations to be used.
\vspace{-.15cm}
\subsection{Supervised LDA}\label{sec:sLDA-model}
Latent Dirichlet allocation (LDA)~\cite{lda} is a hierarchical generative model for topic modeling of text documents or images represented in a bag-of-visual-words format~\cite{ldaimage}. It assumes $k$ different \emph{topics} with topic-word distributions $\bm\mu_1,\cdots,\bm\mu_k\in\Delta^{V-1}$, where $V$ is the vocabulary size and $\Delta^{V-1}$ denotes the probability simplex of a $V$-dimensional random vector.
For a document, LDA models a
topic mixing vector $\bm h\in\Delta^{k-1}$
as a probability distribution over the $k$ topics. A conjugate Dirichlet prior with parameter $\bm \alpha$ is imposed on the topic mixing vectors. A bag-of-words model is then adopted, which first generates a topic indicator for each word $z_j \sim \textrm{Multi}(\bm h)$ and then generates the word itself as $w_j \sim \textrm{Multi}(\bm \mu_{z_j})$.
Supervised latent Dirichlet allocation (sLDA)~\cite{slda} incorporates an extra response variable $y\in\mathbb R$ for each document.
The response variable is modeled by a linear regression model $\bm\eta\in\mathbb R^k$
on either the topic mixing vector $\bm h$ or the averaging topic assignment vector $\bar{\bm z}$,
where $\bar z_i = \frac{1}{M}\sum_{j=1}^M {1_{[z_j=i]}}$ with $M$ being the number of words in the document and $1_{[\cdot]}$ being the indicator function (i.e., equals to 1 if the predicate holds; otherwise 0).
The noise is assumed to be Gaussian with zero mean and $\sigma^2$ variance.
Fig.~\ref{slda} shows the graph structure of the sLDA model using $\bm h$ for regression.
Although previous work has mainly focused on the model using averaging topic assignment vector $\bar{\bm z}$, which is convenient for collapsed Gibbs sampling and variational inference with $\bm h$ integrated out due to the conjugacy between a Dirichlet prior and a multinomial likelihood,
we consider using the topic mixing vector $\bm h$ as the features for regression because it will considerably simplify our spectral algorithm and analysis.
One may assume that whenever a document is not too short, the empirical distribution of its word topic assignments should be close to the document's topic mixing vector.
Such a scheme was adopted to learn sparse topic coding models~\cite{Zhu:stc11}, and has demonstrated promising results in practice.
Our results also prove that this is an effective strategy.
\iftrue
\begin{figure}[t]\vspace{-.1cm}
\centering
\begin{tikzpicture}
\tikzstyle{main}=[circle, minimum size = 5mm, thick, draw =black!80, node distance = 6mm]
\tikzstyle{connect}=[-latex, thick]
\tikzstyle{box}=[rectangle, draw=black!100]
\node[main, fill = white!100] (alpha) [label=center:$\alpha$] { };
\node[main] (h) [right=of alpha,label=center:h] { };
\node[main] (z) [right=of h,label=center:z] {};
\node[main] (mu) [above=of z,label=center:$\bm{\mu}$] { };
\node[main, fill = black!10] (x) [right=of z,label=center:x] { };
\node[main, fill = black!10] (y) [below= of h, label=center:y]{ };
\node[main] (eta) [below=of alpha, label=center:$\bm{\eta}$]{};
\node (exp) [below= of z]{$y = \bm{\eta}^{\top}\bm{h} + \epsilon $};
\path (alpha) edge [connect] (h)
(h) edge [connect] (z)
(z) edge [connect] (x)
(mu) edge [connect] (x)
(h) edge [connect] (y)
(eta) edge [connect] (y);
\node[rectangle, inner sep=-0.9mm, fit= (z) (x),label=below right:M, xshift=5mm] {};
\node[rectangle, inner sep=3.6mm,draw=black!100, fit= (z) (x)] {};
\node[rectangle, inner sep=1mm, fit= (z) (x) (y) ,label=below right:N, xshift=5mm] {};
\node[rectangle, inner sep=5mm, draw=black!100, fit = (h) (z) (x) (y) ] {};
\end{tikzpicture}\vspace{-.3cm}
\caption{A graphical illustration of supervised LDA.
See text for details.}
\label{slda} \vspace{-.4cm
\end{figure}
\fi
\vspace{-.15cm}
\subsection{High-order tensor product and orthogonal tensor decomposition}
Here we briefly introduce something about tensors,
which mostly follow the same as in \cite{a:tensordecomp}.
A real \emph{$p$-th order tensor} $A\in\bigotimes_{i=1}^p{\mathbb R^{n_i}}$ belongs to the tensor product of Euclidean spaces $\mathbb R^{n_i}$.
Without loss of generality, we assume $n_1=n_2=\cdots=n_p=n$. We can identify each coordinate of $A$ by a $p$-tuple
$(i_1,\cdots,i_p)$, where $i_1,\cdots,i_p\in [n]$.
For instance, a $p$-th order tensor is a vector when $p=1$ and a matrix when $p=2$. We can also consider a $p$-th order tensor $A$ as a multilinear mapping.
For $A\in\bigotimes^p\mathbb R^n$ and matrices $X_1,\cdots,X_p\in\mathbb R^{n\times m}$,
the mapping $A(X_1,\cdots,X_p)$ is a $p$-th order tensor in $\bigotimes^p\mathbb R^m$, with
$[A(X_1,\cdots,X_p)]_{i_1,\cdots,i_p}
\triangleq \sum_{j_1,\cdots,j_p\in [n]}{A_{j_1,\cdots,j_p}[X_1]_{j_1,i_1}[X_2]_{j_2,i_2}\cdots [X_p]_{j_p,i_p}}$.
Consider some concrete examples of such a multilinear mapping. When $A$, $X_1$ and $X_2$ are matrices, we have $A(X_1,X_2) = X_1^\top AX_2$. Similarly, when $A$ is a matrix and $x$ is a vector, we have $A(I, x) = Ax$.
An orthogonal tensor decomposition of a tensor $A\in\bigotimes^p\mathbb R^n$ is a collection of orthonormal vectors $\{\bm v_i\}_{i=1}^k$
and scalars $\{\lambda_i\}_{i=1}^k$ such that
$A = \sum_{i=1}^k{\lambda_i\bm v_i^{\otimes p}}$, where we use $\bm v^{\otimes p} \triangleq \bm v\otimes \bm v\otimes\cdots\otimes \bm v$ to denote the $p$-th order tensor
generated by a vector $\bm v$.
Without loss of generality, we assume $\lambda_i$ are nonnegative when $p$ is odd since we can change the sign of $\bm{v_i}$ otherwise.
Although orthogonal tensor decomposition in the matrix case
can be done efficiently by singular value decomposition (SVD),
it has several delicate issues in higher order tensor spaces~\cite{a:tensordecomp}.
For instance, tensors may not have unique decompositions,
and an orthogonal decomposition may not exist for every symmetric tensor~\cite{a:tensordecomp}.
Such issues are further complicated when only noisy estimates of the desired tensors are available.
For these reasons, we need more advanced techniques to handle high-order tensors. In this paper, we will apply \emph{robust tensor power} methods~\cite{a:tensordecomp} to recover robust eigenvalues and eigenvectors of an (estimated) third-order tensor. The algorithm recovers eigenvalues and eigenvectors up to an absolute error $\varepsilon$,
while running in polynomial time w.r.t the tensor dimension and $\log(1/\varepsilon)$.
Further details and analysis of the robust tensor power method are in Appendix~A.2 and \cite{a:tensordecomp}.
\iffalse
\begin{figure}[t]\vspace{-.3cm}
\centering
\subfigure[$y_d=\bm\eta_d^\top\bm h_d + \varepsilon_d$]{
\scalebox{0.8}{
\begin{tikzpicture}
\node[obs] (x) {$x$} ; %
\node[latent, left=of x] (z) {$z$} ; %
\node[latent, left=of z] (h) {$\bm h$}; %
\node[const, left=of h] (alpha) {$\bm\alpha$};
\node[latent, right=of x] (mu) {$\bm\mu$};
\node[const, below=of alpha] (eta) {$\bm\eta,\sigma$};
\node[obs, below=of h, right=of eta] (y) {$y$};
\node[const, below=of mu] (beta) {$\bm\beta$};
\edge {alpha} {h};
\edge {h} {z};
\edge {z} {x};
\edge {mu} {x};
\edge {h} {y};
\edge {eta} {y};
\edge {beta} {mu};
\plate {plate1} { (z) (x) } {$M$};
\plate {plate2} { (h) (y) (plate1) } {$N$};
\plate {plate3} { (mu) } {$k$};
\end{tikzpicture}
}
\label{fig_slda_h}
}
\subfigure[$y_d=\bm\eta_d^\top\bar{\bm z}_d + \varepsilon_d$]{
\scalebox{0.8}{
\begin{tikzpicture}
\node[obs] (x) {$x$} ; %
\node[latent, left=of x] (z) {$z$} ; %
\node[latent, left=of z] (h) {$\bm h$}; %
\node[const, left=of h] (alpha) {$\bm\alpha$};
\node[latent, right=of x] (mu) {$\bm\mu$};
\node[const, below=of alpha] (eta) {$\bm\eta,\sigma$};
\node[obs, below=of h, right=of eta] (y) {$y$};
\node[const, below=of mu] (beta) {$\bm\beta$};
\edge {alpha} {h};
\edge {h} {z};
\edge {z} {x};
\edge {mu} {x};
\edge {z} {y};
\edge {eta} {y};
\edge {beta} {mu};
\plate {plate1} { (z) (x) } {$M$};
\plate {plate2} { (h) (y) (plate1) } {$N$};
\plate {plate3} { (mu) } {$k$};
\end{tikzpicture}
}
\label{fig_slda_z}
}
\vspace{-.2cm}
\caption{Plate notations for two variants of sLDA}\label{fig_slda}\vspace{-.4cm}
\end{figure}
\fi
\iffalse
\begin{figure}
\centering
\includegraphics[width = 15cm]{images/slda.png}
\caption{Plate notations for two variants of sLDA}
\label{slda}
\end{figure}
\fi
\vspace{-.15cm}
\subsection{Notations}
We use $\|\bm v\| = \sqrt{\sum_i{v_i^2}}$ to denote the Euclidean norm of vector $\bm v$,
$\|M\|$ to denote the spectral norm of matrix $M$, $\|T\|$ to denote the operator norm of a high-order tensor, and
$\|M\|_F = \sqrt{\sum_{i,j}{M_{ij}^2}}$ to denote the Frobenious norm of $M$. We use an one-hot vector $\bm x\in\mathbb R^V$ to represent a word in a document (i.e., for the $i$-th word in a vocabulary, only $x_i = 1$ all other elements are $0$).
In the two-stage spectral method, we use $O \triangleq (\bm\mu_1,\bm\mu_2,\cdots,\bm\mu_k)\in\mathbb R^{V\times k}$ to denote
the topic distribution matrix, and $\widetilde O\triangleq (\widetilde{\bm\mu}_1,\widetilde{\bm\mu}_2,\cdots,\widetilde{\bm\mu}_k)$
to denote the canonical version of $O$, where $\widetilde{\bm\mu}_i = \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}}\bm\mu$ with
$\alpha_0\triangleq \sum_{i=1}^k{\alpha_i}$. For the joint spectral method, we combine the topic distribution $\bm{\mu}_i$ with its
regression parameter to form a joint topic distribution vector $\bm{v}_i \triangleq [\bm{\mu}_i',\eta_i]'$. We use notation $O^* \triangleq (\bm{v}_1, \bm{v}_2, ... ,
\bm{v}_k) \in \mathbb{R}^{(V+1) \times k} $ to denote the joint topic distribution matrix and $\widetilde{O}^* \triangleq [\widetilde{\bm{v}}_1, \widetilde{\bm{v}}_2, ...,
\widetilde{\bm{v}}_k]$ to denote its canonical version where $\widetilde{\bm{v}}_i \triangleq \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0 + 1)}} \bm{v}_i$.
\vspace{-.15cm}
\section{A Two-stage Spectral Method}
We first present a two-stage spectral method to recover the parameters of sLDA.
The algorithm consists of two key components---an orthogonal tensor decomposition of observable moments to recover the topic distribution matrix $O$ and a power update method to recover the regression model $\bm\eta$. We present these techniques and a rigorous theoretical analysis below.
\vspace{-.15cm}
\subsection{Moments of observable variables}
Our spectral decomposition methods recover the topic distribution matrix $O$ and the linear regression model $\bm\eta$ by manipulating moments of observable variables. In Definition~\ref{def_moment}, we define a list of moments on random variables from the underlying sLDA model.
\begin{deff} We define the following moments of observable variables:
{\small \setlength\arraycolsep{1pt} \begin{eqnarray}
M_1 &=& \mathbb E[\bm x_1],\quad
M_2 = \mathbb E[\bm x_1\otimes\bm x_2] - \frac{\alpha_0}{\alpha_0+1}M_1\otimes M_1,\\
M_3 &=& \mathbb E[\bm x_1\otimes\bm x_2\otimes\bm x_3] - \frac{\alpha_0}{\alpha_0+2}(\mathbb E[\bm x_1\otimes\bm x_2\otimes M_1] \nonumber \\
& &+ \mathbb E[\bm x_1\otimes M_1\otimes \bm x_2] + \mathbb E[M_1\otimes \bm x_1\otimes \bm x_2])\nonumber\\
& &+ \frac{2\alpha_0^2}{(\alpha_0+1)(\alpha_0+2)}M_1\otimes M_1\otimes M_1,\\
M_y &=& \mathbb E[y\bm x_1\otimes \bm x_2] - \frac{\alpha_0}{\alpha_0+2}(\mathbb E[y] \mathbb E[\bm x_1\otimes\bm x_2]
+ \mathbb E[\bm x_1]\otimes\mathbb E[y\bm x_2] \nonumber\\
& & + \mathbb E[y\bm x_1]\otimes\mathbb E[\bm x_2]) + \frac{2\alpha_0^2}{(\alpha_0+1)(\alpha_0+2)}\mathbb E[y]M_1\otimes M_1.
\end{eqnarray}}\label{def_moment}
\end{deff}\vspace{-.25cm}
Note that the moments $M_1$, $M_2$ and $M_3$ were also defined in~\cite{a:twosvd,a:tensordecomp} for recovering the parameters of LDA models. For sLDA, we need to define a new moment $M_y$ in order to recover the linear regression model $\bm\eta$.
The moments are based on observable variables in the sense that they can be estimated from i.i.d. sampled documents.
For instance, $M_1$ can be estimated by computing the empirical distribution of all words,
and $M_2$ can be estimated using $M_1$ and word co-occurrence frequencies. Though the moments in the above forms look
complicated, we can apply elementary calculations based on the conditional independence structure of sLDA to significantly simplify them and more importantly to get them connected with the model parameters to be recovered, as summarized in Proposition~\ref{prop_moments}, whose proof is elementary and deferred to Appendix C for clarity.
\begin{prop}The moments can be expressed using the model parameters as:\\[-.6cm]
{\small \setlength\arraycolsep{1pt} \begin{eqnarray}
M_2 &=& \frac{1}{\alpha_0(\alpha_0+1)}\sum_{i=1}^k{\alpha_i\bm\mu_i\otimes\bm\mu_i}, \\
M_3 &=& \frac{2}{\alpha_0(\alpha_0+1)(\alpha_0+2)}\sum_{i=1}^k{\alpha_i\bm\mu_i\otimes\bm\mu_i\otimes\bm\mu_i},\\
M_y &=& \frac{2}{\alpha_0(\alpha_0+1)(\alpha_0+2)}\sum_{i=1}^k{\alpha_i\eta_i\bm\mu_i\otimes\bm\mu_i}.
\end{eqnarray}}\label{prop_moments}
\end{prop
\iffalse
\begin{prof*}
All the three equations can be proved in a similar mannar. We only give the proof to the equation on $M_y$.
In sLDA the topic mixing vector $\bm{h}$ follows a Dirichlet prior distribution with parameter $\bm{\alpha}$.
Therefore, we have:
\begin{equation*}
\begin{aligned}
& \mathbb{E}[h_i] = \frac{\alpha_i}{\alpha_0}, \mathbb{E}[h_i,hj] = \left \{ \begin{array}{ll}
\frac{(\alpha_i + 1)\alpha_i}{\alpha_0(\alpha_0 + 1)} & if\ i = j \\
\frac{\alpha_i\alpha_j}{\alpha_0(\alpha_0 + 1)} & if \ i \not = j \end{array}\right . \\
&\mathbb{E}[h_ih_jh_k] = \left \{ \begin{array}{ll}
\frac{\alpha_i(\alpha_i + 1)(\alpha_i + 2)}{\alpha_0(\alpha_0 + 1)(\alpha_0 + 2)} & if \ i= j = k\\
\frac{\alpha_j\alpha_i(\alpha_i+1)}{\alpha_0(\alpha_0 + 1)(\alpha_0 + 2)} & if \ i = j \not = k\\
\frac{\alpha_i\alpha_j\alpha_k}{\alpha_0(\alpha_0 + 1)(\alpha_0 + 2)} & if \ i \not = j \not = k\\
\end{array}\right .
\end{aligned}
\end{equation*}
Next, note that:
\begin{equation*}
\begin{aligned}
&\mathbb{E}[y|\bm{h}] = \bm{\eta}'\bm{h}, \mathbb{E}[\bm{x}_1|\bm{h}] = \sum\limits_{i=1}^{k}h_i\bm{\mu}_i \\
& \mathbb{E}[\bm{x}_1 \otimes \bm{x}_2 | \bm{h}] = \sum\limits_{i,j =1}^{k}h_ih_j\bm{\mu}_i\bm{\mu}_j\\
& \mathbb{E}[y\bm{x}_1\otimes \bm{x}_2 | \bm{h}] = \sum\limits_{i,j,l=1}^{k} h_ih_jh_l\cdot \eta_i\bm{\mu}_j \otimes
\bm{\mu}_l
\end{aligned}
\end{equation*}
the proposition can then be easily proved by taking expectation over the topic mixing vector $\bm{h}$. \
\hfill\QED
\end{prof*}
\fi
\begin{algorithm}[t]
\caption{spectral parameter recovery algorithm for sLDA. Input parameters: $\alpha_0, L, T$.}
\centering
\begin{algorithmic}[1]
\State Compute empirical moments and obtain $\widehat M_2, \widehat M_3$ and $\widehat M_y$.
\State Find $\widehat W\in\mathbb R^{n\times k}$ such that $\widehat M_2(\widehat W, \widehat W) = I_k$.
\State Find robust eigenvalues and eigenvectors $(\widehat\lambda_i,\widehat{\bm v}_i)$ of $\widehat M_3(\widehat W,\widehat W,\widehat W)$
using the robust tensor power method \cite{a:tensordecomp} with parameters $L$ and $T$.
\State Recover prior parameters: $\widehat\alpha_i \gets \frac{4\alpha_0(\alpha_0+1)}{(\alpha_0+2)^2\widehat\lambda_i^2}$.
\State Recover topic distributions: \vspace{-.3cm}
$$\widehat{\bm\mu}_i\gets \frac{\alpha_0+2}{2}\widehat\lambda_i(\widehat W^+)^\top\widehat{\bm v}_i.$$\vspace{-.55cm}
\State Recover the linear regression model: \vspace{-.2cm}
$$\widehat\eta_i \gets \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i.$$\vspace{-.55cm}
\State \textbf{Output:} $\widehat{\bm\eta}$, $\widehat{\bm\alpha}$ and $\{\widehat{\bm\mu}_i\}_{i=1}^k$.
\end{algorithmic}
\label{alg1}
\end{algorithm}
\vspace{-.7cm}
\subsection{Simultaneous diagonalization}
Proposition~\ref{prop_moments} shows that the moments in Definition \ref{def_moment} are all the weighted sums of tensor products of $\{\bm\mu_i\}_{i=1}^k$ from the underlying sLDA model.
One idea to reconstruct $\{\bm\mu_i\}_{i=1}^k$ is to perform \emph{simultaneous diagonalization} on tensors of different orders.
The idea has been used in a number of recent developments of spectral methods for latent variable models~\cite{a:twosvd,a:tensordecomp,specregression}.
Specifically, we first whiten the second-order tensor $M_2$ by finding a matrix $W\in\mathbb R^{V\times k}$ such that
$W^\top M_2 W = I_k$.
This whitening procedure is possible whenever the topic distribuction vectors $\{\bm\mu_i\}_{i=1}^k$ are linearly independent
(and hence $M_2$ has rank $k$).
This is not always correct since in the ``overcomplete" case~\cite{a:overcomplete},
it is possible that the topic number $k$ is larger than vocabulary size $V$. However, the
linear independent assumption gives us a more compact representation for the topic model and
works well in practice. Hence we simply assume that the whitening procedure is possible.
The whitening procedure and the linear independence assumption also
imply that $\{W\bm\mu_i\}_{i=1}^k$ are orthogonal vectors (see Appendix A.2 for details),
and can be subsequently recovered by performing an orthogonal
tensor decomposition on the simultaneously whitened third-order tensor $M_3(W,W,W)$.
Finally, by multiplying the pseudo-inverse of the whitening matrix $W^+$ we obtain the topic distribution vectors $\{\bm\mu_i\}_{i=1}^k$.
It should be noted that Jennrich's algorithm \cite{jennrich1,jennrich2,ankur-review} could recover $\{\bm\mu_i\}_{i=1}^k$ directly from the 3-rd order tensor $M_3$ alone when $\{\bm\mu_i\}_{i=1}^k$ is linearly independent.
However, we still adopt the above simultaneous diagonalization framework because the intermediate vectors $\{W\bm\mu_i\}_{i=1}^k$
play a vital role in the recovery procedure of the linear regression model $\bm\eta$.
\vspace{-.1cm}
\subsection{The power update method}
Although the linear regression model $\bm\eta$ can be recovered in a similar manner by performing simultaneous diagonalization on $M_2$ and $M_y$, such a method has several disadvantages, thereby calling for novel solutions. First, after obtaining entry values $\{\eta_i\}_{i=1}^k$ we need to match them to the topic distributions $\{\bm\mu_i\}_{i=1}^k$ previously recovered.
This can be easily done when we have access to the true moments,
but becomes difficult when only estimates of observable tensors are available
because the estimated moments may not share the same singular vectors due to sampling noise.
A more serious problem is that when $\bm\eta$ has duplicate entries the orthogonal decomposition of $M_y$ is no longer unique.
Though a randomized strategy similar to the one used in~\cite{a:twosvd} might solve the problem,
it could substantially increase the sample complexity~\cite{a:tensordecomp} and render the algorithm impractical.
In \cite{spechmm}, the authors provide a method for the matching problem by reusing eigenvectors.
We here develop a power update method to resolve the above difficulties with a similar spirit. Specifically, after obtaining the whitened (orthonormal) vectors $\{\bm v_i\} \triangleq c_i\cdot W^{\top}\bm\mu_i$ \footnote{$c_i \triangleq \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}} $ is a scalar coefficient that depends on $\alpha_0$ and $\alpha_i$. See Appendix A.2 for details.} we recover the entry $\eta_i$ of the linear regression model directly by computing a power update $\bm v_i^\top M_y(W,W)\bm v_i$. In this way, the matching problem is automatically solved because we know what topic distribution vector $\bm\mu_i$ is used when recovering $\eta_i$. Furthermore, the singular values (corresponding to the entries of $\bm \eta$) do not need to be distinct because we are not using any unique SVD properties of $M_y(W,W)$. As a result, our proposed algorithm works for any linear model $\bm\eta$.
\vspace{-.1cm}
\subsection{Parameter recovery algorithm}
Alg.~\ref{alg1} outlines our parameter recovery algorithm for sLDA (Spectral-sLDA). First, empirical estimations of the observable moments in Definition \ref{def_moment} are computed from the given documents.
The simultaneous diagonalization method is then used
to reconstruct the topic distribution matrix $O$ and its prior parameter $\bm\alpha$. After obtaining $O=(\bm\mu_1,\cdots,\bm\mu_k)$,
we use the power update method introduced in the previous section to recover the linear regression model $\bm\eta$. We can also recover the noise level parameter $\sigma^2$ with the other
parameters in hand by estimating $\mathbb{E}[y]$ and $\mathbb{E}[y^2]$ since
$\mathbb{E}[y] = \mathbb{E}[\mathbb{E}[y|\bm{h}]] = \bm{\alpha}^{\top}\bm{\eta}$ and
$\mathbb{E}[y^2] = \mathbb{E}[\mathbb{E}[y^2|\bm{h}]] =
\bm{\eta}^{\top}\mathbb{E}[\bm{h}\otimes \bm{h}]\bm{\eta} + \sigma^2$, where the term $\mathbb{E}[\bm{h}\otimes \bm{h}]$ can be computed in an analytical form using the model parameters, as detailed in Appendix C.1.
Alg.~1 admits three hyper-parameters $\alpha_0$, $L$ and $T$. $\alpha_0$ is defined as the sum of all entries in the prior parameter $\bm\alpha$. Following the conventions in~\cite{a:twosvd,a:tensordecomp}, we assume that $\alpha_0$ is known a priori and use this value to perform parameter estimation. It should be noted that this is a mild assumption, as in practice usually a homogeneous vector $\bm\alpha$ is assumed and the entire vector is known~\cite{lsa}. The $L$ and $T$ parameters are used to control the number of iterations in the robust tensor power method. In general, the robust tensor power method runs in $O(k^3LT)$ time.
To ensure sufficient recovery accuracy, $L$ should be at least a linear function of $k$ and
$T$ should be set as $T = \Omega(\log(k) + \log\log(\lambda_{\max}/\varepsilon))$,
where $\lambda_{\max} = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$
and $\varepsilon$ is an error tolerance parameter.
Appendix~A.2 and~\cite{a:tensordecomp} provide a deeper analysis into the choice of $L$ and $T$ parameters.
\vspace{-.1cm}
\subsection{Sample Complexity Analysis}
\vspace{-.1cm}
We now analyze the sample complexity of Alg.~\ref{alg1} in order to achieve $\varepsilon$-error with a high probability. For clarity, we focus on presenting the main results, while deferring the proof details to Appendix~A, including the proofs of important lemmas that are needed for the main theorem.
\begin{thm
Let $\sigma_1(\widetilde O)$ and $\sigma_k(\widetilde O)$ be the largest and the smallest singular values of the canonical topic distribution matrix $\widetilde O$.
Define $\lambda_{\min} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\max}}}$
and $\lambda_{\max} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$'∫?
with $\alpha_{\max}$ and $\alpha_{\min}$ the largest and the smallest entries of $\bm\alpha$.
Suppose $\widehat{\bm\mu}$, $\widehat{\bm\alpha}$ and $\widehat{\bm\eta}$ are the outputs of Algorithm 1,
and $L$ is at least a linear function of $k$.
Fix $\delta\in(0,1)$. For any small error-tolerance parameter $\varepsilon > 0$,
if Algorithm \ref{alg1} is run with parameter $T = \Omega(\log(k) + \log\log(\lambda_{\max}/\varepsilon))$
on $N$ i.i.d. sampled documents (each containing at least 3 words) with
$N\geq \max(n_1, n_2, n_3)$, where\vspace{-.2cm}
{\small
\begin{equation*}
\begin{aligned}
n_1 &= C_1\cdot \left(1+\sqrt{\log(6/\delta)}\right)^2\cdot \frac{\alpha_0^2(\alpha_0+1)^2}{\alpha_{\min}}, \nonumber \\
n_2 &= C_2\cdot \frac{(1+\sqrt{\log(15/\delta)})^2}{\varepsilon^2\sigma_k(\widetilde O)^4} \\
&\cdot\max\left(\left( \|\bm\eta\| - \sigma \Phi^{-1}\left( \frac{\delta}{60}\right)\right)^2, \alpha_{\max}^2\sigma_1(\widetilde O)^2\right), \nonumber \\
n_3 &= C_3\cdot \frac{(1+\sqrt{\log(9/\delta)})^2}{\sigma_k(\widetilde O)^{10}}\cdot\max\left(\frac{1}{\varepsilon^2},
\frac{k^2}{\lambda_{\min}^2}\right), \nonumber
\end{aligned}
\end{equation*} }
and $C_1, C_2$ and $C_3$ are universal constants,
then with probability at least $1-\delta$, there exists a permutation $\pi:[k]\to [k]$ such that for every topic $i$, the following holds:\vspace{-.2cm}
\setlength\arraycolsep{-1pt} {\small \begin{eqnarray}
&&|\alpha_i-\widehat{\alpha}_{\pi(i)}| \!\leq\! \frac{4\alpha_0(\alpha_0 \!+\! 1)(\lambda_{\max} \!+\! 5\varepsilon)}{(\alpha_0 \!+\! 2)^2\lambda_{\min}^2(\lambda_{\min} \!-\! 5\varepsilon)^2}\cdot 5\varepsilon,~if~\lambda_{\min} > 5\varepsilon \nonumber \\
&&\|\bm\mu_i - \widehat{\bm\mu}_{\pi(i)}\| \!\leq\! \left(3\sigma_1(\widetilde O)\left(\frac{8\alpha_{\max}}{\lambda_{\min}} + \frac{5(\alpha_0+2)}{2}\right) + 1\right)\varepsilon \nonumber \\
&&|\eta_i - \widehat\eta_{\pi(i)}| \!\leq\! \left(\frac{\|\bm\eta\|}{\lambda_{\min}} + (\alpha_0+2)\right) \varepsilon . \nonumber
\end{eqnarray}}\vspace{-.4cm}
\label{thm_main}
\end{thm}
In brevity, the proof is based on matrix perturbation lemmas (see Appendix~A.1) and analysis to the orthogonal tensor decomposition methods (including SVD and robust tensor power method) performed on inaccurate tensor estimations (see Appendix~A.2).
The sample complexity lower bound consists of three terms, from $n_1$ to $n_3$.
The $n_3$ term comes from the sample complexity bound for the robust tensor power method~\cite{a:tensordecomp};
the $(\|\bm\eta\|- \sigma \Phi^{-1}(\delta/60))^2$ term in $n_2$ characterizes the recovery accuracy for the linear regression model $\bm\eta$,
and the $\alpha_{\max}^2\sigma_1(\widetilde O)^2$ term arises when we try to recover the topic distribution vectors $\bm\mu$;
finally, the term $n_1$ is required so that some technical conditions are met.
The $n_1$ term does not depend on either $k$ or $\sigma_k(\widetilde O)$,
and could be largely neglected in practice.
\begin{rem}
An important implication of Theorem~\ref{thm_main} is that it provides a sufficient condition
for a supervised LDA model to be identifiable, as shown in Remark~\ref{rem_identifiability}.
To some extent, Remark~\ref{rem_identifiability} is the best identifiability result possible under our inference framework,
because it makes no restriction on the linear regression model $\bm\eta$, and the linear independence assumption is unavoidable
without making further assumptions on the topic distribution matrix $O$.
\end{rem}
\begin{rem} Given a sufficiently large number of i.i.d. sampled documents with at least 3 words per document, a supervised LDA model $\mathcal M=(\bm\alpha,\bm\mu,\bm\eta)$ is identifiable if $\alpha_0 = \sum_{i=1}^k{\alpha_i}$ is known and $\{\bm\mu_i\}_{i=1}^k$ are linearly independent.
\label{rem_identifiability}
\end{rem}
We now take a close look at the sample complexity bound in Theorem~\ref{thm_main}.
It is evident that $n_2$ can be neglected when the number of topics $k$ gets large,
because in practice the norm of the linear regression model $\bm\eta$ is usually assumed to be small in order to avoid overfitting.
Moreover, as mentioned before, the prior parameter $\bm\alpha$ is often assumed to be homogeneous with $\alpha_i = 1/k$ \cite{lsa}.
With these observations, the sample complexity bound in Theorem~\ref{thm_main}
can be greatly simplified.
\begin{rem}
Assume $\|\bm\eta\|$ and $\sigma$ are small and $\bm\alpha = (1/k, \cdots, 1/k)$.
As the number of topics $k$ gets large, the sample complexity bound in Theorem~\ref{thm_main} can be simplified as\vspace{-.3cm}
\setlength\arraycolsep{1pt}\begin{equation}
N \geq \Omega\left( \frac{\log(1/\delta)}{\sigma_k(\widetilde O)^{10}}\cdot\max(\varepsilon^{-2}, k^3)\right).
\end{equation}\label{rem_simplebound1}
\end{rem}\vspace{-.6cm}
The sample complexity bound in Remark~\ref{rem_simplebound1} may look formidable
as it depends on $\sigma_k(\widetilde O)^{10}$.
However, such dependency is somewhat necessary because we are using third-order tensors to recover the underlying model parameters.
\section{Joint Parameter Recovery}
The above two-stage procedure has one possible disadvantages, that is, the recovery of the topic distribution matrix does not
use any supervision signal, and thus the recovered topics are often not good enough for prediction tasks, as shown in experiments.
The disadvantage motivates us to develop a joint spectral method with theoretical guarantees.
We now describe our single-phase algorithm.
\subsection{Moments of Observable Variables}
We first define some moments based on the observable variables including the information that we need to recover the model parameters.
Since we aim to recover the joint topic distribution matrix $O^*$, we combine the word vector $\bm{x}$ with the response variable $y$ to form a joint vector $ \bm{z} = [ \bm{x}', y]'$ and define the following moments:
\iffalse
\begin{figure*}[t]
\centering
\begin{tikzpicture}[scale=0.3]
\tikzstyle{ann} = [fill=white, font=\footnotesize, inner sep =1pt]
\foreach [count=\i]\x in {1,...,5}
\foreach [count=\j]\y in {6,...,2}
\foreach [count=\k]\z in {2,...,6}{
\drawbox{\x}{\y}{\z}{blue!25};
}
\foreach [count=\i]\y in {6,...,2}{
\drawbox{6}{\y}{1}{yellow!25};
}
\foreach [count=\i]\x in {1,...,5}{
\drawbox{\x}{1}{1}{yellow!25};
}
\foreach [count=\i]\y in {6,...,2}
\foreach [count=\j]\z in {2,...,6}{
\drawbox{6}{\y}{\z}{green!25};
}
\foreach [count=\i]\x in {1,...,5}
\foreach [count=\j]\z in {2,...,6}{
\drawbox{\x}{1}{\z}{green!25};
}
\drawbox{6}{1}{1}{red!25}
\foreach [count=\i]\z in {2,...,6}{
\drawbox{6}{1}{\z}{yellow!25};
}
\foreach [count= \i]\x in {16,...,21}
\foreach [count =\j]\y in {7,...,3} {
\drawbox{\x}{\y}{1}{green!25};
}
\foreach [count=\i]\x in {16,...,21}
\foreach [count=\j]\y in {7,...,3}
\foreach [count=\k]\z in {3,...,7}{
\drawbox{\x}{\y}{\z}{blue!25};
}
\foreach [count=\i]\y in {7,...,3}{
\drawbox{23}{\y}{1}{yellow!25};
}
\foreach [count=\i]\x in {16,...,21}{
\drawbox{\x}{1}{1}{yellow!25};
}
\foreach [count=\i]\y in {7,...,3}
\foreach [count=\j]\z in {3,...,7}{
\drawbox{23}{\y}{\z}{green!25};
}
\foreach [count=\i]\x in {16,...,21}
\foreach [count=\j]\z in {3,...,7}{
\drawbox{\x}{1}{\z}{green!25};
}
\drawbox{23}{1}{1}{red!25}
\foreach [count=\i]\z in {3,...,7}{
\drawbox{23}{1}{\z}{yellow!25};
}
\draw[arrows=<-,line width=1pt](15.8,8)--(13.5,8.5);
\draw[arrows=<-,line width=1pt](27.2,8)--(29.0,8.5);
\draw[arrows=<-,line width=1pt](27.2,3)--(29.0,3.5);
\draw[arrows=<-,line width=1pt](24.2,0.7)--(26.0,0.7);
\node[ann] at (13,9.2) {$\mathbb{E}[\bm{x}_1 \otimes \bm{x}_2 \otimes \bm{x}_3]$};
\node[ann] at (32,8.7) {$\mathbb{E}[y\bm{x}_1 \otimes \bm{x}_2]$};
\node[ann] at (32,3.7) {$\mathbb{E}[y^2\bm{x}_1]$};
\node[ann] at (28,0.7) {$\mathbb{E}[y^3]$};
\end{tikzpicture}
\caption{Interaction between response variable $y$ and word vector $\bm{x}$ in the 3rd-order tensor $M_3$.}
\label{fig10}
\end{figure*}
\fi
\begin{deff} (Centerized Moments
{\small \begin{equation}
\begin{aligned}
& N_1 = \mathbb{E}[\bm{z}_1] \\
& N_2 = \mathbb{E}[ \bm{z}_1 \otimes \bm{z}_2] - \dfrac{\alpha_0}{\alpha_0 + 1 } N_1 \otimes N_1 - \sigma^2\bm{e} \otimes \bm{e} \\
& N_3 = \mathbb{E}[ \bm{z}_1 \otimes \bm{z}_2 \otimes \bm{z}_3] - \dfrac{\alpha_0}{\alpha_0 + 1}( \mathbb{E}[\bm{z}_1 \otimes \bm{z}_2 \otimes N_1] \\
& \ \ \ \ + \mathbb{E}[\bm{z}_1 \otimes N_1 \otimes \bm{z}_2] + \mathbb{E}[N_1 \otimes \bm{z}_1 \otimes \bm{z}_2] ) \\
& \ \ \ \ + \dfrac{2\alpha_0^2}{(\alpha_0 + 1)(\alpha_0 +2)}N_1 \otimes N_1 \otimes N_1 \\
& \ \ \ \ + 3\sigma^2N_{1_{V+1}}\bm{e} \otimes \bm{e} \otimes \bm{e}
+ \dfrac{1}{\alpha_0 + 1} \sigma^2 (\bm{e} \otimes \bm{e} \otimes N_1 \\
& \ \ \ \ + \bm{e} \otimes N_1 \otimes \bm{e} + N_1 \otimes \bm{e} \otimes \bm{e} ),
\end{aligned}
\end{equation}}
where $\bm{e}$ is the $(V$+$1)$-dimensional vector with the last element equaling to $1$ and all others zero, $N_{1_{V+1}}$ is the $V+1$-th element of $N_1$.
\end{deff}
The intuition for such definitions is derived from
an important observation that once the latent variable $\bm{h}$ is given, the mean value of $y$ is a weighted combination of the regression parameters and $\bm{h}$
(i.e., $\mathbb{E}[y|\bm{h}] = \sum_{i = 1}^{k}\eta_i h_i$), which has the same form as for $\bm{x}$
(i.e., $\mathbb{E}[\bm{x}|\bm{h}] = \sum_{i=1}^{k} {\bm \mu}_i h_i$). Therefore, it is natural to regard $y$ as an additional dimension of the word vector $\bm{x}$, which gives the new vector $\bm{z}$.
This combination leads to some other terms involving the high-order moments of $y$, which introduce the variance parameter $\sigma$ when we centerize the moments.
Although we can recover $\sigma$ in the two-stage method,
recovering it jointly with the other parameters seems to be hard. Thus we treat $\sigma$ as a hyper-parameter.
One can determine it via a cross-validation procedure.
\begin{figure}[t
\centering
\includegraphics[width = .5\textwidth]{images/tensor2.png}\vspace{-.2cm}
\caption{Interaction between response variable $y$ and word vector $\bm{x}$ in the 3rd-order tensor $M_3$.}
\label{fig10}\vspace{-.3cm}
\end{figure}
As illustrated in Fig.~\ref{fig10}, our 3rd-order moment can be viewed as a centerized version of the combination of $N_3' \triangleq \mathbb{E}[\bm{x}_1 \otimes \bm{x}_2 \otimes \bm{x}_3] $,
$N_y' \triangleq \mathbb{E}[y\bm{x}_2\otimes \bm{x}_2]$ and some high-order statistics of the response variables.
Note that this combination has already aligned the regression parameters with the corresponding topics. Hence, we do not need an extra matching step.
In practice, we cannot get the exact values of those moments. Instead, we estimate them from the i.i.d. sampled documents.
Note that we only need the moments up to the third order, which means any document consisting of at least three words can be used in this estimation.
Furthermore, although these moments seem to be complex, they can be expressed via the model parameters in a graceful manner, as
summarized in Proposition~\ref{p1} which can be proved by expanding the terms by definition, similar as in the proof of Proposition 1.
\begin{prop} \label{p1}
The moments in Definition 2 can be expressed by using the model parameters as follows:\vspace{-.2cm}
\begin{equation}
\begin{aligned}
& N_2 = \dfrac{1}{\alpha_0(\alpha_0 + 1)} \sum\limits_{i =1}^{k} \alpha_i \bm{v}_i \otimes \bm{v}_i \\
& N_3 = \dfrac{2}{\alpha_0(\alpha_0 +1)(\alpha_0 +2)} \sum\limits_{i=1}^{k} \alpha_i \bm{v}_i \otimes \bm{v}_i \otimes \bm{v}_i,
\end{aligned}\vspace{-.2cm}
\end{equation}
where $\bm{v}_i$ is the concatenation of the $i$-th word-topic distribution $\bm{\mu}_i$ and
regression parameter $\eta_i$.
\end{prop}
\iffalse
\begin{prof*}
Similar to proposition $1$, the equations can be proved via taking the expectation over the topic
mixing vector $\bm{h}$ and noting that:
\begin{equation*}
\begin{aligned}
&\mathbb{E}[\bm{z}_1|\bm{h}] = \sum\limits_{i=1}^{k} h_i\bm{v}_i \\
&\mathbb{E}[\bm{z}_1 \otimes \bm{v}_2|\bm{h}] = \sum\limits_{i,j = 1}^{k} h_ih_j\bm{v}_i \otimes \bm{v}_j
+ \sigma^2 \bm{e}_{V+1} \otimes \bm{e}_{V+1}\\
&\mathbb{E}[\bm{z}_1 \otimes \bm{z}_2 \otimes \bm{z}_3 | \bm{h}] = \sum\limits_{i,j,l=1}^{k} h_ih_jh_l
\bm{v}_i \otimes \bm{v}_j \otimes \bm{v}_l \\
&\ \ \ \ \ \ \ + 3\sigma^2 \bm{\eta}'\bm{h}\cdot \bm{e}\otimes \bm{e} \otimes \bm{e} +
\sigma^2 \sum\limits_{i=1}^{k} h_i(\bm{e} \otimes \bm{e} \otimes \bm{v}_i \\
& \ \ \ \ \ \ \ + \bm{e}\otimes \bm{v}_i \otimes \bm{e} +
\bm{v}_i \otimes \bm{e} \otimes \bm{e})\\
\end{aligned}
\end{equation*}
\hfill\QED
\end{prof*}
\fi
\subsection{Robust Tensor Decomposition}
Proposition~\ref{p1} shows that the centerized tensors are weighted sums of the tensor products
of the parameters $\{\bm{v}_i\}_{i=1}^k$ to be recovered.
A similar procedure as in the two-stage method can be followed in order to develop our joint spectral method, which consists
of {\it whitening } and {\it robust tensor decomposition} steps.
First, we whiten the 2nd-order tensor $N_2$ by finding a matrix $W \in \mathbb{R}^ {(V +1) \times k}$
such that $W^{\top}N_2W = I_k$.
This whitening procedure is possible whenever the joint topic distribution vectors $\{\bm{v}_i\}^k_{i=1}$
are linearly independent, that is, the matrix has rank $k$. The whitening procedure and the linear independence
assumption also imply that $\{W^{\top}\bm{v}_i\}^k_{i=1}$
are orthogonal vectors and
can be subsequently recovered by performing an orthogonal tensor decomposition on the simultaneously
whitened third-order tensor $N_3(W, W, W)$ as summarized in the following proposition.
\begin{prop}
Define $\bm{\omega}_i = \sqrt{\dfrac{\alpha_i}{\alpha_0(\alpha_0 +1)}}W^{\top}\bm{v}_i$. Then:
\begin{itemize}
\item
$\{\bm{\omega}\}_{i=1}^{k}$ is an orthonormal basis.
\item
$N_3(W,W,W)$ has pairs of robust eigenvalue and eigenvector $(\lambda_i, \bm{v}_i)$
with $\lambda = \dfrac{2}{\alpha_0 + 2} \sqrt{\dfrac{\alpha_0(\alpha_0+1)}{\alpha_i}} $
\end{itemize}
\end{prop}
Finally, by multiplying the pseudo-inverse of the whitening matrix $W^+$ we obtain the joint topic distribution vectors $\{\bm{v}_i\}^k_{i=1}$.
\begin{algorithm}[t]
\caption{a joint spectral method to recover sLDA parameters. Input parameters: $\alpha_0$, $L$, $T$, $\sigma$ }\label{a1}
\begin{algorithmic}[1]
\State Compute empirical moments and obtain $\widehat{N}_2, \widehat{N_3}$.
\State Find $\widehat{W} \in \mathbb{R}^{V + 1\times k}$ such that $\widehat{N}_2(\widehat{W},\widehat{W}) = I_k $.
\State Find robust eigenvalues and eigenvectors $(\widehat{\lambda}_i, \widehat{\bm{v}}_i)$
of $\widehat{N}_3(\widehat{W},\widehat{W},\widehat{W})$ using the robust tensor
power method with parameters $L$ and $T$.
\State Recover prior parameters: $\widehat{\alpha}_i \gets \frac{4\alpha_0(\alpha_0 + 1)}{(\alpha_0+2)^2\widehat{\lambda}_i^2}.$
\State Recover topic distribution: $$\widehat{\bm{v}}_i \gets \frac{\alpha_0 + 2}{2}\widehat{\lambda}_i(W^+)^{\top}\widehat{\bm{\omega}}_i.$$
\State {\bf Output: } model parameters $\widehat{\alpha}_i$, $\widehat{\bm{v}}_i$ $i = 1,...,k$
\end{algorithmic}
\label{alg2}
\end{algorithm}
We outline our single-phase spectral method in Alg.~\ref{a1}. Here we assume that the noise variance $\sigma$ is given. Note that in the two-stage spectral method, it does
not need the parameter $\sigma$ because it does not use the information of the variance of prediction error.
Although there is a disadvantage that we need to tune it, the introduction of $\sigma$ sometimes increases
the flexibility of our methods on incorporating some prior knowledge (if exists)
We additionally need three hyper-parameters $\alpha_0, L $ and $T$, similar as in the two-stage method. The parameter $\alpha_0$ is defined as the summation of all the entries of the prior parameter $\bm{\alpha}$.
$L$ and $T$ are used to control the number of iterations in robust tensor decomposition. To ensure a sufficiently high recovery accuracy, $L$ should
be at least a linear function of $k$, and $T$ should be set as $T = \Omega(\log(k) + \log\log(\lambda_{max}/\epsilon))$,
where $\lambda_{max} = \frac{2}{\alpha_0 + 2}\sqrt{\frac{\alpha_0(\alpha_0 + 1)}{\alpha_{min}}}$ and $\epsilon$ is the error
rate.
\vspace{-.1cm}
\subsection{Sample Complexity Analysis}
\vspace{-.1cm}
We now analyze the sample complexity in order to achieve $\epsilon$-error with a high probability.
For clarity, we defer proof details to Appendix B.
\begin{thm}
Let $ \sigma_1(\widetilde{O}^*)$ and $\sigma_k(\widetilde{O}^*)$ be the largest and smallest singular
values of the joint canonical topic distribution matrix $\widetilde{O}^*$. Let
$\lambda_{max} \triangleq \frac{2}{\alpha_0 + 2}\sqrt{\frac{\alpha_0(\alpha_0 +1)}{\alpha_{min}}}$ where
$\alpha_{min}$ is the smallest element of $\bm{\alpha}$;
$\lambda_{min} \triangleq \frac{2}{\alpha_0 + 2}\sqrt{\frac{\alpha_0(\alpha_0 +1)}{\alpha_{max}}}$ where
$\alpha_{max}$ is the largest element of $\bm{\alpha}$.
For any error-tolerance parameter $\epsilon > 0$, if Algorithm $2$ runs at least
$T \!=\! \Omega(\log(k) \!+\! \log\log(\lambda_{max}/\epsilon))$
iterations on $N$ i.i.d. sampled documents with $N \geq (n_1^\prime, n_2^\prime, n_3^\prime)$, where:\vspace{-.3cm}
{\small \begin{equation*}
\begin{aligned}
& n_1^\prime = K_1 \cdot \dfrac{\alpha_0^2(\alpha_0+1)^2C^2(\delta/36N)\cdot(2 + \sqrt{2\log(18/\delta)})^2 }{\alpha_{min}^2} \\
& n_2^\prime = K_2 \cdot C^2(\delta/144N)(2 + \sqrt{2 \log(72/\sigma)})^2 \\
& n_3^\prime = K_3 \cdot \dfrac{C^2(\delta/36N)(2+\sqrt{2\log(18/\sigma)})^2}
{\sigma_k(\widetilde{O}^*)^{10}}\cdot\max(\dfrac{1}{\epsilon^2}, \dfrac{k^2}{\lambda_{min}^2}),\\
\end{aligned}
\end{equation*}}\vspace{-.3cm}
\noindent $C(x)$ is a polynomial of inverse CDF of normal distribution and the norm of regression parameters $\| {\bm \eta}\|$; $K_1,K_2,K_3$ are some universal constants.
Then with probability at least $1 - \delta$, there exist a permutation $\pi: [k] \to [k]$
such that the following holds for every $i \in [k]$:\vspace{-.4cm}
\begin{equation*}
\begin{aligned}
& |\alpha_i - \widehat{\alpha}_{\pi(i)} |
\leq \dfrac{4\alpha_0(\alpha_0 + 1) }{(\alpha_0 + 2)^2\lambda_{min}^2(\lambda_{min} - 5 \epsilon)^2} \cdot 5\epsilon \\
& \|\bm{v}_i - \widehat{\bm{v}}_{\pi(i)} \| \leq \left ( \sigma_1(\widetilde{O}^*)(\alpha_0 + 2)(\frac{7}{2} +
\frac{8}{\lambda_{min}})\right )\cdot \epsilon.
\end{aligned}
\end{equation*}\vspace{-.1cm}
\label{thm_main2}
\end{thm}
\vspace{-.5cm}
Similar to Theorem 1, the sample complexity bound consists of three terms.
The first and second terms do not
depend on the error rate $\epsilon$, which are required so that some technical conditions
are met. Thus they could be largely neglected in practice.
The third $n_3^\prime$ term comes from the sample complexity bound for the robust tensor power method~\cite{a:tensordecomp}.
\begin{rem}
Note the RHS of the requirements of $N$ includes a
function of $N$~(i.e., $C(1/N)$). As mentioned above, $C(x)$ is polynomial of inverse CDF of normal
distribution with low degree. Since the inverse CDF grows very slowly~(i.e., $|\Phi^{-1}(1/N)| = o(\log(N))$).
We can omit it safely.
\end{rem}
\begin{rem}
Following the above remark and assume that $\|\bm{\eta}\|$ and $\sigma$ are small and $\bm{\alpha}$ are homogeneous, the
sample complexity can be simplified as (a function of $k$):\vspace{-.3cm}
$$
N = O\left( \dfrac{\log(1/\sigma)}{\sigma_k(\widetilde{O}^*)^{10}} \cdot \max(\dfrac{1}{\epsilon^2}, k^3)\right).
$$\label{rem_simplebound2} \vspace{-.45cm}
\end{rem}\vspace{-.1cm}
The factor $1/\sigma_k(\widetilde{O}^*)^{10}$ is large, however, such a factor is necessary since we use the third order tensors. This factor roots in the tensor
decomposition methods and one can expect to improve it if we have other better methods to decompose $\widehat{N}_3$.
\subsection{Sample Complexity Comparison}
\label{compp}
As mentioned in Remark \ref{rem_simplebound1} and Remark \ref{rem_simplebound2}, the joint spectral method shares the
same sample complexity as the two-stage algorithm in order to achieve $\epsilon$ accuracy, except two minor differences.
First, the sample complexity depends on the smallest singular value of (joint) topic distribution $\sigma_k(\widetilde{O})$. For the joint method, the joint topic distribution
matrix consists of the original topic distribution matrix and one extra row of the regression parameters. Thus from Weyl's inequality~\cite{a:weyl}, the smallest singular value
of the joint topic distribution matrix is larger than that of the original topic distribution matrix,
and then the sample complexity of the joint method is a bit lower than that of the two-stage method, as empirically justified in experiments.
Second, different from the two-stage method, the errors of topic distribution $\bm{\mu}$ and regression parameters $\bm{\eta}$ are estimated together in the joint method
(i.e., $\bm{v}$), which can potentially give more accurate estimation of regression parameters considering that the number of
regression parameters is much less than the topic distribution.
\section{Speeding up moment computation}
We now analyze the computational complexity and present some implementation details to make the algorithms more efficient.
\subsection{Two-Stage Method}
In Alg.~1, a straightforward computation of the third-order tensor $\widehat M_3$ requires
$O(NM^3)$ time and $O(V^3)$ storage, where $N$ is corpus size, $M$ is the number of words per document and $V$ is the vocabulary size.
Such time and space complexities are clearly prohibitive for real applications, where the vocabulary usually contains tens of thousands of terms.
However, we can employ a trick similar as in~\cite{speedup} to speed up the moment computation.
We first note that only the whitened tensor $\widehat M_3(\widehat W,\widehat W,\widehat W)$ is needed in our algorithm, which only takes $O(k^3)$ storage.
Another observation is that the most difficult term in $\widehat M_3$ can be written as
$\sum_{i=1}^r{c_i\bm u_{i,1}\otimes \bm u_{i,2}\otimes\bm u_{i,3}}$,
where $r$ is proportional to $N$ and $\bm u_{i,\cdot}$ contains at most $M$ non-zero entries.
This allows us to compute $\widehat M_3(\widehat W,\widehat W,\widehat W)$ in $O(NMk)$ time by computing
$\sum_{i=1}^r{c_i\bm (W^\top\bm u_{i,1})\otimes (W^\top\bm u_{i,2})\otimes (W^\top\bm u_{i,3})}$.
Appendix~C.2 provides more details about this speed-up trick.
The overall time complexity is $O(NM(M+k^2)+V^2+k^3LT)$
and the space complexity is $O(V^2+k^3)$.
\subsection{Joint Method}
For the single-phase algorithm, a straightforward computation of the third-order tensor $\widehat N_3$ has the same complexity of $O(NM^3)$ as in the two-stage method.
And a much higher time complexity is needed for computing $\widehat{N}_3(\widehat{W},\widehat{W},\widehat{W})
, which is prohibitive.
Similar as in the two-stage method, since we only need $\widehat{N}_3(\widehat{W},\widehat{W},\widehat{W})$ in Alg.~\ref{a1}, we turn to compute this term directly.
We can then use the trick mentioned above to do this.
The key idea is to decompose the third-order tensor into different parts based on the occurrence of words and compute them respectively.
The same time comlexity and space complexity is needed for the single-phase method.
Sometimes $N_2$ and $N_3$ are not ``balanced" (i.e., the value of some elements are much larger than the others).
This situation happens when either
the vocabulary size is too large or the range of $\bm{\eta}$ is too large. One can image that
if we have a vocabulary consisting of one million words while $\min{\eta}_i = 1$, then the energy
of the matrix $N_2$ concentrates on $(N_2)_{V+1,V+1}$.
As a consequence, the SVD performs badly when the matrix is ill-conditioned.
A practical solution
to this problem is that we scale the word vector $\bm{x}$ by a constant, that is, for the $i$-th word in the dictionary, we set $x_i = C , x_j = 0, \forall i \not = j$, where $C$ is a constant.
The main effect is that we can make the matrix more stable after this manipulation. Note that when we fix $C$,
this makes no effect on the recovery accuracy. Such a trick is primarily for computational stability. In our experiments, $C$ is set to be $100$.
\subsection{Dealing with large vocabulary size $V$}
One key step in the whitening procedure of both methods is to perform SVD on the second order moment $M_2 \in \mathbb{R}^{V \times V}$ (or $N_2 \in \mathbb{R}^{(V+1) \times (V+1)}$).
A straightforward implementation of SVD has complexity $O(k V^2)$,\footnote{It is not $O(V^3)$ as we only need top-$k$ truncated SVD.} which is unbearable when the vocabulary size $V$ is large.
We follow the method in \cite{a:nystrom} and perform random projection to reduce dimensionality. More precisely, let $S \in \mathbb{R}^{\widetilde{k}}$ where $\widetilde{k} < V$ be
a random matrix and then define $C = M_2 S$ and $\Omega = S^{\top}M_2S$. Then a low rank approximation of $M_2$ is given by $ \widetilde{M_2} = C\Omega^{+}C^{\top} $.
Now we can obtain the whitening matrix without directly performing an SVD on $M_2$ by appoximating $C^{-1}$ and $\Omega$ separately.
The overall algorithm is provided in Alg.~\ref{alg3}. In practice, we set $\widetilde{k} = 10 k$ to get a sufficiently accurate approximation.
\begin{algorithm}[t]
\caption{Randomized whitening procedure. Input parameters: second order moment $M_2$ (or $N_2$).}
\centering
\begin{algorithmic}[1]
\State Generate a random projection matrix $S \in \mathbb{R}^{V\times \widetilde{k}}$.
\State Compute the matrices $C$ and $\Omega$:\vspace{-.3cm}
$$C = M_2S \in \mathbb{R}^{V \times \widetilde{k}},~\textrm{and}~ \Omega = S^{\top}M_2S \in \mathbb{R}^{ \widetilde{k}\times \widetilde{k}}.$$\vspace{-.7cm}
\State Do SVD for both $C$ and $\Omega$:\vspace{-.3cm}
$$C = U_C \Sigma_C D_C^\top,~\textrm{and}~ \Omega = U_\Omega \Sigma_\Omega D^\top_\Omega$$\vspace{-.7cm}
\State Take the rank-$k$ approximation: $U_C \leftarrow U_C(\colon ,1\colon k)$ \vspace{-.3cm}
$$\Sigma_C \leftarrow \Sigma_C(1\colon k,1\colon k), D_C \leftarrow D_C(\colon,1\colon k)$$ \vspace{-.6cm}
$$ D_{\Omega} \leftarrow D_{\Omega}(1\colon k,1\colon k),
\Sigma_{\Omega} \leftarrow \Sigma_{\Omega}(1\colon k,1 \colon k)$$\vspace{-.7cm}
\State Whiten the approximated matrix: \vspace{-.3cm}
$$W = U_C\Sigma_C^{-1}D_C^{\top}D_{\Omega}\Sigma_{\Omega}^{1/2}. $$\vspace{-.7cm}
\State \textbf{Output:} Whitening matrix $W$.
\end{algorithmic}
\label{alg3}
\end{algorithm}
\section{Experiments}\label{sec:exp}
We now present experimental results on both synthetic and two real-world datasets.
For our spectral methods,
the hyper-parameters $L$ and $T$ are set to be $100$, which is sufficiently large for our experiment settings.
Since spectral methods can only recover the underlying parameters, we first run them to
recover those parameters in training and then use Gibbs sampling to infer the topic mixing vectors $\bm{h}$ and topic assignments for each word $t_i$ for testing.
Our main competitor is sLDA with a Gibbs sampler (Gibbs-sLDA), which is asymptotically accurate and often outperforms variational methods.
We implement an uncollapsed Gibbs sampler, which alternately draws samples from the local
conditionals of $\bm{\eta}$, $\bm z$, $\mathbf{h}$, or $\bm{\mu}$,
when the rest variables are given. We monitor the behavior of the
Gibbs sampler by observing the relative
change of the training data log-likelihood, and terminate when the average change
is less than a given threshold (e.g., $1e^{-3}$) in the last $10$ iterations.
The hyper-parameters of the Gibbs sampler are set to be the same as our methods, including topic numbers and $\bm{\alpha}$.
We evaluate a hybrid method that uses the parameters $\bm{\mu},\bm{\eta}, \bm{\alpha}$ recovered by our joint spectral method
as initialization for a Gibbs sampler.
This strategy is similar to that in~\cite{a:meetem}, where
the estimation of a spectral method is used to initialize an EM method for further refining. In our hybrid method, the Gibbs sampler plays the similar role of refining.
We also compare with MedLDA~\cite{medlda}, a state-of-the-art topic model for classification and regression, on real datasets. We use the Gibbs sampler with data augmentation~\cite{gibbsmedlda}, which is more accurate than the original variational methods, and adopts the same stopping condition as above.
On the synthetic data, we first use $L_1$-norm
to measure the difference between the reconstructed parameters
and the underlying true parameters.
Then we compare the prediction accuracy and per-word likelihood on both synthetic and real-world datasets.
The quality of the prediction on the synthetic dataset is measured by mean squared error (MSE) while
the quality on the real-word dataset is assessed by predictive $R^2$ ($pR^2$), a normalized version of MSE, which is defined as
$pR^2 = 1 - \frac{ \sum_i(y_i - \widehat{y}_i)^2 }{ \sum_i(y_i - \bar{y})^2 },
where $\bar{y}$ is the mean of
testing data and $\widehat{y}_i$ is the estimation of $y_i$. The per-word log-likelihood
is defined as $\log p(\omega|\hm{h},O) = \log \sum_{j=1}^{k}p(\omega|t =j, O)p(t = j|\bm{h})$.
\begin{figure*}[t]\vspace{-.1cm
\centering
\includegraphics[width=4.9cm]{images/250_alpha.png}
\includegraphics[width=4.9cm]{images/250_eta.png}
\includegraphics[width=4.9cm]{images/250_mu.png}\vspace{-.2cm}
\caption{Reconstruction errors of two spectral methods when each document contains $250$ words.
$X$ axis denotes the training size $n$ in log domain with base $2$ (i.e., $n = 2^k, k \in \{8,...,15\}$). Error bars denote the standard deviations measured on 3 independent trials under each setting.}
\label{fig_convergence1}\vspace{-.1cm}
\end{figure*}
\begin{figure*}[t
\centering
\includegraphics[width=4.9cm]{images/500_alpha.png}
\includegraphics[width=4.9cm]{images/500_eta.png}
\includegraphics[width=4.9cm]{images/500_mu.png}\vspace{-.3cm}
\caption{Reconstruction errors of two spectral methods when each document contains $500$ words.
$X$ axis denotes the training size $n$ in log domain with base $2$ (i.e., $n = 2^k, k \in \{8,...,15\}$). Error bars denote the standard deviations measured on 3 independent trials under each setting.}
\label{fig_convergence2}\vspace{-.45cm}
\end{figure*}
\vspace{-.15cm}
\subsection{Synthetic Dataset}\label{sec:synthetic}
We generate our synthetic dataset following the generative process of sLDA, with a vocabulary of
size $V = 500$ and topic number $k = 20$.
We generate the topic distribution matrix $O$ by first sampling each entry from a uniform distribution and then normalizing
every column of it. The linear regression model $\bm{\eta}$ is sampled from a standard Gaussian distribution.
The prior parameter $\bm{\alpha}$ is assumed to be homogeneous, i.e., $\bm{\alpha} = (1/k,... , 1/k)$.
Documents and response variables are then generated from the sLDA model specified in Section~\ref{sec:sLDA-model}.
We consider two cases where the length of each document is set to be $250$ and $500$ repectively.
The hyper-parameters are set to be the same as the ones that used to
generate the dataset
\footnote{The methods are insensitive to
the hyper-parameters in a wide range. e.g., we still get high accuracy even we set the hyper-parameter $\alpha_0$ to be twice as large as the true value.}.
\vspace{-.15cm}
\subsubsection{Convergence of estimated model parameters}
Fig.~\ref{fig_convergence1} and Fig.~\ref{fig_convergence2} show the $L_1$-norm reconstruction errors of $\bm{\alpha}$, $\bm{\eta}$ and $\bm{\mu}$ when each document contains different number of words.
Note that due to the unidentifiability of topic models, we only get a permutated estimation of the underlying parameters. Thus we run a bipartite
graph matching to find a permutation that minimizes the reconstruction error. We can find that as the sample size increases, the reconstruction errors for all parameters decrease consistently to zero in both methods, which verifies the correctness of our theory.
Taking a closer look at the figures, we can see that the empirical convergence rates for $\bm{\alpha}$ and $\bm{\mu}$ are
almost the same for the two spectral methods. However, the convergence rate for regression parameters $\bm{\eta}$ in the joint
method is much higher than the one in the two-stage method, as mentioned in the comparison
of the sample complexity in Section~\ref{compp} , due to the fact that the joint method can bound the
estimation error of $\bm{\eta}$ and $\bm{\mu}$ together.
Furthermore, though Theorem~\ref{thm_main} and Theorem~\ref{thm_main2} do not involve the number of words per document,
the simulation results demonstrate a significant improvement when more words are observed in each document,
which is a nice complement for the theoretical analysis.
\vspace{-.15cm}
\subsubsection{Prediction accuracy and per-word likelihood}
Fig.~\ref{fig_prediction1} shows that both spectral methods consistently outperform Gibbs-sLDA.
Our methods also enjoy the advantage of being less variable,
as indicated by the curve and error bars.
Moreover, when the number of training documents is sufficiently large, the performance of the reconstructed model is very close to the true model\footnote{Due to the randomness in the data generating process, the true model has a non-zero prediction error.}, which implies that our spectral methods
can correctly identify an sLDA model from its observations, therefore supporting our theory.
The performances of the two-stage spectral method and the joint one are comparable this time, which is largely because of the
fact the when giving enough training data, the recovered model is accurate enough. The Gibbs method is easily caught in a local
minimum so we can find as the sample size increases, the prediction errors do not decrease monotonously.
\begin{figure}[t]\vspace{-.1cm}
\centering
\includegraphics[width=4cm]{images/250_mse.png}
\includegraphics[width=4cm]{images/250_likeli.png}
\includegraphics[width=4cm]{images/500_mse.png}
\includegraphics[width=4cm]{images/500_likeli.png}\vspace{-.2cm}
\caption{
Mean square errors and negative per-word log-likelihood of Alg.~\ref{alg1} and Gibbs sLDA.
Each document contains $M=500$ words. The $X$ axis denotes the training size ($\times 10^3$).
The ''ref. model" denotes the one with the underlying true parameters.
}
\label{fig_prediction1}\vspace{-.2cm}
\end{figure}
\iffalse
\begin{figure}[ht!
\centering
\includegraphics[width=4.2cm]{images/hotel_pr2_alpha_0_1_small.png}
\includegraphics[width=4.2cm]{images/hotel_likeli_alpha_0_1_med_out_small.png}\vspace{-.2cm}
\caption{$pR^2$ scores and negative per-word log-likelihood on the Hotel Review dataset. The $X$ axis indicates the number of topics. Error bars
indicate the standard deviation of $5$-fold cross-validation. Vocabulary size $V = 5,000$}
\label{hotel}\vspace{-.3cm}
\end{figure}
\begin{figure}[ht!
\centering
\includegraphics[width=4.1cm]{images/hotel_pr2_alpha_0_1_tall.png}
\includegraphics[width=4.1cm]{images/hotel_likeli_alpha_0_1_med_out_tall.png}\vspace{-.2cm}
\caption{$pR^2$ scores and negative per-word log-likelihood on the Hotel Review dataset. The $X$ axis indicates the number of topics. Error bars
indicate the standard deviation of $5$-fold cross-validation. Vocabulary size $V = 5,000$}
\label{hotel}\vspace{-.3cm}
\end{figure}
\fi
\begin{figure}[ht!
\centering
\includegraphics[width=7cm]{images/hotel_pr2_alpha_0_1_fat.png}
\includegraphics[width=7cm]{images/hotel_likeli_alpha_0_1_med_out_fat.png}\vspace{-.2cm}
\caption{$pR^2$ scores and negative per-word log-likelihood on the Hotel Review dataset. The $X$ axis indicates the number of topics. Error bars
indicate the standard deviation of $5$-fold cross-validation.
\label{hotel}\vspace{-.4cm}
\end{figure}
\subsection{Hotel Reviews Dataset}
For real-world datasets, we first test on a relatively small Hotel Review dataset, which consists of $15,000$ documents for training and $3,000$ documents for
testing that are randomly sampled from TripAdvisor website. Each document is associated with
a rating score from $1$ to $5$ and our task is to predict it.
We pre-process the dataset by shifting the review scores so that they have zero mean and unit variance as in~\cite{gibbsmedlda}.
Fig.~\ref{hotel} shows the prediction accuracy and per-word likelihood when the vocabulary size
is $5,000$ and the mean level of $\bm{\alpha}$ is $\hat{\alpha} = 0.1$. As MedLDA adopts a quite different objective from sLDA, we only compare on the prediction accuracy. Comparing with
traditional Gibbs-sLDA and MedLDA, the two-stage spectral method is much worse, while
the joint spectral method is comparable at its optimal value. This result is not surprising
since the convergence rate of regression parameters for the joint method is faster than
that of the two-stage one.
The hybrid method (i.e., Gibbs sampling initialized with the joint spectral method) performs as
well as the state-of-the-art MedLDA. These results show that spectral methods are
good ways to avoid stuck in relatively bad local optimal solution.
\vspace{-.1cm}
\subsection{Amazon Movie Reviews Dataset}
Finally, we report the results on a large-scale real dataset, which is built on Amazon movie reviews~\cite{a:amazon}, to demonstrate
the effectiveness of our spectral methods on improving the prediction accuracy as well as finding discriminative topics.
The dataset consists of $7,911,684$ movie reviews written by $889,176$ users from Aug $1997$ to Oct $2012$. Each review is accompanied
with a rating score from $1$ to $5$ indicating how a user likes a particular movie. The median number of words per
review is $101$. We consider two cases where a vocabulary with $V = 5,000$ terms or $V = 10,000$ is built by selecting high frequency words and deleting
the most common words and some names of characters in movies.
When the vocabulary size $V$ is small (i.e., $5,000$), we run exact SVD for the
whitening step; when $V$ is large (i.e., $10,000$), we run the randomized SVD to approximate the result.
As before, we also pre-process the dataset by shifting the review scores so that they have zero mean and unit variance.
\begin{figure*}[ht!]\vspace{-.1cm
\centering
\iffalse
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_1_2.png}
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_01_2.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_1_2.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_01_2.png}
\fi
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_1_med.png}
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_01_med.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_1_3.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_01_2.png}\vspace{-.2cm}
\caption{$pR^2$ scores and negative per-word log-likelihood on Amazon dataset. The $X$ axis indicates the number of topics. Error bars
indicate the standard deviation of $5$-fold cross-validation. Vocabulary size $V = 5,000$}
\label{fig1}
\end{figure*}
\begin{figure*}[ht!]\vspace{-.1cm
\centering
\iffalse
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_1_1w_2.png}
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_01_V_1w_2.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_1_1w_2.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_01_V1w_2.png}
\fi
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_1_1w_med.png}
\includegraphics[width=7.3cm]{images/amazon_pr2_alpha_0_01_V_1w_med.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_1_1w_med_out.png}
\includegraphics[width=7.3cm]{images/amazon_likeli_alpha_0_01_V_1w_med_out.png}\vspace{-.2cm}
\caption{$pR^2$ scores and negative per-word log-likelihood on Amazon dataset. The $X$ axis indicates the number of topics. Error bars
indicate the standard deviation of $5$-fold cross-validation. Vocabulary size $V = 10,000$}
\label{fig2}\vspace{-.4cm}
\end{figure*}
\vspace{-.15cm}
\subsubsection{Prediction Performance}
Fig.~\ref{fig1} shows the prediction accuracy and per-word log-likelihood when $\bar{\alpha}$ takes different values and the vocabulary size $V = 5,000$, where
$\bar{\alpha} = \alpha_0/k$ denotes the mean level for $\bm{\alpha}$.
We can see that comparing to the classical Gibbs sampling method, our spectral method is
a bit more sensitive to the hyper-parameter $\bm{\alpha}$.
But in both cases, our joint method alone outperforms the Gibbs sampler and the two-stage spectral method.
MedLDA is also sensitive to the hyper-parameter $\bm{\alpha}$.
When $\bm{\alpha}$ is set properly, MedLDA achieves the best result comparing with the other
methods, however, the gap between our joint method and MedLDA is small.
This result is significant for spectral methods, whose practical performance was often much inferior to likelihood-based estimators.
We also note that if $\bar{\alpha}$ is not set properly (e.g., $\bar{\alpha} = 0.01$), a hybrid method that initializes a Gibbs sampler by the results of our spectral methods
can lead to high accuracy, outperforming the Gibbs sampler and MedLDA
with a random initialization.
We use the results of the joint method for initialization because this
gives better performance compared with the two-stage method.
Fig.~\ref{fig2}
shows the results when the vocabulary size $V = 10,000$.
This time the joint spectral method gets the best result while the two-stage method
is comparable with Gibbs sampling but worse than MedLDA.
The hybrid method is comparable with the joint method, demonstrating that this strategy works
well in practice again.
An interesting phenomenon is that the spectral method
gets good results when the topic number is only $3$ or $4$, which means the spectral method can fit the data using fewer topics.
Although there is a rising trend on prediction accuracy
for the hybrid method, we cannot verify this because we cannot get the results of spectral methods when $k$ is large.
The reason is that when $k > 9$, the spectral method fails in the robust
tensor decomposition step, as we get some negative eigenvalues. This phenomenon can be explained by the nature of our methods --- one crucial step in Alg.~\ref{alg1} and Alg.~\ref{a1}
is to whiten $\widehat{M}_2$ which can be done when the underlying topic matrix $O$ ( or joint topic matrix $\widehat{O}^*$) is of full rank. For the Amazon
review dataset, it is impossible to whiten it with more than $9$ topics. This fact can be used for model selection to avoid using too many extra topics.
There is also a rising trend in the Gibbs sampling when $V = 10,000$ as we measure the $pR^2$ indicator,
it reaches peak when topic size $k = 40$ which is about $0.22$ no matter $\alpha_0$ is $0.1$ or $0.01$.
The results may indicate that with a good initialization, the Gibbs sampling method could get much better performance.
Finally, note that Gibbs sampling and the hybrid Gibbs sampling methods get better log-likelihood values. This result is not
surprising because Gibbs sampling is based on MLE while spectral methods do not. Fig.~\ref{fig1} shows that the hybrid Gibbs
sampling achieves the best per-word likelihood. Thus if one's main goal is to maximize likelihood, a hybrid technique is
desirable.
\vspace{-.15cm}
\subsubsection{Parameter Recovery}
We now take a closer investigation of the recovered parameters for our spectral methods.
Table $1$ shows the estimated regression parameters of both methods, with $k=8$ and $V=5,000$.
We can see that the two methods have different ranges of the possible predictions ---
due to the normalization of ${\bm h}$, the range of the predictions by a model with estimate $\widehat{ {\bm \eta} }$ is $[\min( \widehat{ {\bm \eta}}), \max(\widehat{{\bm \eta}})]$.
Therefore, compared with the range provided by the two-stage method (i.e., $[-0.75,0.83]$), the
joint method gives a larger one (i.e. $[-2.00,1.12]$) which better matches the range of the true labels (i.e., $[-3, 1]$) and therefore
leads to more accurate predictions as shown in Fig.~\ref{fig1}.
We also examine the estimated topics by both methods.
For the topics with the large value of $\eta$, positive words (e.g., ``great")
dominate the topics in both spectral methods because the frequencies for them are much higher than negative ones (e.g., ``bad"). Thus we mainly focus on the ``negative" topics where the
difference can be found more expressly.
Table 2 shows the topics correspond to the smallest value of ${\bm \eta}$ by each method.
To save space, for the topic in each method we show the non-neutral words from the top $200$ with highest probabilities.
For each word, we show its probability as well as the rank (i.e., the number in bracket)
in the topic distribution vector.
\begin{wraptable}{r}{.23\textwidth} \vspace{-.5cm}
\caption{Estimated $\boldsymbol \eta$ by the two spectral methods (sorted for ease of comparison).}\label{table:param} \vspace{-.2cm}
\centering
\begin{tabular}{|c|c|}
\hline\hline
Two-stage & Joint \\
\hline
-0.754 & -1.998 \\
-0.385 & -0.762 \\
-0.178 & -0.212 \\
-0.022 & -0.098 \\
0.321 & 0.437 \\
0.522 & 0.946 \\
0.712 & 1.143 \\
0.833 & 1.122 \\
\hline
\end{tabular}\vspace{-.3cm}
\end{wraptable}
We can see that the negative words (e.g., ``bad", ``boring") have a higher rank (on average) in the topic by the joint spectral method than in the topic by the two-stage method,
while the positive words (e.g., ``good", ``great") have a lower rank (on average) in the topic by the joint method than in the topic by the two-stage method.
This result suggests that this topic in the joint method is more strongly associated with the negative reviews, therefore yielding a
better fit of the negative review scores when combined with the estimated ${\bm \eta}$.
Therefore, considering the supervision information can lead to improved topics.
Finally, we also observe that in both topics some positive words (e.g., ``good") have a rather high rank.
This is because the occurrences of such positive words are much frequent than the negative ones.
\begin{table}[t]\vspace{-.15cm}
\caption{Probabilities and ranks (in brackets) of some non-neutral words. N/A means that the word does not appear in the top $200$ ones with highest probabilities.}\vspace{-.2cm}
\centering
\begin{tabular}{|c|l|l|}
\hline\hline
words & Two-stage spec-slda & Joint spec-slda \\
\hline
bad & $ 0.006905 ~(14) $ & $ 0.009864 ~(8) $ \\
boring & $ 0.002163 ~(101) $ & $ 0.002433 ~(63) $ \\
stupid & $ 0.001513 ~(114) $ & $ 0.001841 ~(93) $ \\
horrible & $ 0.001255 ~(121) $ & $ 0.001868 ~(89) $ \\
terrible & $0.001184 ~(136) $ & $0.001868 ~(88) $ \\
waste & $0.001157 ~(140) $ & $ 0.001896 ~(84) $ \\
disappointed & $ 0.000926 ~(171)$ & $0.001282 ~(127) $ \\
\hline
good & $0.012549 ~(7) $ & $0.012033 ~(5) $ \\
great & $0.012549 ~(11) $ & $0.003568 ~(37) $ \\
love & $0.007513 ~(12) $ & $0.003137 ~(46)$ \\
funny & $ 0.004334 ~(26) $ & $ 0.003346 ~(39) $ \\
enjoy & $ 0.002163 ~(77) $ & $ 0.001317 ~(123) $ \\
awesome & $0.001208 ~(132) $ & N/A \\
amazing & $0.001199 ~(133) $ & N/A \\
\hline
\end{tabular}\vspace{-.4cm}
\end{table}
\vspace{-.15cm}
\subsection{Time efficiency}
Finally, we compare the time efficiency with Gibbs sampling.
All algorithms are implemented in C++.
Our methods are very time efficient
because they avoid the time-consuming iterative steps in traditional variational inference and Gibbs sampling methods.
Furthermore, the empirical moment computation, which is the most time-consuming part in Alg.~\ref{alg1} and Alg.~\ref{alg2} when dealing with large-scale datasets,
consists of only elementary operations and can be easily optimized.
Table~\ref{table:time} shows the running time on the synthetic dataset with various sizes in the setting where the topic number is $k=10$, vocabulary size is $V=500$ and document length is $100$.
We can see that both spectral methods are much faster than Gibbs sampling, especially when the data size is large.
Another advantage of our spectral methods is that we can easily parallelize the
computation of the low-order moments over multiple compute nodes, followed by a single step of synchronizing the local moments.
Therefore, the communication cost will be very low, as compared to the distributed algorithms for topic models~\cite{a:yahoolda}
which often involve intensive communications in order to synchronize the messages for (approximately) accurate inference.
\begin{table}[t]\vspace{-.1cm}
\caption{Running time (seconds) of our spectral learning methods and Gibbs sampling.}
\label{table:time}\vspace{-.2cm}
\centering
\begin{tabular}{lllllll}
\hline
$n(\times 5 \times 10^3)$ & 1 & 2 & 4 & 8 & 16 & 32 \\
\hline
Gibbs sampling& 47 & 92 & 167 & 340 & 671 & 1313\\
Joint spec-slda& 11 & 15 & 17 & 28 & 45 & 90\\
Two-stage spec-slda& 10& 13& 15& 22 & 39 & 81\\
\hline
\end{tabular}\vspace{-.3cm}
\end{table}
As a small $k$ is sufficient for the Amazon review dataset, we report the results with different $k$ values on a synthetic dataset where the vocabulary size $V=500$, the document length $m=100$ and the document size $n=1,000,000$. As shown in Fig.~\ref{fig:distribute}, the distributed implementation of our spectral methods (both two-stage and joint) has almost ideal (i.e., linear) speedup
with respect to the number of threads for moments computing.
The computational complexity of the tensor decomposition step is $O(k^{5+\delta})$ for a third-order tensor $T\in \mathbb{R}^{k \times k \times k}$, where $\delta$ is small~\cite{a:tensordecomp}.
When the topic number $k$ is large (e.g., as may be needed in applications with much larger datasets), one can follow the
recent developed stochastic tensor gradient descent (STGD) method to compute the
eigenvalues and eigenvectors~\cite{a:sgdtd}, which
can significantly reduce the running time in the tensor decomposition stage.
\begin{figure}[t
\centering
\includegraphics[width=0.36\textwidth,height=0.25\textwidth]{images/time_new.png}\vspace{-.3cm}
\caption{Running time of our method w.r.t the number of threads. Both $x$ and $y$ axes are plotted in log scale with base $e$.}
\label{fig:distribute}\vspace{-.5cm}
\end{figure}
\vspace{-.15cm}
\section{Conclusions and Discussions}
We propose two novel spectral decomposition methods to recover the parameters of supervised LDA models from labeled documents.
The proposed methods enjoy a provable guarantee of model reconstruction accuracy and are highly efficient
and effective.
Experimental results on real datasets demonstrate that
the proposed methods, especially the joint one, are superior to existing methods.
This result is significant for spectral methods, which were often inferior to MLE-based methods in practice.
For further work, it is interesting to recover parameters when the regression model is non-linear.
\section*{Acknowledgements}
This work is supported by the National 973 Basic Research Program of China (Nos. 2013CB329403, 2012CB316301), National NSF of China (Nos. 61322308, 61332007), and Tsinghua Initiative Scientific Research Program (No. 20141080934).
\bibliographystyle{plain}
\section{Proof to Theorem 1}
In this section, we prove the sample complexity bound given in Theorem 1. The proof consists of three main parts.
In Appendix A.1, we prove perturbation lemmas that bound the estimation error of the whitened tensors
$M_2(W,W), M_y(W,W)$ and $M_3(W,W,W)$ in terms of the estimation error of the tensors themselves.
In Appendix A.2, we cite results on the accuracy of SVD and robust tensor power method when performed on estimated tensors,
and prove the effectiveness of the power update method used in recovering the linear regression model $\bm\eta$.
Finally, we give tail bounds for the estimation error of $M_2,M_y$ and $M_3$ in Appendix A.3
and complete the proof in Appendix A.4.
We also make some remarks on the indirect quantities (e.g. $\sigma_k(\widetilde O)$) used in Theorem 1
and simplified bounds for some special cases in Appendix A.4.
All norms in the following analysis, if not explicitly specified, are 2 norms in the vector and matrix cases and the operator norm in the high-order tensor case.
\subsection{Perturbation lemmas}
We first define the canonical topic distribution vectors $\widetilde{\bm\mu}$ and estimation error of observable tensors,
which simplify the notations that arise in subsequent analysis.
\begin{deff}[canonical topic distribution]
Define the canonical version of topic distribution vector $\bm\mu_i$, $\widetilde{\bm\mu}_i$, as follows:
\begin{equation}
\widetilde{\bm\mu}_i \triangleq \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}}\bm\mu_i.
\label{def_can_mu}
\end{equation}
We also define $O,\widetilde O\in\mathbb R^{n\times k}$ by $O = [\bm\mu_1,\cdots,\bm\mu_k]$ and $\widetilde{O} = [\widetilde{\bm\mu_1},\cdots,\widetilde{\bm\mu_k}]$.
\end{deff}
\begin{deff}[estimation error]
Assume
\begin{eqnarray}
\|M_2-\widehat M_2\| &\leq& E_P,\\
\|M_y-\widehat M_y\| &\leq& E_y,\\
\|M_3-\widehat M_3\| &\leq& E_T.
\end{eqnarray}
for some real values $E_P,E_y$ and $E_T$, which we will set later.
\end{deff}
The following lemma analyzes the whitening matrix $W$ of $M_2$.
Many conclusions are directly from \cite{speclda}.
\begin{lem}[Lemma C.1, \cite{speclda}]
Let $W, \widehat W\in\mathbb R^{n\times k}$ be the whitening matrices such that $M_2(W,W) = \widehat M_2(\widehat W,\widehat W) = I_k$. Let $A = W^\top\widetilde O$ and $\widehat A = \widehat W^\top\widetilde O$. Suppose $E_P \leq \sigma_k(M_2)/2$. We have
\begin{eqnarray}
\|W\| &=& \frac{1}{\sigma_k(\widetilde O)},\\
\|\widehat W\| &\leq& \frac{2}{\sigma_k(\widetilde O)},\label{eq6_pert}\\
\|W-\widehat W\| &\leq& \frac{4E_P}{\sigma_k(\widetilde O)^3},\label{eq_wwhat}\\
\|W^+\| &\leq& 3\sigma_1(\widetilde O),\\
\|\widehat W^+\| &\leq& 2\sigma_1(\widetilde O),\\
\|W^+ - \widehat W^+\| &\leq& \frac{6\sigma_1(\widetilde O)}{\sigma_k(\widetilde O)^2}E_P,\\
\|A\| &=& 1,\\
\|\widehat A\| &\leq & 2,\\
\|A - \widehat A\| &\leq& \frac{4E_P}{\sigma_k(\widetilde O)^2},\\
\|AA^\top - \widehat A\widehat A^\top\| &\leq& \frac{12E_P}{\sigma_k(\widetilde O)^2}.\label{eq4_pert}
\end{eqnarray}
\end{lem}
\begin{proof}
Proof to Eq. (\ref{eq_wwhat}):
Let $\widehat W^\top\widehat M_2\widehat W = I$ and $\widehat W^\top M_2\widehat W = BDB^\top$, where $B$ is orthogonal and $D$ is a positive definite diagonal matrix.
We then see that $W = \widehat WBD^{-1/2}B^\top$ satisfies the condition $W M_2 W^\top = I$.
Subsequently, $\widehat W = WBD^{1/2}B^\top$.
We then can bound $\|W-\widehat W\|$ as follows
$$ \|W-\widehat W\| \leq \|W\|\cdot \|I-D^{1/2}\| \leq \|W\|\cdot \|I-D\| \leq \frac{4E_P}{\sigma_k(\widetilde O)^3},$$
where the inequality $\|I-D\|\leq \frac{4E_P}{\sigma_k(\widetilde O)^2}$ was proved in \cite{speclda}.
Proof to Eq. (\ref{eq4_pert}):
$\|AA^\top-\widehat A\widehat A^\top\| \leq \|AA^\top-A\widehat A^\top\| + \|A\widehat A^\top - \widehat A\widehat A^\top\|
\leq \|A-\widehat A\|\cdot (\|A\|+\|\widehat A\|) \leq \frac{12E_P}{\sigma_k(\widetilde O)^2}$.
All the other inequalities come from Lemma C.1, \cite{speclda}.
\end{proof}
We are now able to provide perturbation bounds for estimation error of whitened moments.
\begin{deff}[estimation error of whitened moments]
Define
\begin{eqnarray}
\varepsilon_{p,w} &\triangleq& \|M_2(W, W) - \widehat M_2(\widehat W,\widehat W)\|,\\
\varepsilon_{y,w} &\triangleq& \|M_y(W, W) - \widehat M_y(\widehat W,\widehat W)\|,\\
\varepsilon_{t,w} &\triangleq& \|M_3(W,W,W) - \widehat M_3(\widehat W,\widehat W,\widehat W)\|.
\end{eqnarray}
\label{def_epsilon}
\end{deff}
\begin{lem}[Perturbation lemma of whitened moments]
Suppose $E_P\leq \sigma_k(M_2)/2$. We have
\begin{eqnarray}
\varepsilon_{p,w} &\leq& \frac{16E_P}{\sigma_k(\widetilde O)^2},\\
\varepsilon_{y,w} &\leq& \frac{24\|\bm\eta\|E_P}{(\alpha_0+2)\sigma_k(\widetilde O)^2} + \frac{4E_y}{\sigma_k(\widetilde O)^2},\\
\varepsilon_{t,w} &\leq& \frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5} + \frac{8E_T}{\sigma_k(\widetilde O)^3}.
\end{eqnarray}
\end{lem}
\begin{proof}
Using the idea in the proof of Lemma C.2 in \cite{speclda}, we can split $\varepsilon_{p,w}$ as
\begin{equation*}
\begin{aligned}
&\varepsilon_{p,w} \\
=& \|M_2(W,W) - M_2(\widehat W,\widehat W) + M_2(\widehat W,\widehat W) - \widehat M_2(\widehat W,\widehat W)\|\\
\leq& \|M_2(W,W) - M_2(\widehat W,\widehat W)\| + \|M_2(\widehat W,\widehat W) - \widehat M_2(\widehat W,\widehat W)\|.
\end{aligned}
\end{equation*}
We can the bound the two terms seperately, as follows.
For the first term, we have
\begin{eqnarray*}
\|M_2(W,W) - M_2(\widehat W,\widehat W)\|
&=& \|W^\top M_2 W-\widehat W^\top\widehat M_2\widehat W\|\\
&=& \|AA^\top - \widehat A\widehat A^\top\|\\
&\leq& \frac{12E_P}{\sigma_k(\widetilde O)^2}.
\end{eqnarray*}
where the last inequality comes from Eq. (\ref{eq4_pert})
For the second term, we have
$$\|M_2(\widehat W,\widehat W)-\widehat M_2(\widehat W,\widehat W)\| \leq \|\widehat W\|^2\cdot \|M_2-\widehat M_2\| \leq \frac{4E_P}{\sigma_k(\widetilde O)^2},$$
where the last inequality comes from Eq. (\ref{eq6_pert}).
Similarly, $\varepsilon_{y,w}$ can be splitted as $\|M_y(W,W)-M_y(\widehat W,\widehat W)\|$ and $\|M_y(\widehat W,\widehat W)-\widehat M_y(\widehat W,\widehat W)\|$, which can be bounded separately.
For the first term, we have
\begin{equation*}
\begin{aligned}
&\|M_y(W,W) - M_y(\widehat W,\widehat W)\| \\
=& \|W^\top M_y W - \widehat W^\top M_y\widehat W\|\\
=& \frac{2}{\alpha_0+2}\|A\mathrm{diag}(\bm\eta)A^\top - \widehat A\mathrm{diag}(\bm\eta)\widehat A^\top\|\\
\leq& \frac{2\|\bm\eta\|}{\alpha_0+2}\cdot \|AA^\top - \widehat A\widehat A^\top\|\\
\leq& \frac{24\|\bm\eta\|}{(\alpha_0+2)\sigma_k(\widetilde O)^2}\cdot E_P.
\end{aligned}
\end{equation*}
For the second term, we have
$$\|M_y(\widehat W,\widehat W)-\widehat M_y(\widehat W,\widehat W)\| \leq \|\widehat W\|^2\cdot\|M_y-\widehat M_y\| \leq \frac{4E_y}{\sigma_k(\widetilde O)^2}.$$
Finally, we bound $\varepsilon_{t,w}$ as below, following the work \cite{specregression}.
\begin{eqnarray*}
\varepsilon_{t,w} &=& \|M_3(W,W,W) - \widehat M_3(\widehat W,\widehat W,\widehat W)\|\\
&\leq& \|M_3\|\cdot \|W-\widehat W\|\cdot (\|W\|^2 + \|W\|\cdot \|\widehat W\|\\
&\ \ +& \|\widehat W\|^2) + \|\widehat W\|^3\cdot \|M_3-\widehat M_3\|\\
&\leq& \frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5} + \frac{8E_T}{\sigma_k(\widetilde O)^3},
\end{eqnarray*}
where we have used the fact that $$\|M_3\| \leq \sum_{i=1}^k{\frac{2\alpha_i}{\alpha_0(\alpha_0+1)(\alpha_0+2)}} = \frac{2}{(\alpha_0+1)(\alpha_0+2)}$$.
\end{proof}
\subsection{SVD accuracy}
The key idea for spectral recovery of LDA topic modeling is the \emph{simultaneous diagonalization} trick,
which asserts that we can recover LDA model parameters by performing orthogonal tensor decomposition
on a pair of simultaneously whitened moments, for example, $(M_2, M_3)$ and $(M_2, M_y)$.
The following proposition details this insight, as we derive orthogonal tensor decompositions for the whitened tensor product
$M_y(W,W)$ and $M_3(W,W,W)$.
\begin{prop}
Define $\bm v_i\triangleq W^\top \widetilde{\bm\mu}_i = \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}}W^\top\bm\mu_i$. Then
\begin{enumerate}
\item $\{\bm v_i\}_{i=1}^k$ is an orthonormal basis.
\item $M_y$ has a pair of singular value and singular vector $(\sigma_i^y, \bm v_i)$ with $\sigma_i^y = \frac{2}{\alpha_0+2}\eta_j$ for some $j \in[k]$.
\item $M_3$ has a pair of robust eigenvalue and eigenvector \cite{tensorpower} $(\lambda_i, \bm v_i)$ with $\lambda_i = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{j'}}}$ for some $j'\in[k]$.
\end{enumerate}
\end{prop}
\begin{proof}
The orthonormality of $\{\bm v_i\}_{i=1}^k$ follows from the fact that $W^\top M_2W = \sum_{i=1}^k{\bm v_i\bm v_i^\top} = I_k$.
Subsequently, we have
\begin{eqnarray*}
M_y(W, W) &=& \frac{2}{\alpha_0+2}\sum_{i=1}^k{\eta_i \bm v_i\bm v_i^\top},\\
M_3(W,W,W) &=& \frac{2}{\alpha_0+2}\sum_{i=1}^k{\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_i}}\bm v_i\otimes\bm v_i\otimes\bm v_i}.
\end{eqnarray*}
\end{proof}
The following lemmas (Lemma \ref{lem_eta} and Lemma \ref{lem_mu})
give upper bounds on the estimation error of $\bm\eta$ and $\bm\mu$
in terms of $|\widehat\lambda_i-\lambda_i|$, $|\widehat{\bm v}_i-\bm v_i|$
and the estimation errors of whitened moments defined in Definition \ref{def_epsilon}.
\begin{lem}[$\eta_i$ estimation error bound]
Define $\widehat\eta_i \triangleq \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i$,
where $\widehat{\bm v}_i$ is some estimation of $\bm v_i$.
We then have
\begin{equation}
|\eta_i - \widehat\eta_i| \leq 2\|\bm\eta\|\|\widehat{\bm v}_i-\bm v_i\| + \frac{\alpha_0+2}{2}(1+2\|\widehat{\bm v}_i-\bm v_i\|)\cdot\varepsilon_{y,w}.
\end{equation}
\label{lem_eta}
\end{lem}
\begin{proof}
First, note that $\bm v_i^\top M_y(W,W)\bm v_i = \frac{2}{\alpha_0+2}\eta_i$ because $\{\bm v_i\}_{i=1}^k$ are orthonormal.
Subsequently, we have
\begin{equation*}
\begin{aligned}
&\frac{2}{\alpha_0+2}|\eta_i - \widehat\eta_i|\\
=& \Big|\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - \bm v_i^\top M_y(W,W)\bm v_i\Big| \\
\leq& \Big| (\widehat{\bm v}_i-\bm v_i)^\top \widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i\Big| + \Big|\bm v_i^\top\left(\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\right)\Big|\\
\leq& \|\widehat{\bm v}_i-\bm v_i\|\|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i\| + \|\bm v_i\|\|\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\|.\\
\end{aligned}
\end{equation*}
Note that both $\bm v_i$ and $\widehat{\bm v}_i$ are unit vectors. Therefore,
\begin{equation*}
\begin{aligned}
&\frac{2}{\alpha_0+2}|\eta_i-\widehat\eta_i|\\
\leq& \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i-\bm v_i\| + \|\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\|\\
\leq& \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i-\bm v_i\| + \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i - \bm v_i\|\\& \ \ \ \ + \|\widehat M_y(\widehat W,\widehat W)-M_y(W,W)\|\|\bm v_i\|\\
\leq& 2\|\widehat{\bm v}_i-\bm v_i\|\left(\frac{2}{\alpha_0+2}\|\bm\eta\| + \varepsilon_{y,w}\right) + \varepsilon_{y,w}.
\end{aligned}
\end{equation*}
The last inequality is due to the fact that $\|M_y(W,W)\| = \frac{2}{\alpha_0+2}\|\bm\eta\|$.
\end{proof}
\begin{lem}[$\bm\mu_i$ estimation error bound]
Define $\widehat{\bm\mu}_i \triangleq \frac{\alpha_0+2}{2}\widehat\lambda_i(\widehat W^+)^\top\widehat{\bm v}_i$,
where $\widehat\lambda_i, \widehat{\bm v}_i$ are some estimates of singular value pairs $(\lambda_i, \bm v_i)$ of $M_3(W,W,W)$.
We then have
\begin{equation}
\begin{aligned}
\|\widehat{\bm\mu}_i - \bm\mu_i\| \leq& \frac{3(\alpha_0+2)}{2}\sigma_1(\widetilde O)|\widehat\lambda_i-\lambda_i| \\
&+ 3\alpha_{\max}\sigma_1(\widetilde O)\|\widehat{\bm v}_i-\bm v_i\|
+ \frac{6\alpha_{\max}\sigma_1(\widetilde O)E_P}{\sigma_k(\widetilde O)^2}.
\end{aligned}
\end{equation}
\label{lem_mu}
\end{lem}
\begin{proof}
First note that $\bm\mu_i = \frac{\alpha_0+2}{2}\lambda_i (W^+)^\top\bm v_i$. Subsequently,
\begin{equation*}
\begin{aligned}
&\frac{2}{\alpha_0+2}\|\bm\mu_i - \widehat{\bm\mu}_i\|\\
=& \|\widehat\lambda_i(\widehat W^+)^\top\widehat{\bm v}_i - \lambda_i(W^+)^\top\bm v_i\|\\
\leq& \|\widehat\lambda_i\widehat W^+ - \lambda_i W^+\|\|\widehat{\bm v}_i\| + \|\lambda_i W^+\|\|\widehat{\bm v}_i-\bm v_i\|\\
\leq& |\widehat\lambda_i-\lambda_i|\|\widehat W^+\| + |\lambda_i|\|\widehat W^+ - W^+\| + |\lambda_i|\|W^+\|\|\widehat{\bm v}_i-\bm v_i\|\\
\leq& 3\sigma_1(\widetilde O)|\widehat\lambda_i-\lambda_i| + \frac{2\alpha_{\max}}{\alpha_0+2}\cdot \frac{6\sigma_1(\widetilde O) E_P}{\sigma_k(\widetilde O)^2}\\
&+ \frac{2\alpha_{\max}}{\alpha_0+2}\cdot 3\sigma_1(\widetilde O)\cdot \|\widehat{\bm v_i} - \bm v_i\|.
\end{aligned}
\end{equation*}
\end{proof}
To bound the error of orthogonal tensor decomposition performed on the estimated tensors $\widehat M_3(\widehat W,\widehat W,\widehat W)$,
we cite Theorem 5.1 \cite{tensorpower}, a sample complexity analysis on the robust tensor power method we used
for recovering $\widehat\lambda_i$ and $\widehat{\bm v}_i$.
\begin{lem}[Theorem 5.1, \cite{tensorpower}]
Let $\lambda_{\max} = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$,
$\lambda_{\min} = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\max}}}$,
where $\alpha_{\min} = \min{\alpha_i}$ and $\alpha_{\max} = \max{\alpha_i}$.
Then there exist universal constants $C_1, C_2 > 0$ such that the following holds:
Fix $\delta'\in(0,1)$.
Suppose $\varepsilon_{t,w}\leq \varepsilon$ and
\begin{eqnarray}
\varepsilon_{t, w} &\leq& C_1\cdot \frac{\lambda_{\min}}{k},\label{eq1_tensorpower}
\end{eqnarray}
Suppose $\{(\widehat\lambda_i, \widehat{\bm v}_i)\}_{i=1}^k$ are eigenvalue and eigenvector pairs
returned by running Algorithm 1 in \cite{tensorpower} with input $\widehat M_3(\widehat W,\widehat W,\widehat W)$ for $L = \text{poly}(k)\log(1/\delta')$ and $N \geq C_2\cdot(\log(k) + \log\log(\frac{\lambda_{\max}}{\varepsilon}))$ iterations.
With probability greater than $1-\delta'$, there exists a permutation $\pi':[k]\to[k]$ such that
for all $i$,
$$\|\widehat{\bm v}_i - \bm v_{\pi'(i)}\| \leq 8\varepsilon/\lambda_{\min},\quad |\widehat\lambda_i - \lambda_{\pi'(i)}| \leq 5\varepsilon.$$
\label{lem_tensorpower}
\end{lem}
\subsection{Tail Inequalities}
\begin{lem}[Lemma 5, \cite{specregression}]
Let $\bm x_1,\cdots,\bm x_N\in\mathbb R^d$ be i.i.d. samples from some distribution with bounded support (i.e., $\|\bm x\|_2\leq B$ with probability 1 for some constant $B$).
Then with probability at least $1-\delta$,
$$\left\|\frac{1}{N}\sum_{i=1}^N{\bm x_i} - \mathbb E[\bm x]\right\|_2 \leq \frac{2B}{\sqrt{N}}\left(1+\sqrt{\frac{\log(1/\delta)}{2}}\right).$$
\label{lem_chernoff}
\end{lem}
\begin{cor}
Let $\bm x_1,\cdots,\bm x_N\in\mathbb R^d$ be i.i.d. samples from some distributions with $\Pr[\|\bm x\|_2 \leq B] \geq 1-\delta'$.
Then with probability at least $1-N\delta'-\delta$,
$$\left\|\frac{1}{N}\sum_{i=1}^N{\bm x_i} - \mathbb E[\bm x]\right\|_2 \leq \frac{2B}{\sqrt{N}}\left(1+\sqrt{\frac{\log(1/\delta)}{2}}\right).$$
\label{cor_chernoff}
\end{cor}
\begin{proof}
Use union bound.
\end{proof}
\begin{lem}[concentraion of moment norms]
Suppose we obtain $N$ i.i.d. samples (i.e., documents with at least three words each and their regression variables in sLDA models).
Define $R(\delta) \triangleq \|\bm\eta\|+ \Phi^{-1}(\delta)$,
where $\Phi^{-1}(\cdot)$ is the inverse function of the CDF of a standard Gaussian distribution.
Let $\mathbb E[\cdot]$ denote the mean of the true underlying distribution
and $\widehat{\mathbb E}[\cdot]$ denote the empirical mean.
Then
\begin{enumerate}
\item $\Pr\left[\|\mathbb E[\bm x_1] - \widehat{\mathbb E}[\bm x_1]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[\bm x_1\otimes\bm x_2\otimes\bm x_3] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y] - \widehat{\mathbb E}[y]\| < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}} \right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y\bm x_1] - \widehat{\mathbb E}[y\bm x_1]\|_F < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[y\bm x_1\otimes\bm x_2]\|_F < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}}\right] \geq 1 - \delta$.
\end{enumerate}
\label{lem_tail_ineq}
\end{lem}
\begin{proof}
Use Lemma \ref{lem_chernoff} and Corrolary \ref{cor_chernoff} for concentration bounds involving the regression variable $y$.
\end{proof}
\begin{cor}
With probability $1-\delta$ the following holds:
\begin{enumerate}
\item $E_P = \|M_2-\widehat M_2\| \leq 3\cdot \frac{2+\sqrt{2\log(6/\delta)}}{\sqrt{N}}$.
\item $E_y = \|M_y-\widehat M_y\| \leq 10R(\delta/60\sigma N)\cdot \frac{2+\sqrt{2\log(15/\delta)}}{\sqrt{N}}$.
\item $E_T = \|M_3-\widehat M_3\| \leq 10\cdot \frac{2+\sqrt{2\log(9/\delta)}}{\sqrt{N}}$.
\end{enumerate}
\label{cor_ep}
\end{cor}
\begin{proof}
Corrolary \ref{cor_ep} can be proved by expanding the terms by definition and then using tail inequality in Lemma \ref{lem_tail_ineq} and union bound.
Also note that $\|\cdot\| \leq \|\cdot\|_F$ for all matrices.
\end{proof}
\subsection{Completing the proof}
We are now ready to give a complete proof to Theorem \ref{thm_main}.
\begin{thm}[Sample complexity bound]
Let $\sigma_1(\widetilde O)$ and $\sigma_k(\widetilde O)$ be the largest and the smallest singular values of the canonical topic distribution matrix $\widetilde O$.
Define $\lambda_{\min} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\max}}}$
and $\lambda_{\max} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$,
where $\alpha_{\max}$ and $\alpha_{\min}$ are the largest and the smallest entries in $\bm\alpha$.
Suppose $\widehat{\bm\mu}$, $\widehat{\bm\alpha}$ and $\widehat{\bm\eta}$ are the outputs of Algorithm 1,
and $L$ is at least a linear function in $k$.
Fix $\delta\in(0,1)$.
For any small error-tolerance parameter $\varepsilon > 0$,
if Algorithm 1 is run with parameter $T = \Omega(\log(k) + \log\log(\lambda_{\max}/\varepsilon))$
on $N$ i.i.d. sampled documents with
$N\geq \max(n_1, n_2, n_3)$, where
\begin{eqnarray}
n_1 &=& C_1\cdot \left(1+\sqrt{\log(6/\delta)}\right)^2\cdot \frac{\alpha_0^2(\alpha_0+1)^2}{\alpha_{\min}}, \label{eq_n1} \\
n_2 &=& C_2\cdot \frac{(1+\sqrt{\log(15/\delta)})^2}{\varepsilon^2\sigma_k(\widetilde O)^4}
\\
&\ &\cdot\max\left((\|\bm\eta\| + \Phi^{-1}(\delta/60\sigma))^2, \alpha_{\max}^2\sigma_1(\widetilde O)^2\right),
\label{eq_n2} \\
n_3 &=& C_3\cdot \frac{(1+\sqrt{\log(9/\delta)})^2}{\sigma_k(\widetilde O)^{10}}\cdot\max\left(\frac{1}{\varepsilon^2}, \frac{k^2}{\lambda_{\min}^2}\right),
\label{eq_n3}
\end{eqnarray}
and $C_1, C_2$ and $C_3$ are universal constants,
then with probability at least $1-\delta$, there exists a permutation $\pi:[k]\to [k]$ such that for every topic $i$, the following holds:\\
\begin{enumerate}
\item $|\alpha_i-\widehat{\alpha}_{\pi(i)}| \leq \frac{4\alpha_0(\alpha_0+1)(\lambda_{\max}+5\varepsilon)}{(\alpha_0+2)^2\lambda_{\min}^2(\lambda_{\min}-5\varepsilon)^2}\cdot 5\varepsilon$, if $\lambda_{\min} > 5\varepsilon$\\;
\item $\|\bm\mu_i - \widehat{\bm\mu}_{\pi(i)}\| \leq \left(3\sigma_1(\widetilde O)\left(\frac{8\alpha_{\max}}{\lambda_{\min}} + \frac{5(\alpha_0+2)}{2}\right) + 1\right)\varepsilon$\\;
\item $|\eta_i - \widehat\eta_{\pi(i)}| \leq \left(\frac{\|\bm\eta\|}{\lambda_{\min}} + (\alpha_0+2)\right) \varepsilon$.
\end{enumerate}
\label{thm_main}
\end{thm}
\begin{proof}
First, the assumption $E_P\leq\sigma_k(M_2)$ is required for error bounds on $\varepsilon_{p,w}, \varepsilon_{y,w}$ and $\varepsilon_{t,w}$.
Noting Corrolary \ref{cor_ep} and the fact that $\sigma_k(M_2) = \frac{\alpha_{\min}}{\alpha_0(\alpha_0+1)}$, we have
$$N = \Omega\left(\frac{\alpha_0^2(\alpha_0+1)^2(1+\sqrt{\log(6/\delta)})^2}{\alpha_{\min}^2}\right).$$
Note that this lower bound does not depend on $k$, $\varepsilon$ and $\sigma_k(\widetilde O)$.
For Lemma \ref{lem_tensorpower} to hold, we need the assumptions that
$\varepsilon_{t,w} \leq \min(\varepsilon, O(\frac{\lambda_{\min}}{k}))$.
These imply Eq. (\ref{eq_n3}),
as we expand $\varepsilon_{t,w}$ according to Definition \ref{def_epsilon}
and note the fact that the first term $\frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5}$ dominates the second one.
The $\alpha_0$ is missing in Eq. (\ref{eq_n3}) because $\alpha_0+1 \geq 1$, $\alpha_0+2 \geq 2$ and we discard them both.
The $|\alpha_i-\widehat{\alpha}_{\pi(i)}|$ bound follows immediately by Lemma \ref{lem_tensorpower}
and the recovery rule $\widehat{\alpha}_i = \frac{\alpha_0+2}{2}\widehat\lambda_i$.
To bound the estimation error for the linear classifier $\bm\eta$, we need to further bound $\varepsilon_{y,w}$.
We assume $\varepsilon_{y,w}\leq\varepsilon$.
By expanding $\varepsilon_{y,w}$ according to Definition \ref{def_epsilon} in a similar manner
we obtain the $(\|\bm\eta\|+\Phi^{-1}(\delta/60\sigma))^2$ term in Eq. (\ref{eq_n2}).
The bound on $|\eta_i-\widehat\eta_{\pi(i)}|$ follows immediately by Lemma \ref{lem_eta}.
Finally, we bound $\|\bm\mu_i-\widehat{\bm\mu}_{\pi(i)}\|$ using Lemma \ref{lem_mu}.
We need to assume that $\frac{6\alpha_{\max}\sigma_1(\widetilde O)E_P}{\sigma_k(\widetilde O)^2} \leq \varepsilon$,
which gives the $\alpha_{\max}^2\sigma_1(\widetilde O)^2$ term in Eq. (\ref{eq_n2}).
The $\|\bm\mu_i-\widehat{\bm\mu}_{\pi(i)}\|$ bound then follows by Lemma \ref{lem_mu} and Lemma \ref{lem_tensorpower}.
\end{proof}
We make some remarks for the main theorem. In Remark~\ref{rem_connection}, we establish links between indirect quantities appeared in Theorem~\ref{thm_main} (e.g., $\sigma_k(\widetilde O)$) and the functions of original model parameters (e.g., $\sigma_k(O)$).
These connections are straightforward following their definitions.
\begin{rem}
The indirect quantities $\sigma_1(\widetilde O)$ and $\sigma_k(\widetilde O)$ can be related to $\sigma_1(O)$, $\sigma_k(O)$ and $\bm\alpha$ in the following way:
{\small \begin{equation*}
\sqrt{\frac{\alpha_{\min}}{\alpha_0(\alpha_0+1)}}\sigma_k(O) \leq \sigma_k(\widetilde O)\leq \sqrt{\frac{\alpha_{\max}}{\alpha_0(\alpha_0+1)}}\sigma_k(O);~~~~
\end{equation*}
\begin{equation*}
\sigma_1(\widetilde O) \leq \sqrt{\frac{\alpha_{\max}}{\alpha_0(\alpha_0+1)}}\sigma_1(O) \leq \frac{1}{\sqrt{\alpha_0+1}}.
\end{equation*}}
\label{rem_connection}
\end{rem}
We now take a close look at the sample complexity bound in Theorem~\ref{thm_main}.
It is evident that $n_2$ can be neglected when the number of topics $k$ gets large,
because in practice the norm of the linear regression model $\bm\eta$ is usually assumed to be small in order to avoid overfitting.
Moreover, as mentioned before, the prior parameter $\bm\alpha$ is often assumed to be homogeneous with $\alpha_i = 1/k$ \cite{lsa}.
With these observations, the sample complexity bound in Theorem~\ref{thm_main}
can be greatly simplified.
\begin{rem}
Assume $\|\bm\eta\|$ and $\sigma$ are small and $\bm\alpha = (1/k, \cdots, 1/k)$.
As the number of topics $k$ gets large, the sample complexity bound in Theorem~\ref{thm_main} can be simplified as
\begin{equation}
N = \Omega\left( \frac{\log(1/\delta)}{\sigma_k(\widetilde O)^{10}}\cdot\max(\varepsilon^{-2}, k^3)\right).
\end{equation}\label{rem_simpbound}
\end{rem}\vspace{-.5cm}
The sample complexity bound in Remark~\ref{rem_simpbound} may look formidable
as it depends on $\sigma_k(\widetilde O)^{10}$.
However, such dependency is somewhat necessary because we are using third-order tensors to recover the underlying model parameters.
Furthermore, the dependence on $\sigma_k(\widetilde O)^{10}$ is introduced by the robust tensor power method to recover LDA parameters,
and the reconstruction accuracy of $\bm\eta$ only depends on $\sigma_k(\widetilde O)^4$ and $(\|\bm\eta\|+\Phi^{-1}(\delta/60\sigma))^2$.
As a consequence, if we can combine our power update method for $\bm \eta$ with LDA inference algorithms that have milder dependence on the singular value $\sigma_k(\widetilde O)$,
we might be able to get an algorithm with a better sample complexity.
Such an extension is discussed in Appendix~C.1.
\section{Moments of Observable Variables}
\subsection{Proof to Proposition 1}
The equations on $M_2$ and $M_3$ have already been proved in \cite{speclda} and \cite{tensorpower}.
Here we only give the proof to the equation on $M_y$.
In fact, all the three equations can be proved in a similar manner.
In sLDA the topic mixing vector $\bm h$ follows a Dirichlet prior distribution with parameter $\bm\alpha$.
Therefore, we have
\begin{equation}
\begin{aligned}
&\mathbb E[h_i] = \frac{\alpha_i}{\alpha_0},\\
&\mathbb E[h_ih_j] = \left\{\begin{array}{ll} \frac{\alpha_i^2}{\alpha_0(\alpha_0+1)}, &\text{if }i=j,\\
\frac{\alpha_i\alpha_j}{\alpha_0^2}, &\text{if }i\neq j\end{array}\right.,\\
&\mathbb E[h_ih_jh_k] = \left\{\begin{array}{ll} \frac{\alpha_i^3}{\alpha_0(\alpha_0+1)(\alpha_0+2)}, & \text{if }i=j=k,\\
\frac{\alpha_i^2\alpha_k}{\alpha_0^2(\alpha_0+1)}, & \text{if }i=j\neq k,\\
\frac{\alpha_i\alpha_j\alpha_k}{\alpha_0^3}, &\text{if }i\neq j, j\neq k, i\neq k\end{array}\right.
\end{aligned}
\end{equation}
Next, note that
\begin{equation}
\begin{aligned}
&\mathbb E[y|\bm h] = \bm\eta^\top\bm h,\\
&\mathbb E[\bm x_1|\bm h] = \sum_{i=1}^k{h_i\bm\mu_i},\\
&\mathbb E[\bm x_1\otimes\bm x_2|\bm h] = \sum_{i, j=1}^k{h_ih_j\bm\mu_i\otimes\bm\mu_j},\\
&\mathbb E[y\bm x_1\otimes\bm x_2|\bm h] = \sum_{i, j, k=1}^k{h_ih_jh_k\cdot \eta_k\bm\mu_j\otimes\bm\mu_k}.
\end{aligned}
\end{equation}
Proposition 1 can then be proved easily by taking expectation over the topic mixing vector $\bm h$.
\subsection{Details of the speeding-up trick}
In this section we provide details of the trick mentioned in the main paper to speed up empirical moments computations.
First, note that the computation of $\widehat M_1$, $\widehat M_2$ and $\widehat M_y$ only requires $O(NM^2)$ time and $O(V^2)$ space.
They do not need to be accelerated in most practical applications.
This time and space complexity also applies to all terms in $\widehat M_3$ except the $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]$ term,
which requires $O(NM^3)$ time and $O(V^3)$ space if using naive implementations.
Therefore, this section is devoted to speed-up the computation of $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]$.
More precisely, as mentioned in the main paper,
what we want to compute is the whitened empirical moment $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3](\widehat W,\widehat W,\widehat W) \in
\mathbb R^{k\times k\times k}$.
Fix a document $D$ with $m$ words.
Let $T \triangleq \widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3|D]$ be the empirical tensor demanded.
By definition, we have
\begin{equation}
T_{i,j,k} = \frac{1}{m(m-1)(m-2)}\left\{\begin{array}{ll}
n_i(n_j-1)(n_k-2),& i=j=k;\\
n_i(n_i-1)n_k,& i=j,j\neq k;\\
n_in_j(n_j-1),& j=k,i\neq j;\\
n_in_j(n_i-1),& i=k,i\neq j;\\
n_in_jn_k,& \text{otherwise};\end{array}\right.
\label{eq_T}
\end{equation}
where $n_i$ is the number of occurrences of the $i$-th word in document $D$.
If $T_{i,j,k} = \frac{n_in_jn_k}{m(m-1)(m-2)}$ for all indices $i,j$ and $k$,
then we only need to compute
$$T(W,W,W) = \frac{1}{m(m-1)(m-2)}\cdot (W\bm n)^{\otimes 3},$$
where $\bm n \triangleq (n_1,n_2,\cdots,n_V)$.
This takes $O(Mk + k^3)$ computational time because $\bm n$ contains at most $M$ non-zero entries,
and the total time complexity is reduced from $O(NM^3)$ to $O(N(Mk+k^3))$.
We now consider the remaining values, where at least two indices are identical.
We first consider those values with two indices the same, for example, $i=j$.
For these indices, we need to subtract an $n_in_k$ term, as shown in Eq. (\ref{eq_T}).
That is, we need to compute the whitened tensor $\Delta(W,W,W)$, where $\Delta\in\mathbb R^{V\times V\times V}$ and
\begin{equation}
\Delta_{i,j,k} = \frac{1}{m(m-1)(m-2)}\cdot \left\{\begin{array}{ll}
n_in_k,& i=j;\\
0,& \text{otherwise}.\end{array}\right.
\end{equation}
Note that $\Delta$ can be written as $\frac{1}{m(m-1)(m-2)}\cdot A\otimes\bm n$,
where $A = \text{diag}(n_1,n_2,\cdots,n_V)$ is a $V\times V$ matrix and $\bm n = (n_1,n_2,\cdots,n_V)$ is defined previously.
As a result, $\Delta(W,W,W) = \frac{1}{m(m-1)(m-2)}\cdot (W^\top AW)\otimes \bm n$.
So the computational complexity of $\Delta(W,W,W)$ depends on how we compute $W^\top AW$.
Since $A$ is a diagonal matrix with at most $M$ non-zero entries, $W^\top AW$ can be computed in $O(Mk^2)$ operations.
Therefore, the time complexity of computing $\Delta(W,W,W)$ is $O(Mk^2)$ per document.
Finally we handle those values with three indices the same, that is, $i=j=k$.
As indicated by Eq. (\ref{eq_T}), we need to add a $\frac{2n_i}{m(m-1)(m-2)}$ term for compensation.
This can be done efficiently by first computing $\widehat{\mathbb E}(\frac{2n_i}{m(m-1)(m-2)})$
for all the documents (requiring $O(NV)$ time),
and then add them up, which takes $O(Vk^3)$ operations.
\iffalse
\section{Discussions}
\subsection{Appendix C.1. Extension to other topic recovery algorithms}
One important advantage of our proposed inference algorithm is its flexibility---the algorithm can be combined with many other LDA inference algorithms to infer supervised LDA model parameters.
More specifically, given access to any algorithm that recovers the topic distribution matrix $O$
and the prior parameter $\bm\alpha$ from i.i.d. sampled documents,
an inference algorithm for a supervised LDA model can be immediately obtained,
as shown in Algorithm \ref{alg2}.
\begin{algorithm}
\caption{sLDA parameter recovery based on an existing LDA inference algorithm $\mathcal A$. Input parameter: $\alpha_0$.}
\centering
\begin{algorithmic}[1]
\State Compute empirical moments and obtain $\widehat M_2, \widehat M_y$. Set $\widehat{\bm\eta} = 0$.
\State Find $\widehat W\in\mathbb R^{n\times k}$ such that $\widehat M_2(\widehat W, \widehat W) = I_k$.
\State Run algorithm $\mathcal A$ with observed documents and parameter $\alpha_0$. Obtain the estimated
topic distribution matrix $\widehat O=(\widehat{\bm\mu}_1,\cdots,\widehat{\bm\mu}_k)$ and prior parameter $\widehat{\bm\alpha}$.
\State Compute $\widehat{\bm v}_i = \sqrt{\frac{\widehat{\alpha}_i}{\alpha_0(\alpha_0+1)}}\widehat W^\top\widehat{\bm\mu}_i$ for each topic $i$.
\State Recover linear classifier: $\widehat\eta_i \gets \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i$.
\State \textbf{Output:} $\widehat{\bm\eta}$, $\bm\alpha$ and $\{\widehat{\bm\mu}_i\}_{i=1}^k$.
\end{algorithmic}
\label{alg2}
\end{algorithm}
The sample complexity of Algorithm \ref{alg2} depends on the sample complexity of the LDA inference algorithm $\mathcal A$.
Although the LDA inference algorithm $\mathcal A$ is free to make any assumptions on the topic distribution matrix $O$,
we comment that the linear independence of topic distribution vectors $\bm\mu$ is still required,
because the power update trick (step 5 in Algorithm 1) is valid only $W\bm\mu_i$ are orthogonal vectors.
\subsection*{Appendix C.2. Going beyond SVD}
The proposed methods are based on spectral decomposition of observable moments
and are provably correct.
However, a major drawback of these methods is their assumption that the topic distributions $\bm\mu$ are linearly independent.
Although standard LDA models assume this topic independence \cite{lda},
in practice there is strong evidence that different topics might be related \cite{correlatelda}.
Therefore, it is important to consider possible extensions.
Recently, there has been much work on provable LDA inference algorithms that do not require the topic independence assumption \cite{practicalalgo,beyondsvd}.
Instead of making the independence assumption, these methods assume a $p$-separability condition on the topic distribution matrix,
that is, there exists an \emph{anchor word} for each topic such that the anchor word only appears in this specific topic,
and its probability (given that topic) is at least $p$.
Here we briefly describe an idea that might lead to an sLDA inference algorithm without assuming topic independence.
First, let $O\in\mathbb R^{V\times k}$ be the topic distribution matrix defined previously and $H = (\bm h_1,\cdots,\bm h_N)\in\mathbb R^{k\times N}$
be the matrix of topic mixing vectors for each document.
Suppose $Q = \mathbb E[\bm x_1\bm x_2^\top]$ is the word co-occurrence frequency matrix
and $Q_y = \mathbb E[y\bm x_1]$ is the co-occurrence frequency matrix between word and regression variable observations.
It is clear that both $Q$ and $Q_y$ can be estimated from the training data, and we have
\begin{equation}
Q = ORO^\top, Q_y = OR\bm\eta,
\end{equation}
where $R \triangleq \mathbb E[HH^\top]$.
Assuming the $p$-separability condition holds and using methods developed in \cite{beyondsvd,practicalalgo},
we can consistently recover the $O$ and $R$.
Now note that $Q_y = (OR)\cdot \bm\eta$.
As a result, knowing both $OR$ and $Q_y$ we reduce the $\bm\eta$ inference problem to solving a linear equation systems.
Future research could be done on (1) determining when the matrix $OR$ has full rank $r$ so that the linear equations can be solved, and
(2) investigating the sample complexity problem for such an inference algorithm.
\subsection*{Appendix C.3. From regression to classification}
A natural extension to our result is to consider supervised LDA models for classification purposes.
The simplest model is a logistic regression classification model, where we assume the response variable $y_d$
for each document $d$ is in $\{+1,-1\}$, and
$$\Pr[y_d = 1|\bm h_d] = \frac{\exp(\bm\eta^\top\bm h_d)}{1+\exp(\bm\eta^\top\bm h_d)},$$
where $\bm\eta$ is a linear classifier.
Though appears simple, such an extension incurs many fundamental problems.
A major obstacle is the fact that the conditional expectation $\mathbb E[y|\bm h]$ is no longer linear
in the topic mixing vector $\bm h$.
As a result, we cannot even evaluate $\mathbb E[y]$ (or higher order tensors like $\mathbb E[y\bm x_1\otimes\bm x_2]$)
in closed forms.
Further more, even if we have accurate estimation to the above moments,
it is rather difficult to infer $\bm\eta$ back due to its non-linearality.
To our knowledge, the classification problem remains unsolved in terms of spectral decomposition methods,
and there is doubt whether a provably correct algorithm exists in such scenarios.
\fi
\section{Proof to Theorem 1}
In this section, we prove the sample complexity bound given in Theorem 1. The proof consists of three main parts.
In Appendix A.1, we prove perturbation lemmas that bound the estimation error of the whitened tensors
$M_2(W,W), M_y(W,W)$ and $M_3(W,W,W)$ in terms of the estimation error of the tensors themselves.
In Appendix A.2, we cite results on the accuracy of SVD and robust tensor power method when performed on estimated tensors,
and prove the effectiveness of the power update method used in recovering the linear regression model $\bm\eta$.
Finally, we give tail bounds for the estimation error of $M_2,M_y$ and $M_3$ in Appendix A.3
and complete the proof in Appendix A.4.
We also make some remarks on the indirect quantities (e.g. $\sigma_k(\widetilde O)$) used in Theorem 1
and simplified bounds for some special cases in Appendix A.4.
All norms in the following analysis, if not explicitly specified, are 2 norms in the vector and matrix cases and the operator norm in the high-order tensor case.
\subsection{Perturbation lemmas}
We first define the canonical topic distribution vectors $\widetilde{\bm\mu}$ and estimation error of observable tensors,
which simplify the notations that arise in subsequent analysis.
\begin{deff}[canonical topic distribution]
Define the canonical version of topic distribution vector $\bm\mu_i$, $\widetilde{\bm\mu}_i$, as follows:
\begin{equation}
\widetilde{\bm\mu}_i \triangleq \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}}\bm\mu_i.
\label{def_can_mu}
\end{equation}
We also define $O,\widetilde O\in\mathbb R^{n\times k}$ by $O = [\bm\mu_1,\cdots,\bm\mu_k]$ and $\widetilde{O} = [\widetilde{\bm\mu_1},\cdots,\widetilde{\bm\mu_k}]$.
\end{deff}
\begin{deff}[estimation error]
Assume
\begin{eqnarray}
\|M_2-\widehat M_2\| &\leq& E_P,\\
\|M_y-\widehat M_y\| &\leq& E_y,\\
\|M_3-\widehat M_3\| &\leq& E_T.
\end{eqnarray}
for some real values $E_P,E_y$ and $E_T$, which we will set later.
\end{deff}
The following lemma analyzes the whitening matrix $W$ of $M_2$.
Many conclusions are directly from \cite{a:twosvd}.
\begin{lem}[Lemma C.1, \cite{speclda}]
Let $W, \widehat W\in\mathbb R^{n\times k}$ be the whitening matrices such that $M_2(W,W) = \widehat M_2(\widehat W,\widehat W) = I_k$. Let $A = W^\top\widetilde O$ and $\widehat A = \widehat W^\top\widetilde O$. Suppose $E_P \leq \sigma_k(M_2)/2$. We have
\begin{eqnarray}
\|W\| &=& \frac{1}{\sigma_k(\widetilde O)},\\
\|\widehat W\| &\leq& \frac{2}{\sigma_k(\widetilde O)},\label{eq6_pert}\\
\|W-\widehat W\| &\leq& \frac{4E_P}{\sigma_k(\widetilde O)^3},\label{eq_wwhat}\\
\|W^+\| &\leq& 3\sigma_1(\widetilde O),\\
\|\widehat W^+\| &\leq& 2\sigma_1(\widetilde O),\\
\|W^+ - \widehat W^+\| &\leq& \frac{6\sigma_1(\widetilde O)}{\sigma_k(\widetilde O)^2}E_P,\\
\|A\| &=& 1,\\
\|\widehat A\| &\leq & 2,\\
\|A - \widehat A\| &\leq& \frac{4E_P}{\sigma_k(\widetilde O)^2},\\
\|AA^\top - \widehat A\widehat A^\top\| &\leq& \frac{12E_P}{\sigma_k(\widetilde O)^2}.\label{eq4_pert}
\end{eqnarray}
\label{lem_whiten}
\end{lem}
\begin{proof}
Proof to Eq. (\ref{eq_wwhat}):
Let $\widehat W^\top\widehat M_2\widehat W = I$ and $\widehat W^\top M_2\widehat W = BDB^\top$, where $B$ is orthogonal and $D$ is a positive definite diagonal matrix.
We then see that $W = \widehat WBD^{-1/2}B^\top$ satisfies the condition $W M_2 W^\top = I$.
Subsequently, $\widehat W = WBD^{1/2}B^\top$.
We then can bound $\|W-\widehat W\|$ as follows
$$ \|W-\widehat W\| \leq \|W\|\cdot \|I-D^{1/2}\| \leq \|W\|\cdot \|I-D\| \leq \frac{4E_P}{\sigma_k(\widetilde O)^3},$$
where the inequality $\|I-D\|\leq \frac{4E_P}{\sigma_k(\widetilde O)^2}$ was proved in \cite{speclda}.
Proof to Eq. (\ref{eq4_pert}):
$\|AA^\top-\widehat A\widehat A^\top\| \leq \|AA^\top-A\widehat A^\top\| + \|A\widehat A^\top - \widehat A\widehat A^\top\|
\leq \|A-\widehat A\|\cdot (\|A\|+\|\widehat A\|) \leq \frac{12E_P}{\sigma_k(\widetilde O)^2}$.
All the other inequalities come from Lemma C.1, \cite{speclda}.
\end{proof}
We are now able to provide perturbation bounds for estimation error of whitened moments.
\begin{deff}[estimation error of whitened moments]
Define
\begin{eqnarray}
\varepsilon_{p,w} &\triangleq& \|M_2(W, W) - \widehat M_2(\widehat W,\widehat W)\|,\\
\varepsilon_{y,w} &\triangleq& \|M_y(W, W) - \widehat M_y(\widehat W,\widehat W)\|,\\
\varepsilon_{t,w} &\triangleq& \|M_3(W,W,W) - \widehat M_3(\widehat W,\widehat W,\widehat W)\|.
\end{eqnarray}
\label{def_epsilon}
\end{deff}
\begin{lem}[Perturbation lemma of whitened moments]
Suppose $E_P\leq \sigma_k(M_2)/2$. We have
\begin{eqnarray}
\varepsilon_{p,w} &\leq& \frac{16E_P}{\sigma_k(\widetilde O)^2},\\
\varepsilon_{y,w} &\leq& \frac{24\|\bm\eta\|E_P}{(\alpha_0+2)\sigma_k(\widetilde O)^2} + \frac{4E_y}{\sigma_k(\widetilde O)^2},\\
\varepsilon_{t,w} &\leq& \frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5} + \frac{8E_T}{\sigma_k(\widetilde O)^3}.
\end{eqnarray}
\end{lem}
\begin{proof}
Using the idea in the proof of Lemma C.2 in \cite{speclda}, we can split $\varepsilon_{p,w}$ as
\begin{equation*}
\begin{aligned}
\varepsilon_{p,w}
=& \|M_2(W,W) - M_2(\widehat W,\widehat W) + M_2(\widehat W,\widehat W) - \widehat M_2(\widehat W,\widehat W)\|\\
\leq& \|M_2(W,W) - M_2(\widehat W,\widehat W)\| + \|M_2(\widehat W,\widehat W) - \widehat M_2(\widehat W,\widehat W)\|.
\end{aligned}
\end{equation*}
We can the bound the two terms seperately, as follows.
For the first term, we have
\setlength\arraycolsep{1pt}\begin{eqnarray*}
\|M_2(W,W) - M_2(\widehat W,\widehat W)\|
&=& \|W^\top M_2 W-\widehat W^\top\widehat M_2\widehat W\|\\
&=& \|AA^\top - \widehat A\widehat A^\top\|\\
&\leq& \frac{12E_P}{\sigma_k(\widetilde O)^2}.
\end{eqnarray*}
where the last inequality comes from Eq. (\ref{eq4_pert}).
For the second term, we have
$$\|M_2(\widehat W,\widehat W)-\widehat M_2(\widehat W,\widehat W)\| \leq \|\widehat W\|^2\cdot \|M_2-\widehat M_2\| \leq \frac{4E_P}{\sigma_k(\widetilde O)^2},$$
where the last inequality comes from Eq. (\ref{eq6_pert}).
Similarly, $\varepsilon_{y,w}$ can be splitted as $\|M_y(W,W)-M_y(\widehat W,\widehat W)\|$ and $\|M_y(\widehat W,\widehat W)-\widehat M_y(\widehat W,\widehat W)\|$, which can be bounded separately. For the first term, we have
\begin{equation*}
\begin{aligned}
\|M_y(W,W) - M_y(\widehat W,\widehat W)\|
=& \|W^\top M_y W - \widehat W^\top M_y\widehat W\|\\
=& \frac{2}{\alpha_0+2}\|A\mathrm{diag}(\bm\eta)A^\top - \widehat A\mathrm{diag}(\bm\eta)\widehat A^\top\|\\
\leq& \frac{2\|\bm\eta\|}{\alpha_0+2}\cdot \|AA^\top - \widehat A\widehat A^\top\|\\
\leq& \frac{24\|\bm\eta\|}{(\alpha_0+2)\sigma_k(\widetilde O)^2}\cdot E_P.
\end{aligned}
\end{equation*}
For the second term, we have
$$\|M_y(\widehat W,\widehat W)-\widehat M_y(\widehat W,\widehat W)\| \leq \|\widehat W\|^2\cdot\|M_y-\widehat M_y\| \leq \frac{4E_y}{\sigma_k(\widetilde O)^2}.$$
Finally, we bound $\varepsilon_{t,w}$ as below, following the work \cite{specregression}.
\begin{eqnarray*}
\varepsilon_{t,w} &=& \|M_3(W,W,W) - \widehat M_3(\widehat W,\widehat W,\widehat W)\|\\
&\leq& \|M_3\|\cdot \|W-\widehat W\|\cdot (\|W\|^2 + \|W\|\cdot \|\widehat W\|
+ \|\widehat W\|^2) + \|\widehat W\|^3\cdot \|M_3-\widehat M_3\|\\
&\leq& \frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5} + \frac{8E_T}{\sigma_k(\widetilde O)^3},
\end{eqnarray*}
where we have used the fact that $$\|M_3\| \leq \sum_{i=1}^k{\frac{2\alpha_i}{\alpha_0(\alpha_0+1)(\alpha_0+2)}} = \frac{2}{(\alpha_0+1)(\alpha_0+2)}.$$
\end{proof}
\subsection{SVD accuracy}
The key idea for spectral recovery of LDA topic modeling is the \emph{simultaneous diagonalization} trick,
which asserts that we can recover LDA model parameters by performing orthogonal tensor decomposition
on a pair of simultaneously whitened moments, for example, $(M_2, M_3)$ and $(M_2, M_y)$.
The following proposition details this insight, as we derive orthogonal tensor decompositions for the whitened tensor product
$M_y(W,W)$ and $M_3(W,W,W)$.
\begin{prop}
Define $\bm v_i\triangleq W^\top \widetilde{\bm\mu}_i = \sqrt{\frac{\alpha_i}{\alpha_0(\alpha_0+1)}}W^\top\bm\mu_i$. Then
\begin{enumerate}
\item $\{\bm v_i\}_{i=1}^k$ is an orthonormal basis.
\item $M_y$ has a pair of singular value and singular vector $(\sigma_i^y, \bm v_i)$ with $\sigma_i^y = \frac{2}{\alpha_0+2}\eta_j$ for some $j \in[k]$.
\item $M_3$ has a pair of robust eigenvalue and eigenvector \cite{a:tensordecomp} $(\lambda_i, \bm v_i)$ with $\lambda_i = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{j'}}}$ for some $j'\in[k]$.
\end{enumerate}
\end{prop}
\begin{proof}
The orthonormality of $\{\bm v_i\}_{i=1}^k$ follows from the fact that $W^\top M_2W = \sum_{i=1}^k{\bm v_i\bm v_i^\top} = I_k$.
Subsequently, we have
\setlength\arraycolsep{1pt}\begin{eqnarray*}
M_y(W, W) &=& \frac{2}{\alpha_0+2}\sum_{i=1}^k{\eta_i \bm v_i\bm v_i^\top},\\
M_3(W,W,W) &=& \frac{2}{\alpha_0+2}\sum_{i=1}^k{\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_i}}\bm v_i\otimes\bm v_i\otimes\bm v_i}.
\end{eqnarray*}
\end{proof}
The following lemmas (Lemma \ref{lem_eta} and Lemma \ref{lem_mu})
give upper bounds on the estimation error of $\bm\eta$ and $\bm\mu$
in terms of $|\widehat\lambda_i-\lambda_i|$, $|\widehat{\bm v}_i-\bm v_i|$
and the estimation errors of whitened moments defined in Definition \ref{def_epsilon}.
\begin{lem}[$\eta_i$ estimation error bound]
Define $\widehat\eta_i \triangleq \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i$,
where $\widehat{\bm v}_i$ is some estimation of $\bm v_i$.
We then have
\begin{equation}
|\eta_i - \widehat\eta_i| \leq 2\|\bm\eta\|\|\widehat{\bm v}_i-\bm v_i\| + \frac{\alpha_0+2}{2}(1+2\|\widehat{\bm v}_i-\bm v_i\|)\cdot\varepsilon_{y,w}.
\end{equation}
\label{lem_eta}
\end{lem}
\begin{proof}
First, note that $\bm v_i^\top M_y(W,W)\bm v_i = \frac{2}{\alpha_0+2}\eta_i$ because $\{\bm v_i\}_{i=1}^k$ are orthonormal.
Subsequently, we have
\begin{equation*}
\begin{aligned}
\frac{2}{\alpha_0+2}|\eta_i - \widehat\eta_i|
=& \Big|\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - \bm v_i^\top M_y(W,W)\bm v_i\Big| \\
\leq& \Big| (\widehat{\bm v}_i-\bm v_i)^\top \widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i\Big| + \Big|\bm v_i^\top\left(\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\right)\Big|\\
\leq& \|\widehat{\bm v}_i-\bm v_i\|\|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i\| + \|\bm v_i\|\|\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\|.\\
\end{aligned}
\end{equation*}
Note that both $\bm v_i$ and $\widehat{\bm v}_i$ are unit vectors. Therefore,
\begin{equation*}
\begin{aligned}
\frac{2}{\alpha_0+2}|\eta_i-\widehat\eta_i|
\leq& \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i-\bm v_i\| + \|\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i - M_y(W,W)\bm v_i\|\\
\leq& \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i-\bm v_i\| + \|\widehat M_y(\widehat W,\widehat W)\|\|\widehat{\bm v}_i - \bm v_i\| + \|\widehat M_y(\widehat W,\widehat W)-M_y(W,W)\|\|\bm v_i\|\\
\leq& 2\|\widehat{\bm v}_i-\bm v_i\|\left(\frac{2}{\alpha_0+2}\|\bm\eta\| + \varepsilon_{y,w}\right) + \varepsilon_{y,w}.
\end{aligned}
\end{equation*}
The last inequality is due to the fact that $\|M_y(W,W)\| = \frac{2}{\alpha_0+2}\|\bm\eta\|$.
\end{proof}
\begin{lem}[$\bm\mu_i$ estimation error bound]
Define $\widehat{\bm\mu}_i \triangleq \frac{\alpha_0+2}{2}\widehat\lambda_i(\widehat W^+)^\top\widehat{\bm v}_i$,
where $\widehat\lambda_i, \widehat{\bm v}_i$ are some estimates of singular value pairs $(\lambda_i, \bm v_i)$ of $M_3(W,W,W)$.
We then have
\begin{equation}
\begin{aligned}
\|\widehat{\bm\mu}_i - \bm\mu_i\| \leq& \frac{3(\alpha_0+2)}{2}\sigma_1(\widetilde O)|\widehat\lambda_i-\lambda_i|
+ 3\alpha_{\max}\sigma_1(\widetilde O)\|\widehat{\bm v}_i-\bm v_i\|
+ \frac{6\alpha_{\max}\sigma_1(\widetilde O)E_P}{\sigma_k(\widetilde O)^2}.
\end{aligned}
\end{equation}
\label{lem_mu}
\end{lem}
\begin{proof}
First note that $\bm\mu_i = \frac{\alpha_0+2}{2}\lambda_i (W^+)^\top\bm v_i$. Subsequently,
\begin{equation*}
\begin{aligned}
\frac{2}{\alpha_0+2}\|\bm\mu_i - \widehat{\bm\mu}_i\|
=& \|\widehat\lambda_i(\widehat W^+)^\top\widehat{\bm v}_i - \lambda_i(W^+)^\top\bm v_i\|\\
\leq& \|\widehat\lambda_i\widehat W^+ - \lambda_i W^+\|\|\widehat{\bm v}_i\| + \|\lambda_i W^+\|\|\widehat{\bm v}_i-\bm v_i\|\\
\leq& |\widehat\lambda_i-\lambda_i|\|\widehat W^+\| + |\lambda_i|\|\widehat W^+ - W^+\| + |\lambda_i|\|W^+\|\|\widehat{\bm v}_i-\bm v_i\|\\
\leq& 3\sigma_1(\widetilde O)|\widehat\lambda_i-\lambda_i| + \frac{2\alpha_{\max}}{\alpha_0+2}\cdot \frac{6\sigma_1(\widetilde O) E_P}{\sigma_k(\widetilde O)^2}
+ \frac{2\alpha_{\max}}{\alpha_0+2}\cdot 3\sigma_1(\widetilde O)\cdot \|\widehat{\bm v_i} - \bm v_i\|.
\end{aligned}
\end{equation*}
\end{proof}
To bound the error of orthogonal tensor decomposition performed on the estimated tensors $\widehat M_3(\widehat W,\widehat W,\widehat W)$,
we cite Theorem 5.1 \cite{a:tensordecomp}, a sample complexity analysis on the robust tensor power method we used
for recovering $\widehat\lambda_i$ and $\widehat{\bm v}_i$.
\begin{lem}[Theorem 5.1, \cite{a:tensordecomp}]
Let $\lambda_{\max} = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$,
$\lambda_{\min} = \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\max}}}$,
where $\alpha_{\min} = \min{\alpha_i}$ and $\alpha_{\max} = \max{\alpha_i}$.
Then there exist universal constants $C_1, C_2 > 0$ such that the following holds:
Fix $\delta'\in(0,1)$.
Suppose $\varepsilon_{t,w}\leq \varepsilon$ and
\begin{eqnarray}
\varepsilon_{t, w} &\leq& C_1\cdot \frac{\lambda_{\min}}{k},\label{eq1_tensorpower}
\end{eqnarray}
Suppose $\{(\widehat\lambda_i, \widehat{\bm v}_i)\}_{i=1}^k$ are eigenvalue and eigenvector pairs
returned by running Algorithm 1 in \cite{a:tensordecomp} with input $\widehat M_3(\widehat W,\widehat W,\widehat W)$ for $L = \text{poly}(k)\log(1/\delta')$ and $N \geq C_2\cdot(\log(k) + \log\log(\frac{\lambda_{\max}}{\varepsilon}))$ iterations.
With probability greater than $1-\delta'$, there exists a permutation $\pi':[k]\to[k]$ such that
for all $i$,
$$\|\widehat{\bm v}_i - \bm v_{\pi'(i)}\| \leq 8\varepsilon/\lambda_{\min},\quad |\widehat\lambda_i - \lambda_{\pi'(i)}| \leq 5\varepsilon.$$
\label{lem_tensorpower}
\end{lem}
\subsection{Tail Inequalities}
\begin{lem}[Lemma 5, \cite{specregression}]
Let $\bm x_1,\cdots,\bm x_N\in\mathbb R^d$ be i.i.d. samples from some distribution with bounded support (i.e., $\|\bm x\|_2\leq B$ with probability 1 for some constant $B$).
Then with probability at least $1-\delta$,
$$\left\|\frac{1}{N}\sum_{i=1}^N{\bm x_i} - \mathbb E[\bm x]\right\|_2 \leq \frac{2B}{\sqrt{N}}\left(1+\sqrt{\frac{\log(1/\delta)}{2}}\right).$$
\label{lem_chernoff}
\end{lem}
\begin{cor}
Let $\bm x_1,\cdots,\bm x_N\in\mathbb R^d$ be i.i.d. samples from some distributions with $\Pr[\|\bm x\|_2 \leq B] \geq 1-\delta'$.
Then with probability at least $1-N\delta'-\delta$,
$$\left\|\frac{1}{N}\sum_{i=1}^N{\bm x_i} - \mathbb E[\bm x]\right\|_2 \leq \frac{2B}{\sqrt{N}}\left(1+\sqrt{\frac{\log(1/\delta)}{2}}\right).$$
\label{cor_chernoff}
\end{cor}
\begin{proof}
Use union bound.
\end{proof}
\begin{lem}[concentration of moment norms]
Suppose we obtain $N$ i.i.d. samples (i.e., documents with at least three words each and their regression variables in sLDA models).
Define $R(\delta) \triangleq \|\bm\eta\| - \sigma\Phi^{-1}(\delta)$,
where $\Phi^{-1}(\cdot)$ is the inverse function of the CDF of a standard Gaussian distribution.
Let $\mathbb E[\cdot]$ denote the mean of the true underlying distribution
and $\widehat{\mathbb E}[\cdot]$ denote the empirical mean.
Then
\iffalse
\begin{enumerate}
\item $\Pr\left[\|\mathbb E[\bm x_1] - \widehat{\mathbb E}[\bm x_1]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[\bm x_1\otimes\bm x_2\otimes\bm x_3] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y] - \widehat{\mathbb E}[y]\| < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}} \right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y\bm x_1] - \widehat{\mathbb E}[y\bm x_1]\|_F < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}}\right] \geq 1-\delta$.
\item $\Pr\left[\|\mathbb E[y\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[y\bm x_1\otimes\bm x_2]\|_F < R(\delta/4\sigma N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}}\right] \geq 1 - \delta$.
\end{enumerate}
\fi
\begin{equation}
\begin{aligned}
& \Pr \left [\|\mathbb E[\bm x_1] - \widehat{\mathbb E}[\bm x_1]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}} \right ] \geq 1-\delta,\\
&\Pr \left [\|\mathbb E[\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2]\|_F < \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}} \right ] \geq 1-\delta,\\
&\Pr \left [\|\mathbb E[\bm x_1\otimes\bm x_2\otimes\bm x_3] - \widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]\|_F< \frac{2+\sqrt{2\log(1/\delta)}}{\sqrt{N}} \right ] \geq 1-\delta,\\
&\Pr \left [\|\mathbb E[y] - \widehat{\mathbb E}[y]\| < R(\delta/4 N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}} \right ] \geq 1-\delta,\\
&\Pr \left [\|\mathbb E[y\bm x_1] - \widehat{\mathbb E}[y\bm x_1]\|_F < R(\delta/4 N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}} \right ] \geq 1-\delta,\\
&\Pr \left [\|\mathbb E[y\bm x_1\otimes\bm x_2] - \widehat{\mathbb E}[y\bm x_1\otimes\bm x_2]\|_F < R(\delta/4 N)\cdot\frac{2+\sqrt{2\log(2/\delta)}}{\sqrt{N}} \right ] \geq 1 - \delta.
\end{aligned}
\end{equation}
\label{lem_tail_ineq}
\end{lem}
\begin{proof}
Use Lemma \ref{lem_chernoff} and Corrolary \ref{cor_chernoff} for concentration bounds involving the regression variable $y$.
\end{proof}
\begin{cor}
With probability $1-\delta$ the following holds:
\begin{enumerate}
\item $E_P = \|M_2-\widehat M_2\| \leq 3\cdot \frac{2+\sqrt{2\log(6/\delta)}}{\sqrt{N}}$.
\item $E_y = \|M_y-\widehat M_y\| \leq 10R(\delta/60 N)\cdot \frac{2+\sqrt{2\log(15/\delta)}}{\sqrt{N}}$.
\item $E_T = \|M_3-\widehat M_3\| \leq 10\cdot \frac{2+\sqrt{2\log(9/\delta)}}{\sqrt{N}}$.
\end{enumerate}
\label{cor_ep}
\end{cor}
\begin{proof}
Corrolary \ref{cor_ep} can be proved by expanding the terms by definition and then using tail inequality in Lemma \ref{lem_tail_ineq} and union bound.
Also note that $\|\cdot\| \leq \|\cdot\|_F$ for all matrices.
\end{proof}
\subsection{Completing the proof}
We are now ready to give a complete proof to Theorem \ref{thm_main}
\iffalse
\jun{the following restatement of the theorem is not necessary. Refer to the theorem in the main text. If restating, make them consistent.}
\begin{thm}[Sample complexity bound]
Let $\sigma_1(\widetilde O)$ and $\sigma_k(\widetilde O)$ be the largest and the smallest singular values of the canonical topic distribution matrix $\widetilde O$.
Define $\lambda_{\min} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\max}}}$
and $\lambda_{\max} \triangleq \frac{2}{\alpha_0+2}\sqrt{\frac{\alpha_0(\alpha_0+1)}{\alpha_{\min}}}$,
where $\alpha_{\max}$ and $\alpha_{\min}$ are the largest and the smallest entries in $\bm\alpha$.
Suppose $\widehat{\bm\mu}$, $\widehat{\bm\alpha}$ and $\widehat{\bm\eta}$ are the outputs of Algorithm 1,
and $L$ is at least a linear function in $k$.
Fix $\delta\in(0,1)$.
For any small error-tolerance parameter $\varepsilon > 0$,
if Algorithm 1 is run with parameter $T = \Omega(\log(k) + \log\log(\lambda_{\max}/\varepsilon))$
on $N$ i.i.d. sampled documents with
$N\geq \max(n_1, n_2, n_3)$, where
\setlength\arraycolsep{1pt} \begin{eqnarray}
n_1 &=& C_1\cdot \left(1+\sqrt{\log(6/\delta)}\right)^2\cdot \frac{\alpha_0^2(\alpha_0+1)^2}{\alpha_{\min}}, \label{eq_n1} \\
n_2 &=& C_2\cdot \frac{(1+\sqrt{\log(15/\delta)})^2}{\varepsilon^2\sigma_k(\widetilde O)^4}
\\
&\ &\cdot\max\left((\|\bm\eta\| + \Phi^{-1}(\delta/60\sigma))^2, \alpha_{\max}^2\sigma_1(\widetilde O)^2\right),
\label{eq_n2} \\
n_3 &=& C_3\cdot \frac{(1+\sqrt{\log(9/\delta)})^2}{\sigma_k(\widetilde O)^{10}}\cdot\max\left(\frac{1}{\varepsilon^2}, \frac{k^2}{\lambda_{\min}^2}\right),
\label{eq_n3}
\end{eqnarray}
and $C_1, C_2$ and $C_3$ are universal constants,
then with probability at least $1-\delta$, there exists a permutation $\pi:[k]\to [k]$ such that for every topic $i$, the following holds:\\
\begin{enumerate}
\item $|\alpha_i-\widehat{\alpha}_{\pi(i)}| \leq \frac{4\alpha_0(\alpha_0+1)(\lambda_{\max}+5\varepsilon)}{(\alpha_0+2)^2\lambda_{\min}^2(\lambda_{\min}-5\varepsilon)^2}\cdot 5\varepsilon$, if $\lambda_{\min} > 5\varepsilon$;\\
\item $\|\bm\mu_i - \widehat{\bm\mu}_{\pi(i)}\| \leq \left(3\sigma_1(\widetilde O)\left(\frac{8\alpha_{\max}}{\lambda_{\min}} + \frac{5(\alpha_0+2)}{2}\right) + 1\right)\varepsilon$;\\
\item $|\eta_i - \widehat\eta_{\pi(i)}| \leq \left(\frac{\|\bm\eta\|}{\lambda_{\min}} + (\alpha_0+2)\right) \varepsilon$.
\end{enumerate}
\label{thm_main}
\end{thm}
\fi
\begin{proof}{(Proof of Theorem ~\ref{thm_main})}
First, the assumption $E_P\leq\sigma_k(M_2)$ is required for error bounds on $\varepsilon_{p,w}, \varepsilon_{y,w}$ and $\varepsilon_{t,w}$.
Noting Corrolary \ref{cor_ep} and the fact that $\sigma_k(M_2) = \frac{\alpha_{\min}}{\alpha_0(\alpha_0+1)}$, we have
$$N = \Omega\left(\frac{\alpha_0^2(\alpha_0+1)^2(1+\sqrt{\log(6/\delta)})^2}{\alpha_{\min}^2}\right).$$
Note that this lower bound does not depend on $k$, $\varepsilon$ and $\sigma_k(\widetilde O)$.
For Lemma \ref{lem_tensorpower} to hold, we need the assumptions that
$\varepsilon_{t,w} \leq \min(\varepsilon, O(\frac{\lambda_{\min}}{k}))$.
These imply $n_3$,
as we expand $\varepsilon_{t,w}$ according to Definition \ref{def_epsilon}
and note the fact that the first term $\frac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde O)^5}$ dominates the second one.
The $\alpha_0$ is missing in the third requirment $n_3$ because $\alpha_0+1 \geq 1$, $\alpha_0+2 \geq 2$ and we discard them both.
The $|\alpha_i-\widehat{\alpha}_{\pi(i)}|$ bound follows immediately by Lemma \ref{lem_tensorpower}
and the recovery rule $\widehat{\alpha}_i = \frac{\alpha_0+2}{2}\widehat\lambda_i$.
To bound the estimation error for the linear classifier $\bm\eta$, we need to further bound $\varepsilon_{y,w}$.
We assume $\varepsilon_{y,w}\leq\varepsilon$.
By expanding $\varepsilon_{y,w}$ according to Definition \ref{def_epsilon} in a similar manner
we obtain the $(\|\bm\eta\|+\Phi^{-1}(\delta/60\sigma))^2$ term in the requirment of $n_2$.
The bound on $|\eta_i-\widehat\eta_{\pi(i)}|$ follows immediately by Lemma \ref{lem_eta}.
Finally, we bound $\|\bm\mu_i-\widehat{\bm\mu}_{\pi(i)}\|$ using Lemma \ref{lem_mu}.
We need to assume that $\frac{6\alpha_{\max}\sigma_1(\widetilde O)E_P}{\sigma_k(\widetilde O)^2} \leq \varepsilon$,
which gives the $\alpha_{\max}^2\sigma_1(\widetilde O)^2$ term.
The $\|\bm\mu_i-\widehat{\bm\mu}_{\pi(i)}\|$ bound then follows by Lemma \ref{lem_mu} and Lemma \ref{lem_tensorpower}.
\end{proof}
We make some remarks for the main theorem. In Remark~\ref{rem_connection}, we establish links between indirect quantities appeared in Theorem~\ref{thm_main} (e.g., $\sigma_k(\widetilde O)$) and the functions of original model parameters (e.g., $\sigma_k(O)$).
These connections are straightforward following their definitions.
\begin{rem}
The indirect quantities $\sigma_1(\widetilde O)$ and $\sigma_k(\widetilde O)$ can be related to $\sigma_1(O)$, $\sigma_k(O)$ and $\bm\alpha$ in the following way:
{\small \begin{equation*}
\sqrt{\frac{\alpha_{\min}}{\alpha_0(\alpha_0+1)}}\sigma_k(O) \leq \sigma_k(\widetilde O)\leq \sqrt{\frac{\alpha_{\max}}{\alpha_0(\alpha_0+1)}}\sigma_k(O);~~~~
\end{equation*}
\begin{equation*}
\sigma_1(\widetilde O) \leq \sqrt{\frac{\alpha_{\max}}{\alpha_0(\alpha_0+1)}}\sigma_1(O) \leq \frac{1}{\sqrt{\alpha_0+1}}.
\end{equation*}}
\label{rem_connection}
\end{rem}
We now take a close look at the sample complexity bound in Theorem~\ref{thm_main}.
It is evident that $n_2$ can be neglected when the number of topics $k$ gets large,
because in practice the norm of the linear regression model $\bm\eta$ is usually assumed to be small in order to avoid overfitting.
Moreover, as mentioned before, the prior parameter $\bm\alpha$ is often assumed to be homogeneous with $\alpha_i = 1/k$ \cite{lsa}.
With these observations, the sample complexity bound in Theorem~\ref{thm_main}
can be greatly simplified.
\begin{rem}
Assume $\|\bm\eta\|$ and $\sigma$ are small and $\bm\alpha = (1/k, \cdots, 1/k)$.
As the number of topics $k$ gets large, the sample complexity bound in Theorem~\ref{thm_main} can be simplified as
\begin{equation}
N = \Omega\left( \frac{\log(1/\delta)}{\sigma_k(\widetilde O)^{10}}\cdot\max(\varepsilon^{-2}, k^3)\right).
\end{equation}\label{rem_simpbound}
\end{rem}\vspace{-.5cm}
The sample complexity bound in Remark~\ref{rem_simpbound} may look formidable
as it depends on $\sigma_k(\widetilde O)^{10}$.
However, such dependency is somewhat necessary because we are using third-order tensors to recover the underlying model parameters.
Furthermore, the dependence on $\sigma_k(\widetilde O)^{10}$ is introduced by the robust tensor power method to recover LDA parameters,
and the reconstruction accuracy of $\bm\eta$ only depends on $\sigma_k(\widetilde O)^4$ and $(\|\bm\eta\|+\Phi^{-1}(\delta/60\sigma))^2$.
As a consequence, if we can combine our power update method for $\bm \eta$ with LDA inference algorithms that have milder dependence on the singular value $\sigma_k(\widetilde O)$,
we might be able to get an algorithm with a better sample complexity.
\iffalse
\section{Discussions}
\subsection{Appendix C.1. Extension to other topic recovery algorithms}
One important advantage of our proposed inference algorithm is its flexibility---the algorithm can be combined with many other LDA inference algorithms to infer supervised LDA model parameters.
More specifically, given access to any algorithm that recovers the topic distribution matrix $O$
and the prior parameter $\bm\alpha$ from i.i.d. sampled documents,
an inference algorithm for a supervised LDA model can be immediately obtained,
as shown in Algorithm \ref{alg2}.
\begin{algorithm}
\caption{sLDA parameter recovery based on an existing LDA inference algorithm $\mathcal A$. Input parameter: $\alpha_0$.}
\centering
\begin{algorithmic}[1]
\State Compute empirical moments and obtain $\widehat M_2, \widehat M_y$. Set $\widehat{\bm\eta} = 0$.
\State Find $\widehat W\in\mathbb R^{n\times k}$ such that $\widehat M_2(\widehat W, \widehat W) = I_k$.
\State Run algorithm $\mathcal A$ with observed documents and parameter $\alpha_0$. Obtain the estimated
topic distribution matrix $\widehat O=(\widehat{\bm\mu}_1,\cdots,\widehat{\bm\mu}_k)$ and prior parameter $\widehat{\bm\alpha}$.
\State Compute $\widehat{\bm v}_i = \sqrt{\frac{\widehat{\alpha}_i}{\alpha_0(\alpha_0+1)}}\widehat W^\top\widehat{\bm\mu}_i$ for each topic $i$.
\State Recover linear classifier: $\widehat\eta_i \gets \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i$.
\State \textbf{Output:} $\widehat{\bm\eta}$, $\bm\alpha$ and $\{\widehat{\bm\mu}_i\}_{i=1}^k$.
\end{algorithmic}
\label{alg2}
\end{algorithm}
The sample complexity of Algorithm \ref{alg2} depends on the sample complexity of the LDA inference algorithm $\mathcal A$.
Although the LDA inference algorithm $\mathcal A$ is free to make any assumptions on the topic distribution matrix $O$,
we comment that the linear independence of topic distribution vectors $\bm\mu$ is still required,
because the power update trick (step 5 in Algorithm 1) is valid only $W\bm\mu_i$ are orthogonal vectors.
\subsection*{Appendix C.2. Going beyond SVD}
The proposed methods are based on spectral decomposition of observable moments
and are provably correct.
However, a major drawback of these methods is their assumption that the topic distributions $\bm\mu$ are linearly independent.
Although standard LDA models assume this topic independence \cite{lda},
in practice there is strong evidence that different topics might be related \cite{correlatelda}.
Therefore, it is important to consider possible extensions.
Recently, there has been much work on provable LDA inference algorithms that do not require the topic independence assumption \cite{practicalalgo,beyondsvd}.
Instead of making the independence assumption, these methods assume a $p$-separability condition on the topic distribution matrix,
that is, there exists an \emph{anchor word} for each topic such that the anchor word only appears in this specific topic,
and its probability (given that topic) is at least $p$.
Here we briefly describe an idea that might lead to an sLDA inference algorithm without assuming topic independence.
First, let $O\in\mathbb R^{V\times k}$ be the topic distribution matrix defined previously and $H = (\bm h_1,\cdots,\bm h_N)\in\mathbb R^{k\times N}$
be the matrix of topic mixing vectors for each document.
Suppose $Q = \mathbb E[\bm x_1\bm x_2^\top]$ is the word co-occurrence frequency matrix
and $Q_y = \mathbb E[y\bm x_1]$ is the co-occurrence frequency matrix between word and regression variable observations.
It is clear that both $Q$ and $Q_y$ can be estimated from the training data, and we have
\begin{equation}
Q = ORO^\top, Q_y = OR\bm\eta,
\end{equation}
where $R \triangleq \mathbb E[HH^\top]$.
Assuming the $p$-separability condition holds and using methods developed in \cite{beyondsvd,practicalalgo},
we can consistently recover the $O$ and $R$.
Now note that $Q_y = (OR)\cdot \bm\eta$.
As a result, knowing both $OR$ and $Q_y$ we reduce the $\bm\eta$ inference problem to solving a linear equation systems.
Future research could be done on (1) determining when the matrix $OR$ has full rank $r$ so that the linear equations can be solved, and
(2) investigating the sample complexity problem for such an inference algorithm.
\subsection*{Appendix C.3. From regression to classification}
A natural extension to our result is to consider supervised LDA models for classification purposes.
The simplest model is a logistic regression classification model, where we assume the response variable $y_d$
for each document $d$ is in $\{+1,-1\}$, and
$$\Pr[y_d = 1|\bm h_d] = \frac{\exp(\bm\eta^\top\bm h_d)}{1+\exp(\bm\eta^\top\bm h_d)},$$
where $\bm\eta$ is a linear classifier.
Though appears simple, such an extension incurs many fundamental problems.
A major obstacle is the fact that the conditional expectation $\mathbb E[y|\bm h]$ is no longer linear
in the topic mixing vector $\bm h$.
As a result, we cannot even evaluate $\mathbb E[y]$ (or higher order tensors like $\mathbb E[y\bm x_1\otimes\bm x_2]$)
in closed forms.
Further more, even if we have accurate estimation to the above moments,
it is rather difficult to infer $\bm\eta$ back due to its non-linearality.
To our knowledge, the classification problem remains unsolved in terms of spectral decomposition methods,
and there is doubt whether a provably correct algorithm exists in such scenarios.
\fi
\section{Proof of Theorem 2}
In this section we give the proof of Theorem 2, following the similar line of the proof of Theorem
1. We bound the estimation errors and the errors introduced by the tensor decomposition step
respectively
\subsection{Definitions}
We first recall the definition of joint canconical topic distribution.
\begin{deff}
(Joint Canconical topic distribution) Define the canconical version of joint topic distribution vector
$\bm{v}_i,\widetilde{\bm{v}}_i$ as follows:
$$
\widetilde{\bm{v}}_i = \sqrt{\dfrac{\alpha_i}{\alpha_0(\alpha_0 +1)}}\bm{v}_i,
$$
where $\bm{v}_i = [ \bm{\mu}_i' , \eta_i]'$ is the topic dictribution vector extended by its regression parameter.
We also define $O^*, \widetilde{O^*} \in \mathbb{R}^{(V+1) \times k}$ by $ O^* = [ \bm{v}_1, ... ,\bm{v}_k] $
and $ \widetilde{O^*} = [ \widetilde{\bm{v}}_1 , ..., \widetilde{\bm{v}}_k]$.
\end{deff}
\subsection{Estimation Errors}
The tail inequatlities in last section \ref{lem_chernoff},\ref{cor_chernoff} gives the following estimation:
\begin{lem}
\label{l1}
Suppose we obtain $N$ i.i.d. samples. Let $\mathbb{E}[\cdot]$ denote the mean of the true underlying distribution and $\hat{\mathbb{E}}[\cdot]$ denote
the empirical mean. Define
\begin{equation}
\begin{aligned}
& R_1(\delta) = \|\bm{\eta}\| - \sigma\Phi^{-1}(\delta),\\
& R_2(\delta) = 2\|\bm{\eta}\|^2 + 2\sigma^2[\Phi^{-1}(\delta)]^2, \\
& R_3(\delta) = 4\|\bm{\eta}\|^3 - 4\sigma^3[\Phi^{-1}(\delta)]^3, \\
\end{aligned}
\end{equation}
where $\Phi^{-1}(\cdot)$ is the inverse function of the CDF of a
standard Gaussian distribution. Then:
\begin{equation}
\begin{aligned}
& \Pr \left [ \| \mathbb{E}[ \mathbf{x_1} \otimes \mathbf{x_2} \otimes \mathbf{x_3}] - \hat{\mathbb{E}}[\mathbf{x_1} \otimes
\mathbf{x_2} \otimes \mathbf{x_3} ] \|_F < \dfrac{2 + \sqrt{2\log(1/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta; \\
& \Pr \left [ \| \mathbb{E}[ y^i ] - \hat{\mathbb{E}}[ y^i ] \|_F < R_i(\delta/4 N )\cdot
\dfrac{2 + \sqrt{2\log(2/\delta)} }{ \sqrt{N} } \right ] \geq 1 - \delta \ \ \ \ i = 1, 2, 3;\\
& \Pr \left [ \| \mathbb{E}[ y^i \mathbf{x_1} ] - \hat{\mathbb{E}}[ y^i \mathbf{x_1} ] \|_F
< R_i(\delta / 4 N) \cdot \dfrac{2 + \sqrt{2\log(2/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta \ \ \ \ i = 1,2;\\
& \Pr \left [ \| \mathbb{E}[ y^i \mathbf{x_1} \otimes \mathbf{x_2} ] - \hat{\mathbb{E}}[ y^i \mathbf{x_1} \otimes \mathbf{x_2} ] \|_F
< R_i(\delta / 4 N) \cdot \dfrac{2 + \sqrt{2\log(2/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta \ \ \ \ i = 1 , 2.\\
\end{aligned}
\end{equation}
\end{lem}
\begin{proof}
This lemma is a direct application of lemma \ref{lem_chernoff} and corollary \ref{cor_chernoff}.
\end{proof}
\begin{cor}
\label{c1}
With probability $1 -\delta$, the following holds:
\begin{equation}
\begin{aligned}
& Pr \left [ \| \mathbb{E}[ \mathbf{z_1}] - \hat{\mathbb{E}}[ \mathbf{z_1}] \| < C_1(\delta / 8 N)
\dfrac{2 + \sqrt{2\log(4/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta, \\
& Pr \left [ \| \mathbb{E}[ \mathbf{z_1} \otimes \mathbf{z_2}] - \hat{\mathbb{E}}[ \mathbf{z_1} \otimes \mathbf{z_2}] \| < C_2(\delta / 12 N)
\dfrac{2 + \sqrt{2\log(6/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta, \\
& Pr \left [ \| \mathbb{E}[ \mathbf{z_1} \otimes \mathbf{z_2} \otimes \mathbf{z_3}] - \hat{\mathbb{E}}[\mathbf{z_1} \otimes
\mathbf{z_2} \otimes \mathbf{z_3} ] \|_F < C_3(\delta/16N) \dfrac{2 + \sqrt{2\log(8/\delta)} }{ \sqrt{N}} \right ] \geq 1 - \delta, \\
\end{aligned}
\end{equation}
where
\begin{equation}
\begin{aligned}
& C_1(\delta) = R_1(\delta) + 1, \\
& C_2(\delta) = R_1(\delta) + R_2(\delta) + 1, \\
& C_3(\delta) = R_1(\delta) + R_2(\delta) + R_3(\delta) + 1. \\
\end{aligned}
\end{equation}
\end{cor}
\begin{proof}
Use lemma \ref{l1} as well as the fact that $\sqrt{a + b} \leq \sqrt{a} + \sqrt{b}, \forall a,b \geq 0$ and $ Pr( X \leq t_1 , Y \leq t_2) \leq Pr( X + Y \leq t_1 + t_2)$.
\end{proof}
\begin{lem}
\label{l3}
(concentration of moment norms) Using notations in lemma \ref{l1} and suppose
$C_3(\delta/144N)\dfrac{2 +\sqrt{2\log(72/\sigma)}}{\sqrt{N}} \leq 1 $
we have:
\begin{equation}
\begin{aligned}
& E_P = \|N_2 - \hat{N}_2\| \leq 3C_2(\delta/36N)) \cdot \dfrac{2 + \sqrt{2 \log(18/\sigma)}}{\sqrt{N}},\\
& E_T = \|N_3 - \hat{N}_3\| \leq 10C_3(\delta/144N)) \cdot \dfrac{2 + \sqrt{2 \log(72/\sigma)}}{\sqrt{N}}.\\
\end{aligned}
\end{equation}
\end{lem}
\begin{proof}
This lemma can be proved by expanding the terms by definition and use lemma \ref{l1} , corollary \ref{c1} as well as union bound.
Also note that $\| \cdot\| \leq \|\cdot \|_F$.
\end{proof}
\begin{lem}
(estimation error of whitened moments)
Define
\begin{equation}
\begin{aligned}
& \epsilon_{p,w} = \| N_2(W,W) - \hat{N}_2(\hat{W},\hat{W})\|, \\
& \epsilon_{t,w} = \| N_3(W,W,W) - \hat{N}_3(\hat{W},\hat{W},\hat{W})\|. \\
\end{aligned}
\end{equation}
Then we have:
\begin{equation}
\begin{aligned}
& \epsilon_{p,w} \leq \dfrac{16E_p}{\sigma_k(\widetilde{O^*})^2}, \\
& \epsilon_{t,w} \leq \dfrac{54E_P}{(\alpha_0+1)(\alpha_0+2)\sigma_k(\widetilde{O^*})^5} +
\dfrac{8E_T}{\sigma_k(\widetilde{O^*})^3}.
\end{aligned}
\end{equation}
\end{lem}
\subsection{SVD accuracy}
We rewrite a part of Lemma ~\ref{lem_whiten} which we will use in the following.
\begin{lem}
\label{l5}
Let $W,\hat{W} \in R^{(V+1)\times k}$ be the whitening matrices such that
$N_2(W,W) = \hat{W}_2(\hat{W},\hat{W}) = I_k$. Further more, we suppose
$ E_P \leq \sigma_k(N_2)/2$. Then we have:
\begin{equation}
\begin{aligned}
& \|W - \hat{W}\| \leq \dfrac{4E_P}{\sigma_k(\widetilde{O^*})^3},\\
& \|W^+\| \leq 3\sigma_1(\widetilde{O^*}), \\
& \|\widetilde{W}^+\| \leq 2\sigma_1(\widetilde{O^*}), \\
& \|W^+ - \hat{W}^+\| \leq \dfrac{6\sigma_1(\widetilde{O^*})E_P}{\sigma_k(\widetilde{O^*})^2}.\\
\end{aligned}
\end{equation}
\end{lem}
Using the above lemma, we now can estimate the error introduced by SVD.
\begin{lem}($\bm{v}_i$ estimation error)
\label{l6}
Define $\hat{\bm{v}}_i = \frac{\alpha_0 + 2}
{2}\hat{\lambda}_i(\hat{W}^+)^{\top}\hat{\bm{\omega}}_i$ where
$(\hat{\lambda}_i,\hat{\omega}_i)$ are some estimations of svd pairs
$(\lambda_i, \omega_i)$ of $ N_3(W,W,W)$. We then have:
\begin{equation}
\begin{aligned}
\|\hat{\bm{v}}_i - \bm{v}_i\| \leq& \dfrac{(\alpha_0 + 1)6\sigma_1(\widetilde{O^*})E_P}{\sigma_k(\widetilde{O^*})^2}
+ \dfrac{(\alpha_0 +2)\sigma_1(\widetilde{O^*})|\hat{\lambda}_i - \lambda_i|}{2}
+3(\alpha_0 +1)\sigma_1(\widetilde{O^*})\|\hat{\bm{\omega}}_i - \bm{\omega}_i\|.
\end{aligned}
\end{equation}
\end{lem}
\begin{proof}
Note that $\bm{v}_i = \frac{}{}\lambda_i(W^+)^{\top}\bm{\omega}_i$. Thus we have:
\begin{equation}
\begin{aligned}
\frac{2}{\alpha_0 + 2 } \|\hat{\bm{v}}_i - \bm{v}_i\|
= &\|\hat{\lambda}_i(\hat{W}^+)^{\top}\hat{\bm{\omega}_i}
- \lambda_i(W^+)^{\top}\bm{\omega}_i \| \\
=& \|\hat{\lambda}_i(\hat{W}^+)^{\top}\hat{\bm{\omega}_i}
- \lambda_i(W^+)^{\top}\hat{\bm{\omega}}_i
+ \lambda_i(W^+)^{\top}\hat{\bm{\omega}}_i
- \lambda_i(W^+)^{\top}\bm{\omega}_i \| \\
\leq& \|\hat{\lambda}_i(\hat{W}^+)^{\top} - \lambda_i(W^+)^{\top}\|
\|\hat{\bm{\omega}}_i\|
+ \|\lambda_i(W^+)^{\top}\|\|\hat{\bm{\omega}}_i - \bm{\omega}_i\|\\
\leq& |\lambda_i|\|\hat{W}^+ - W^+\|
\|\hat{\bm{\omega}}_i\| +
|\hat{\lambda}_i - \lambda_i|\|\hat{W}^+\|
\|\hat{\bm{\omega}}_i\|
+ |\lambda_i|\|W^+\|\|\hat{\bm{\omega}}_i - \bm{\omega}_i\|\\
\leq& \dfrac{2(\alpha_0 +1)}{\alpha_0 + 2}\dfrac{6E_P\sigma_1(\widetilde{O^*})}{\sigma_k(\widetilde{O^*})^2}
+ 3\sigma_1(\widetilde{O^*})|\hat{\lambda}_i - \lambda_i| +
\dfrac{2(\alpha_0 + 1)}{\alpha_0 + 2}\cdot
3\sigma_1(\widetilde{O^*})\cdot \|\hat{\bm{\omega}}_i - \bm{\omega}_i \|.
\end{aligned}
\end{equation}
\end{proof}
\iffalse
We now cite an lemma from \cite{a:tensor} that provides a bound for the estimation error of $\bm{\omega}_i$
and $\lambda_i$.
\begin{lem}
\label{l7}
Let $\lambda_{max} =
\frac{2}{\alpha_0 + 2}\sqrt{\dfrac{\alpha_0(\alpha_0 + 1)}{\alpha_{min}}}$ and
$\lambda_{min} =
\frac{2}{\alpha_0 + 2}\sqrt{\dfrac{\alpha_0(\alpha_0 + 1)}{\alpha_{max}}}$, where
$\alpha_{max} = \max \alpha_i ,\alpha_{min} = \min \alpha_i$. Then there exist universal
constants $C_1,C_2 > 0$ such that the following hods: Fix $\delta' \in ( 0,1)$. Suppose
$\epsilon_{t,w} \leq \epsilon$ and
$$
\epsilon_{t,w} \leq C_1 \cdot \dfrac{\lambda_{min}}{k}
$$
Suppose $\{(\hat{\lambda}_i, \hat{\bm{v}}_i\}_{i=1}^{k}\}$ are eigenvalue and eigenvector pairs
returned by running Algorithm $1$ in \cite{a:tensor} with input $\hat{W}_3(\hat{W},\hat{W},\hat{W}) $ for
$L = poly(k)\log(1/\delta')$ and $N \geq C_2\cdot (\log(k) +
\log\log(\frac{\lambda_{max}}{\epsilon}) $ iterations. With probability at least $1 - \delta'$,
there exists a permutation $\pi':[k]\to[k]$ such that for all i:
$$
\|\hat{\bm{\omega}}_i - \bm{\omega}_i\| \leq 8\epsilon / \lambda_{min}, |\hat{\lambda}_i - \lambda_i | \leq 5 \epsilon.
$$
\end{lem}
\fi
\subsection{Completeing the proof}
We are now ready to complete the proof of Theorem ~\ref{thm_main2}.
\iffalse
\begin{theorem}
(Sample complexity bound)
Let $ \sigma_1(\widetilde{O^*})$ and $\sigma_k(\widetilde{O^*})$ be the largest and smallest singular
values of the joint canonical topic dictribution matrix $\widetilde{O^*}$.
For any error-tolerance parameter $\epsilon > 0$, if Algorithm $1$ runs at least
$T = \Omega(\log(k)) + \log\log(\dfrac{\lambda_{max}}{\epsilon})$ iterations on $N$ i.i.d. sampled documents with $N$
satisying:
\begin{equation}
\begin{aligned}
& N \geq K_1 \cdot \dfrac{\alpha_0^2(\alpha_0+1)^2C_2^2(\delta/36N)\cdot(2 + \sqrt{2\log(18/\delta)})^2 }{\alpha_{min}^2} \\
& N \geq K_2 \cdot C_3^2(\delta/144N)(2 + \sqrt{2 \log(72/\sigma)})^2 \\
&N \geq K_3 \cdot \dfrac{C_3^2(\delta/36N)(2+\sqrt{2\log(18/\sigma)})^2}
{\sigma_k(\widetilde{O^*})^{10}}\cdot\max(\dfrac{1}{\epsilon^2}, \dfrac{k^2}{\lambda_{min}^2})\\
\end{aligned}
\end{equation}
then with probability at least $1 - \delta$, there exist a permutation $\pi: [k] \to [k]$
such that the following holds for every $i \in [k]$:
\begin{equation}
\begin{aligned}
& |\alpha_i - \hat{\alpha}_{\pi(i)} |
\leq \dfrac{4\alpha_0(\alpha_0 + 1) }{(\alpha_0 + 2)^2\lambda_{min}^2(\lambda_{min} - 5 \epsilon)^2} \cdot 5\epsilon \\
& \|\bm{v}_i - \hat{\bm{v}}_{\pi(i)} \| \leq \left ( \sigma_1(\widetilde{O^*})(\alpha_0 + 2)(\frac{7}{2} +
\frac{8}{\lambda_{min}})\right )\cdot \epsilon \\
\end{aligned}
\end{equation}
\end{theorem}
\fi
\begin{proof}{(Proof of Theorem~\ref{thm_main2})}
We check out the conditions that must be satisfied for $\epsilon$ accuracy.
First $E_P \leq \sigma_k(N_2)/2$ is required in lemma \ref{l5}. Noting that $\sigma_k(N_2) = \dfrac{\alpha_{min}}{\alpha_0(\alpha_0 + 1)} $, we have:
$$
N \geq O\left ( \dfrac{\alpha_0^2(\alpha_0+1)^2C_2^2(\delta/36N)\cdot(2 + \sqrt{2\log(18/\delta)})^2 }{\alpha_{min}^2}\right ).
$$
In lemma \ref{l3}, we need that $C_3(\delta/144N)\dfrac{2\log(72/\sigma)}{\sqrt{N}} \leq 1$, which means:
$$
N \geq O \left ( C_3^2(\delta/144N)(2 + \sqrt{2 \log(72/\sigma)})^2 \right ).
$$
For lemma \ref{lem_tensorpower} to hold, we need the assumption that $\epsilon_{t,w} \leq C_1 \cdot \frac{\lambda_{min}}{k}$, which implies:
$$
N \geq O \left ( \dfrac{C_3^2(\delta/36N)(2+\sqrt{2\log(18/\sigma)})^2}{\sigma_k(\widetilde{O^*})^{10}}\cdot\max(\dfrac{1}{\epsilon^2},
\dfrac{k^2}{\lambda_{min}^2}) \right ).
$$
From the recovery rule $\hat{\alpha_i} = \dfrac{4\alpha_0(\alpha_0 + 1)}{(\alpha_0 + 2)^2 \hat{\lambda_i}^2}$, we have:
\begin{equation}
\begin{aligned}
|\hat{\alpha}_i - \alpha_i |
\leq \dfrac{4\alpha_0(\alpha_0 + 1) }{(\alpha_0 + 2)^2}
\left |\dfrac{1}{\hat{\lambda}_i^2} - \dfrac{1}{\lambda_i^2} \right |
\leq \dfrac{4\alpha_0(\alpha_0 + 1) }{(\alpha_0 + 2)^2\lambda_{min}^2(\lambda_{min} - 5 \epsilon)^2} \cdot 5\epsilon. \\
\end{aligned}
\end{equation}
Finally, we bound $\|\hat{\bm{v}}_i - \bm{v}_i\|$, which is a direct consequence of lemma \ref{l6} and lemma \ref{lem_tensorpower}.
We additional assume the following condition holds: $\dfrac{6E_P}{\sigma_k(\widetilde{O^*})^2} \leq \epsilon $. In fact, this condition is already satisfied
if the above requirements for $N$ hold.
Under this condition, we have:
$$
\|\hat{\bm{v}}_i - \bm{v}_i \| \leq \left ( \sigma_1(\widetilde{O^*})(\alpha_0 + 2)(\frac{7}{2} +
\frac{8}{\lambda_{min}})\right )\cdot \epsilon.\\
$$
\end{proof}
\section{Moments of Observable Variables}
\subsection{Proof to Proposition 1}
The equations on $M_2$ and $M_3$ have already been proved in \cite{speclda} and \cite{a:tensordecomp}.
Here we only give the proof to the equation on $M_y$.
In fact, all the three equations can be proved in a similar manner.
In sLDA the topic mixing vector $\bm h$ follows a Dirichlet prior distribution with parameter $\bm\alpha$.
Therefore, we have
\begin{equation}
\begin{aligned}
&\mathbb E[h_i] = \frac{\alpha_i}{\alpha_0},\\
&\mathbb E[h_ih_j] = \left\{\begin{array}{ll} \frac{\alpha_i^2}{\alpha_0(\alpha_0+1)}, &\text{if }i=j,\\
\frac{\alpha_i\alpha_j}{\alpha_0^2}, &\text{if }i\neq j\end{array}\right.,\\
&\mathbb E[h_ih_jh_k] = \left\{\begin{array}{ll} \frac{\alpha_i^3}{\alpha_0(\alpha_0+1)(\alpha_0+2)}, & \text{if }i=j=k,\\
\frac{\alpha_i^2\alpha_k}{\alpha_0^2(\alpha_0+1)}, & \text{if }i=j\neq k,\\
\frac{\alpha_i\alpha_j\alpha_k}{\alpha_0^3}, &\text{if }i\neq j, j\neq k, i\neq k\end{array}\right. .
\end{aligned}
\end{equation}
Next, note that
\begin{equation}
\begin{aligned}
&\mathbb E[y|\bm h] = \bm\eta^\top\bm h,\\
&\mathbb E[\bm x_1|\bm h] = \sum_{i=1}^k{h_i\bm\mu_i},\\
&\mathbb E[\bm x_1\otimes\bm x_2|\bm h] = \sum_{i, j=1}^k{h_ih_j\bm\mu_i\otimes\bm\mu_j},\\
&\mathbb E[y\bm x_1\otimes\bm x_2|\bm h] = \sum_{i, j, k=1}^k{h_ih_jh_k\cdot \eta_k\bm\mu_j\otimes\bm\mu_k}.
\end{aligned}
\end{equation}
Proposition 1 can then be proved easily by taking expectation over the topic mixing vector $\bm h$.
\subsection{Details of the speeding-up trick}
In this section we provide details of the trick mentioned in the main paper to speed up empirical moments computations.
First, note that the computation of $\widehat M_1$, $\widehat M_2$ and $\widehat M_y$ only requires $O(NM^2)$ time and $O(V^2)$ space.
They do not need to be accelerated in most practical applications.
This time and space complexity also applies to all terms in $\widehat M_3$ except the $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]$ term,
which requires $O(NM^3)$ time and $O(V^3)$ space if using naive implementations.
Therefore, this section is devoted to speed-up the computation of $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3]$.
More precisely, as mentioned in the main paper,
what we want to compute is the whitened empirical moment $\widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3](\widehat W,\widehat W,\widehat W) \in
\mathbb R^{k\times k\times k}$.
Fix a document $D$ with $m$ words.
Let $T \triangleq \widehat{\mathbb E}[\bm x_1\otimes\bm x_2\otimes\bm x_3|D]$ be the empirical tensor demanded.
By definition, we have
\begin{equation}
T_{i,j,k} = \frac{1}{m(m-1)(m-2)}\left\{\begin{array}{ll}
n_i(n_j-1)(n_k-2),& i=j=k;\\
n_i(n_i-1)n_k,& i=j,j\neq k;\\
n_in_j(n_j-1),& j=k,i\neq j;\\
n_in_j(n_i-1),& i=k,i\neq j;\\
n_in_jn_k,& \text{otherwise};\end{array}\right.
\label{eq_T}
\end{equation}
where $n_i$ is the number of occurrences of the $i$-th word in document $D$.
If $T_{i,j,k} = \frac{n_in_jn_k}{m(m-1)(m-2)}$ for all indices $i,j$ and $k$,
then we only need to compute
$$T(W,W,W) = \frac{1}{m(m-1)(m-2)}\cdot (W\bm n)^{\otimes 3},$$
where $\bm n \triangleq (n_1,n_2,\cdots,n_V)$.
This takes $O(Mk + k^3)$ computational time because $\bm n$ contains at most $M$ non-zero entries,
and the total time complexity is reduced from $O(NM^3)$ to $O(N(Mk+k^3))$.
We now consider the remaining values, where at least two indices are identical.
We first consider those values with two indices the same, for example, $i=j$.
For these indices, we need to subtract an $n_in_k$ term, as shown in Eq. (\ref{eq_T}).
That is, we need to compute the whitened tensor $\Delta(W,W,W)$, where $\Delta\in\mathbb R^{V\times V\times V}$ and
\begin{equation}
\Delta_{i,j,k} = \frac{1}{m(m-1)(m-2)}\cdot \left\{\begin{array}{ll}
n_in_k,& i=j;\\
0,& \text{otherwise}.\end{array}\right.
\end{equation}
Note that $\Delta$ can be written as $\frac{1}{m(m-1)(m-2)}\cdot A\otimes\bm n$,
where $A = \text{diag}(n_1,n_2,\cdots,n_V)$ is a $V\times V$ matrix and $\bm n = (n_1,n_2,\cdots,n_V)$ is defined previously.
As a result, $\Delta(W,W,W) = \frac{1}{m(m-1)(m-2)}\cdot (W^\top AW)\otimes \bm n$.
So the computational complexity of $\Delta(W,W,W)$ depends on how we compute $W^\top AW$.
Since $A$ is a diagonal matrix with at most $M$ non-zero entries, $W^\top AW$ can be computed in $O(Mk^2)$ operations.
Therefore, the time complexity of computing $\Delta(W,W,W)$ is $O(Mk^2)$ per document.
Finally we handle those values with three indices the same, that is, $i=j=k$.
As indicated by Eq. (\ref{eq_T}), we need to add a $\frac{2n_i}{m(m-1)(m-2)}$ term for compensation.
This can be done efficiently by first computing $\widehat{\mathbb E}(\frac{2n_i}{m(m-1)(m-2)})$
for all the documents (requiring $O(NV)$ time),
and then add them up, which takes $O(Vk^3)$ operations.
\iffalse
\section{Discussions}
\subsection{Appendix C.1. Extension to other topic recovery algorithms}
One important advantage of our proposed inference algorithm is its flexibility---the algorithm can be combined with many other LDA inference algorithms to infer supervised LDA model parameters.
More specifically, given access to any algorithm that recovers the topic distribution matrix $O$
and the prior parameter $\bm\alpha$ from i.i.d. sampled documents,
an inference algorithm for a supervised LDA model can be immediately obtained,
as shown in Algorithm \ref{alg2}.
\begin{algorithm}
\caption{sLDA parameter recovery based on an existing LDA inference algorithm $\mathcal A$. Input parameter: $\alpha_0$.}
\centering
\begin{algorithmic}[1]
\State Compute empirical moments and obtain $\widehat M_2, \widehat M_y$. Set $\widehat{\bm\eta} = 0$.
\State Find $\widehat W\in\mathbb R^{n\times k}$ such that $\widehat M_2(\widehat W, \widehat W) = I_k$.
\State Run algorithm $\mathcal A$ with observed documents and parameter $\alpha_0$. Obtain the estimated
topic distribution matrix $\widehat O=(\widehat{\bm\mu}_1,\cdots,\widehat{\bm\mu}_k)$ and prior parameter $\widehat{\bm\alpha}$.
\State Compute $\widehat{\bm v}_i = \sqrt{\frac{\widehat{\alpha}_i}{\alpha_0(\alpha_0+1)}}\widehat W^\top\widehat{\bm\mu}_i$ for each topic $i$.
\State Recover linear classifier: $\widehat\eta_i \gets \frac{\alpha_0+2}{2}\widehat{\bm v}_i^\top\widehat M_y(\widehat W,\widehat W)\widehat{\bm v}_i$.
\State \textbf{Output:} $\widehat{\bm\eta}$, $\bm\alpha$ and $\{\widehat{\bm\mu}_i\}_{i=1}^k$.
\end{algorithmic}
\label{alg2}
\end{algorithm}
The sample complexity of Algorithm \ref{alg2} depends on the sample complexity of the LDA inference algorithm $\mathcal A$.
Although the LDA inference algorithm $\mathcal A$ is free to make any assumptions on the topic distribution matrix $O$,
we comment that the linear independence of topic distribution vectors $\bm\mu$ is still required,
because the power update trick (step 5 in Algorithm 1) is valid only $W\bm\mu_i$ are orthogonal vectors.
\subsection*{Appendix C.2. Going beyond SVD}
The proposed methods are based on spectral decomposition of observable moments
and are provably correct.
However, a major drawback of these methods is their assumption that the topic distributions $\bm\mu$ are linearly independent.
Although standard LDA models assume this topic independence \cite{lda},
in practice there is strong evidence that different topics might be related \cite{correlatelda}.
Therefore, it is important to consider possible extensions.
Recently, there has been much work on provable LDA inference algorithms that do not require the topic independence assumption \cite{practicalalgo,beyondsvd}.
Instead of making the independence assumption, these methods assume a $p$-separability condition on the topic distribution matrix,
that is, there exists an \emph{anchor word} for each topic such that the anchor word only appears in this specific topic,
and its probability (given that topic) is at least $p$.
Here we briefly describe an idea that might lead to an sLDA inference algorithm without assuming topic independence.
First, let $O\in\mathbb R^{V\times k}$ be the topic distribution matrix defined previously and $H = (\bm h_1,\cdots,\bm h_N)\in\mathbb R^{k\times N}$
be the matrix of topic mixing vectors for each document.
Suppose $Q = \mathbb E[\bm x_1\bm x_2^\top]$ is the word co-occurrence frequency matrix
and $Q_y = \mathbb E[y\bm x_1]$ is the co-occurrence frequency matrix between word and regression variable observations.
It is clear that both $Q$ and $Q_y$ can be estimated from the training data, and we have
\begin{equation}
Q = ORO^\top, Q_y = OR\bm\eta,
\end{equation}
where $R \triangleq \mathbb E[HH^\top]$.
Assuming the $p$-separability condition holds and using methods developed in \cite{beyondsvd,practicalalgo},
we can consistently recover the $O$ and $R$.
Now note that $Q_y = (OR)\cdot \bm\eta$.
As a result, knowing both $OR$ and $Q_y$ we reduce the $\bm\eta$ inference problem to solving a linear equation systems.
Future research could be done on (1) determining when the matrix $OR$ has full rank $r$ so that the linear equations can be solved, and
(2) investigating the sample complexity problem for such an inference algorithm.
\subsection*{Appendix C.3. From regression to classification}
A natural extension to our result is to consider supervised LDA models for classification purposes.
The simplest model is a logistic regression classification model, where we assume the response variable $y_d$
for each document $d$ is in $\{+1,-1\}$, and
$$\Pr[y_d = 1|\bm h_d] = \frac{\exp(\bm\eta^\top\bm h_d)}{1+\exp(\bm\eta^\top\bm h_d)},$$
where $\bm\eta$ is a linear classifier.
Though appears simple, such an extension incurs many fundamental problems.
A major obstacle is the fact that the conditional expectation $\mathbb E[y|\bm h]$ is no longer linear
in the topic mixing vector $\bm h$.
As a result, we cannot even evaluate $\mathbb E[y]$ (or higher order tensors like $\mathbb E[y\bm x_1\otimes\bm x_2]$)
in closed forms.
Further more, even if we have accurate estimation to the above moments,
it is rather difficult to infer $\bm\eta$ back due to its non-linearality.
To our knowledge, the classification problem remains unsolved in terms of spectral decomposition methods,
and there is doubt whether a provably correct algorithm exists in such scenarios.
\fi
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,339 |
Q: Mean.js Node.js background processes I've got a mean.js app I'm writing and I am conceptually confused about background processes.
I need to have some processes that run continuously in the background that operate on the mongodb database and do stuff like cleanup, email, tweets, etc.
I need a lot of the same functionality and libraries I have in my web app available to these background procs.
What is the best way to do this? Do I start with a brand new source base and treat these worker procs like a separate app? Or do I create, say, a daemon folder and fork the background procs when I start the server.js with grunt?
I think I am confusing myself and probably making this more complicated than it should be. I have looked at node daemon and child_processes and simple_daemon. But I'm not sure what path to take.
Thanks for your help.
A: You can use setInterval() to run scheduled or repeating tasks within the mean.js app. Because of the way node.js works, as long as node is running your app, any callback defined to run in setInterval() or setTimeout() will run once loaded. This means you can keep your background logic inside your controllers/models or in an adjacent file. You can include your background script, e.g. require()-ing it from the main app.js file, or from anywhere within your controllers, models, etc.
e.g.
app.js:
require('tasks/doStuff');
require('express');
/* express/app stuff here */
tasks/doStuff.js:
require('mongoose');
require('some/other/stuff');
setInterval( function() {
console.log('interval happened');
}, 1000);
This approach does require some design/architectural considerations. Namely, your tasks are now tied to the successful execution of your node mean.js app. If your mean.js app crashes/dies, your tasks will have also died.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 349 |
Hawaii is among the smallest states in America, but that doesn't mean the archipelago isn't growing over time.
The Big Island's volcanic eruption this month, for instance, covered 2,400 acres of existing land with new lava. More could be added to the island's coast as lava reaches to ocean, cooling and hardening, though the United States Geological Survey isn't willing to say how much yet.
Kilauea is the Big Island's most active volcano, and has expanded the island for millennia. Its 1960 Kapoho eruption added nearly 500 acres of fresh turf to the Big Island's southeastern tip. Between 1983 and 2013, Kilauea's lava flows added another 500 acres—marking the prolific thirty-year-long eruption of Pu'u 'Ō'ō, a volcanic cone that's still active today. And an additional five acres were added to the island's shoreline in 2016.
So, here's question for the modern era: Who owns all that new land?
The short answer? The state.
"'New' land belongs to the state, such as land formed by cooling and therefore hardening lava spilling into the ocean," David Callies, a law professor at the University of Hawaii at Manoa who specializes in land use, told me in an email.
In 2008, this precedent was cited to determine that a Big Island man unlawfully built a pavillion on a lava extension in Puna on the southeast coastline. Furthermore, lands covered by fresh lava flows don't change hands. So the dozens of Big Island land owners whose homes were tragically destroyed this month won't lose their properties, for example.
But culturally speaking, it's not so cut and dry. The modern idea of property ownership is relatively new to Hawaii, and Native Hawaiians saw (and still see) themselves as stewards, not masters, of their environment. Ahupua'a, or land divisions, stretched from the ocean to the mountains (under a chieftain's rule), ensuring the people's access to a multitude of resources.
Land was not an object to be bought or sold, but was a responsibility to be cared for in perpetuity...In the mid-1840s, King Kauikeaouli instituted a process that imposed fee simple land ownership. Culminating in the Māhele of 1848 and Kuleana Act of 1850, even these earliest distributions of land were subject to the rights of the maka'āinana, or native tenants, to continue subsistence lifestyles.
In 1893, Hawaii's Queen Lili'uokalani was overthrown by businessmen and sugar barons with the backing of the American military, and in 1959 Hawaii officially became part of the United States.
So, no: Someone can't just stake a claim to the land created by a volcano. People have tried, and failed. Still, it'd be kind of cool, though. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,357 |
Canadian Drug Summary: Cocaine
Canadian Drug Summary Cocaine 2022
Author: Canadian Centre on Substance Use and Addiction
Summarizes data on the use of cocaine, including the harms and effects, legal status and access to treatment. The prevalence of cocaine use was two per cent among people living in Canada in 2019. Cocaine contributes to the second-highest criminal justice costs of any substance in Canada after alcohol.
Presentation on the Impacts of Methamphetamine Use in Canada to the House of Commons Standing Committee on Health
Statement by Rita Notarandrea, CEO, on International Day Against Drug Abuse and Illicit Trafficking
All related news from CCSA
'Dangerous' to think B.C.'s decriminalization plan will reduce OD deaths: researcher
What you need to know about the decriminalization of possessing illicit drugs in B.C.
All related news from Addiction News Daily | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,798 |
.\" Copyright (c) 1986, 1993
.\" The Regents of the University of California. All rights reserved.
.\"
.\" Redistribution and use in source and binary forms, with or without
.\" modification, are permitted provided that the following conditions
.\" are met:
.\" 1. Redistributions of source code must retain the above copyright
.\" notice, this list of conditions and the following disclaimer.
.\" 2. Redistributions in binary form must reproduce the above copyright
.\" notice, this list of conditions and the following disclaimer in the
.\" documentation and/or other materials provided with the distribution.
.\" 3. All advertising materials mentioning features or use of this software
.\" must display the following acknowledgement:
.\" This product includes software developed by the University of
.\" California, Berkeley and its contributors.
.\" 4. Neither the name of the University nor the names of its contributors
.\" may be used to endorse or promote products derived from this software
.\" without specific prior written permission.
.\"
.\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
.\" ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
.\" SUCH DAMAGE.
.\"
.\" @(#)socketpair.c 8.1 (Berkeley) 6/8/93
.\"
#include <sys/types.h>
#include <sys/socket.h>
#include <stdio.h>
#define DATA1 "In Xanadu, did Kublai Khan . . ."
#define DATA2 "A stately pleasure dome decree . . ."
/*
* This program creates a pair of connected sockets then forks and
* communicates over them. This is very similar to communication with pipes,
* however, socketpairs are two-way communications objects. Therefore I can
* send messages in both directions.
*/
main()
{
int sockets[2], child;
char buf[1024];
if (socketpair(AF_UNIX, SOCK_STREAM, 0, sockets) < 0) {
perror("opening stream socket pair");
exit(1);
}
if ((child = fork()) == -1)
perror("fork");
else if (child) { /* This is the parent. */
close(sockets[0]);
if (read(sockets[1], buf, 1024, 0) < 0)
perror("reading stream message");
printf("-->%s\en", buf);
if (write(sockets[1], DATA2, sizeof(DATA2)) < 0)
perror("writing stream message");
close(sockets[1]);
} else { /* This is the child. */
close(sockets[1]);
if (write(sockets[0], DATA1, sizeof(DATA1)) < 0)
perror("writing stream message");
if (read(sockets[0], buf, 1024, 0) < 0)
perror("reading stream message");
printf("-->%s\en", buf);
close(sockets[0]);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,275 |
Q: How can I login to amazon ec2 with root directly in PuTTY or WinSCP?
*
*How can I login to Amazon EC2 with root directly in PuTTY??
Each time I need to su - root first and change to root... Is there any other ways??
*How can I login to Amazon EC2 with root directly in WinSCP or Filezilla??
Since I want to "push" the file to some directly from my computer, but I cannot do this (e.g. create dir when login with ec2-user)
A: Change the PermitRootLogin setting from no to yes in your sshd_config file, then restart your SSH server.
A: When setting up a server, the simplest solution, really, is to just upload the file as root – probably not the best idea from a security perspective, but it really does save a lot of effort when you are copying configuration files back and forth between machines.
On the EC2 instance run:
# visudo (or edit /etc/sudoers)
Comment out Defaults requiretty (line 55)
(or change to Defaults !requiretty)
In WinSCP:
Under Session:
Set the username to same username you login with (the default is ec2-user)
Change the File protocol to 'SCP'
Under Environment > SCP/Shell:
Change the shell to 'sudo su -' (available in the dropdown)
Login and you should find yourself in /root. Once you no long need to use SCP as root, it is advisable to re-enable requiretty.
Source: http://www.thatsgeeky.com/2011/10/connect-to-amazons-linux-via-winscp-as-root/
A:
*With WinSCP you should first identify the location of the sftp-server
You can run the following command:
find / -name sftp-server
In my case, the location was: /usr/libexec/openssh/sftp-server
The previous location should be configured in WinSCP in your session: Advanced Site Setting > SFTP > SFTP Server
sudo /usr/libexec/openssh/sftp-server
A: I use Filezilla to login, and follow the steps below.
To connect to a running Amazon EC2 instance with Filezilla:
*
*Edit -> Settings -> Connection -> SFTP
Click "Add keyfile"
*Browse to the location of your .pem file and select it.
*A message box will appear asking your permission to convert the file into ppk format. Click Yes, then give the file a name and store it somewhere.
*If the new file is shown in the list of Keyfiles, then continue to the next step. If not, then click "Add keyfile..." and select the converted file.
*File -> Site Manager
*Add a new site wih the following paramerters:
Host: Your public dns name of ec2 instance
Protocol: SFTP
Logon Type: Normal
User: ec2-user
Press Connect Button
Video Tutorial: Click Here
| {
"redpajama_set_name": "RedPajamaStackExchange"
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THEORY OF MIND AND LITERATURE
# THEORY OF MIND AND LITERATURE
EDITED BY
PAULA LEVERAGE, HOWARD MANCING,
RICHARD SCHWEICKERT, AND JENNIFER MARSTON WILLIAM
Purdue University Press
West Lafayette, Indiana
Copyright 2011 by Purdue University. All rights reserved.
Printed in the United States of America.
**Library of Congress Cataloging-in-Publication Data**
Theory of mind and literature / edited by Paula Leverage ... [et al.].
p. cm.
Includes bibliographical references and index.
ISBN 978-1-55753-570-2
1. Literature--History and criticism--Theory, etc. 2. Philosophy of mind in literature. I. Leverage, Paula, 1970-
PN441.T45 2010
801'.92--dc22
2010009895
To our loving families.
## Contents
**Acknowledgments**
**Introduction**
**1: Theory of Mind Now and Then: Evolutionary and Historical Perspectives**
[Theory of Mind and Theory of Minds in Literature
_Keith Oatley_](part01chapter01.html)
[Social Minds in _Little Dorrit_
_Alan Palmer_](part01chapter02.html)
[The Way We Imagine
_Mark Turner_](part01chapter03.html)
[Theory of Mind and Fictions of Embodied Transparency
_Lisa Zunshine_](part01chapter04.html)
**2: Mind Reading and Literary Characterization**
[Theory of the Murderous Mind: Understanding the Emotional Intensity of John Doyle's Interpretation of Sondheim's Sweeney Todd
_Diana Calderazzo_](part02chapter05.html)
[Distraction as Liveliness of Mind: A Cognitive Approach to Characterization in Jane Austen
_Natalie Phillips_](part02chapter06.html)
[Sancho Panza's Theory of Mind
_Howard Mancing_](part02chapter07.html)
[Is Perceval Autistic?: Theory of Mind in the _Conte del Graal_
_Paula Leverage_](part02chapter08.html)
**3: Theory of Mind and Literary / Linguistic Structure**
[Whose Mind's Eye? Free Indirect Discourse and the Covert Narrator in Marlene Streeruwitz's _Nachwelt_
_Jennifer Marston William_](part03chapter09.html)
[Attractors, Trajectors, and Agents in Racine's "Récit de Théramène"
_Allen G. Wood_](part03chapter10.html)
[The Importance of Deixis and Attributive Style for the Study of Theory of Mind: The Example of William Faulkner's Disturbed Characters
_Ineke Bockting_](part03chapter11.html)
**4: Alternate States of Mind**
[Alternative Theory of Mind for Artificial Brains: A Logical Approach to Interpreting Alien Minds
_Orley K. Marron_](part04chapter12.html)
[Reading Phantom Minds: Marie Darrieussecq's _Naissance des fantômes_ and Ghosts' Body Language
_Mikko Keskinen_](part04chapter13.html)
[Theory of Mind and Metamorphoses in Dreams: _Jekyll & Hyde_, and _The Metamorphosis_
_Richard Schweickert and Zhuangzhuang Xi_](part04chapter14.html)
[Mother/Daughter Mind Reading and Ghostly Intervention in Toni Morrison's _Beloved_
_Klarina Priborkin_](part04chapter15.html)
**5: Theoretical, Philosophical, Political Approaches**
[Changing Minds: Theory of Mind and Propaganda in Egon Erwin Kisch's _Asien gründlich verändert_
_Seth Knox_](part05chapter16.html)
[Functional Brain Imaging and the Problem of Other Minds
_Dan Lloyd, Vince Calhoun, Godfrey Pearlson, and Robert Astur_](part05chapter17.html)
[How is it Possible to Have Empathy? Four Models
_Fritz Breithaupt_](part05chapter18.html)
[Theory of Mind and the Conscience in _El casamiento engañoso_
_José Barroso Castro_](part05chapter19.html)
**Contributors**
**Index**
## Acknowledgments
As most books do, _Theory of Mind and Literature_ has accumulated a significant number of debts in the course of its development, not least because it has been a collaborative enterprise on several levels since its inception. The book emerges from the exciting interdisciplinary exchanges that took place at a conference of the same name at Purdue University at the end of 2007. Our first debt of gratitude is to the seventy participants in the conference who met over the course of four days to explore intently the importance of Theory of Mind in reading and understanding literature. The conversations, both formal and informal, were exciting, original, and always productive. Many of the essays submitted for consideration in this volume were reworked in light of insights culled from the conference.
Neither the conference, nor this book would have been possible without the unstinting moral and financial support of Purdue University, especially the College of Liberal Arts, under the direction of former Dean John Contreni; the Department of Foreign Languages and Literatures, under the direction of former Head Paul Dixon; and the Department of Psychological Sciences, under the direction of former Head Howard Weiss. We would also like to thank the following programs and departments for additional generous support: Classics, Film Studies, Jewish Studies, the Department of English, Comparative Literature, and Medieval and Renaissance Studies. We are extremely grateful to Howard Weiss and Christopher Agnew (former and current Head of the Department of Psychological Sciences, respectively) for subsidizing the preparation of the index, and to Tom Berndt (interim Head of the Department of Foreign Languages and Literatures) for additional support.
For permission to reprint the essays by Mark Turner and Lisa Zunshine, we are grateful to Oxford University Press and the British Academy, to Ohio State University Press, and especially to Ilona Roth for her assistance in reprinting Mark Turner's essay. For careful reading and astute suggestions for revision, we are grateful to the reviewers of the individual essays and the book as a whole. Finally, we would like to thank Charles Watkinson, the Director of Purdue University Press, and the staff of Purdue University Press, especially Katherine Purple and Becki Corbin, for guiding this book patiently and expertly to publication.
## Introduction
Theory of Mind (ToM) is what enables us to "put ourselves in another's shoes." It is mind reading, empathy, creative imagination of another's perspective: in short, it is simultaneously a highly sophisticated ability, and a very basic necessity for human communication. ToM is central to such commercial endeavors as market research and product development, but it is also just as important in maintaining human relations over a cup of coffee. Not surprisingly, it is a critical tool in reading and understanding literature, which abounds with characters, situations, and "other people's shoes." Furthermore, it is becoming increasingly apparent that reading literature also hones these critical mind reading skills (Oatley 2009; Mar, Djikic, Oatley 2008; Zunshine 2006).
In her popular book _Mindreading: An Investigation into How We Learn to Love and Lie_ (1997), the primatologist-journalist Sanjida O'Connell defines ToM as:
> > the mechanism we use to understand what is going on in other people's heads. How we react to one another socially is the most important aspect of our lives. Without an understanding of what people think, what they want and what they believe about the world, it is impossible to operate in any society. Theory of Mind is the name given to this understanding of others. It is the basic necessity of humanity and is understood the same way the world over. (2)
O'Connell's use of vocabulary here reflects terms we use frequently in everything we say and write: understand, think, want, believe. She suggests that these verbs have "become invisible" and that "We interpret people's actions using words that describe their mental states so often that we cease to think about what it is we are actually doing" (3). ToM is the core tenet of what is often called _folk psychology_ (or belief-desire psychology): the default understanding that other people are (largely) autonomous agents, that they have mental states commonly called beliefs and desires, and that they are motivated by these mental states. When we rely on our folk psychology, we tend to understand, define, and describe people on the basis of their perceived (or understood) beliefs, desires, feelings, values, experiences, and intentions. It is because we understand people's actions in terms of these mental states that we explain to ourselves and to each other why people have done certain things, and predict what they might do in certain contexts.
When we read a work of literature, we treat characters as if they were real people, and we ascribe to them a ToM. We could not understand a novel or a poem if we did not do this:
> Without our ability to form a theory of mind, human culture would not be possible. Much of the world of literature, drama, and humor relies on the supreme ability of humans not only to create theories about each character's mind but also to imagine simultaneously how each of these imaginary minds might view the minds of other characters. The tragic nature of Shakespeare's _Romeo and Juliet_ , for instance, comes from a series of misconceptions among the characters that only the audience is aware of. Romeo's suicide is the result of his thinking that Juliet has died, and the audience is aware that if Romeo knew what they knew, this suicide would not have to have happened. To an audience of monkeys, however, Romeo's actions would make no sense, because they wouldn't be able to distinguish between their own beliefs and his.
In this way William F. Allman lays out the basic premise that ToM is as essential for understanding a literary work as it is for understanding a verbal utterance or another human being's motives (68-69).
Psychologist and anthropologist Robin Dunbar attributes to the human ToM the "crucial ability to step back from ourselves and look at the rest of the world with an element of disinterest" (101). Dunbar describes three levels of ToM: 1) the ability to be aware of our own thoughts; 2) the ability to understand other people's feelings; and 3) the ability "to imagine how someone who does not actually exist might respond to particular situations" (101-2). Elaborating on this third level of ToM, Dunbar states its obvious implication: "we can begin to create literature, to write stories that go beyond a simple description of events as they occurred to delve more and more deeply into why the hero should behave in the way he does, into the feelings that drive him ever onwards in his quest" (102).
While psychologists and anthropologists have long recognized the importance of ToM to literature, it was not until very recently, with the publication of the first monograph on the subject, that literary scholars began to take seriously the implications of ToM. Lisa Zunshine's seminal _Why We Read Fiction: Theory of Mind and the Novel_ (2006) argues forcefully and eloquently that ToM "makes literature as we know it possible" (10). The title of this work announces the author's basic premise that ToM is not only an essential prerequisite of the reading experience, but also a motivation for reading. Zunshine argues that reading engages and improves mind reading abilities. This presents an argument for an evolutionary bias of the aesthetic experience, which situates literature squarely within the parameters of the essential human experience and not on the periphery as a distraction for people of leisure and means.
However, the history of scientific inquiry into ToM does not start with human beings, much less with the supremely human creation that is literature. The term itself was coined to describe the inferences and predictions made by chimpanzees. In their classic paper, "Does the chimpanzee have a theory of mind?" Premack and Woodruff say, "An individual has a theory of mind if he imputes mental states to himself and others. A system of inferences of this kind is properly viewed as a theory because such states are not directly observable, and the system can be used to make predictions about the behavior of others" (115). Their subject, Sarah, was a fourteen-year-old chimpanzee. Sarah was highly trained and famous for earlier work in which she learned to communicate with plastic shapes placed in order. When Sarah was shown videos of a human confronting a problem, her responses indicated she was imputing knowledge and intentions to the human. Premack and Woodruff concluded that the chimpanzee Sarah had a ToM. If questions about ToM were confined to animals, the topic would be of interest only to specialists. But before long, investigators began to ask whether a human necessarily has a ToM. Wimmer and Perner found that a child less than six years old typically seems to believe that if he or she knows something, then everyone else knows it too. Very young children seem not to have developed a ToM, and it also appears that ToM is underdeveloped in autistic people, who often have impaired socialization, imagination, and communication. Baron-Cohen, Leslie, and Frith found that most autistic children are unable to impute beliefs to others, perhaps a source of difficulties in predicting behavior of others.
Investigators suspect that something in the brain changes when ToM develops. In a recent review, Gallagher and Frith conclude that ToM ability is mediated by the anterior paracingulate cortex, with activity of the superior temporal sulcus and temporal poles in a supportive role. Details remain to be settled, but it is remarkable that we are close to discovering how brain activity differs depending on whether or not mental states must be imputed to others.
Complex stimuli are needed for learning how people apprehend the mental world of others. In experiments on ToM, stories are told, videos are shown, and people act out skits. These materials, richer than usual for cognitive scientists, are minimal compared with what literary theorists study, but on the verge. Cognitive scientists notice this, of course, but are reluctant to follow up on their own. As Premack and Woodruff say in their study of mental states, "there are other, more exotic ones [mental states], belonging to the novelist; we will not be concerned with them in this paper" (515).
As we have seen above, the "exotic" mental states Premack and Woodruff impute to the novelist are beginning to be investigated, along with the mental states of the readers who make sense of the novels. For example, Alan Palmer's _Fictional Minds_ proposes that readers construct an embedded narrative for characters from references scattered throughout novels that then permit readers to enter their story world and minds.
Studies in ToM and literature belong to a relatively new area of investigation that brings an understanding of cognitive scientific studies and neuroscience to the reading and interpretation of literature. This "cognitive turn" in literary theory started during the 1980s and was influenced by work in cognitive linguistics (Lakoff and Johnson). In very recent years the field has expanded rapidly, as more scholars seek new perspectives beyond post-structuralist and social constructionist models of literature.
_Theory of Mind and Literature_ is a timely contribution to the field in advancing current research on reading mental states in fictional characters. However, it is important to recognize that while mental states in novels may appear "exotic" in comparison to any mental states imputed to or by chimpanzees, there is, in fact, nothing even vaguely exotic about mind reading in literature. As Mark Turner has so convincingly argued in _The Literary Mind_ , the cognitive mechanisms for understanding literature are precisely those we employ in quotidian tasks. Lakoff and Johnson's work on metaphors likewise demonstrated before Turner that human thought processes are largely metaphorical. Literature, as observed above, does not belong to the category of decorative vanities with which we populate our world, but is germane to our human condition. As such, perhaps it may be considered to be a form of distributed cognition, or an agent of what Alan Palmer calls "intermental thought."
It was in order to explore the implications of ToM for literature and literary theory that the international conference on "Theory of Mind and Literature" was held at Purdue University in November 2007. The idea for this conference emerged during the Literature and Cognitive Science Conference at the University of Connecticut in spring 2006, which was the first North American conference to offer a forum to scholars working at the intersection of two disciplines with increasingly good reason to communicate across disciplinary boundaries. The two most popular areas of research at the Literature and Cognitive Science conference were conceptual blending (a general theory of cognition proposed by Mark Turner and Gilles Fauconnier to describe the "blending" that underlies thought and language) and ToM. The aim in bringing a conference on Theory of Mind and Literature to Purdue University was to maintain the momentum of the first conference's intellectual stimulation, while offering opportunities to explore more deeply one particular subdomain of cognitive literary theory through a tightly focused theme.
The academic collaboration and discussion that set in motion a cascade of ideas and exciting research at the first conference on cognitive literary studies at the University of Connecticut, and the first conference on Theory of Mind and Literature at Purdue, illuminate the essays presented within this volume, which hopefully conveys the excitement of pioneering work in a new area of research.
_Theory of Mind and Literature_ is a collection of nineteen essays by prominent scholars working in the field of cognitive literary studies. The essays, by literary scholars, linguists, cognitive scientists, and philosophers, range widely across national literatures (Spanish, French, German, British, American), genres (theatre, poetry, science fiction, novel), and historical periods (from the Middle Ages to the functional brain imaging of the twenty-first century), illuminating the central, enduring importance of ToM to our human condition.
_Theory of Mind and Literature_ aims to explore the functionality of ToM as a concept in literary studies, and the scope of its usefulness across a wide range of contexts. Among our interests in assembling this collection of essays are the following objectives: to introduce readers to the myriad ways in which ToM can illuminate critical readings of literature; to present models of how to approach literary texts from a cognitive perspective; to extend the scope of the Theory of Mind and Literature Conference beyond the four days of the conference itself by sharing the insights and perspectives of the new research presented there with a wider audience; and to promote cognitive literary studies in general.
The book is structured thematically, with the nineteen essays organized into five sections. The first section, "Theory of Mind Now and Then: Evolutionary and Historical Perspectives," features essays by four of the leading scholars currently working in the field: Keith Oatley, Alan Palmer, Mark Turner, and Lisa Zunshine. The following four sections each contain four essays (with the exception of the section on Theory of Mind and Literary / Linguistic Structure with three essays), which cohere thematically under the following headings: "Mind Reading and Literary Characterization"; "Theory of Mind and Literary / Linguistic Structure"; "Alternate States of Mind"; and "Theoretical, Philosophical, Political Approaches."
The opening section, "Theory of Mind Now and Then: Evolutionary and Historical Perspectives," introduces some of the central concerns in current discussions about ToM and literature, and in cognitive literary studies in general: evolution (Oatley, Turner, and Zunshine), embodied cognition (Zunshine), distributed cognition (Palmer), and blending theory (Turner). Keith Oatley's "Theory of Mind and Theory of Minds in Literature" reviews work on evolution, narrative, and ToM. Despite the advantages to having a ToM, data show that humans are "good at theory of mind, but not that good." For millennia, humans have supplemented meager facts about thoughts and feelings of others with knowledge gained from experience in worlds of narrative. Fiction simulates an entire social world, presenting a "theory of minds," not only a "Theory of Mind." It is natural to claim that social understanding benefits from reading fiction, and recent data support this.
Alan Palmer's "Social Minds in _Little Dorrit_ " continues Oatley's emphasis on pluralities of mind within fictional social worlds by exploring the workings of social or public minds and intermental thought in the story world of Charles Dickens's _Little Dorrit_. Alan Palmer contends that while traditional narrative theory privileges an "internalist" perspective on fiction, it is important to recognize that novels are largely invested in social minds, and that an "externalist" perspective, which studies the aspects of the mind that are "outer, active, public, social, behavioral, evident, embodied, and engaged," uncovers much unrecognized subtlety. Through careful, close readings, he demonstrates how characters form intermental units that are capable of communicating through facial expressions, looks, and signs, and how Dickens's prose represents socially distributed cognition. He also draws attention to Dickens's awareness of _physically_ distributed cognition. In conclusion, he theorizes that eventually, after many more analyses of minds in novels, it may be possible to identify differences among novels, novelists, and literary periods in their emphasis on inner, passive, introspective, individual thought on the one hand, and outer, social, intermental thought on the other, and to attribute these differences to historical, cultural factors.
The essays by Mark Turner and Lisa Zunshine both emphasize cognitive processes integral to everyday living that are manifest in the ways in which we read and construct literature and art: in Turner's essay, "conceptual, and double-scope, integration"; in Zunshine's, the physical body's relation, and perceived relation, to the mind and mental states. In "The Way We Imagine," Mark Turner describes the mental operation of "conceptual integration," a cognitive procedure by which conceptual arrays are combined to produce new mental spaces. Particularly important is "double-scope" integration, a hallmark characteristic of human beings. Turner proposes that this ability arose some fifty thousand years ago in the Upper Paleolithic age, and discusses how these double-scope blendings are constructed, and the implications of recognizing this process.
Zunshine's essay, "Theory of Mind and Fictions of Embodied Transparency," elucidates how the human body functions as a text to be read by others through brief moments of "embodied transparency"—unintentional, physical indications of emotion. Zunshine provides a short overview of ToM before outlining the double perspective that embodied transparency implies: the body might appear to be the window to the mind, but it does not always provide accurate information regarding mental states. Through a variety of examples from literature, painting, and film, Zunshine situates representations of context-dependent embodied transparencies within our cognitive evolutionary heritage. She concludes the essay by suggesting that the double view of the body may be used to productively interpret how other cultural media, such as the disembodied but extremely social network of the Internet, are shaped.
Following the essays by Keith Oatley, Alan Palmer, Mark Turner, and Lisa Zunshine, the section on "Mind Reading and Literary Characterization" illustrates how bringing ToM to literary studies can augment our understanding of the traditional terms of literary analysis such as characterization. This section thus bridges the divide between more traditional forms of literary criticism and a cognitive theoretical approach. Diana Calderazzo's essay on the 2005 Broadway production of _Sweeney Todd_ , "Theory of the Murderous Mind: Understanding the Emotional Intensity of John Doyle's Interpretation of Sondheim's _Sweeney Todd_ ," analyzes the processes of mirroring and empathy, which bring the minds of the audience unconsciously into an intense relationship with those of the production's macabre characters. Her introduction affords a useful overview of the legend of the murderous barber from the nineteenth to the twenty-first centuries, demonstrating the ageless, universal appeal of the grizzly tale, which she attributes to the biological basis of primary emotional response. Drawing on Antonio Damasio's insights into the body as a theater of the emotions, and Bruce McConachie's cognitive analysis of theatrical reception, she addresses aspects of storytelling, concepts of obsession and compulsion, and the function of laughter as a foil to fear within Doyle's production.
Natalie Phillips's essay on "Distraction as Liveliness of Mind: A Cognitive Approach to Characterization in Jane Austen" continues the discussion of the reception of characters, but in relation to characterization in the nineteenth-century novel. She contrasts the distractible and flexible Elizabeth Bennet in _Pride and Prejudice_ with her focused and concentrated sister Mary. She argues that the reader's ToM is stimulated unequally by the two characters, and that Jane Austen uses this imbalance to impress upon the reader the relative complexity of the mind of Elizabeth. Phillips's analysis shows that understanding a major character in literature requires extending a comparative ToM to minor characters.
The last two essays in this section by Howard Mancing and Paula Leverage continue to investigate ToM in relation to characterization, but focus less on the reader's use of ToM in constituting the characters, than on the ToM of the characters themselves. Both of these essays analyze literature from a much earlier period: from the early seventeenth and late twelfth centuries, respectively. In "Sancho Panza's Theory of Mind," Howard Mancing examines a scene from early in the second part of Cervantes's _Don Quixote_ (1615) in which Sancho uses his chivalric ToM developed throughout the earlier parts of the novel to deceive Don Quixote and make him believe that a peasant woman is the incomparable (and imaginary) beauty Dulcinea. Mancing further argues that the subtlety and sophistication of this scene (and others from Renaissance Spanish literature) refute the claim that the presentation of such subtle mind reading did not appear before the time of Jane Austen.
In "Is Perceval Autistic?: Theory of Mind in the _Conte del Graal_ ," Paula Leverage reexamines the characterization of Perceval in Chrétien de Troyes's twelfth-century _Conte du Graal_ and in the romance's reception from the Middle Ages to the present. She argues that the dominant perceptions in the Arthurian tradition of Perceval as a naïve, ignorant young man are constructions from later works, which misinterpret the first literary incarnation of Perceval, who, in Chrétien's romance, displays the behavioral characteristics of an autistic individual. Leverage's analysis of Perceval as autistic affords a critical perspective that is relevant to many scenes of the romance, which in the past have been interpreted separately, using in each case quite different models to explain Perceval's behavior. She further argues that ToM is an important concern of the romance as a whole, which explores communication and its failures.
The third section "Theory of Mind and Literary / Linguistic Structure" continues to probe the relationship between traditional literary criticism and that informed by cognitive science by considering how stylistic elements of written texts such as Free Indirect Discourse (FID) and attributive style mediate the minds of characters, their narrators, and the minds of readers. Jennifer Marston William's essay "Whose Mind's Eye? Free Indirect Discourse and the Covert Narrator in Marlene Streeruwitz's _Nachwelt_ " argues for a reframing of the narrative phenomenon of FID within the context of the cognitive "mind's eye" metaphor. Through the example of Marlene Streeruwitz's 1999 novel _Nachwelt_ , which is written primarily in the mode of FID, William illustrates how FID shifts the deictic center and renders the third-person narrator temporarily covert while representing the mind's-eye view of the protagonist. The unmediated thoughts, motivations, and emotions of the character conveyed by FID prompt the reader's mind reading processes. At the same time, particularly in a strongly autobiographical text such as _Nachwelt_ , FID allows the author to convey his or her own perspective while maintaining a certain distance from his or her narrator, so that the reader is not encouraged to equate the two positions completely.
In "Attractors, Trajectors, and Agents in Racine's 'Récit de Théramène,'" Allen G. Wood provides a close reading of Théramène's famous _récit_ passage in Racine's tragedy _Phèdre_. By applying the cognitive poetic concepts of textual attractors, trajectors, and agents, Wood demonstrates how Racine foregrounds certain aspects to elicit an emotional response in the spectator or reader. Wood contends that the _récit_ functions for Théramène, the guardian of the young prince Hippolyte, as a mind reading device by which he aims to "update" the beliefs of Thésée, Hippolyte's father, who believes his son to be in love with stepmother Phèdre. At the same time, the passage becomes a mental conduit between the play's cast and its audience members, who receive insights into the mind of the character Théramène, the first-person narrator for the _récit_.
The third and final essay in this section, by Ineke Bockting, "The Importance of Deixis and Attributive Style for the study of Theory of Mind: The Example of William Faulkner's Disturbed Characters," is a linguistically informed analysis that elucidates how deixis and attributive clauses work to present a fictional character's ToM. Through a consideration of Faulkner's characters with cognitive problems, such as the autistic Benjy or the apparently schizophrenic Quentin in _The Sound and the Fury_ , Bockting examines how an unanchored or absent deictic projection can result in a challenging narrative style, but one that reflects the confusion and communicative obstacles faced by these literary figures. Bockting also discusses the significance of attributive verb clauses, which directly reveal characters' mental experiences, raw sensations, and temporal perspectives.
The fourth section of the book, "Alternate States of Mind," introduces robots, ghosts, and dream personae, thus expanding the limits of ToM. Orley Marron proposes that we consider what she calls a Theory of Artificial Mind (ToAM) in "Alternative Theory of Mind for Artificial Brains: A Logical Approach to Interpreting Alien Minds." She bases this ToAM on the stories about human-robot interactions in Isaac Asimov's _I Robot_ and on the adventures of Alice in the books by Lewis Carroll. We use our ToM in natural, nonconscious ways when dealing with other people, but this is not adequate for understanding the cognitive processes of nonstandard human minds. When dealing with robots or the fantasy inhabitants of Carroll's worlds, Marron suggests, we must more consciously be aware of the types of logical, deductive steps one must go through in our mind reading processes.
Mikko Keskinen's essay "Reading Phantom Minds: Marie Darrieussecq's _Naissance des fantômes_ and Ghosts' Body Language" introduces the concept of mind reading in relation to ghosts, which is taken up again in the fourth essay of this section by Klarina Priborkin, who writes about the ghost in Toni Morrison's _Beloved_. Keskinen is exploring the limits of mind reading in a situation in which body language has been silenced. In Marie Darrieussecq's novel _Naissance des fantômes_ (1999), the female narrator's husband has disappeared. It is unclear whether he has died or simply disappeared. This ambiguity is the premise of the novel as the wife tries to impute intention and emotion to her absent, ghostly husband through ToM. The novel recounts the narrator's efforts to read her husband's mind, basing her inferences on memories, since in her husband's corporeal absence, she cannot read his body language. Keskinen draws a comparison between the narrator's reading of the paranormal with reading fictional characters in general, since in both cases "Theory of Phantom Mind" equals "Practice of Absent Body," and suggests that we supplement Fludernik's "natural narratives" and Richardson's "unnatural narration" with the supernatural aspects of narrative fiction.
Continuing the exploration of alternative Theories of Mind in ephemeral beings such as ghosts, psychologists Richard Schweickert and Zhuangzhuang Xi present their research on ToM in dreams. Their contribution, "Theory of Mind and Metamorphoses in Dreams, _Jekyll & Hyde_, and _The Metamorphosis_ ," looks at metamorphoses in dreams and in two works informed by dreams, _Dr. Jekyll and Mr. Hyde_ , by Stevenson, and _The Metamorphosis_ , by Kafka. When a metamorphosis occurs, presumably the inner state of one character is replaced by that of another. But is this noted by the dreamer or author? Schweickert and Xi ask whether an increase in ToM attributions accompanies a metamorphosis.
Klarina Priborkin's "Mother / Daughter Mind Reading and Ghostly Intervention in Toni Morrison's _Beloved_ " returns to ghosts, as she shows how Beloved, the ghost of the murdered child in Morrison's novel, enables the character Denver to acquire the mind reading skills necessary to understand the perspective of her mother, Sethe. Priborkin first traces how intermental thinking shapes the environment in which Denver is raised, contributing to a repression of past black slavery as well as to silence about the infanticide committed by Sethe. Priborkin then analyzes the interactions of Beloved and Denver in the context of ToM, illuminating how Denver's empathy for her mother did not materialize suddenly and without explanation, but can be interpreted rather as a result of Beloved's mediation.
The final section of _Theory of Mind and Literature_ , "Theoretical, Philosophical, Political Approaches," proposes working models for ToM-based analysis developed through functional brain imaging or philosophical inquiry. This section also extends the discussion beyond literature to consider ToM in the political arena, where its role in propagandist agendas is a serious concern. The first essay, Seth Knox's "Changing Minds: Theory of Mind and Propaganda in Egon Erwin Kisch's _Asien gründlich verändert_ ," discusses literary propaganda through analysis of the travel book _Asien gründlich verändert_ ( _Changing Asia_ ), by German writer Egon Erwin Kisch. Specifically, Knox studies the way in which Kisch employs two mind reading techniques: 1) "anchoring" strategies that resonate with the sympathetic reader's values; and 2) a blurring of the reader's "source-tracking" ability. With reference to these techniques, Knox adapts Simon Baron-Cohen's concept of a "Shared Attention Mechanism" to construct a graphic model of propagandist-propagandee relationships. In the three essays that follow, the authors propose further models for investigating ToM, which focus particularly on the relationship between narration, empathy, and mind reading. In "Functional Brain Imaging and the Problem of Other Minds," Dan Lloyd and his co-authors Vince Calhoun, Godfrey Pearlson, and Robert Astur point out that subjectivity would not be noticed were it not for differences between individuals. This presents a problem for many contemporary attempts to learn about subjective experiences with neuroimaging, because the typical method of averaging over subjects eliminates differences. They discuss an approach in their neuroimaging work in which similar brain images from the same subject at different times are clustered into categories. The passage of the brain from one category to another over time becomes the object of study, and they propose that techniques from narratology may become useful for further development of this approach. Fritz Breithaupt's "How is it Possible to have Empathy? Four Models" compares sources of empathy using evidence arising in narrative fiction. He notes that three commonly proposed sources are not contradictory, namely (a) empathy from similarity of oneself to others, (b) expectation that empathy extended will be reciprocated, and (c) empathy as a basis for understanding the emotions or predicting the behavior of another. He proposes combining the three in a fourth, (d) empathy arising in a narrative scene. With this proposal, narration enables layers of empathy.
José Barroso Castro's "Theory of Mind and the Conscience in _El casamiento engañoso_ " opens with a statement by Cervantes from the prologue to his 1613 collection of short fictions, _Novelas ejemplares_ ( _Exemplary Stories_ ). Cervantes claims to have metaphorically placed a billiard table in the public square so that all readers might entertain themselves as they choose with his tales. The author then proceeds to study the mental processes of the two main characters in the story _El casamiento engañoso_ ( _The Deceitful Marriage_ ), Peralta and Campuzano. Taking a historical perspective, Barroso Castro places his inquiry within the Scholastic and Spanish Golden Age contexts by examining the implications of theologians such as Augustine and Thomas as well as writers contemporary to Cervantes.
The editors of this collection are four Purdue University faculty members who organized the Theory of Mind and Literature Conference at Purdue in November 2007. Paula Leverage, Howard Mancing, and Jennifer Marston William are from the Department of Foreign Languages and Literatures, and Richard Schweickert is from the Department of Psychological Sciences. The editors are the founding members of Purdue University's Center for Cognitive Literary Studies.
### **W ORKS CITED**
Allman, William F. _The Stone Age Present: How Evolution Has Shaped Modern Life— From Sex, Violence, and Language to Emotions, Morals, and Communities_. New York: Simon and Schuster, 1994. Print.
Baron-Cohen, Simon, Alan M. Leslie, and Uta Frith. "Does the Autistic Child Have a 'Theory of Mind'?" _Cognition_ 21 (1985): 37-46. Print.
Dunbar, Robin. _Grooming, Gossip, and the Evolution of Language_. Cambridge: Harvard University Press, 1996. Print.
Fauconnier, Gilles, and Mark Turner. _The Way We Think: Conceptual Blending and the Mind's Hidden Complexities_. New York: Basic Books, 2002. Print.
Gallagher, Helen L., and Christopher D. Frith. "Functional Imaging of 'Theory of Mind.'" _Trends in Cognitive Sciences_ 7 (2003): 77-83. Print.
Holland, Norman N. _The Brain of Robert Frost_ : _A Cognitive Approach to Literature_. New York: Routledge, 1988. Print.
Lakoff, George, and Mark Johnson. _Metaphors We Live By_. Chicago: University of Chicago Press, 1980. Print.
Mar, Raymond, Maja Djikic, and Keith Oatley. "Effects of Reading on Knowledge, Social Abilities, and Selfhood." In _Directions in Empirical Literary Studies: In Honor of Willie van Peer_. Eds. Sonia Zyngier, Marisa Bortolussi, Anna Chesnokova, and Jan Auracher. Amsterdam: Benjamins, 2008. 127-37. Print.
Oatley, Keith. "Changing our minds." _Greater Good_ 5 (2009). Print.
Palmer, Alan. _Fictional Minds_. Frontiers of Narrative. Lincoln: University of Nebraska Press, 2004. Print.
Premack, David and Guy Woodruff. "Does the Chimpanzee have a Theory of Mind?" _Behavioral and Brain Sciences_ 4 (1978): 515-26. Print.
Turner, Mark. _The Literary Mind_. New York: Oxford University Press, 1996. Print.
Wimmer, Heinz, and Josef Perner. "Beliefs About Beliefs: Representation and Constraining Function of Wrong Beliefs in Young Children's Understanding of Deception." _Cognition_ 13 (1983): 103-28. Print.
Zunshine, Lisa. _Why We Read Fiction: Theory of Mind and the Novel_. Columbus: The Ohio State University Press, 2006. Print.
## Theory of Mind and Theory of Minds in Literature
KEITH OATLEY
### PART 1. THEORY OF MIND
Not far into _The Way by Swann's_ (the title of the rather good new translation by Lydia Davis of the first book of Proust's _À la recherche du temps perdu_ ) readers find this:
> A real human being, however profoundly we sympathize with him, in large part is perceived through our senses, that is to say remains opaque to us, offers a dead weight that our sensibilities cannot lift. If a calamity should strike him, it is only in a small part of the total notion that we have of him that we will be able to be moved by this. (87)
Because of the opacity of others, human beings need a theory or model, or simulation, or _image_ (Proust's term) of mind to infer what a real human being might be thinking and feeling. Theory of Mind, in this sense, has become an important topic in developmental psychology. Thus psychologists have discovered that the soul enters the body not at conception or at birth, but at age four. It is at this age that a child can start to experience herself or himself as a thinking and feeling being who is separate from others. It is at this age that the child can begin to peer through the opacity of others, to infer what they might be thinking and feeling. The same applies to the child's own self: it is at this age that a child begins to know that what she or he thinks and feels now can be different from what she or he had thought and felt in the past (see, for instance, Gopnik and Astington 1988).
Proust was also onto the comparison with other versions of oneself that Gopnik and Astington described. He says: "Even more, it is only in a part of the total notion he has of himself that he will be able to be moved by himself" ( _The Way by Swann's_ 87).
So, being able to know others and oneself requires imagination. The response of psychologists to these issues has been to conduct many experiments to find out what kinds of inferences children make about others, and at what ages they can make them. There has been considerable progress. But we might think that in literature, progress has been yet greater, for as Proust says:
> The novelist's happy discovery was to have the idea of replacing these parts [of the real human being], impenetrable to the soul, by an equal quantity of immaterial parts, that is to say parts which our soul can assimilate . . . [the novelist] sets loose in us all possible happinesses and all possible unhappinesses, just a few of which we would spend years of our lives coming to know. ( _The Way by Swann's_ 87)
Having a Theory of Mind is like having a watch or a clock. These devices are models of the rotation of the earth. The positions of sun and stars are indications of time, but they are difficult to judge accurately, and heavenly bodies are not always visible. A model—a watch or clock—is much better. It allows us to synchronize our activities with each other, as in the following passage from Simenon's _Les vacances de Maigret,_ in which the Commissioner has gone on holiday with his wife to the seaside town of Sables-d'Olonne. Recovering from an emergency operation for appendicitis, Mme Maigret is in a clinic run by a strict order of nuns. Maigret is allowed to visit her each day for just half an hour, at three in the afternoon.
> Before putting his foot on the door-step, Maigret took his watch from his pocket: it said three o'clock. At the same moment he heard a bell call lightly from the chapel, then, above the roofs of the little houses of the town, another one, more serious, from Notre Dame. (trans. K. Oatley. _Les vacance_ s _de Maigret_ 5-6)
After his visit to his wife, Maigret returns to the _brasserie,_ where he takes his meals. There he sits, smoking his pipe, sipping a glass of white wine. There, every afternoon, toward the back of the restaurant, some of the town's wealthy men—the old money—who include a ship-owner, a property developer, and an eminent doctor, gather to play bridge. Just as the position of the sun can be difficult to judge, so can facial expression.
> The doctor was not a man to shock anyone. Not a trace of anything on his face. He glanced at his hand and said:
>
> "Two clubs."
>
> Then, as the game progressed, for the first time he began to examine Maigret, mentally undressing him. It was scarcely noticeable. His inspections were so brief, he made them as he glanced casually around . . .
>
> Rarely had Maigret seen eyes that were at the same time so eager and so dry, a man who was master of his nerves, capable of completely concealing his feelings. (trans. K. Oatley. _Les vacance_ s _de Maigret_ 24-25)
### _T HE REQUIREMENT HYPOTHESIS_
Without a Theory of Mind—which psychologists now often call "mind-reading" (see, for example, Ickes 2003)—one would scarcely know what this passage from Simenon was about. With a Theory of Mind, the game of bridge primes us for inferences about what others may know, and for possibilities of deception. The juxtaposition of bridge with Maigret's and the doctor's attempts at mind reading is a metonymic figure in the sense of Jakobson (1988). Maigret and the doctor both have reasons for hiding their thoughts and feelings as they try to comprehend each other. Each tries to assess the other man's character, to know what he is up to.
We can call the relation of mind reading to literature the _Requirement Hypothesis._ Theory of Mind is a _requirement_ to understand this kind of narrative. As Lisa Zunshine (2006) has further proposed, we seem to enjoy being able to exercise this skill. So that may account, at least partly, for our enjoyment of detective stories: working out what others keep hidden.
Simenon is a rather good example here, because in the two kinds of books for which he is most famous—Maigrets and psychological novels—the focus is on a particular person. Although in each story in which he appears Maigret seems to set out to solve a crime, really he is making a mental model of another person's character in a circumstance in which this person has, as it were, gone beyond his or her usual limits. So when Maigret works to understand each clue, it is, as in _Les vacance_ s _de Maigret_ , a clue to character, to the mind of the person who is being analyzed, to some matter that the person prefers to keep private. If such a clue also helps solve the crime, so much the better, but for Maigret (and for Simenon) that aspect is almost secondary.
What about the opacity of which Proust wrote, and with which Maigret and the doctor struggle? What has been made of that empirically? Stinson and Ickes (1992) invited to their laboratory pairs of male participants who were either friends or strangers. Paired participants sat together in a waiting room where they chatted and were covertly video-recorded. Subsequently they were told about the recording. They then watched the tape of themselves, stopped it when they remembered having a specific thought or feeling, and wrote what it was. They also watched the recording of the other man and guessed what he was thinking or feeling at the points where he had stopped the tape. The mean content accuracy between friends was 48%, and this was 50% higher than accuracy between strangers. In a different kind of study, Oatley and Larocque (1995) asked people to record in structured diaries errors that occurred in plans that they had explicitly arranged with another person in their everyday lives. The researchers also asked the other person to recall the error and the emotions surrounding it. Participants were quite good at knowing, after the error, what the other person was feeling when that feeling was anger (73% correct), but they were less good at knowing when the other person felt happiness (27% correct), fear (24% correct), sadness (43% correct), shame (10% correct), or guilt (42% correct).
So, in the quotidian world we adults are good at Theory of Mind, but not that good. Alan Palmer (2004) proposes that fiction offers a fuller disclosure than we can obtain elsewhere of the contents of human minds. Although we are not conscious in our everyday lives of everything in the minds of others or even in our own minds, we can become more completely conscious of the minds of Emma Woodhouse or Emma Bovary . . . or of the people investigated by Commissioner Maigret.
Often the idea of Theory of Mind refers to knowing what another person may be thinking or feeling in the moment, but with people whom we are likely to interact with on future occasions, we gather information to build mental models of them over the long term, from their utterances, from their behavior, and from what other people say about them. Such sources of information are exactly those that a detective uses in a mystery novel. From them we can make a mental model of the person, including socially unacceptable aspects (the criminal, the unconscionable, the unconscious). And, as the study by Stinson and Ickes (1992) implies, the better our mental model of people over the long term (as in friendship), the better we can know what they are thinking and feeling on a particular occasion.
Overall, Theory of Mind has a projective quality: we cast a model onto another person in ordinary life, or onto a character in fiction. The model can be good, but as we interact more with the person or character, we sometimes find our model to be mistaken in some important respect: an opportunity to improve it, engaging in fiction but usually painful in real life.
### _E VOLUTIONARY ORIGINS OF MENTAL MODELS OF OTHERS_
What do we know about the origins and neurological basis of this model-making function? The first clue was discovered by Robin Dunbar (1992) who recognized that primates—an order that includes lemurs, monkeys, apes, and humans—not only tend to live in social groups (an adaptation against predation), but that the size of these groups varies among different species. Dunbar surveyed group size and brain size across the whole order of primates: for example the maximum size of the social group for lemurs is about 9, for cebus monkeys about 18, for chimpanzees about 50, and for humans about 150. (This number in humans is of individuals with whom personal relationships are maintained, not just people who can be recognized by sight.) Dunbar found a close correlation between size of the social group and the size of the cortex, so for example in lemurs the cortex is about 1.2 times the size of the rest of the brain, in cebus monkeys about 2.4 times, in chimpanzees about 3.2 times, and in humans about 4.1 times.
The primate system of maintaining trust in relationships is by means of grooming: sitting quietly with another individual, stroking, and picking through their fur for insects and twigs. In chimpanzees this activity takes about 20% of each individual's waking time. Dunbar (1993, 2003) has proposed that as group size kept increasing in our forebears, a problem was reached when the amount of grooming time needed to maintain an increasing number of relationships reached about 30%. The problem was that to devote any more time to grooming would not allow enough time to forage and do the other things that primates need to do. In the line that led to humans, the problem was solved by language emerging to replace grooming. So, contrary to what one might have thought, it was not to talk about where to find food or how to make flint tools that language came into being. It emerged in order to converse, and the primary function of conversation is to maintain relationships. Conversation is verbal grooming. Though we never lose the influence of touch, of cuddling, and other such ways of expressing affectionate emotions that our cousins the chimpanzees have, language emerged to supplement them. By means of conversation, we humans can maintain many more relationships than chimpanzees could. One can converse in groups, one can converse when foraging, or when working. Dunbar's hypothesis is known as the social brain hypothesis. It is that the progressive enlargement of the cortex, which has occurred in the evolution of lemurs, monkeys, apes, and humans, has taken place because the brain needs to house the neural machinery to contain mental models of progressively more individuals in the social group, and that language has taken over from grooming as the primary means for maintaining relationships in the group. Although the social brain hypothesis was at first greeted skeptically, it is now regarded as the best hypothesis we have about the origins of language.
And with the new, more efficient, relationship-maintaining method of conversation, one can see in the graphs of hominid evolution that the straight line relating brain-cortex size to group size changed direction about 500,000 years ago. It became steeper than that for other primates. This is the point, argues Dunbar (1993, 2003), at which language-based conversation took over from grooming as the primary way—the more efficient way—of maintaining a large number of relationships.
So what do people converse about? Dunbar has found that some 70% of conversation in a university refectory is about the social lives and plans of ourselves and those we know, the trusted and the less trustworthy. We can pass little fragments of mental models to each other verbally: "She always does what she says she will" or "You wouldn't tell him anything you didn't want everyone else to know." Dunbar has proposed that music and laughter similarly function to promote social bonding: music, when it is shared, is cohesive. Later in human evolution music and laughter get taken, along with language, to a new level in art (the emergence of which I discuss below).
There is another contributor to human brain size. It is Theory of Mind. In terms of models, we can make mental models of others, and recursively, mental models of others' mental models. Chimpanzees are only on the edge of being able to do this (Tomasello 1999; Herrmann et al. 2007). For instance, although chimpanzees follow the line of another individual's sight, they are not good at knowing what the other is looking at. Even quite young children can do this. Chimpanzees do not have the Theory of Mind abilities that four-year-old human children possess. Tomasello and his group have proposed the cultural intelligence hypothesis. It is not so much that we have more general intelligence than our primate cousins. Rather, we are more socially intelligent, and this already shows in young children. So as well as the number of people of whom we maintain mental models, yet more neural capacity has become necessary to allow Theory of Mind (models of other people's models). In fact we humans can maintain several layers of models—of what philosophers call "intentional states" for different other people. (An intentional state is a mental state that is about something: believing something, wanting something, feeling something, etc.)
What does this have to do with literature? First, it is from conversation that literature developed (Oatley 2004). Conversation is the principal kind of event depicted in three of the most important kinds of literature: plays, novels, and short stories, although it is usually less important in three others: action in epic poetry, inward thought in lyric poetry, action and scenic juxtaposition in films. Second, Dunbar has enumerated the layers of intentional states required for certain kinds of literature. Thus, for _Othello,_ Shakespeare " _intended_ [1] that his audience _realize_ [2] that the eponymous moor _believed_ [3] that his servant Iago was being honest when he claimed to _know_ [4] that his beloved Desdemona _loved_ [5] Cassio" (Dunbar 2003, 162).
In terms of the _Requirement Hypothesis:_ to understand _Othello,_ the audience is required to maintain four layers of intentional states in their models, and to write the play Shakespeare required five layers. Following Dunbar's typographical conventions for these layers, we can say that Theory of Mind or mind reading is _guessing_ [1] what another person might be _thinking_ or _feeling_ [2]. It reaches just the second of these layers. Ability to maintain a third layer may have been reached in our hominid ancestors at about the same time that conversational language emerged. Certainly for us modern humans this third layer is easy. So in conversation we might _think_ [1] and say: "You wouldn't tell him anything you didn't _want_ [2] everyone else to _know_ [3]." Chimpanzees can manage about one and a half layers.
### _T HE SOCIAL-IMPROVEMENT HYPOTHESIS_
Almost since the invention of written fiction there have been claims that reading it is good for you. We call this the _Social_ - _Improvement Hypothesis._ Important versions of it have been stated by Adam Smith (1759), George Eliot (1883), and Martha Nussbaum (1995), who have proposed that reading improves our social abilities, especially in such realms as social justice, because it gives us practice in projecting ourselves into the minds of others, sympathizing with them, empathizing with them, understanding the world from their point of view. At least two versions of this idea have been tested empirically.
One version involves shifts of attitude: people exposed to fictional characters who exhibit a certain kind of behavior—a child being stoical about separation from parents, a resistance fighter during an enemy occupation, a member of a minority standing up for equality—will shift their attitudes positively toward such behavior although it is not the norm. There have been a number of experimental studies that show this kind of effect. For instance, Flerx, Fidler, and Rogers (1976) found that five-year-olds shifted their views toward more egalitarian norms for women's roles when they were read stories in which such egalitarian roles were portrayed by a protagonist, as compared with stories in which more traditional sex roles were portrayed.
Hakemulder (2000) had European participants read either a chapter of a novel about the difficult life of an Algerian woman or an essay on women's rights in Algeria. As compared with those who read the essay, those who read the fictional piece said they would be less likely to accept current Algerian norms for relationships between men and women. Hakemulder's hypothesis was that by depicting individuals in the fictional portrayal (note the parallel to making mental models of others we know) as compared with discussing generalities in the essay, readers are encouraged to project themselves into the role of a character. This made the readers of the fictional story more empathetic to people in the ordinary world whose roles were comparable to those of the fictional character. In addition, Batson et al. (2002) showed that empathy for a drug addict was shown by participants who listened to a recorded interview with him, and these participants were more likely to allocate funds to an agency to help addicts. A similar though smaller effect was found when people heard the same interview but were told it was fiction based on the lives of addicts. (See also Lamm, Batson, and Decety [2007] for the neural basis of empathetic abilities.)
In a second version of the idea, we can hypothesize that reading fiction in general strengthens abilities of accurate empathy and other social skills. It rests on the parallel between comprehending the minds of characters in a Theory of Mind kind of way and comprehending the minds of people who make up our ordinary social world. This parallel was tested by Mar, Oatley, Hirsh, dela Paz, and Peterson (2006) by examining exposure to fiction and non-fictional texts as measured by Keith Stanovich and Richard West's Author Recognition Tests (see, for example, Stanovich, West, and Harrison 1995) and investigating whether exposure predominantly to fiction as compared with non-fiction would predict performance on measures of social acumen, including empathy. It did. Mar et al. found that the more fiction people read, the more accurate they were in a test of empathy, and the better they were at being able to understand what was going on socially in fifteen video-recorded scenes.
### PART 2. THEORY OF MINDS
Theory of Minds is different from Theory of Mind: it is the idea that narrative fiction is a model or simulation of the social world (Oatley 1999)—as William Shakespeare, Samuel Taylor Coleridge, and Robert Louis Stevenson called it, a dream; the dream is of unfolding events among several characters in interaction.
As has been understood since the work of Kenneth Craik (1943), what the mind does is to make models. Language is well endowed with words for this function: allegory, analogy, hypothesis, metaphor, model, schema, simile, theory, and so on. The more recent research of Dunbar, Tomasello, et al. (reviewed above) is that for us humans, who are very social beings, although we can and do make models of the physical world, and although we make models of other individuals, we also make models of the complex social world. As Bruner (1986) put it, narrative is a distinct mode of thought about human intentions and their vicissitudes. Narrative is based on a process of modeling how intentions unfold over time, and the repercussions of these intentions with those of other people. And, as Mark Turner (1996) has argued, because we can assimilate so much of human life to a set of narrative schemas, to understand narrative in this way is to understand mind at a deep level.
To approach an understanding of narrative, let us first think about art in general. To make art is to make something that is both itself and something else. For instance, I can make marks on paper and these are also something else: a story. Art can be thought of as the more or less conscious externalization of models or metaphors. In prehistory, it emerged, as Mithen (1996) showed, between 50,000 and 30,000 years ago. Mithen argues that, before this, domains of knowledge were separate in the mind-brain: the social world / plants and foods / finding one's way about / tool-making / and so on. Then these domains began to overlap. Metaphor became possible: a certain herb is a friend.
From 43,000 years ago in Slovenia comes the world's earliest known musical instrument (Huron 2003): both a piece of hollow wood and a flute. Then from around 30,000 years ago, there have been found ornamentation (both stones and a necklace); human burial (with presumed stories of people who were both dead and living in some other place); and cave paintings (both marks on a wall and animals). One of the earliest cave paintings is from Chauvet, 150 kilometers northwest of Marseille, a rhinoceros, painted 31,000 years ago (Chauvet, Deschamps, and Hillaire 1996). Also, about 50,000 years ago, there was an enormous elaboration in the kinds of tools that were made, not just flint scrapers, but knives, harpoons, needles, and arrows. So in round numbers, language emerged as conversation 500,000 years ago; elaborate tool making, stories, and other art about 50,000 years ago; writing about 5,000 years ago; print about 500 years ago. We can take it that by 30,000 years ago the modern mind was in place with metaphors, stories, and cave paintings, and that later, by means of the technologies of writing and printing, it enhanced itself, with papyrus and paper as externalized memory and a new possibility for communication at a distance. And, as Herman (2003) argues, stories become tools for thinking about the social world.
Shakespeare was a member of the first generations to be educated in writing during the age of print. For seven years from 1571, he is thought to have attended the grammar school in Stratford, modeled after specifications created by Erasmus around 1520.
In or about December 1594, Shakespeare is thought to have read or re-read _Praise of Folly_ by Erasmus, and I have proposed (Oatley 2001) that, at this time, he conceived the idea of theater as a model of the social world. He then not only used the idea for his subsequent plays from _A Midsummer Night's Dream_ onward (dream = simulation-model), but embedded the idea recursively in some of his plays to explain it to the audience, for instance in the play within the play in _Hamlet,_ the transposition to the Forest of Arden in _As You Like It,_ Iago's improvised play-simulation to deceive Othello , and Prospero's speech:
> . . . These our actors,
>
> As I foretold you, were all spirits and
>
> Are melted into air, into thin air.
>
> And, like the baseless fabric of this vision,
>
> The cloud capped towers, the gorgeous palaces,
>
> The solemn temples, the great globe itself,
>
> Yea, all which it inherit, shall dissolve,
>
> And like this insubstantial pageant faded,
>
> Leave not a rack behind. We are such stuff
>
> As dreams are made on, and our little life
>
> Is rounded with a sleep. ( _The Tempest_ 4, 1, 148-58)
This—I think we can agree—is literary art, in which both creator and reader share in an "insubstantial vision" a "dream." And in a further metaphorical shift, theater becomes a metaphor for life, which closes at the end of the performance.
Now back to the chimpanzees for a few moments. Part of Jane Goodall's (1986) genius in her work with the chimpanzees of Gombe was to realize that in order to understand them she had to recognize them individually, make mental models of each one (as they do themselves), and give them names. Understanding them became partly ethology with particular chimpanzees—for example, following an individual for many days and counting instances of inter-individual aggression in which it was involved— but partly the recounting of narratives about their lives with each other. Recounting a narrative is what Frans de Waal does in _Chimpanzee Politics_ (1982) for instance, in the power struggle in which over a period of two months Luit displaces Yeroen from his alpha status. The narrative genre is not too different from that of ancient epics, or even from of that in which Prospero is displaced from his dukedom. The story of Luit and Yeroen went something like this (my version):
> Once, in a group of two-dozen chimpanzees who lived in Arnheim, there was an alpha male whose name was Yeroen. From time to time he would run full tilt at other chimpanzees as they sat quietly: his hair erect, stamping his feet, bearing his teeth. He made a terrible noise. Mothers picked up their youngsters, and everyone would scatter. Then as suddenly as it had been interrupted, peace would return. Yeroen would sit. Everyone would come up to him to make deferential greetings. Then, one day, the younger male, Luit, who shared sleeping quarters with Yeroen, stopped offering him these deferential greetings . . .
Though the chimpanzees knew what went on, and knew the other individuals in their group, only we can understand such events as a narrative. As John Donne put it in his remark about Theory of Mind in one of his sermons given at the beginning of the seventeenth century: "The beast does but know, but the man knows that he knows" (225).
Plays, films, short stories, and novels are about people with intentions who interact with each other. People are good at understanding processes one step at a time, but less good at understanding interactions of such processes. In thinking about the weather, for instance, we can understand that winds blow from areas of high atmospheric pressure to areas of low pressure, but what happens when other factors operate at the same time? Does the simple understanding hold in the same way when a warm air mass is blown toward a cold mass? Does it operate in the same way over land and over water? To give a weather forecast, we need to enter into a computer simulation both the wind-producing effects of different atmospheric pressures and also many other processes that interact with them. When you look at a weather forecast on the television or in the newspaper, you are looking at the output of a computer simulation. A narrative is also a kind of simulation, one that runs on minds (Oatley 1999). It binds a complex of intentions, outcomes, and emotions together into a comprehensible whole. If Alice thwarts Beatrice in an intention, we know that Beatrice is likely to be angry. But what happens when Beatrice is Alice's mother? What if she is Alice's employer? What if she is Alice's lover? What happens when the interactions of Alice and Beatrice unfold over time? It is for such matters that we need not just a Theory of Mind, but a Theory of Minds. It is for such matters that we need narrative simulations.
### _T HE SELF-IMPROVEMENT HYPOTHESIS_
Just as there have been claims that reading is good for people, there have been claims that it can promote personal change: see for instance the title of Alain de Botton's (1997) book, _How Proust Can Change Your Life_ (see also Schutte and Malouff 2006). We can call this the _Self-Improvement Hypothesis._
Whereas Theory of Mind has a projective quality, we can think of Theory of Minds (and self-improvement) as reciprocating: we project our self into a story, and the self comes back to us somewhat changed. "Improvement" here should be taken very provisionally: when a person changes the improvement is according to the person's own criteria, not necessarily according to anyone else's. As with the Social-Improvement Hypothesis, until recently empirical evidence for changes of selfhood has been meager.
Consider a short story, a simulation that runs on minds: Anton Chekhov's "The Lady with the Little Dog" (1899), thought of as one of the world's best short stories. The story starts with a casual holiday romance between Anna Sergeevna and Gurov in the Russian seaside town of Yalta. In a hotel room, they make love for the first time.
> "It's not good," she said. "You'll be the first one not to respect me now."
>
> There was a watermelon on the table in the hotel room. Gurov cut himself a slice and unhurriedly began to eat it.
Anna is appalled at what she has done. Gurov is bored by her protestations, but speaks softly to her. . . . On the next page, they take a cab to Oreanda, where the weather is not just a metaphor for the simulations of literature. It enters the story.
> In Oreanda they sat on a bench not far from the church, looked down on the sea, and were silent. Yalta was barely visible through the morning mist, white clouds stood motionless on the mountain tops. The leaves of the trees did not stir, cicadas called, and the monotonous dull sound of the sea, coming from below, spoke of the peace, of the eternal sleep that awaits us. So it had sounded below when neither Yalta or Oreanda were there, so it sounded now, and would go on sounding with the same dull indifference when we are no longer here. (trans. R. Pevear and L. Volokhonsky 365-66)
At Oreanda the weather and sea are weather and sea. But as in other art, they are also something else. Different levels in literature have been recognized from very early times in Hebrew interpretations of scripture, and they were recognized by Dante (for instance in the _Convivio,_ written in the early years of the fourteenth century). Dante proposed there are four levels: the literal, the allegorical, the moral, and the anagogical (having to do with the relationship to God).
In this passage about Oreanda, the weather and the sea are themselves (the literal). At the same time they are reminders that concerns that are almost overwhelming to us humans are ephemeral (the allegorical). Matters that are of great importance in human society (the moral) are of no concern to the physical world but, with the church not far from the bench on which Anna and Gurov sit, these matters may be of concern in the eyes of God (the anagogical).
In a story such as Chekhov's "The Lady with the Little Dog," we imagine ourselves into other lives, we identify with—that is to say empathize with—the characters in their plans, their interactions, their predicaments. As we run characters' plans on our own planning processors in the story-simulation, we experience emotions: our own emotions. Thus the extended version of our self in the simulated world can experience things, and feel things, that in the everyday world it would not. We project ourselves into the simulated world, and we may come back changed.
One of Proust's ideas was that by understanding the faults of others we can recognize them in ourselves, and thus be in a better position to correct them. Can we see in Gurov anything of ourselves, when a companion is in mental turmoil, and we do not use our Theory of Mind to enter into their mental state? So the Theory of Minds includes our own mind.
Djikic, Oatley, Zoeterman, and Peterson (in press) found that people who read the Chekhov story, with its weather and sea-sounds, experienced more changes in emotions and personality traits as a result of reading the story than those who read a control text: an account based on a set of (supposed) divorce proceedings from a courtroom that contained exactly the same information as the Chekhov story, which was exactly the same length and reading difficulty, and which people found just as interesting. So, as a result of such experiences in art, everyday selves can change.
Let us end, as we started, with Proust:
> In reality each reader, when he is reading, is uniquely reading himself. The writer's work is only a kind of optical instrument which he offers the reader to enable him to discern what without this book he might not have seen in himself. ( _Finding Time Again_ 219-20)
### ACKNOWLEDGMENTS
I am warmly grateful to my collaborators, Laurette Larocque, Maja Djikic, and Raymond Mar. For grant support I thank the Social Science and Humanities Research Council of Canada.
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## Social Minds in _Little Dorrit_
ALAN PALMER
### THEORETICAL CONTEXT
This essay will consider the subject of Theory of Mind in the novel in terms of the notion of _social minds._ After explaining what is meant by this phrase, I will attempt to illustrate the importance of social minds in the novel by analyzing their functioning in an example text, Charles Dickens's _Little Dorrit_. I hope to show that it is not possible to understand this novel without an awareness of these minds as they operate within its storyworld. They are the chief means by which the plot is advanced. If you were to take all of the social thought out of _Little Dorrit_ , very little, I would argue, would be left. So, given the importance of this subject to the study of the novel, it seems to me that it is necessary to find room for it at the centre of narrative theory.
Speaking very broadly, there are two perspectives on the mind: the internalist and the externalist. These two perspectives form more of a continuum than an either/or dichotomy, but the distinction is, in general, a valid one.
> An internalist perspective on the mind stresses those aspects that are inner, introspective, private, solitary, individual, psychological, mysterious, and detached.
>
> An externalist perspective on the mind stresses those aspects that are outer, active, public, social, behavioral, evident, embodied, and engaged.
The _social mind_ and the _public mind_ are the synonyms that I will use to describe those aspects of the whole mind that are revealed through the externalist perspective.
It seems to me that the traditional narratological approach to the representation of fictional consciousness is an internalist one that stresses those aspects that are inner, passive, introspective, and individual. This undue emphasis on private, solitary, and highly verbalized thought at the expense of all the other types of mental functioning has resulted in a preoccupation with such concepts as free indirect discourse, stream of consciousness, and interior monologue. As a result, the _social_ nature of fictional thought has been neglected. But, as the neuroscientist Antonio Damasio suggests, "the study of human consciousness requires both internal and external views" (2000, 82), and so an externalist perspective is required as well, one that stresses the public, social, concrete, and located aspects of mental life in the novel.
An important part of the social mind is our capacity for _intermental thought_ (Palmer 2005), which is joint, group, shared, or collective thought, as opposed to _intramental_ , or individual or private thought. It is also known as _socially distributed, situated_ , or _extended cognition_ , and also as _intersubjectivity_. Intermental thought is a crucially important component of fictional narrative because much of the mental functioning that occurs in novels is done by large organizations, small groups, work colleagues, friends, families, couples, and other intermental units. It could plausibly be argued that a large amount of the subject matter of novels is the formation, development, and breakdown of these intermental systems. Much of the mental functioning that goes on within these units may be termed _Intermental Theory of Mind_. _Little Dorrit_ , in common with many other novels, contains a number of general or universal statements about the typical behavior of intermental units. For example, when Blandois arrives one evening at a French inn, "There had been that momentary interruption of the talk about the stove, and that temporary inattention to and distraction from one another, which is usually inseparable in such a company from the arrival of a stranger" (167-68). These sorts of statements, though taken little account of in narrative theory because of its preference for the internalist perspective, are quite common in fictional discourse. (For work by postclassical narrative theorists on distributed cognition, see Margolin [1996 and 2000] and Herman [2003a and 2003b].)
Within the real-mind disciplines of psychology and philosophy there is a good deal of interest in _the mind beyond the skin_ : the realization that mental functioning cannot be understood merely by analyzing what goes on inside the skull but can only be fully comprehended once it has been seen in its social and physical context. For example, social psychologists routinely use the terms _mind_ and _mental action_ not only about individuals, but also about groups of people working as intermental units. So, it is appropriate to say of groups that they think or that they remember. As the psychologist James Wertsch puts it, a _dyad_ (that is, two people working as a cognitive system) can carry out such functions as problem solving on an intermental plane (1991, 27). You may be asking what is achieved by talking in this way, instead of simply referring to individuals pooling their resources and working in cooperation together. The advocates of the concept of distributed cognition such as the theoretical anthropologists Gregory Bateson (1972) and Clifford Geertz (1993), the philosophers Andy Clark and David Chalmers (1998) and Daniel Dennett (1996), and the psychologists Edwin Hutchins (1995) and James Wertsch all stress that the purpose of the concept is increased explanatory power. They argue that the way to delineate a cognitive system is to draw the limiting line so that you do not cut out anything which leaves things inexplicable (Bateson 1972, 465). For example, Wertsch tells the story of how his daughter lost her shoes and he helped her to remember where she had left them. Wertsch asks: Who is doing the remembering here? He is not, because he had no prior knowledge of where they were, and she is not, because she had forgotten where they were. It was the intermental unit formed by the two of them that remembered (Sperber and Hirschfeld 1999, cxxiv).
The basis of this essay is _attribution theory_ (Palmer 2007): how narrators, characters, and readers attribute states of mind such as emotions, dispositions, and reasons for action to characters and, where appropriate, also to themselves. How do heterodiegetic narrators attribute states of mind to their characters? By what means do homodiegetic narrators attribute states of mind to themselves and also to other characters? And, with regard to the issue of characterization, how does an attribution of a mental state by a narrator help to build up in the reader a sense of the whole personality of the character who is the subject of that attribution? Attribution theory rests on the concept of _Theory of Mind_ , the term used by philosophers and psychologists to describe our awareness of the existence of other minds, our knowledge of how to interpret our own and other people's thought processes, our mind reading abilities in the real world. Readers of novels have to use their Theory of Mind in order to try to follow the workings of characters' minds. Otherwise, they will lose the plot. The only way in which the reader can understand a novel is by trying to follow the workings of characters' minds and thereby by attributing states of minds to them. (For more on Theory of Mind and the novel, see Zunshine 2006.) Of particular importance to the concept of the social mind is the fact that this mind reading also involves readers trying to follow characters' attempts to read other characters' minds. A basic level of minimal mind reading is required for characters to understand each other in order to make life possible. At the next level up, characters who know each other well form intermental pairs and small groups. To put the point simply, they are more likely to know what the other is thinking than total strangers will. These small groups will obviously vary greatly in the quantity and quality of their intermental thought. In addition, individuals may be part of larger groups that will also have a tendency to think together on certain issues. In all of these units, large and small, the individuals that belong to them will, of course, frequently think separately as well. It is this balance between public and private thought, intermental and intramental functioning, social and individual minds, that novels are preoccupied with, and _Little Dorrit_ is no exception.
It is common, when reading discussions of the sustained inside views of characters' private minds in the novels of, say, George Eliot or Henry James, to be told that, by contrast, characters in novels by Charles Dickens are really only ever seen from the outside. We only see their surface. The effect is often to sound rather patronizing about Dickens's achievement. "Brilliant novelist in his way, of course, but without the _depth_ of Eliot or James!" I would like to reverse that perspective. In cognitive terms, nearly all of your life is spent on the surface, on the outside, in the sense that all of the minds with which you are involved (with the admittedly rather important exception of your own) are only ever experienced on the surface, and from the outside. From this point of view, it is not surprising that Oscar Wilde said that "It is only shallow people who do not judge by appearances. The true mystery of the world is the visible, not the invisible." Dickens is the novelist of appearances, and of the visible, and his achievement can only be fully appreciated from the externalist perspective.
An emphasis on social minds will inevitably question these twin assumptions: first, that the workings of our minds are not accessible to others; and second, that the workings of our own minds are unproblematically accessible to ourselves. This essay will question the first assumption but will make almost no reference to the equally questionable second.
### PRIVATE THOUGHT
There are very few extended passages of inside views of intramental or private thought in _Little Dorrit_. One such is a long passage of Clennam's inner speech regarding his love for Minnie and his concern about growing old: "And he had plenty of unsettled subjects to meditate upon. . . . First, there was the subject seldom absent from his mind, the question, what he was to do henceforth in life" (231). In addition, there is a good deal of _contextual thought report_ (Palmer 2004, 209): short, unobtrusive sentences, phrases, or even single words that describe characters' mental functioning. Much of the contextual thought report in the novel is related to private thinking. For example, when Fanny cries with Little Dorrit after she tells her that she is engaged, "It was the last time Fanny ever showed that there was any hidden, suppressed, or conquered feeling in her on the matter" (654). From that time onward, Fanny's feelings will be hidden from others.
This intramental thought benefits from an externalist perspective, though, just as intermental thought does. To illustrate, I will draw attention to three features of intra-mental thought that are undervalued by the internalist perspective. First, it is worth questioning for a moment the apparent inaccessibility of private thought to others. How much mental functioning is there in the novel that at least one other character is not aware of, albeit in general terms? The answer that I would suggest is: not much. Regarding the two passages referred to just now, Little Dorrit for one is aware of Clennam's feelings. She knows immediately that something is wrong with Clennam after he gives up thoughts of Minnie and she knows about his anxieties about growing old: "He never thought that she saw in him what no one else could see" (432). Little Dorrit would also have known of Fanny's subsequently concealed feelings, having been shown them once. When going through the characters one by one asking this question ("Does another character know about their minds?"), I am struck by how public the thought in the novel tends to be. Miss Wade is an obvious exception. Mrs. Clennam, possibly? Flintwinch, perhaps? But there are few others, it seems to me.
Second, as Bakhtin (1984) has shown, intramental thought is intensely dialogic. In a splendidly Bakhtinian phrase that the narrator uses of Mrs. Clennam, "It was curious how she seized the occasion to argue with some invisible opponent" (407). In a phrase that brings to mind Daniel Dennett's notion of "mind-ruts" (1991, 300), Mrs. General is described as having "a little circular set of mental grooves or rails on which she started little trains of other people's opinions" (503). The important point to note here is that it is _other people's_ opinions that are running along her mind-grooves. The private thoughts of Mrs. Clennam and Mrs. General, in common with all the characters in the novel, are filled with the thoughts of others.
Third, the presence of intermental thought is often concealed within descriptions of intramental functioning: "From the days of their honeymoon, Minnie Gowan felt sensible of being usually regarded as the wife of a man who had made a descent in marrying her" (541). At first reading, this sounds like a simple example of intramental contextual thought report. However, Minnie's dialogic anticipation of the feelings of others contains an intermental component that is disguised within the passive construction ("regarded as"). It may be decoded as follows: the group of people who know her regard her (Minnie thinks) as that sort of wife. (For more on the use of the passive voice to construct intermental functioning, see Palmer 2005.) When Miss Wade is part of the quarantine party at the beginning of the novel, the narrator makes this remark: "And yet it would have been as difficult as ever to say, positively, whether she avoided the rest, or was avoided" (62). Although this is a statement about Miss Wade's disposition to be unsociable, it is also a statement about the intermental functioning of "the rest": their awareness of her disposition and their resulting behavior in avoiding her. Similarly, it is said of Little Dorrit that "She passed to and fro in it shrinkingly now, with a womanly consciousness that she was pointed out to every one" (118). This is a description of her state of mind, but it also refers to the thinking of the prison population who consider Little Dorrit as the Child of the Marshalsea. The externalist perspective is required in order to tease out the intermental element in sentences that appear to be simply presentations of private thought.
Before discussing in detail the readability of fictional minds, I will briefly refer here to unreadability, or, at the very least, the possibility of incomplete, inaccurate, or otherwise defective reading of other minds. As I have said, some characters tend to be difficult to read. Much is made of Flintwinch's impassivity and impenetrability. Despite being physically pushed about by Blandois, he "brought himself up with a face completely unchanged in its stolidity" (602). When Mr. Casby is questioned by Clennam about Miss Wade, being determined to tell him nothing of what he knows about her, he "knew his strength lay in silence" (594). When Miss Wade herself is similarly intent on revealing nothing of her mind, she "stood by the table so perfectly composed and still after this acknowledgement of his remark that Mr. Meagles stared at her under a sort of fascination, and could not even look to Clennam to make another move" (376). This is a vivid illustration of the importance of publicly available cues when reading other minds. When those cues are missing, as when Miss Wade deliberately reduces them to order to keep the workings of her mind hidden, then Meagles is at a loss. Miss Wade herself thinks that she is able to read other minds accurately: "From a very early age I have detected what those about me thought they hid from me" (725). However, it is clear from the context that she was frequently wrong, simply misinterpreting genuine kindness. On a more amusing level, the workings of Mr. F.'s Aunt's mind are, mercifully perhaps, completely opaque: "Mr. F.'s Aunt may have thrown in these observations on some system of her own, and it may have been ingenious, or even subtle: but the key to it was wanted" (199).
### THE EXTERNALIST PERSPECTIVE
Notwithstanding the caveat in the previous paragraph, there are copious examples in this novel of the fact that much of the thought of its characters is public and easily available to others. Even a very solipsistic character like Mr. Dorrit is able to notice when, for example, Merdle is out of sorts (674). The fairly unobservant Young John Chivery knows that Little Dorrit is in love with Clennam, and can make a remark to Clennam such as: "'I _see_ you recollect the room, Mr. Clennam?'" (791, my emphasis). Also, the reserved characters referred to earlier such as Miss Wade are not always able to conceal all of their thoughts. During a discussion with Clennam, "She heard him with evident surprise, and with more marks of suppressed interest than he had _seen_ in her" (719, my emphasis). This is a novel in which the visibility of thought is frequently and pointedly emphasized. When Pancks gives a glass of wine to Blandois, it is "not without a _visible_ conflict of feeling on the question of throwing it at his head" (814, my emphasis). The narrator mocks the efforts made by Merdle and Lord Decimus to keep their thoughts secret. They move about at their dinner party, "each with an absurd pretence of not having the other on his mind, which could not have been more transparently ridiculous though his real mind had been chalked on his back" (624).
In addition to the specific mental events that occur in the minds of others, Theory of Mind also relates to dispositions that persist over time and that form part of another's character or personality. I stress this point because the hugely important area of characters' dispositions has been neglected by traditional narrative theory. For example, Blandois's selfishness is clear from the way he moves around a room soiling the furniture (402). In a particularly vivid image, when Clennam watches Gowan when he is unawares, "There was something in his way of spurning [stones] out of their places with his heel, and getting them into the required position, that Clennam thought had an air of cruelty in it" (245). There are a large number of examples of characters passing judgments, often spiteful but accurate, on other characters' personalities. Fanny says of her loved one, Sparkler, "'If it's possible—and it generally is—to do a foolish thing, he is sure to do it'" (664). Fanny is again perceptive, this time about Mrs. General: "' _I_ know her sly manner of feeling her way with those gloves of hers'" (666). Flintwinch is another character whose disposition is to be unsparing about the failings of others. He says of Clennam's father, "'He was an undecided, irresolute chap, who had everything but his orphan life scared out of him when he was young'" (224). At the end of the novel, he shouts to Mrs. Clennam: "'But that's the way you cheat yourself'" (851). The importance of the concept of dispositions is that it links specific mental events and actions (doing foolish, sly, irresolute, or dishonest things) with those characters' stable, long-lasting personalities (being a foolish, sly, irresolute, or dishonest person). Occasionally, the insight that one character has into another extends beyond the aspects of their personality that I have labeled dispositions and encompasses their whole mind. A character may feel that another person knows them completely, all about how their mind works. Miss Wade says of Gowan that "He understood the state of things at a glance, and he understood me. He was the first person I had ever seen in my life who had understood me. . . . He accompanied every movement of my mind" (732).
I will now discuss three specific ways in which the social minds in the novel communicate with each other: the face, sign language, and the look.
### THE FACE
One of the obvious ways in which thought is made public is by means of the face. As we do in real life, characters pick up cues about the mental functioning of others by reading facial expressions. Of course, this is only one means amongst others. At one point, Blandois knows what Clennam is thinking simply by watching the back of his head, as when,
> Though Clennam's back was turned while [Blandois] spoke . . . he kept those glittering eyes of his . . . upon him, and evidently saw in the very carriage of the head . . . that he was saying nothing which Clennam did not already know. (819)
Nevertheless, the face is a particularly important source of knowledge about the minds of others and there is a continual stream of references in the novel to the face and its role in the public nature of thought. Some relate to individual mental events. Mrs. Clennam says to Little Dorrit: "'You love Arthur. (I can see the blush upon your face.)'" (859). Clennam "saw a shade of disappointment on [Mrs. Plornish's] face, as she checked a sigh, and looked at the low fire" (178). "There was an expression in his face [Blandois] as he released his grip of his friend's [Cavalletto] jaw, from which his friend inferred that . . . [etc]" (174). "'My God!' said Bar, starting back, and clapping his hand upon the other's breast. '. . . I see it in your face'" (773). On other occasions, the face is seen as an indicator of long-term dispositions. When Clennam was a child, Mrs. Clennam could see him looking at her "with his mother's face" (859) and therefore knows that he will, as she thinks, take after her. "Little Dorrit stopped. For there was neither happiness nor health in the face that turned to her" (857). The narrator comments of Mr. Chivery that "As to any key to his inner knowledge being to be found in his face, the Marshalsea key was as legible as an index to the individual characters and histories upon which it was turned" (346). Changes in the flow of events can be signaled by changes in characters' faces. In the climax of the novel, Mrs. Clennam's face is, at first, as inscrutable as ever: "Her face neither acquiesced or demurred" (837), and "Her face was ever frowning, attentive, and settled" (839). However, as events unravel and get beyond her control, "Mrs. Clennam's face had changed. There was a remarkable darkness of color on it, and the brow was more contracted" (841).
Many of the most powerful moments in the novel involve descriptions of facial expressions. In the scene in which Little Dorrit is meditating on London Bridge and is caught unawares by Young John Chivery,
> She started and fell back from him, with an expression in her face of fright and something like dislike that caused him unutterable dismay. . . . It was but a momentary look, inasmuch as she checked it. . . . But she felt what it had been, as he felt what it had been; and they stood looking at one another equally confused. (260)
Here, her facial expression inadvertently reveals her true feelings, and both she and Young John are shocked as a result. In marked contrast, facial expressions can also serve a dramaturgical function. Characters self-consciously use them to present to the world the sort of self that they want the world to see.
> With these words, and with a face expressive of many uneasy doubts and much anxious guardianship, he [Mr. Dorrit] turned his regards upon the assembled company in the Lodge; so plainly indicating that his brother was to be pitied for not being under lock and key. (269)
However, these efforts can be unsuccessful: "Do what he could to compose his face, he could not convey so much of an ordinary expression into it, but that the moment she saw it, she dropped her work and cried, 'Mr. Clennam! What's the matter?'" (465). Characters are continually attempting to read faces as cues to action. Clennam "suffered a few people to pass him in whose face there was no encouragement to make the inquiry" (118). This face-reading may be only partially successful. A character may find out something of what another is thinking, but not the whole story. For instance, "As she [Little Dorrit] looked at him [Clennam] silently, there was something in her affectionate face that he did not quite comprehend: something that could have broken into tears in a moment, and yet that was happy and proud" (885). Sometimes, the facial expression serves almost as a kind of sign language, as discussed in the next section. Clennam has difficulty following Mrs. Plornish's train of thought because she is rather inarticulate: "He was at a loss to understand what she meant; and by expressing as much in his looks, elicited her explanation" (178).
### SIGN LANGUAGE
In addition to the face, another way in which social minds are made publicly available to each other is sign language. There is a surprisingly large amount of this type of communication in the novel. It generally occurs between characters who know each other well and who therefore form an intermental unit. An example is the Dorrit family. Little Dorrit "looked in amazement at her sister and would have asked a question, but that Fanny with a warning frown pointed to a curtained doorway of communication with another room" (284). Tip gives Fanny "a slight nod and a slight wink; in acknowledgement of which, Miss Fanny looked surprised, and laughed and reddened" (536).Within such a unit, nonverbal communication is an efficient supplement to speech. "In answer to Cavalletto's look of inquiry, Clennam made him a sign to go; but he added aloud, 'unless you are afraid of him.' Cavalletto replied with a very emphatic finger-negative, 'No, master'" (821). This is a cooperative, beneficial unit. Fanny and Mrs. General form a conflicted, competitive unit but the sign language is just as efficacious. When "Miss Fanny coughed, as much as to say, 'You are right'" (661), Mrs. General knows exactly what she means.
The choreography of the sign language in the novel is beautifully judged and often extremely subtle. So much so that the absence of any sign can sometimes be sign language enough. Within people of the same social class, who understand each other well, the significance of doing nothing can be well understood. "Ferdinand Barnacle looked knowingly at Bar as he strolled upstairs and gave him no answer at all. 'Just so, just so,' said Bar, nodding his head" (614). In contrast, a refusal to admit to an understanding of sign language is highly significant. Plornish, "having intimated that he wished to speak to her [Little Dorrit] privately, in a series of coughs so very noticeable" (326) that Mr. Dorrit must be aware of their significance, Mr. Dorrit nevertheless refuses to admit that he understands Plornish's sign language. To do so would be an admission that he knows that Little Dorrit works to support him. At the end of the novel, Mrs. Clennam is reduced to a cruel parody of communication in her use of sign language: "Except that she could move her eyes and faintly express a negative and affirmative with her head, she lived and died a statue" (863).
One example of the absence of a sign is worth dwelling on because it raises interesting issues relating to the concept of action. Meagles tactlessly praises Gowan's connections with the Barnacles in Doyce's presence and so "Clennam looked at Doyce, but Doyce knew all about it beforehand, and looked at his plate, and made no sign, and said no word" (248). It is clear from the rhythm of the prose, the emphasis of each clause, that there is a conscious decision by Doyce to do nothing and take no action. But this decision differs very little, in cognitive terms, from a decision to perform a physical action. In addition, this is a social situation in which certain actions are expected such as showing interest by nodding, smiling, and agreeing, and so Doyce's refusal to do any of these things is a non-action that has a dramaturgical function. He is demonstrating that he is unhappy with Meagles's remarks. But who is he demonstrating this disapproval to? Clennam, who is looking at him and will presumably be expecting disapproval-action? Or, less likely perhaps, Meagles himself, who will perhaps insensitively be expecting approval-action? Is he doing nothing because of these conflicting expectations? In any case, the non-action performs the same function as an action. What these questions suggest is that the boundary between actions and non-actions is certainly very blurred and that the refusal to engage in expected social action _is_ perhaps an action after all.
It will be illuminating, I think, to consider the role of sign language within a single intermental unit, the one comprising Clennam and Pancks. At first, these two appear to be a distinctly unlikely pair for such a possibility. I referred above to characters with inscrutable faces and Pancks is one.
> With his former doubt whether this dry hard personage were quite in earnest, Clennam again turned his eyes attentively upon his face. It was as scrubby and dingy as ever, and as eager and quick as ever, and he could see nothing lurking in it that was at all expressive of a latent mockery that had seemed to strike upon his ear in the voice. (322)
When Pancks speaks highly of Casby, "Arthur for his life could not have said with confidence whether Pancks really thought so or not" (462). The difficulty that Clennam has in reading Pancks's mind becomes particularly important when Clennam does not know if Pancks's interest in the Dorrit family is well meant or not. His activities "caused Arthur Clennam much uneasiness at this period" (367), and "awakened many wondering speculations in his mind" (323). However, over time, Clennam comes to know Pancks's mind better.
> Between this eccentric personage and Clennam, a tacit understanding and accord had been always improving. . . . Though he had never before made any profession or protestation to Clennam, and though what he had just said was little enough as to the words in which it was expressed, Clennam had long had a growing belief that Mr. Pancks, in his own odd way, was becoming attached to him. (637)
It is important to note the emphasis in this passage on the fact that their relationship does not rely on words. Indeed, during the conversation in which Pancks tempts Clennam to speculate (638-43), much is understood, or partly understood, between the two men, but very little is actually said. Indeed, it is not even explicitly stated that Pancks has been successful in persuading Clennam to speculate.
Over time, the understanding between Clennam and Pancks becomes sufficiently intermental for them to be able to communicate by signs.
> Mr. Pancks in shaking hands merely scratched his eyebrow with his left fore-finger and snorted once, but Clennam, who understood him better now than of old, comprehended that he had almost done for the evening and wished to say a word to him outside. (594)
When outside, "Arthur thought he received his cue to speak to him as one who knew pretty well what had just now passed" (595). However, the resulting conversation demonstrates that the efficiency of intermental thought should not be overestimated. Mis-readings can occur. When Pancks says that he wants to take a razor to Casby (in fact, to cut his hair), Clennam thinks that he wants to cut his throat!
### THE LOOK
The importance of the look is continually stressed in the text. Clearly, for minds to be public and available, it is necessary for characters to look attentively at each other in order to pick up the sorts of cues that I have been describing. This may even require staring at the other person. In fact, there is a noticeable preponderance in the text of those three words— _attentive, look_ , and _stare_. The word _attention_ is used in _Little Dorrit_ 60 times, _attentive_ 21 times, and _attentively_ 11 times (total: 92). As a control, the comparable numbers for _Middlemarch_ (about the same length text) are these: 42, 4, and 0 (total: 46), and for _The Ambassadors_ (about half the length): 19, 4, and 1 (total: 24). The occurrences in _Little Dorrit_ of the words _look_ , _looks_ , _looked_ , and _looking_ total 995 (766 in _Middlemarch_ and 291 in _The Ambassadors_ ). The comparable totals for the words _stare_ , _stares_ , _stared_ , and _staring_ are 64, 22, and 27, respectively. Clearly, the difference in the usage of these words is significant. However, whilst it may be tempting to ascribe the variation to Dickens's interest in social thought as compared to the more internalist perspective of George Eliot and, in particular, Henry James, further research on the issue is required before any claims of this sort can be made with confidence.
In _Little Dorrit_ , the act of looking fulfils a number of very different functions. The list that follows is a selection. Some of the quotations from the text that are used elsewhere in this essay reveal other uses of the look.
> _Information-seeking_ : "Monsieur Rigaud's eyes . . . were so drawn in that direction that [Cavalletto] more than once followed them to and back from the pavement in some surprise" (46).
>
> _Information-giving_ : When Mrs. Clennam talks to Clennam, "Her emphasis had been derived from her eyes quite as much as from the stress she laid on her words" (747). "Their looks met. Something thoughtfully apprehensive in [Minnie's] large, soft eyes, had checked Little Dorrit in an instant" (544).
>
> _Warning_ : "To Arthur's increased surprise, Mistress Affery, stretching her eyes wide at himself, as if in warning that this [Blandois] was not a gentleman for him to interfere with" (599).
>
> _Thanking_ : "Mother, with a look which thanked Clennam in a manner agreeable to him" (581).
>
> _Expressing curiosity_ : Blandois and Mrs. Clennam "looked very closely at one another. That was but natural curiosity" (403).
>
> _Bonding_ : "There was a silent understanding between them [Little Dorrit and Minnie] . . . She looked at Mrs. Gowan with keen and unabated interest" (544).
>
> _Intimidating_ : Flintwinch says to Mrs. Clennam: "'Now, I know what you mean by opening your eyes so wide at me'" (850).
>
> _Controlling_ : "As Mrs. Clennam never removed her eyes from Blandois . . . so Jeremiah never removed his from Arthur" (602).
The use of the look can be an important element in a character's personality. Little Dorrit looks at Clennam "with all the earnestness of her soul looking steadily out of her eyes" (214). The look is often expressive of the attitude of the looker towards the "lookee." For example, "The visitor [Miss Wade] stood looking at her [Tattycoram] with a strange attentive smile. It was wonderful to see the fury of the contest in the girl, and the bodily struggle she made as if she were rent by the Demons of old" (65); "Mrs. Clennam and Jeremiah had exchanged a look; and had then looked, and looked still, at Affery" (834). However, it can sometimes be that the accusation of staring is more informative about the uneasy state of mind of the "staree" than it is about the alleged starer. Fanny unfairly reproaches Little Dorrit for staring at her (665), and Mr. Dorrit thinks that the Chief Butler looks at him "in a manner that Mr. Dorrit considered questionable" (678). When Mr. Dorrit's mind is collapsing and "his daughter had been observant of him with something more than her usual interest," he demands peevishly, "'Amy, what are you looking at?'" (701). Miss Wade refers to someone in her past who "had a serious way with her eyes of watching me" (727).
The mechanics of the look are interesting. It may be combined with the sign language discussed above: Clennam, "more with his eyes than by the slight impulsive motion of his hand, entreated her [Little Dorrit] to be reassured and to trust him" (121). Characters sometimes see significance in an exchange of looks by others: When Pancks comes to break the news of the Dorrit wealth, "The excitement of this strange creature was fast communicating itself to Clennam. Little Dorrit with amazement, saw this, and observed that they exchanged quick looks" (437). Characters are frequently uncomfortably aware of being the subject of a stare. When Mr. Dorrit goes to see Mrs. Clennam, "he felt that the eyes of Mr. Flintwinch and of Mrs. Clennam were on him. He found, when he looked up, that this sensation was not a fanciful one" (686). Mrs. Clennam (perhaps the major starer in the novel) "sat looking at her until she attracted her attention. Little Dorrit colored under such a gaze, and looked down" (390). Flintwinch comments dryly to Clennam: "'You'll be able to take my likeness, the next time you call, Arthur, I should think'" (744). Clennam is comically uncomfortable with a look from Flora: "In his ridiculous distress, Clennam received another of the old glances without in the least knowing what to do with it" (194). Occasionally, a look can be so compelling that the lookee has to return it: "Throughout he [Blandois] looked at her [Little Dorrit]. Once attracted by his peculiar eyes, she could not remove her own, and they had looked at each other all the time" (546). Mr. F.'s Aunt "looked at Clennam with an expression of such intense severity that he felt obliged to look at her in return, against his personal inclinations" (590). "'None of your eyes at me,'" she scolds him.
### CONCLUSION
I stressed at the beginning of this essay that both perspectives on fictional minds, the internalist and the externalist, are required. The narrator of _Little Dorrit_ recognizes this truth. Employing the internalist perspective on those aspects of the mind that are inner, introspective, solitary, private, individual, psychological, mysterious, and detached, the narrator remarks of Mr. Dorrit that "Only the wisdom that holds the clue to all hearts and all mysteries, can surely know to what extent a man, especially a man brought down as this man had been, can impose upon himself" (275). Employing the externalist perspective that stresses those aspects of the mind that are outer, active, public, social, behavioral, evident, embodied, and engaged, the narrator comments of Mr. Chivery (as quoted above in the section on the face) that "As to any key to his inner knowledge being to be found in his face, the Marshalsea key was as legible as an index to the individual characters and histories upon which it was turned" (346).
Nevertheless, within this balance, I have emphasized social minds for two reasons. One is that they have been neglected by traditional narrative theory. The other is that, in my view, the social minds in this particular novel are more important than the solitary or private minds. It is not possible to understand _Little Dorrit_ without an understanding of the public minds that operate within its storyworld. It is difficult to think of any important aspect of the novel that is left out of an externalist analysis of it. A good deal of the significance of the thought in the novel is lost if only the internalist perspective is employed. My intention in quoting so frequently from the novel was to show that these social minds are woven into the fabric of its discourse. As to whether or not the conclusions reached here regarding the social minds in this novel can also be applied to other novels, more research is required.
### WORKS CITED
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Bateson, Gregory. _Steps to an Ecology of Mind: A Revolutionary Approach to Man's Understanding of Himself_. New York: Ballantine, 1972. Print.
Clark, Andy and David J. Chalmers. "The Extended Mind." _Analysis_ 58 (1998): 7-19. Print.
Damasio, Antonio. _The Feeling of What Happens: Body, Emotion and the Making of Consciousness_. London: Heinemann, 2000. Print.
Dennett, Daniel C. _Consciousness Explained_. Harmondsworth: Penguin, 1991.
\---. _Kinds of Minds: Towards an Understanding of Consciousness_. London: Weidenfeld and Nicholson, 1996. Print.
Dickens, Charles. _Little Dorrit_. Ed. John Holloway. Harmondsworth: Penguin, 1857 [1967]. Print.
Geertz, Clifford. _The Interpretation of Cultures: Selected Essays_. London: Fontana, 1993. Print.
Herman, David. "Stories as a Tool for Thinking." _Narrative Theory and the Cognitive Sciences_. Ed David Herman. Stanford: CSLI, 2003a. 163-92. Print.
\---. "Regrounding Narratology: The Study of Narratively Organized Systems for Thinking." _What is Narratology: Questions and Answers Regarding the Status of a Theory_. Eds. Tom Kindt and Hans-Harald Muller. Berlin: de Gruyter, 2003b. 303-32. Print.
Hutchins, Edwin. _Cognition in the Wild_. Cambridge, MA: MIT Press, 1995. Print.
Margolin, Uri. "Telling Our Story: On 'We' Literary Narratives." _Language and Literature_ 5.2 (1996): 115-33. Print.
\---. "Telling in the Plural: From Grammar to Ideology." _Poetics Today_ 21.3 (2000): 591-618. Print.
Palmer, Alan. "Attribution Theory." _Contemporary Stylistics_. Ed. by Marina Lambrou and Peter Stockwell. London: Continuum, 2007. Print.
\---. _Fictional Minds_. Lincoln: University of Nebraska Press, 2004. Print.
\---. "Intermental Thought in the Novel: The Middlemarch Mind." _Style_ 39.4 (2005): 427-39. Print.
Sperber, Dan and Lawrence Hirschfeld. "Culture, Cognition, and Evolution." _The MIT Encyclopedia of the Cognitive Sciences_. Eds. Robert Wilson and Frank Keil. Cambridge, MA: MIT Press, 1999. cxi-cxxxii. Print.
Wertsch, James V. _Voices of the Mind: A Sociocultural Approach to Mediated Action_. Cambridge, MA: Harvard University Press, 1991. Print.
Zunshine, Lisa. _Why We Read Fiction: Theory of Mind and the Novel_. Columbus: The Ohio State University Press, 2006. Print.
## The Way We Imagine1
MARK TURNER
### BLENDING AND THE HUMAN MIND
Fifty thousand years ago, more or less, during the Upper Paleolithic, unmistakable archaeological evidence began to accumulate of a remarkable set of human singularities: art, science, religion, refined tool use, advanced music and dance, fashions of dress, language, and mathematics. Human beings began to demonstrate an unprecedented ability to be imaginative in whatever they encountered. Cognitively modern human beings throughout the world since that time have demonstrated this remarkable ability, as a routine part of what it means to be human.
In _The Way We Think_ , Gilles Fauconnier and I proposed that this change happened in the following way (Fauconnier and Turner 2002). The basic mental operation of conceptual integration, also known as "blending," has been present and evolving in various species for a long time. Modern human beings evolved not an entirely different kind of mind, but instead the capacity for the strongest form of conceptual integration, known as "double-scope" blending. It is the engine of the human imagination.
What is blending and why is it so important?2 Let us begin with an example. A man is participating in a wedding. He is consciously enacting a familiar mental story, with roles, participants, a plot, and a goal. But while he is fulfilling his role in the wedding story, he is remembering a different story, which took place a month before off the Cycladic island of Despotico, where he and his girlfriend, who is not present at the wedding, went diving in the hopes of retrieving sunken archaeological treasures from the newly discovered Temple of Apollo and Artemis. Why, cognitively, should he be able mentally to activate and interweave these two stories? There are rich possibilities for confusion, but in all the central ways, he remains unconfused. He does not mistake the bride for his girlfriend, for the treasure, for the fish, for the temple, or even for Artemis. He does not swim down the aisle, or speak as if through a snorkel.
Human beings go beyond merely imagining stories or concepts that run counter to the present environment. We can also connect them and blend them to make a third mental array. The man at the wedding can make analogical connections between his girlfriend and the bride and between himself and the groom, and blend these counterparts into a daydream in which it is he and his girlfriend who are being married at this particular ceremony. This blended story is manifestly false, and he should not make the mistake, as he obediently discharges his duties at the real wedding, of thinking that he is in the process of marrying his girlfriend. But he can realize that he likes the blended story, and so formulate a plan of action to make it real. Or, in the blended story, when the bride is invited to say "I do," she might say, "I would never marry you!" Her response might reveal to him a truth he had sensed intuitively but not recognized.
### DOUBLE-SCOPE BLENDING
The most imaginative blending networks are double-scope networks. In a double-scope network, the two inputs have different (and often clashing) organizing frames, and the blend has an organizing frame that receives projections from each of those organizing frames. The blend also has emergent structure of its own that cannot be found in any of the inputs. Sharp differences between the organizing frames of the inputs offer the possibility of rich clashes. Far from blocking the construction of the network, such clashes offer challenges to the imagination. The resulting blends can turn out to be highly imaginative.
The imagination is not arbitrary or accidental. It has elaborate constitutive principles and governing principles (Fauconnier and Turner 2002). It cannot be explained as the sparks of crossed wires, the mere loss of separation between supposedly modular capacities. It is much too systematic for such an explanation to be plausible. In addition, imaginative blending operates with as much systematicity within individual conceptual domains as it does across domains.
The ability for highly imaginative double-scope blending seems to be available to children very early. For example, in Crockett Johnson's (1983) _Harold and the Purple Crayon,_ written for three-year-olds, Harold uses his purple crayon to draw, and whatever he draws is real, although the result is clearly a child's sketch.
His world is a blend of spatial reality and its representation. In the blend, the representation is fused with what it represents. When Harold wants light to go for a walk, he draws the moon, and so he has moonlight. The moon stays with him as he moves. This blend has two inputs. One input has elements of the real spatial world as we experience it and perceive it. One of those elements is the moon. The other input to the blend has conventional knowledge about drawing. In the input with the real moon, the moon cannot be created by drawing and it does not come into existence at someone's will. In the input with drawing, a drawn moon cannot emit moonlight, or float along in the sky as the artist's companion. But in the blend, there is a special blended moon with special emergent properties.
The mechanisms of blending that give us this special blended moon work generally throughout _Harold and the Purple Crayon_. When he needs to walk, he draws a path, and then sets off on his walk. When Harold wants to return home, he draws a window around the moon, positioning the moon where it would appear in his window if he were in his bedroom, and so he is automatically in fact in his bedroom and can go to sleep (Figure 8.1).
_Figure 8.1. Blending network for Harold and the Purple Crayon._
Child Harold's blended world has new kinds of causality and event shape that are unavailable from either the domain of drawing or the domain of spatial living. Blends of this sort are found widely throughout art and literature.
### DOUBLE-SCOPE BLENDING AND ATTRIBUTIONS OF MIND
Imaginative double-scope blending is equally the mainstay of everyday thought and understanding. Consider, as an example of routine blending, our perception of a seal. The eyes of a seal are remarkably like the eyes of a human being. When we see a seal at the seashore, it is impossible to resist the conclusion that we and the seal share a category. Compelling and evident analogies leap out at us, between the seal's appearance and ours, between the seal's motion and ours. Our human eyes align toward an object as our limbs propel our bodies toward it, and it seems to be no different for the seal.
We immediately forge a mental blend of ourselves and the seal. The result is a conception of a seal that has not only all of the seal's appearance and motion, but additionally a feature we know only of ourselves—the possession of a mind (Figure 8.2).
In the mental blend, we conceive of a seal as having a mind something like ours, lying behind its appearance and motion. In the mental blend, the seal's eyes are not merely open, round, clear, and active, but also alert, intelligent, inquisitive, and perceptive. It _inspects_ us with wide-eyed, penetrating _attention_. It _intends_ to _pursue_ an object. It has perception, appetite, and memory. We believe in this blend completely, long before we have any refined scientific evidence for it.
This is the sort of blend we assemble unconsciously, from early childhood, for any other human being. In the standard blend that we use for conceiving of another human being, the human being has not only all the organismic appearance and movement that we routinely perceive when we pay attention to the person, but also something we project to it of ourselves—the possession of a mind. It has perception, sensation, and intention behind its appearance and movements, just as we have perception, sensation, and intention behind ours. In the blend, the person whom we watch has mental states that accord with what we see.
_Figure 8.2. Blending network for seal with a mind._
We are adept at varying the conceptual structure that we project to the blend. When we perceive that someone is in a situation or condition that is not identical to ours, we can project their situation or condition to the blend, giving their perspective to the blend, with consequences for the thoughts we imagine them to have. When we see that someone's behavior is unlike our own even in identical conditions, we adjust the projection to the blend accordingly.
The projection of mind to the seal automatically gives the seal some viewpoint, but we can vary the specific details. We can choose specific details that belong to the conditions of the seal. Alternatively, we can mix in elements that belong to our own condition. At the one extreme, the seal has a viewpoint very different from ours in both location and disposition, and we apprehend the blended seal-with-a-mind from a distance, as a strange and foreign species. At the other extreme, the blend can be given our own first-person viewpoint, and we can see, in the blend, through the seal's eyes. (Try it: in imagination, be the seal looking at you, the human being, watching the seal from the seashore. Do you suddenly feel a little wet? Do you feel yourself trying to keep yourself afloat?) The nature of the mind possessed by the seal in the blend can also be varied: we can imagine what it is like to be a seal with seal-like abilities and preferences, or we can imagine what it is like to be something like us clothed in seal form. Such blending apparently outstrips by a vast distance anything a non-human species can perform.
Children routinely perform such acts of blending in conceiving of other human beings, and perhaps equally routinely in conceiving of animals, with the result that talking animals are the mainstay of the human nursery. Other species show no disposition to make dolls of other species and then attribute to them their own vocalizations, but the creative projection done by the human child can easily produce a seal who talks, who makes friends with us, who invites us to come swimming for his birthday party, and who winks at us collusively as we engage in adventures.
These are again "double-scope" blending networks, with inputs to the blend that have different (and often clashing) organizing frames and an organizing frame for the blend that includes parts of each of those organizing frames and has emergent structure of its own. _The Way We Think_ presents the details of double-scope blending in examples drawn from mathematics, science, grammar, counterfactual reasoning, causal reasoning, humor, the construction of identity, category extension, artifacts, and so on.
In all such cases, there is emergent structure in the blend. In the case of the talking seal, the creature in the blend has specific properties that belong to neither the human being nor the seal, that is, to neither of the two mental concepts that feed the blend. Consider the speech of the talking seal. It might have a sound system for its language that includes barks and growls, and a grammar bizarre for a human being. No human being has such speech, and of course no seal has speech at all, but the talking seal has just this emergent style of speech.
It is an open cognitive scientific question how the modern human being, from infant to adult, activates these conceptions of other minds, and how these blended conceptions differ. One of the most important variations is whether the blend has a counterpart in our conception of our reality. The seal that we see at the beach has an immediate counterpart in our notion of our reality, while Donald Duck does not. We are not deluded, and the difference is strong.
Similarly, there is the question of how the human mind evolved so as to be able to make immediate recognition of other minds. Consider a range of hypotheses:
> • Possibility 1: Human beings have "Swiss Army Knife" minds, whose different capacities are like different tools, unrelated in their mechanisms and evolution, and operating separately but with some coordination in the brain. On this view, a separate module evolved to recognize other human beings as having minds, and that module has no particular computational relation to any other.
>
> • Possibility 2: Human beings have the operation of double-scope blending, and put the notion of another mind together from scratch through blending every time they encounter a person or invent an imaginary being.
These are surely straw-man possibilities. Possibility 1 seems implausible because human beings do double-scope blending across many different and perhaps all conceptual domains, and do it for non-human beings. Possibility 2 seems implausible because it offers no place for efficiency and entrenchment. But there are two plausible possibilities:
> • Possibility 3: People have double-scope blending that can achieve attributions of mind for non-human beings, but extremely early in life put together blending templates that serve them thereafter for dealing with people. These templates are quickly entrenched, and people "live in the blend," never aware of the work that went into the template, but able to open it back up actively and on-line when they want to do new work. We never need to construct these templates afresh again. Instead, we activate the blend directly, just like that. This possibility allows for adjustment, so that the newborn who regards the voice-activated mobile as an intentional agent could refine its reactions later.
>
> • Possibility 4: Genetic assimilation has picked up some of the work of double-scope blending in the case of human beings, so that human beings now have a head start in achieving the blending templates for other minds.
Possibilities 3 and 4 could combine: double-scope blending is responsible for our evolution of concepts of robust other minds and accordingly for our outstanding abilities for social cognition. Indeed, the adaptiveness of social cognition contributes to the much greater overall adaptiveness of double-scope blending. Modern human beings continue to do active double-scope blending with robust on-line construction when we assemble imaginary beings such as intelligent robots, or a river that rises up in a Japanese animated film to express, in exotic ways, its pain at being polluted. But the basic human reaction to another human being does not need to be assembled from scratch.
### IMAGINATIVE BLENDS: THE SELKIE
Let us consider a remarkable blend that has no counterpart in our conception of reality: a blend of _seal_ and _human being_. This is the concept of a _selkie_. Selkies have new properties. In the folklore of the Orkney Islands, they are shape-shifting beings. When in seal form, a selkie can shed its coat to become a human being, or rather, something deceptively like a human being. When in human form, it can converse and mate with a human being. Selkies shed their coats in the moonlight and dance on the level shore. A prudent selkie hides its coat carefully before cavorting. Here we see a case where the emergent meaning in the blend includes not new properties for a seal but in fact a new species that falls into the category of neither human being nor seal.
In the selective projection to the blend, the selkie when out of its coat has the anatomical parts and proportions of a human being but the sleek and lithe movements of the seal. Accordingly, when out of their coats, selkies are sexually irresistible to human beings. In the Orkney legends, a man sometimes steals the coat of a female selkie to compel her to agree to marry him if she ever wants to regain her coat.
But male selkies also shed their coats and slip into villages to mate with deliriously grateful women. Selkies have a relation to their coats that is a blend of a seal's relation to its skin and a human being's relation to clothes. Selkies take off their clothes to have fun, and are vulnerable when thus "naked."
### BLENDS OF BLENDS
It is common in art to work with many blends, and to make blends of blends. A particularly elaborate development of the selkie legend is offered in a well-known modern tale for older children titled "Aunt Charlotte and the NGA Portraits" (Turner, M. W. 1995). NGA is the acronym for "National Gallery of Art," the one in Washington, DC. It sits next to the Capitol Building on the National Mall.
"Aunt Charlotte and the NGA Portraits" presents a character named Olga Weathers. Halfway into the story, the reader discovers that Olga is a selkie lacking her coat. The word "selkie" never occurs in the story, and no prior knowledge of selkies is required to understand the story.
Olga has the mental character of a woman but, understandably, no native taste for the human world. She would prefer a life of swimming in the water. Wearing her coat, she would have something like the body of a seal, naturally. But not quite, since her coat can be removed, and when it is, she becomes a woman. But not quite, since, when she is a woman, she retains her knowledge of the sea and retains, too, the remarkable instinctive capabilities of a marine mammal.
Olga Weathers has features possessed by no seal. Neither seal nor woman can lose its skin, or assume the skin of another species. Neither seal nor woman can be transformed into a member of another species. And it is not only Olga who is different in the world of this story. A real man in our world cannot obtain a wife by stealing the skin of a seal, but in the story, a man can try to get a wife by stealing the coat of a female selkie.
These things are possible in Olga's world. She switches from species to species according to whether she is wearing her coat. She is never either woman or seal, but always something different, and this difference counts in the story as her "magic." A mean man, it turns out, did steal her coat. He hoped she would marry him in order to regain it but she refused to marry him, because she knew he would have kept her coat forever, and she would never have been free. He thought she was helpless and had no choice, but she was not powerless, and she had a few friends who helped her make a home on Ocracoke, a seaside town in North Carolina, where she earned her living by helping the fishermen. Olga can tell where the fish are, and she can foretell the weather, and she has a sense for the conditions of the sea. The fishermen therefore pay her for advice. In Olga's World, it seems, boats have an intentional nature, too, or at least, they can hear a selkie, and they are happy to comply with her requests. She can call the boats home when they are lost. As luck would have it, the man who stole her coat was injured while hunting narwhals. The wound turned septic and he died without telling her where he had hidden her coat. So there is Olga, beached on Ocracoke.
In the blending network that produces this selkie, the shape and movement that are projected from the human being input and from the seal input to Olga do not make her lithe and frisky. On the contrary, her body is massive, like a seal's, and she has relative difficulty moving on land, as a seal might. She is hefty. Her long hair combs out in perfect waves. She is herself a kind of undulation whenever she passes over the sand.
Olga's world, inhabited by selkies, is brushed to that extent by the magic of shape-shifting, the magic of moving from one category to another, of blending incompatible elements such as woman and seal to make not just a mental blend but elements in the world to which the blend refers.
### PICTURE WORLDS
There is another magic pattern of blending in Olga's world, and this additional magic is also based on a very familiar pattern of blending, one that concerns our everyday, entirely pedestrian concept of representation. Human beings effortlessly understand the concept of "representation." As a matter of straightforward practice, we routinely put something and its representation into mental correspondence. The representational element is understood as "representing" a world or a scene or an element in a scene (Figure 8.3).
_Figure 8.3. A Representation Vital Relation connecting elements in different inputs._
It is extremely common for us to blend these two related spaces and in so doing to compress the "outer-space" relation of _representation_ between them into a unique object in the blend, as in Figure 8.4.
For example, a person and a photograph are two quite different things, but we can blend the photographic element and the person. In the blend, the person is fused with the photographic element. We point at the picture and say, "This is John." Of course, we are not deluded in the least: we know that, in the contributing space with the person, John is three-dimensional and moves. We know that the photograph is two-dimensional and does not move. But the conceptual blend in which the representation of John is fused with John is extremely useful. Most blends of this sort, but not all, have outer-space connections not only of representation but also of analogy between the representation and the element represented. That is, the visual image that is the representation of John is visually and topologically analogous to John himself: there are two images of eyes in the representation and two eyes in John's head; the two images of eyes in the representation are above the image of the nose in the representation just as the John's actual two eyes are above John's actual nose; and so on (Figure 8.5).
_Figure 8.4. Blend of represented element and representing element._
_Figure 8.5. Blend of represented element and representing element, with an additional Analogy Vital Relation._
This is, of course, just the kind of blending network we saw in _Harold and the Purple Crayon_. In addition to the contributing mental space that contains the representation and the other contributing mental space that contains what is represented, there is yet another contributing mental space that can be brought into this common "compression of representation" network. A window, or a gap in the curtains or the fence, or a portal of any sort gives us a view. When we catch only a punctual view through a portal, we get a framed glimpse. Many vital conceptual relations create a connection between a framed representation and a framed glimpse. For example, a photograph of a person and a framed glimpse of the same person are strongly analogous not only in the content of what is viewed but also in the fact that there is a viewer of that content. It seems to be conventional to blend the photograph of a person with a framed glimpse of the person, so that, in the blend, the representation of the person is the person while the entire representation is a framed glimpse of the part of the visual field the photograph represents, including the person. (So if you show someone a mug shot against a wall, and ask "what's to the left?" the answer is "the rest of the person," even though the picture does not contain these elements. They are projected instead from our conception of the image as a framed glimpse.) We often say not only that the photograph gives us a "glimpse," but also that a true visual glimpse is a "snapshot."
We project much of what we know about a framed glimpse to the blended space in which the representation is fused with what it represents. Principally, we know that what we see in a framed glimpse is only part of a world that has spatial and temporal extent not directly represented in the framed glimpse. The ability to conceive of a framed glimpse as part of a larger spatial and temporal world may be common across all mammals that have vision, and this tendency to take a limited percept as implying a larger context may extend across all sensory modalities. For example, if we feel something small in the dark, we take it immediately to be part of a larger spatial and temporal world. Human beings, and perhaps dogs and dolphins, are very adept at conceiving of rich dynamic scenes with full spatial and temporal continuity on the basis of very partial perceptions.
When we blend a framed representation such as a photograph with our concept of a framed glimpse, the photograph thereby becomes a spatially and temporally limited part of a rich dynamic world with temporal and spatial continuities and changes that are not directly represented. Our conception of the conditions of photography leads us to project to the blend many elements from the represented scene that have no visible counterpart in the representation itself. If we see a photograph of the middle of a bridge, we routinely and naturally conceive of the bridge as extending beyond the frame. We conceive of the person on the bridge as obscuring elements behind her.
But of course we can do just the same thing with a painting or a sketch. We project to the blend elements that correspond to our understanding of the reality of a framed glimpse even though there is no visible counterpart of them in the representation itself, and we do this even when we know that the representation is fictitious. In the blend, the painted woman on the painted bridge is obscuring something from our view even if the painter invented both her and the bridge.
In the case of Olga, the blend that derives from all three contributing spaces— that is, the representation, what it represents, and a framed glimpse of what it represents—is reified: this blend is "true" in her world, in the following way. Olga's world has paintings, in fact the same paintings that exist in our world. But in Olga's world, those paintings actually are rich dynamic worlds of their own, and the few elite viewers in the world who in fact see properly, that is, with intelligence and open-minded insight, can indeed see the dynamic painted worlds that the paintings present. When they look at paintings, the people move, the sea rolls, the wind blows, vehicles enter and depart from the scene.
This emergent structure in the blend—rich, dynamic worlds inside the paintings— comes in part from projection of what we know about a glimpse through a window or portal. We know that if we prolong the glimpse to a stare, we might, looking through the portal, see change, dynamism, movement. Just so, in Olga's world, if you are talented and trained and you stare at the painting, you might see change, dynamism, movement. Why do you see it? In Olga's world, the answer is straightforward: because it is _there_.
Olga, of course, is one of those who see properly. When she looks at a painting, or at least a certain painting, she can see the people in it move, breathe, and act, because in the blend they in fact do, and in Olga's world, the blend is real. In the reified mental array that blends the painting, what it represents, and a prolonged view of what it represents, the representation of a person is not merely a person but indeed a person who has received very full projections from our notion of staring through a window at a world: the blended painted person can move, converse, think, plan, become hungry, eat, and so on. Yet the projections from the space of the person are not complete: in this blend, these painted people do not age. With a few exceptions, they are unaware of anything outside the world of the particular painting they inhabit.
### INTO AND OUT OF THE FRAME
We are all familiar with a basic "Picture World" blending template. We use this template routinely. In it, there is a further set of correspondences between the representation and what it represents, as follows. Suppose that in our real world we have a bridge over a canal. Well, something can literally be part of that scene. If the physics works out right, that thing can literally be put into the scene, or removed from the scene. For example, we can row a real gondola into the real scene. Over in the representation, that is, the picture, there can be individual representations that can be created there; they can be erased, or otherwise made invisible. So, for example, the painter can do something with paint and a paintbrush that results in the existence of a representation of a gondola as part of the represented scene. There is an outer-space correspondence, that is, a correspondence between two of the contributing spaces, that connects two acts: putting something into a real scene, and taking some action that results in a new element in a representation. This correspondence connects two caused changes and their visible results. That outer-space correspondence can be compressed in our routine "Picture World" blending template to yield, in the blend, a blended causality. The blend fuses these two caused changes, so that the performing of actions that result in a representational element in the representation is fused with "putting" what it represents "into" the "scene." We say, "the painter put a gondola into the painting." Of course, we are not at all deluded: while a gondola must exist before it can be rowed into the Grand Canal, the exact flat composition of paint that represents the boat in the representation does not in fact exist as such to be "put" into the picture until the artist is quite finished taking the artistic action. We use such expressions all the time, as when we say that the artist "put some flowers into the sketch" but then "took them out," or "Hey! You forgot to put Grandma into your sketch." The blend that fuses the representation with what it represents gives a very natural way to think and speak about representations at human scale. It lets us recruit for the purpose of talking about representations and their creation the deeply understood logic of manipulating objects. So our notion of the creation of a representation already has some structure that can be projected to the blend to support the idea of "putting" an element "into" the representation. This blend has some remarkable emergent structure: we can for example "put" a "mountain" into the picture. Indeed, we can "put" "the moon" or "the sun" into the picture, even though in the real world we cannot perform the corresponding action.
There is yet further useful structure in what we know of looking through a portal onto a scene. One of the things we know about a window or portal on a real scene is that we can throw things through it or go through it ourselves, and then be part of what we previously only saw. We can project to the blend this action of "entrance," to give, in the blend, the possibility of moving something from our world into the picture world.
Olga's world reifies the "Picture World" blend and additionally provides the possibility of moving an object from our world into the picture world. The man who stole Olga's coat as a means of compelling her to marry him hid it not under a rock, or up a tree, or in any other normal locale in our world, but instead inside a painting, of Venice, by Canaletto. Within the logic of Olga's world, he literally "hid" it "in" Canaletto's painting. The coat is there, in the painting, in the exact sense that it was here, outside the picture, where any actual coat ought to be, but he moved it from here to there, and now it is there and not here. Olga's coat, in accordance with the physics of our world, can be in only one location. But in the hyper-blend that comes of blending Olga's Selkie World network and her Picture World network, this location can literally be inside a painting, and there is a means of moving things from locations outside the painting to locations inside the painting.
Olga's ignorance of the fact that her suitor hid her coat in the Canaletto painting presented her with a difficulty, but she overcame it, partly by studying art history to help her locate its hiding place. When she has located the painting, she faces a far greater challenge. As the suitor-thief had anticipated, it is not so easy for the land-bound selkie to take the coat out of the painting. To get to it, you must first physically enter the painting. But Olga cannot do that; she is too stout to squeeze through the frame. More daunting still, there is nothing but water across the bottom of the frame. Olga, massive and cumbersome, would fall into the water, and, unable to swim in her present unfortunate form, would drown. It is extremely witty to manipulate a selkie through fear of drowning.
Olga's world is a blend of two blending networks. One is the blend in which there are selkies, Selkie World . The other is a Picture World blend in which representation is compressed: in this Picture World network, the outer-space representation link between two separate mental inputs is compressed into uniqueness in the blend, so that the representation and what it represents are fused there into a single element. In this Picture World blend, the painting of the water really is water, for example, even as it is part of the painting. The Picture World blending network in Olga's World has, as mentioned, yet another input to the blend: the concept of a portal, such as a window, on a real scene. Projecting _portal_ to the Picture World blend results in a Picture World that is much fuller than our view of it, a Picture World into which we can insert elements from the external world, such as a coat.
In the blend of blends that comes from blending these two blending networks, there is a selkie, and her coat can be hidden inside a painting (Figure 8.6). In Olga's world, although the separate Picture Worlds that correspond to individual pictures are rich, they do not possess anything like the completeness of our world. Someone who enters this world from the outside finds that the world inside the Picture World fades out. If you are in the Canaletto Picture World, and you walk through the marketplace to where the side streets lead deeper into the city, and you open yet one more door, you are then confronted with "impenetrable grey mist." There is not anything there, you see. As a character in the story says, "It's a painting. It only goes so far." There are other ways in which the Picture World in Olga's world is unlike our world, some of them influenced by projections from the representation input. For example, one can enter the Canaletto Picture World and take a chicken away from its marketplace and dine on it. But when you go back, the chicken is still there. "It's a painting," the story explains. "When you go back, everything will be just exactly as it was before you came."
_Figure 8.6. Blend of Selkie World and Picture World._
Well, not exactly. Olga recruits a young girl to help her. This girl, Charlotte, is a talented but solitary child, adept at solving puzzles. She is visiting Ocracoke in November with her adequate but bored mother and adequate but distracted father. Through Olga's tacit coaching, Charlotte learns to see the people moving in the Canaletto painting. It is not clear to the reader that even a talented and motivated human child could perceive the Picture World of Canaletto's painting of Venice without both Olga's training and influence: Charlotte encounters the painting in Olga's home, which is perhaps a magical place itself. Olga sings, puts her arm around Charlotte, and unpins her own long hair so its smooth waves brush across Charlotte's bare arm. Charlotte sees, and a few minutes later goes through the frame to retrieve the coat. Charlotte is much smaller than Olga, and she can swim very well.
In general, "Picture World" is a generic double-scope conceptual integration network, which we have at our mental disposal to apply to any picture. It guides the mental act of blending and offers options. It does not dictate all the details of the particular picture world. For example, some picture worlds can be entered by outsiders, others cannot. In some picture worlds, a visitor from outside the picture can be perceived by those who inhabit the picture, but in other cases, the visitor is invisible to the natives.
In Olga's world, the picture worlds can be entered, at least by those who have the knack to see that they are picture worlds. Insight brings new possibilities, a central theme of this story. In Olga's world, the usual blend of understanding and seeing that we all know and deploy ("I _see_ what you are saying") undergoes remarkable conceptual development. Charlotte must work on her literal ability to see. She must strive to attain advanced vision. As she looks at the painting, its elements and their movement become clearer and clearer. Seeing better, she understands deeply. She already had, as Olga knew, talent in that direction: solitary and friendless Charlotte spent her time on jigsaw puzzles, looking at each piece, seeing its significance, perceiving its place. After their first meeting, Olga and Charlotte worked together on jigsaw puzzles, and Charlotte improved. The idea that Olga might have superior perceptual abilities is naturally projected to her from our knowledge of a seal: we are familiar with the idea that many animals have perceptual abilities we lack—the ability to hear sounds we cannot hear, to see patterns we cannot. Olga explains to Charlotte why she attends to people: "I like to piece together their actions in order to understand their thoughts." Charlotte has honed the identical knack through inspecting her mother and father. She turns it on Olga, too, and figures her out. She sees. She looks at the Canaletto painting, and she sees.
Charlotte hooks her foot on the frame, which turns out to be as solid as a rock banister (because it is a rock banister), and throws herself over the banister through the frame into the Grand Canal. Most of the rest of the story presents the adventures of the real girl, Charlotte, inside the Picture World that happens to contain the hidden coat of a selkie.
### DOUBLE-SCOPE BLENDING AND EVERYDAY LANGUAGE
All of these networks rely on an important power of blending, to compress outer-space relations to inner-space elements in blends. It is a useful virtue of compression that it can result in human-scale elements in the blend that can then be expressed through existing basic human-scale grammatical constructions. While a full integration network, with its network of outer-space relations, might be quite difficult to express without using language that is extensive, discursive, or periphrastic, compression can save the day. The compressed human-scale mental array in the blend can often fit into available grammatical forms. For example, "Mom could make us work" seems entirely prosaic. But the workaday grammar underlying that sentence is readily available for a wonderful blend that is compressed in the right way: "Honey, you could make a blind man see." Here is an example that involves a cascade of blending compressions, but that can be fit into everyday grammatical constructions because the blending compressions result in a conceptual array that suits the grammar:
> A halloween costume that limits sight or movement is an accident lurking in disguise. (National Public Radio warning a few days before Halloween, October 2000)
Let us consider one of these examples in some detail. It is an advertisement broadcast on the radio a few days before a three-day holiday weekend:
> At South Shore Lumber, get no sales tax Friday, Saturday, Sunday, and Monday!
In one mental space, there is a person buying something at South Shore Lumber during Friday, Saturday, Sunday, or Monday. The buyer pays a price. In another mental space that is highly analogous but somewhat disanalogous to the first, the buyer pays the price and an additional amount, namely, sales tax on the price. In the blend, the buyer pays the price. But the blend now contains an element that is available in neither of the inputs; it is a compression of the outer-space disanalogy between the two spaces. That new imaginative element in the blend is an event, a transitive event in which what is transferred to the buyer has a certain property, namely, the property of being the absence of something that is in one of the inputs. The buyer receives both what he buys and something extra, the absence of sales tax. This conceptual structure is very compressed and familiar, and can be expressed in an existing common grammatical construction, a transitive verb phrase whose noun phrase has a quantifier in the determiner sequence: "get no sales tax." One customer "gets a screwdriver"; another "gets a sheet of plywood"; and they both "get no sales tax." This last phrase prompts us to construct an entire blending network in order to understand the disanalogy between the two input spaces, and, motivated by that disanalogy, to go to South Shore Lumber (Figure 8.7).
Now consider a somewhat different example. I overheard someone ask in the Sangre de Cristo Mountains, "At what altitude do the deer turn into elk?" Here, we actually notice a compression pattern that is highly productive but usually unremarked. We notice that the analogy and disanalogy between the two mental spaces is compressed. In the blend, the outer-space analogy has been compressed to an inner-space category (the animals), and the outer-space disanalogy is compressed to a change for that category (the animals turn from deer to elk). Of course, no one is fooled or deluded. We know perfectly well that one species of animal does not change into a different species of animal. But the compressed blend has human-scale structure that can be expressed in an existing and relatively simple clausal construction. "At what altitude do the deer turn into elk?" is striking, but the same conceptual and grammatical patterns sound perfectly straightforward in expressions like "The new theory is that dinosaurs turned into birds." In the outer-space, diffuse array of inputs, dinosaurs did not turn into birds. Instead, organisms were born and died over very long stretches of time. None of those organisms itself changed in any of the relevant ways. Instead, there were analogies and disanalogies across these organisms in a long line of descent, and cause–effect connections between ancestors and descendents. Those analogies and disanalogies are compressed into change for a category in the blend: dinosaurs turned into birds. These compressions make it possible to use existing grammatical forms to evoke the blend and hence the integration network. New ideas usually do not require new grammatical constructions. Instead, what they require for their expression is imaginative compression into a blend that fits existing grammatical.
_Figure 8.7. Blending network for Get No Sales Tax._
Now consider a third phrase, heard on a National Public Radio broadcast:
> We are eating the food off our children's plates. When we overfish, we eat not only today's fish, but tomorrow's fish, too.
In one mental space, we have a certain amount of fishing. This mental space has reference to present reality. In another mental space, we have a lower level of fishing that would lead causally to an acceptable amount of fish reproduction later in time (Figure 8.8).
_Figure 8.8. Mental space networks for fishing and stock, under two different initial conditions._
The disanalogy between these two amounts of fish is compressed so that, in the blend for the present moment, a portion of present fishing is now "overfishing" (Figure 8.9).
_Figure 8.9. Blending network for "overfishing."_
Similarly, a portion of present consumption of fish can be thought of as "over-consumption" (Figure 8.10).
There are cause–effect vital relations between the overfishing and the eating of the fish, and there is also a long-range cause–effect vital relation between the overfishing and the smaller amount of fish reproduction and accordingly the smaller number of fish in the future. A blending network is created for a hypothetical future. In it, the disanalogy between the small number of fish that will be available in the future as a consequence of the overfishing and the number of fish that would be there under appropriate fishing is compressed: from both of those future spaces we project the category _fish_ to the hypothetical blend for the future, and the outer-space disanalogy between the inputs is compressed to a particular property: _missing_. Now we have _missing fish_ in the blend for the hypothetical future (Figure 8.11).
_Figure 8.10. Blending network for "over consumption."_
_Figure 8.11. Blending network for "missing" fish._
The Time vital relation between now and the future is compressed by scaling it down to a day, so that, in the blend for the hypothetical future, the future is tomorrow. The _missing fish_ in the blend for the future are now part of tomorrow's fish, the missing part. Then there is another compression: the cause–effect relations between eating fish now, fishing a lot now, and the missing part of tomorrow's fish are compressed into one scene in which the fish are missing tomorrow because we are eating them now (Figure 8.12).
_Figure 8.12. Blending network for Harvesting/Eating over time._
This structure fits a standard frame of indulgence. We all know the normative stricture against raiding the icebox and eating up food reserved for tomorrow. Dramatically, there is an additional compression that brings even the one-day lag down to the immediate moment. Those missing fish are not only in the space for tomorrow. Now the future fish are compressed with the food that is on our children's plates. But the food is "missing." Why? Because we are taking it from them and eating it ourselves. We are eating the food off our children's plates. This compression produces a highly human-scale scene, with a strong judgemental framing (Figure 8.13). (Note, incidentally, that we are taking the food off our children's plates, not the fish off their plates, because fish is not, for the relevant audience, the mainstay of child fare. It is also more exhaustive and serious to take their food than their fish.)
Again, no one is fooled by the blend. The fishing that is happening today out in the oceans is completely unlike taking away the food your children are eating today in order to eat it yourself. But the compressed blend, expressible in basic grammatical forms, evokes the entire and elaborate integration network, with the appropriate normative inferences.
_Figure 8.13. Blending network for Fishing/Eating food off our children's plates._
Human beings, all of them, are geniuses at double-scope blending. They produce imaginative compressions inconceivable to other species. But they do it so routinely that we notice the performance only very rarely, such as when a writer places it explicitly on stage, as it were, drawing attention to it. Even then we miss the extraordinary systematicity of the operation and sophistication of the operation. It is only now that cognitive science is beginning to unearth the real powers of the imagination to show how different from their ancestors human beings really are.
### NOTES
1 This essay also appears in _Imaginative Minds_ , ed. Ilona Roth (London: British Academy & Oxford University Press, 2007). Reproduced by permission from The British Academy. We are grateful to Ilona Roth for her help in bringing this chapter to fruition. The essay draws on Mark Turner, "The Origin of Selkies," _Journal of Consciousness Studies_ 11(5–6) (2004): 90–115.
2 Technical introductions to the nature and mechanisms of blending can be found in Fauconnier 1997; Fauconnier and Turner 1998, 2002; Turner 2001, 2003. See also Goguen.
### WORKS CITED
Fauconnier, Gilles. _Mappings in Thought and Language._ Cambridge: Cambridge University Press, 1997. Print.
\---, and Mark Turner. _The Way We Think: Conceptual Blending and the Mind's Hidden Complexities_. New York: Basic, 2002. Print.
\---. "Conceptual integration networks." _Cognitive Science_ , 22 (2) (April–June 1998): 133–87. Print.
Goguen, Joseph. "An Introduction to Algebraic Semiotics, with Application to User Interface Design." In _Computation for Metaphor, Analogy, and Agents._ C. Nehaniv, ed. Berlin: Springer-Verlag, 1999. 242–91. Print.
Johnson, Crockett. _Harold and the Purple Crayon_. New York: Harper & Row, 1983 [1955]. Print.
Turner, Mark. _Cognitive Dimensions of Social Science: The Way We Think About Politics, Economics, Law, and Society_. New York: Oxford University Press, 2001. Print.
\---. The Blending Website. Web. <<http://blending.stanford.edu>> 1999-2007.
Turner, M. W. "Aunt Charlotte and the NGA Portraits." _Instead of Three Wishes._ New York: Greenwillow Books, 1995. 45-71. Print.
## Theory of Mind and Fictions of Embodied Transparency1
LISA ZUNSHINE
### 1. UPPING THE AGONY
It's another day at _The Office_ —a British mock documentary about the "boss from hell," David Brent, regional manager of a fictitious paper company. David (played by Ricky Gervais) is about to interview two candidates for the position of secretary, and he has just put his receptionist, Dawn Tinsley (Lucy Davis), in an embarrassing situation, flaunting his own lack of professionalism and making Dawn seem somewhat complicit in his offensive behavior. Used to her boss's ways yet painfully ashamed of his antics in front of the two strangers and the camera, Dawn stands next to him silently, now smoothing her hair nervously, now checking her nails, and trying to avoid any eye contact (see Fig. 1).
_The Office_ cultivates such scenes of recorded unease. The documentary format allows the film crew to focus pitilessly on people's faces just when they would rather not be seen, encouraging the kind of staring that would be considered rude in real life. As Gervais, who co-directed the miniseries with Stephen Merchant, puts it, one "advantage" of having the ever-watchful camera in _The Office_ is "that it would up the agony" (Commentary). And although we may share some of that agony as we watch Dawn cringe and squirm, we remain glued to our screens.
Such moments of embodied transparency—we can see what Dawn feels even if she does not want us to see it—appear to arise naturally out of the unique makeup of this particular show: its genre (documentaries have a complex relationship with voyeurism), its setting (offices build strange interfaces between intimacy and bureaucracy), and the peculiar sensibilities of the directors and actors.
However, I want to consider yet another factor behind _The Office's_ obvious fascination with putting people into such trying emotional situations that they can't control their behavior and so their feelings are written all over their bodies: a factor grounded in our cognitive evolutionary heritage and, specifically, our Theory of Mind (ToM).
_Figure 1. The Office._
Why turn to evolutionary history? We know so much about the immediate cultural history of _The Office_ that it is not at all clear why we must also put Dawn's squirming into a hundred-thousand-year perspective. As this essay will argue, however, by cultivating moments of transparency, _The Office_ participates in a particular representational tradition that extends well beyond its immediate genre, but to uncover this fascinating pedigree we need that hundred-thousand-year-old evolutionary perspective.
I begin with a brief overview of ToM,2 drawing on the work of cognitive evolutionary psychologists and cognitive neuroscientists. I then spell out the key assumptions underlying my argument, first, that ToM is a "hungry" adaptation that constantly needs to process thoughts, feelings, and intentions, and, second, that the body occupies a double position in relation to this cognitive hunger, figuring as both the best and the worst source of information about the mind. I show further how this double perspective informs our cultural representations—novels, paintings, and moving images— in which bodies are temporarily forced into functioning as direct conduits to mental states. I conclude by speculating about further applications of this cognitive-evolutionary view of the body to analyses of cultural media, including the Internet. Prompted by _The Office_ 's commitment to "agony," my essay thus seeks to explain why our cultural representations repeatedly invent new contexts for making characters reveal their true feelings, often against their will, and why these moments of embodied transparency must be brief to be convincing.
### 2. THEORY OF MIND
Theory of Mind, also known as mind reading, is the term used by psychologists and philosophers to describe our ability to explain behavior in terms of underlying thoughts, feelings, desires, and intentions. We attribute states of mind to ourselves and others all the time3(e.g., we see somebody reaching for a cup of coffee and assume that he is thirsty). Our attributions are frequently incorrect (the person who we thought was thirsty might have actually wanted to read the name of the manufacturer on the bottom of the cup).4 Still, making them is the default way by which we construct and navigate our social environment. When ToM is impaired, as it is in varying degrees in the case of autism and schizophrenia, communication breaks down.
Cognitive evolutionary psychologists believe that mind-reading adaptations must have developed during the "massive neurocognitive evolution" that took place during the Pleistocene (1.8 million to 10,000 years ago). The emergence of these adaptations was evolution's answer to the "staggeringly complex" challenge faced by our ancestors, who needed to make sense of the behavior of other people in their group, which could include up to two hundred individuals. As Simon Baron-Cohen points out, "attributing mental states to a complex system (such as a human being) is by far the easiest way of understanding it," that is, of "coming up with an explanation of the complex system's behavior and predicting what it will do next" (21). In other words, mind reading is both predicated on the intensely social nature of our species _and_ makes this intense social nature possible. (Lest this argument appear circular, think of our legs: their shape is both predicated upon the evolution of our species's locomotion _and_ makes locomotion possible.)
Note that the words _theory_ in ToM and _reading_ in mind reading are potentially misleading because they seem to imply that we attribute states of mind intentionally and consciously. In fact, it might be difficult for us to appreciate just how much mind reading takes place on a level inaccessible to our consciousness. For it seems that while our perceptual systems "eagerly" register the information about people's bodies and their facial expressions, these systems do not necessarily make all that information available to us for our conscious interpretation. Think of the intriguing functioning of "mirror neurons." Studies of imitation in monkeys and humans have discovered a "neural mirror system that demonstrates an internal correlation between the representations of perceptual and motor functionalities" (Borenstein and Ruppin 229). What this means is that "an action is understood when its observation causes the motor system of the observer to 'resonate.'" So when you observe someone else grasping a cup, the "same population of neurons that control the execution of grasping movements becomes active in [your own] motor areas" (Rizzolatti et al 662). At least on some level, your brain does not seem to distinguish between you doing something and a person that you observe doing it.5
In other words, our neural circuits are powerfully attuned to the presence, behavior, and emotional display of other members of our species. This attunement begins early (some form of it is already present in newborn infants), and it takes numerous nuanced forms as we grow into our environment. We are intensely aware of the body language and facial expressions of other people, even if the full extent and significance of such awareness escape us. As cognitive neuroscientists working with ToM speculate,
> [Mirror] neurons provide a neural mechanism that may be a critical component of imitation and our ability to represent the goals and intentions of others. Although the early functional imaging studies have mostly focused on understanding how we represent the _simple actions_ of others . . . recent articles have proposed that similar mechanisms are involved in understanding the _feelings and sensations_ of others. . . . The growing interest in the phenomenon of empathy has led to the recent emergence of imaging studies investigating sympathetic or empathetic reactions in response to others making emotional facial expressions or telling sad versus neutral stories. (Singer et al _xv–xvi_ )6
Cognitive scientists thus begin to enter the territory that has been extensively charted by philosophers and literary critics exploring mimesis (from Aristotle's _Poetics_ , David Hume's "Of Tragedy," and Erich Auerbach's _Mimesis_ and Walter Kauffmann's _Tragedy and Philosophy_ , to the recent rethinking of mimesis and performativity in cultural studies), phenomenology (such as George Butte's compelling reintroduction of Maurice Merleau-Ponty into literary and film studies, _I Know That You Know That I Know_ ), and intentionality (such as Martha Nussbaum's critique of the tradition of correlating "an emotion and a discernible physical state" [96]). Although the work on mirror neurons is still in a relatively early stage, one can see exciting possibilities emerging at the intersection of traditionally humanistic research and the inquiry into the neural basis of interpersonal subjectivity.
### 3. TWO UNDERLYING ASSUMPTIONS
The first assumption underlying my argument is that cognitive adaptations for mind reading are promiscuous, voracious, and proactive. Their very condition of being is a constant stimulation delivered either by direct interactions with other people or by imaginary approximations of such interactions, which include countless forms of representational art and narrative.
To clarify this assumption, it is useful to compare our adaptations for mind reading with our adaptations for seeing. Because our species evolved to take in so much information about our environment visually, we cannot help seeing once we open our eyes in the morning,7 and the range of cultural practices grounded in the particularities of our system of visual adaptations is staggering. Similarly, as cognitive evolutionary psychologist Jesse M. Bering observes, after a certain age people "cannot turn off their mind-reading skills even if they want to. All human actions are forevermore perceived to be the products of unobservable mental states, and every behavior, therefore, is subject to intense sociocognitive scrutiny" (12). Hence, although we are far from grasping the full extent to which our lives are structured by adaptations for mind reading, we should be prepared that the cultural effect of those adaptations may prove just as profound and far-ranging as that of being able to see.
The second assumption is a paradox. We perceive people's observable behavior as both a highly informative and at the same time quite unreliable source of information about their minds. This double perspective is fundamental and inescapable, and it informs all of our social life and cultural representations.
To appreciate the power of this double perspective, consider the reason we remain suspicious of each other's body language. When I am speaking to somebody, she counts on my registering information conveyed by her face, movements, and appearance. That is, she cannot know what particular grin or shrug or tattoo I will notice and consider significant at a given moment; indeed, I don't know either. Our evolutionary past ensures, however, that she intuitively expects me to "read" her body as indicative of her thoughts, desires, and intentions. Moreover, the same evolutionary past ensures that I intuitively know that she expects me to read her body in this fashion. This means that I have to constantly negotiate between trusting this or that bodily sign of hers more than another.
Were I to put this negotiation in words—which will sound funny because we do _not_ consciously articulate it to ourselves—it might go as follows: "Did she smile because she liked what I said or because she wanted me to think that she liked what I said, or because she was thinking of how well she handled an argument yesterday, or was she thinking of something altogether unrelated?"
Thus, we treat with caution the information about the person's state of mind inferred from her observable behavior precisely because we can't help treating her observable behavior as a highly valuable source of information about her mind— _and we both know it_. Because the body is _the_ text that we read throughout our evolution as a social species, we are now stuck with cognitive adaptations that forcefully focus our attention on that particular text.
Nor would we want to completely distrust the body—our quick and far-from-perfect reading of each other is what gets us through the day. Still, as we unreflexively interpret each other's observable behavior in terms of underlying mental states, on some level we keep active the hypothesis that the observable behavior is misleading. (Note, too, that it does not have to be intentionally misleading: If I meet a person whose natural expression is a frown, I may incorrectly assume that he does not like me. The body may misrepresent the mind.)
What all this adds up to is that we are in a bind. We have the hungry ToM that needs constant input in the form of observable behavior indicative of unobservable mental states. And we have the body that our ToM evolved to focus on in order to get that input. And that body, the object of our ToM's obsessive attention, is a privileged and, as such, potentially misleading source of information about the person's mental state.
The research on ToM complements our own discipline's insight about the body as a site of performance. Because we are drawn to each other's bodies in our quest to figure out each other's thoughts and intentions, we end up _performing_ our bodies (not always consciously or successfully) to shape other people's perceptions of our mental states. A particular body thus can be viewed only as a time-and-place-specific cultural construction, that is, as an attempt to influence others into perceiving it in a certain way.
Cognitive evolutionary research thus lends strong support to theorists in cultural studies who seek to expand the meaning of performativity, such as Joseph Roach, who argues that performance, "though it frequently makes references to theatricality as the most fecund metaphor for the social dimensions of social production, embraces a much wider range of human behaviors. Such behaviors may include what Michel de Certeau calls 'the practice of everyday life,' in which the role of spectator expands into that of participant" (46). Indeed, work on ToM indicates that our everyday mind reading turns each of us into a performer and a spectator, whether we are aware of it or not.
A closely related implication of the studies on ToM is that they encourage us to think of a broad variety of cultural institutions and social practices as both reflecting our overarching need to attribute minds _and_ remaining subject to the instabilities inherent to our mind-reading processes. For example, our social infrastructure seems to be chock-full of devices designed to bypass our fakeable, performable, constructable body in reading the person's mind. We use blood and hair samples, credit and medical histories, fingerprinting and polygraph tests to avoid the situation in which we have to make an important decision based on information provided solely by the person's immediate observable behavior.
Some of these devices succeed better than others and none are perfect. We may not yet be living in the future depicted in _Gattaca_ (1997), whose protagonist (played by Ethan Hawke) fakes his blood and hair samples to deceive others about his intentions, but that sci-fi moment does capture an important sociocognitive feature of our world: there is a constant arms race going on between cultural institutions trying to claim some aspects of the body as essential, unfakeable, and intentionality-free, and individuals finding ways to perform even those seemingly unperformable aspects of the body.
Again, compare this cognitive-evolutionary insight with the work done by cultural theorists ranging from Judith Butler to Peggy Phelan, who have written extensively on the body as a constantly receding signified, a perennially contested depository of reliable meanings. Think, for example, of Phelan's observation that whereas "the living performing body is the center of semiotic crossings, which allows one to perceive, interpret and document the performance event," we long to "return to some place where language is not needed," an "Imaginary Paradise" where there are no "linguistic and visual distinctions between who one is and what one sees" (15, 29). Think, too, that some of the resistance to the view of the body as always constructed and always performed can come from our hoping against all hope that it _must_ be possible to carve some zones of certainty in the exasperating world where our favorite source of information, the body, is often untrustworthy in direct proportion to the extent to which we trust it.
### 4. TO KNOW AND KNOW NOT
With such a peculiar setup in place, what should we expect from our cultural representations? Or, to put it differently, how will our worldview change if we think about our culture as enmeshed with paradoxes and instabilities of our greedy ToM? Of course, this big question cannot be answered in one essay. But as a starting point, let us consider step-by-step what it means to live in a world in which we know and at the same time don't know what other people are thinking.8
First, we _know_ that there must be a mental state behind an observable behavior. Say you see somebody jumping up in the middle of a meeting. Try making sense of his action without talking about his presumed mental state; for example, he had an idea; he remembered something suddenly; he wanted to see how high he could jump; he felt something sharp on the seat beneath him; he saw a snake and was terrified; he wanted to check if everybody was awake.
Our belief that there must be a mental state behind a behavior is itself a cognitive artifact that reflects the way we perceive people. The question of whether my colleague over there _truly and really_ had some thought, feeling, or emotion that prompted him to jump is relatively irrelevant.9 What is relevant is that for you and me and every other human being with fully functional ToM, his jump signals an underlying mental state.10
Second, even though we know that there must be a mental state behind a behavior, we don't _really_ know what that state is. That is, there is always a possibility that something else is going on behind even the most seemingly transparent behavior. We can remember situations when our thoughts did not fit the circumstances, and no observable behavior could reveal them to people around us, or so we hope. On these occasions we say to ourselves, "Thank God, we can't read each other's minds, so that they have no way of knowing what goes through my head."
Third, even though we can't really know what other people are thinking, we conduct our daily lives on the assumption that we do, more or less. To borrow from a related discussion by the cognitive literary critic Ellen Spolsky, our everyday mind-attributions are "good enough."11 Obviously, I don't know what that man is really thinking as he strides purposively toward that particular weightlifting machine, but it has served me well in the past and is likely to serve me well in the future to assume that he wants to use it right away, which means that for the next five minutes I'd better turn to a different machine. Such rough-and-ready interpretations get us through the day. To quote the cognitive evolutionary anthropologist Dan Sperber, in "our everyday striving to understand others, we make do with partial and speculative interpretations (the more different from us the others, the more speculative the interpretation). For all their incompleteness and uncertainty, these interpretations help us—us individuals, us peoples—to live with one another" (38).
Were we to stop and try to figure out what the people around us are really thinking, we would become socially incapacitated, overwhelmed with possible interpretations, and unable to commit to any course of action. Perhaps the reason that we even notice our moments of "Thank God, we can't read each other's minds!" is because they stand out amid our daily unreflective mind-attribution. They interrupt its course. They force us to juxtapose a _good enough_ mind-attribution—that is, what people could be expected to think in such a situation—with an _exact_ and unexpected mind-attribution: what I really thought in that situation.
Fourth, because we go around knowing that there must be a mental state behind the behavior, and because we don't really know what that state is, even as we act as if we know, our cultural representations exploit this precarious state of knowing and not knowing. One important distinction between daily life and fiction is that we generally get through the day with our far-from-perfect attributions of intentionality, but authors are less interested in such "good enough" attributions.12 Works of fiction magnify and vivify various points on the continuum of our imperfect mutual knowledge: Spectacular feats and failures of mind reading are the hinges on which many a fictional plot turns.
### 5. EMBODIED TRANSPARENCY
Here is one specific argument we can make approaching cultural representations from the cognitive perspective outlined above. There seems to be a representational tradition, which manifests itself differently in different genres and individual works, of putting protagonists in situations in which their bodies spontaneously reveal their true feelings, sometimes against their wills. Such moments are carefully foregrounded within the rest of the narrative. In each case an author builds up a context in which brief access to a character's mental state via her body language stands out sharply against the relative opacity of other characters or of the same character a moment ago.
Every moment of transparency is thus entirely relative and context-dependent, but the wish to create and behold such moments seems to be perennial, grounded in our evolutionary history as a social species. Representations of embodied transparency regale us with something that we hold at a premium in our everyday lives and never get much of: perfect access to other people's minds via their observable behavior. As such, they must be immensely flattering to our ToM adaptations, which evolved to read minds through bodies but have to constantly contend with the possibility of misreading and the resulting social failure. The pleasure derived from moments of embodied transparency is thus largely a social pleasure—a titillating illusion of superior social discernment and power.
To see what forms such illusions of perfect access may assume, consider the three following examples, which deal respectively with novels, genre paintings, and moving images.
### _A. A NGRY BODIES AND SADISTIC BENEFACTORS_
At first glance, novels seem to be an unlikely setting for cultivating special moments of embodied transparency because they are _already_ in the business of revealing their characters' thoughts and feelings. Alan Palmer captures this when he observes in his remarkable _Fictional Minds_ that one "of the pleasures of reading novels is the enjoyment of being told what a variety of fictional people are thinking. . . . This is a relief from the business of real life, much of which requires the ability to decode accurately the behavior of others" (10).
Yet there can be an important difference between telling how the character feels using omniscient narration and making the character show his or her true feelings. Thus in the first proposal scene of Austen's _Pride and Prejudice_ (1813), we hear of Elizabeth's anger but we _see_ Mr. Darcy's anger. Here is Elizabeth's listening to Mr. Darcy's confession that he "struggled . . . in vain" to repress his love for her:
> In spite of her deeply-rooted dislike, she could not be insensible to the compliment of such a man's affections, and though her intentions did not vary for an instant, she was first sorry for the pain he was about to receive; till, roused to resentment by his subsequent language, she lost all compassion in anger. She tried, however, to compose herself to answer him with patience, when he should have done. (129)
Elizabeth does not fully succeed in composing herself. Toward the end of Mr. Darcy's speech, "the colour [rises] into her cheeks." Still, heightened color can be indicative of a variety of mental states, some flattering to the suitor, whereas Mr. Darcy's subsequent bodily reaction to Elizabeth's disdainful answer provides direct and unequivocal access to his feelings:
> Mr. Darcy, who was leaning against the mantelpiece with his eyes fixed on her face, seemed to catch her words with no less resentment than surprise. His complexion became pale with anger, and the disturbance of his mind was visible in every feature. He was struggling for the appearance of composure, and would not open his lips, till he believed himself to have attained it." (129)
Note the contrasts that went into constructing Mr. Darcy's transparency. Not only is he now more readable than Elizabeth, but he is also more readable than himself earlier in the novel and a moment before. In the first sentence of the last quoted passage, he is described as _seeming_ to catch her words with no less resentment than surprise. That is, there is still some possibility of misinterpreting his body language at that point— he _seems_ to be resentful and surprised, but he might not actually be so. Then the next sentence ("His complexion became pale with anger" [ibid.]) leaves no doubt that his body reflects his mind fully and faithfully. These contrasts convey a strong impression that this moment of perfect mental access cannot last long, sharpening our appreciation for the vision of the body caught in spontaneous emotion.13
Here is an example from a more recent work, Helen Fielding's _Bridget Jones: The Edge of Reason_ (1999). Fielding's treatment of embodied transparency is particularly interesting because, written from the first-person point of view and obsessed with the issue of gender and communication, her novel provides an apparently exhaustive report of Bridget's feelings and those of women surrounding her. Men's minds remain strategically obscured—in the tradition of Austen, whose _Pride and Prejudice_ and _Persuasion_ inspired "Bridget Jones" duology.
But even in the novel in which transparency (at least for women) seems to be the default narrative mode, moments of embodied transparency are still presented as rare and valuable flashes of insight. For example, there is the scene at a ski resort in which Bridget is talking to her boyfriend, Mark Darcy, and an attractive woman, Rebecca, who is trying to steal Mark from Bridget. Rebecca invites Mark and Bridget to a skiing party, where she would have more opportunities to flirt with Mark, especially if Bridget, a poor skier, could be separated from him:
> "Oh, it's so exhilarating," said Rebecca, putting her goggles on her head and laughing into Mark's face. "Listen, do you both want to have supper with us tonight? We are going to have a fondue up the mountain, then a torchlight ski down—oh sorry, Bridget, but you could come down in the cable car."
>
> "No," said Mark abruptly, "I missed Valentine's Day so I'm taking Bridget for a Valentine's dinner."
>
> The good thing about Rebecca is there is always a split second when she gives herself away by looking really pissed-off.
>
> "Okey-dokey, whatever, have a fun time," she said, flashed the toothpaste advert smile, then put her goggles on and skied off with a flourish towards the town. (73)
Observe all the careful framing that goes into foregrounding Rebecca's embodied transparency. First, her involuntary giving "herself away by looking really pissed-off" is contrasted with her fake spontaneity one moment earlier, when she appears unable to contain her good spirits buoyed by skiing, and her fake friendliness right after, when she smiles broadly to show that she does not mind Mark's rejection. Second, Rebecca's bodily display of feelings is contrasted with Mark's opacity—his answer is "abrupt," which means that nothing in his body language has prepared the two women for what he is about to say. Third, Fielding has Bridget actively draw our attention to both the value of this revelatory moment ("the good thing about Rebecca . . .") and to its transience: it lasts only a "split second," so one is lucky to catch it.
It may seem, based on my examples from Austen and Fielding, that embodied transparency always finds an appreciative spectator within the story itself (for example, when Mr. Darcy looks angry, Elizabeth is there to observe him and interpret his body language), but the situation is more complex. Many fictional characters remain oblivious of others' momentary transparency. In Henry Fielding's _Tom Jones_ (1749), Tom ignores Lawyer Dowling's uncomfortable twitching during their conversation about Tom's parentage. Similarly, Mr. Allworthy remains blind to Doctor Blifil's grimacing attempts to conceal his laughter in response to Allworthy's heartfelt "sermon" about love and marriage (62).14 Sometimes such obliviousness is gently glossed over by the narrative; sometimes it is strategically foregrounded to illustrate the protagonist's unfortunate lack of awareness or his tactful reluctance to pry into other people's thoughts.
But if characters may notice or miss moments of embodied transparency, we readers are always made to notice them. We may perceive such moments as perceived by characters in the story or as _not_ perceived by them, but, either way, they offer us a dazzling possibility of an escape from our double perspective of the body as a highly privileged and yet unreliable source of information about the mind.
This is why Bridget Jones's insight is our gain but so is Mr. Allworthy's blindness. We look at Doctor Blifil and we see what Mr. Allworthy does not see—a body that desperately does not want to be read and thus _is_ readable in this desire not to be read. So when I speak of representations of embodied transparency and their effects, I mean primarily their effects on the ToM of _readers_. We are the ultimate appreciative audience for such representations.
Hence a fascinating sub-tradition in the novelistic treatment of embodied transparency, which I call a tradition of sadistic benefaction. Some fictional characters do not seem to be content with serendipitously observing (as Elizabeth and Bridget do) other people's spontaneous reactions. Instead, they want to _script_ such reactions, forcing others into revelatory body language and appropriating, in effect, the privileged positions of readers as ultimate observers of moments of embodied transparency. The ethics of these situations are extremely ambiguous. The characters are presented as believing that their actions will ultimately benefit the people whom they are forcing into transparency, but the impression of emotional sadism lingers.15
Consider, for example, Sarah Fielding's _The History of Ophelia_ (1760), in which a rich man sends a poor man on an emotional rollercoaster to enjoy the spectacle of his feelings against his will. The novel's protagonist, Lord Dorchester, comes across a starving half-pay soldier, Captain Traverse, and decides to help him. Through his connections at the court, he secretly procures Traverse the choice of two jobs and then proceeds to torture the man by telling him first only of the job that he knows the Captain will not be able to take due to family circumstances. The poor Captain, unwilling to appear ungrateful, receives "this News with as much Gratitude as if it had been the very Thing he wished" and turns it down politely. Lord Dorchester then expresses his disappointment in such terms as to drive the Captain to break down in tears when he thinks nobody is watching (1: 252)—when, in fact, everybody is.
Not yet content with this show of emotions, Lord Dorchester then reveals the Captain's family waiting in the next room and urges him again to take the first job. In response, the Captain "faint[s] away instantly," terrifying his wife and making the onlookers fear for his life. When the Captain comes to, Lord Dorchester augments "the general Joy" that his recovery occasions by telling him of the second job, one that is completely acceptable and will save the whole family from starving. The joy now increases "to a great Degree of Extacy," raising to a "Height that must have been painful." The Captain and his wife look upon the "Lord with Adoration, and [give] way to Raptures that would have forced a Heart the most insensible to the Sensations of others, to partake of theirs" (1: 254–55).
Lord Dorchester is a grotesque embodiment of the eighteenth-century obsession with private philanthropy, social class, and sentimental discourse. His privileged social standing allows him to torture such people as Captain Traverse before helping them. Fielding thus draws on recognizable cultural contexts of her day to construct a scene in which the poor man's feelings are rendered transparent against his desires and observed hungrily by the rich benefactor.
Lest we think that the eighteenth-century sentimental novel holds the exclusive right to such scenes, consider Chuck Palahniuk's _Fight Club_ (1996), in which Tyler Durden holds a gun to a head of a man he has just met (who turns out to be a twenty-three-year-old college dropout) and extracts from him the promise that he will go back to school to finish his degree in veterinary medicine. Tyler wants to impress on "Raymond Hessel" (the name he reads off his driver's license) an important life lesson: Death can strike any minute, so study hard and follow your dreams.
But before Tyler gets to the salutary follow-your-dreams part, he uses every grisly cliché to convince Raymond of his imminent and terrible demise. He explains to the crying man how he will "cool" down, passing from a "person" to an "object" and how his "Mom and Dad would have to call old doctor whoever and get [their son's] dental records because there wouldn't be much left of [his] face." The scene is so structured that Tyler seems merely to report the images arising in his victim's mind. Raymond is forced into embodied transparency: half-paralyzed with fear, crying harder and harder, following meekly Tyler's orders, thinking of things that Tyler tells him to think of.
When Tyler finally lets Raymond go, his thoughts stay on him as he muses with great satisfaction: "Raymond fucking Hessel, your dinner is going to taste better than any meal you've ever eaten, and tomorrow will be the most beautiful day of your entire life" (155). Like Lord Dorchester, Tyler apparently believes that to intensify some-body's happiness you have to first make him truly miserable. More importantly, like Lord Dorchester, Tyler treasures every moment of transparency he can wring out of his victim. Tyler enjoys knowing exactly what Raymond is thinking now and what he will be thinking tomorrow.
Or does he? Scenes of sadistic benefaction reveal an asymmetry in novels' treatment of embodied transparency. Readers are indeed occasionally allowed to transcend the double position of the body and enjoy the illusion of perfect readability (Mr. Darcy looks angry and he _is_ angry; Rebecca looks pissed off, and she _is_ pissed off), but when characters themselves try to create contexts of such direct access, the results are often mixed. Subversion lurks. Because there is a potential for a gap between the forced emotion and the actual emotion, the performing body may reassert itself.
Look again at the scene in _Fight Club_ in which Tyler reads his victim's body as an open book:
> "You [are] going to cool, the amazing miracle of death. One minute, you're a person, the next minute you're an object, and Mom and Dad would have to call old doctor whoever and get your dental records because there wouldn't be much left of your face, and Mom and Dad, they'd always expected so much more from you and, no, life wasn't fair and now it was come to this." (153)
If we return to this scene after we have finished the novel and found out that Tyler is what we call an unreliable narrator,16 we may start seeing problems with this assured interpretation of Raymond's body. We may notice, for example, that Tyler's account of Raymond's thoughts draws on conventional images provided by crime dramas. That is, we cannot know exactly what really goes through a person's head when he has a gun pressed to his cheek. After all, he might be thinking that now he will not have to go through his scheduled biopsy to find out what that scary lump is. We do, however, have a visual repertoire of scenes associated with violent deaths, for example: cut to the bereaved family; cut to the corpse in the morgue; cut to the dentist, or some other doctor, confirming the victim's identity; and so forth. This is the repertoire that Tyler dips into for his "report" from Raymond's head.
Similarly, when Tyler confidently foretells what Raymond will be thinking even after he is out of Tyler's clutches ("tomorrow will be the most beautiful day of your entire life"), I see his point. I can certainly imagine how tomorrow Raymond might feel almost unbearably happy to be alive and thankful for every crumb that passes his lips and for every leaf that he sees trembling in the wind. I can also imagine Raymond racing to school, profoundly grateful for the opportunity to "work [his] ass off" (154) on various hard subjects, just as Tyler told him he should. Finally, I can imagine Raymond eventually becoming a successful veterinarian, loved by his family, respected by his neighbors, and remembering now and then with wonder and gratitude that fateful moment when a stranger with a gun forced him to turn his life around and make the most of it.
But then I can also imagine Raymond falling into a profound depression soon after his encounter with Tyler, thinking obsessively that his life depends on the whim of some jerk with a gun, and killing himself one day after school.
In other words, Raymond's body remains transparent and his mind accessible as long as we consider this scene in isolation from the tradition of unreliable narration. For within this tradition, when a first-person narrator reports another character's thoughts, he is almost immediately suspect and his reporting must be scrutinized for signs of inconsistency, vested interests, or even madness. So Palahniuk's readers believe that Tyler really knows what Raymond is thinking only as long as they are not aware that Tyler is an unreliable narrator. Once they are aware of it, they have an option— which, of course, they may choose not to follow—of assuming that they have learned very little about Raymond's actual feelings on the occasion.
But even if we do not consider this episode in relation to the tradition of unreliable narration, something else in it alerts us to the possible gap between Tyler's assured interpretation of Raymond's body and Raymond's actual mental state. The scene's timing is off. Tyler keeps reading Raymond's mind for three straight pages, and somewhere along the way embodied transparency is bound to become a performance. The same applies to the protracted exchange between Lord Dorchester and Captain Traverse. Tyler and Lord Dorchester may convince themselves, egomaniacs that they are, that they can sustain their victims in the state of spontaneous emoting for as long as they choose. The longer they keep it going, however, the more willing we are to entertain the possibility that after a while Raymond and Captain Traverse intuit something about the twisted psychology of their torturers / benefactors and start performing their despair and gratitude at the top of their lungs.
In contrast, think again about Helen Fielding's emphasis on transience as she describes Rebecca's body language: her give-away "pissed off" expression is visible for only a "split second."17 Think, too, how quickly Austen's Mr. Darcy regains control of his features and ceases to be transparent. True, Austen emphasizes that time slows down for Elizabeth as she watches his internal struggle—"the pause was to Elizabeth's feelings dreadful" (130)—but we know that this pause could not really have lasted very long. To be credible, embodied transparency has to be brief and spontaneous. Brevity makes it ethically defensible, too:18 we do not want to think that Elizabeth actually enjoys watching Mr. Darcy at his most transparent.19
There is yet another side to the issue of spontaneity. It is not enough if a specific moment of embodied transparency in a specific work of fiction is unexpected and short. To be convincing, it also has to look unconventional in the larger context of its genre. I said earlier that there is a constant arms race going on between cultural institutions trying to claim some aspects of the body as performance-free and individuals finding ways to perform even these seemingly "essential" aspects of the body. The history of cultural representations that create contexts of transparency must be viewed in the relation to this larger arms race. Writers, artists, and movie directors have to keep inventing _new_ ways of forcing the body into a state of transparency because as soon as one way of doing it emerges as an established convention, it becomes vulnerable to subversion and parody. The mind retreats further, leaving the body as a front going through the expected motions of "revealing" the "true" states of mind. The double perspective of the body returns with vengeance.
As an example of such a subverted context of transparency, consider the eighteenth-century sentimental novel with its loving attention to blushing, crying, panting, fainting bodies. In Samuel Richardson's _Pamela_ (1739), such bodily displays still stand for real feelings,20 but in his next novel, _Clarissa_ (1747–48), they are already consciously faked for the benefit of naïve observers.21 The term "sentimental" itself undergoes a change between 1740 and 1820. Originally neutral, "characterized by sentiment," or positive, "characterized by or exhibiting refined and elevated feeling," it acquires a pejorative meaning of "addicted to indulgence in superficial emotion."22 In other words, keep looking at the emoting body in hopes that it will keep providing direct access to the person's mental states, and you will soon be treated to "superficial emotion," performed for your viewing pleasure. So when Lord Dorchester tries to script Captain Traverse's embodied transparency in 1760, not only is his script too long, but it is also quickly becoming outdated.
### _B. P ROBLEM PICTURES AND PROPOSAL COMPOSITIONS_
At first glance, genre paintings seem another unlikely candidate for bracketing off special moments of embodied transparency. Paintings that depict people engaged in everyday activities rely as heavily on our ToM as do novels. To make sense of such scenes we have to attribute intentions to each character, or to a group of characters if the painting encourages us to construct them as sharing a certain attitude. Still, compare two kinds of genre paintings, partially overlapping in time but pointedly different in their treatment of transparency: problem pictures and proposal compositions.
Problem pictures, writes Pamela M. Fletcher in her _Narrating Modernity: The British Problem Picture, 1895–1914_ , "were an extraordinarily popular feature of the Edwardian Royal Academy. The term referred to ambiguous, and often slightly risqué, paintings of modern life which invited multiple, equally plausible interpretations" (1). Consider John Collier's _A Confession_ (1902), which depicts a "couple engaged in an emotional conversation," in which the woman's face is "in shadow, while the man, leaning forward with his elbows on his knees, brings his face into the light of the fire," staring down and slightly to the viewer's right (62; Fig. 2).
According to Collier, he received "many inquiries" about the picture's subject. In one extant letter, the "writer pleads: ' _Oh!_ Honourable John, I want to know _very_ badly which ( _please_ tell me) is confessing in your Royal Academy picture—the man or the woman'" (62, original emphasis). In response to such queries, the artist apparently offered "multiple interpretations," cultivating "oracular ambiguity" along the lines of "The woman did it and the man confessed it" (63). In 1913, he revisited the subject with his _Fallen Idol_ , in which a "woman kneels at a man's feet, her upper body resting on his knees and her head bowed in an attitude of grief or shame. The man holds one of her hands in his, and stares directly out of the canvas, his face illuminated by a shaft of light" (129). As Fletcher reports,
_Figure 2. The Confession._
> Critical responses in the press were almost equally divided between those who read the story as completely open to interpretation, and those who assumed that the woman had "fallen." The _Daily Mirror_ , _Queen_ , _Reynolds's_ , and the _Daily Sketch_ all read the picture as ambiguous, predicting, "Lots of stories will be woven around this picture, and probably none of them the right one." The _Daily Mirror_ made the point by "quoting" two visitors: "'Of course, he's just confessed something he's done,' said one woman yesterday confidently. 'She's just been found out' said the next comer with equal assurance." (130–31)
The spatial arrangement of the figures in problem pictures and the pattern of lighting do not consistently single out one gender as more mysterious than the other: the man is just as likely as the woman to be "in the position of the 'problem'" (62). Contrast this with the pattern identified by Stephen Kern in his study _Eyes of Love: The Gaze in English and French Paintings and Novels, 1840–1900_. As Kern puts it, when "French and English artists . . . depicted a man and a woman in the same composition [they] typically rendered the face and eyes of the woman with greater detail and in more light. Most important, the men are in profile, while the women are frontal" (7). These works exemplify what Kern calls a "proposal composition":
> Such a composition highlights the woman's moment of decision after the man has proposed that the relationship move to some higher level of intimacy. At such moments she must respond, whether it be to his searching look or friendly inquiry, or more significantly, to his seductive offer or proposal of marriage. Her eyes convey an impending answer to the question _Will she or won't she_? And because she is thinking about many possible consequences of her answer, her expression is especially intriguing. In contrast, the man has done his thinking and said what is on his mind. He wants to hear a _Yes_ , so his face bears a more predictable and less interesting expression. (7)
Kern makes a convincing case against the accepted critical view that visual representations of women—especially beautiful women—always objectify them. He argues that in proposal compositions, such as, for example, William Midwood's _At the Crafter's Wheel_ (Fig. 3), women "are not objectified by the male gaze but retain a commanding subjectivity that, in comparison to the man's more erotically focused purpose and expression, conveys a wider range of thoughts and emotions" (228).
_More_ interesting, _less_ predictable, _especially_ intriguing, conveying a _wider_ range of emotions—contrasts and degrees are at the heart of proposal compositions. Of course, any art that appeals to the eye must cultivate gradients. As Ernst Gombrich observed, "even newly hatched chickens classify their impressions according to relationships" (298). Still, what is important for the purpose of the present argument is that using both our interest in contrasts and the familiar social script of courtship, proposal compositions construct the context in which the body of one protagonist is maneuvered into embodied transparency. We know what the man is thinking, his "erotically focused" purpose made even more obvious because it is contrasted with the "intriguing" thought processes of the woman.
In her study, Fletcher demonstrates that interpretations of problem pictures reflected some of the "most pressing issues of the early twentieth century, including the nature of modern marriage and motherhood, the emergence and definition of the new professional classes, and the existence of a specifically feminine morality" (1). No doubt many of the same issues were at play in proposal compositions, but observe the crucial difference between their respective constructions of protagonists' subjectivity.
Whereas a problem picture leaves their feelings largely open to interpretation and only somewhat constrains them within broad categories, such as "distress" or "surprise," the proposal composition constructs one participant as more transparent than another.
Moreover, it turns out that the issue of time, so important in fictional representations of embodied transparency, is just as important in paintings. In a proposal composition, we know what a man is thinking, but this moment of transparency cannot last. For, depending on the woman's reaction, the man will soon adopt a different posture, attempting perhaps to conceal his disappointment if she says _no_ or hesitates for too long. The same cultural narrative—the courtship narrative—that makes the instance of transparency convincing ensures that it is but an instance, serendipitously "caught" by the artist.
_Figure 3. At the Crafter's Wheel._
Something else might be at work in sustaining this illusion of a serendipitously caught moment of transparency: it steals upon the spectator unexpectedly. After all, "proposal compositions" were not known as such to their contemporaries. Whereas "problem picture" is a recognizable historical term referring to a particular sub-genre associated with a specific style, "proposal composition" is a term introduced by Kern to describe not so much a genre as a recurrent compositional pattern and interpersonal dynamic that can be found across different schools, styles, and representational traditions of the second part of the nineteenth century.
Note too that the titles of such paintings rarely indicate that we are witnessing a scene of romantic inquiry and hesitation (in contrast to the straightforward titles of problem pictures, e.g., _A Confession_ ). Very few titles are leading (e.g., _The Proposal_ , _Pleading_ , or _Showing a Preference_ ); the majority are all over the place, referring to a setting, to a prominent artifact present on the scene, or to the main protagonist (e.g., _A Dance in the Country_ , _Nameless and Friendless_ , _Waiting for the Ferry_ , _Effie Dean_ , _A Rest by the Seine_ , _Blossom Time_ , _The Picnic_ , _The Umbrellas_ , _At the Crafter's Wheel_23). Because there is neither an explicit genre affiliation nor a title that would mark a proposal composition as such, viewers have to infer on their own in the case of each specific painting that it contains a deliberation induced by a proposal of marriage or romantic liaison.
So as contemporary spectators approached a painting, say, Renoir's _A Dance in the_ _Country_ ,24 Osborn's _Nameless and Friendless_ ,25 or Horsley's _Blossom Time_ ,26 they did not know beforehand that the body of one of its protagonists was supposed to be strikingly readable (even though they could see it right away). Had it been known—that is, had the proposal composition indeed emerged as an established sub-genre with its own set of typical titles—the man's transparency would have eventually become a convention and as such would have required an extra effort to be rendered convincing. (The fate of French eighteenth-century absorptive paintings, as discussed by Michael Fried in _Absorption and Theatricality_ , is a vivid example of what conventions and expectations can do to destroy the illusion of immediate access to a represented subject's mental state.27) This did not happen, however: the impression of serendipity was not marred by the thought that in this artistic sub-genre serendipity _is_ a convention.
I am not making any teleological claims about the history of proposal compositions, such as that late nineteenth-century artists and art critics consciously avoided recognizing a new sub-genre in their midst in order to be able to continue constructing compelling narratives of embodied (male) transparency. Rather it seems that authors of proposal compositions differed widely from each other in their styles and sensibilities and did not give much thought to this particular common denominator, and neither did their audiences. We needed Kern's book to finally see this common denominator, and we need research on ToM to see why we intuitively value so much the moments when bodies reveal minds so vividly.
### _C. M OCK DOCUMENTARIES_
Let us now return to the scene of social torment that opened this essay. Observe that what _The Office_ does here (and for the purposes of this discussion I am thinking of this mock documentary as representing moving images at large28) is similar to what novels and paintings do when they build their moments of transparency.
_The Office_ cultivates contrasts. That is, Gervais and Merchant constantly prod us to gage one character's embodied transparency as contrasted to that of other characters or to her own embodied transparency a moment ago. For example, in the scene in which David makes a fool of himself in front of Dawn and the two job candidates, both interviewees must be unpleasantly surprised by David's behavior. Still, the scene is shot so as to emphasize the contrast between what we may infer about their respective feelings and what we may infer about Dawn's feelings. We can see that Dawn goes through the agony of embarrassment _and_ tries to conceal her disapproval of David's actions. Because we can imagine how much mental energy this must take, we conclude that it is highly unlikely that Dawn is capable of thinking of anything else at this moment (e.g., about her fiancé, Lee, her co-worker, Tim, or the book that she was reading earlier). Just now Dawn is strikingly transparent.
Not so Karen and Stuart, the interviewees. Karen smiles a lot, and at one point she also shoots a curious glance at Dawn. We may infer that she is taken aback by David's lack of professionalism but determined to get through the interview. Other than making this tentative inference, however, we have no way of knowing what goes through her head as she listens to David's harangue.
Stuart is even less transparent than Karen. We may guess that he is disappointed and angry, and that he has already given up on this job and is now keeping up the appearance of polite interest because the camera is present. These speculations, however, draw more on the context of the scene than on his body language. For he wears a small, noncommittal smile throughout and otherwise betrays no emotions.
The contrast propels the story. We take in the characters' body language—almost all at once—and then align it along the continuum of transparency. This subconscious process of comparing, contrasting, and aligning builds both on the cues planted by the directors (e.g., the actor playing Stuart must have been told to maintain a poker face) and on the particularities of our information processing (i.e., to make sense of our world we need to construct it in relational terms). As such _The Office_ is similar to thousands of other television shows and feature movies—but also to novels and paintings—whose makers intuitively cultivate such hierarchies of transparency to charge their tableaux with inner dynamism.
Then there is also the issue of time. We have seen already how important it is for writers and artists to convince their audiences that the moment of transparency is transient and hence particularly valuable. Did the makers of _The Office_ do something special to keep the instances of embodied transparency brief?
It turns out that they did, even if they did not think about it in these terms. In their commentary to _The Office_ , Merchant and Gervais explain that they wanted to avoid the feel of "situation comedy" and to keep their material jumpy and raw. To achieve that, they made a point of cutting abruptly from one scene to another. Note, however, one important side effect of this editing technique: it ensures that whenever characters are forced into a state of embodied transparency, we never stay with them long enough for them to seize control of the situation and start performing their feelings.
This really is an effective narrative trick. _The Office_ never dwells on the same person for a long time, yet it leaves us under a strong illusion that it does, by making us aware that we stare at the protagonists when they are embarrassed or making fools of themselves for much longer than would be polite in real life. So we are made to feel that we have seen too much when, in fact, we have seen just enough to be convinced that the protagonists were caught off-guard and did not have time to rally their spirits and put on a suitable performance.
With its treatment of embodied transparency, a mock documentary occupies a peculiar position in relation to its grandparent genre of regular documentary and its immediate progenitors, cinéma vérité and direct cinema.29 Cinéma vérité thrived on the spontaneous embodiment of emotions—indeed actively looked for contexts that would allow for maximum transparency. As Hope Ryden, who wrote, directed, and produced documentary films from 1961 through 1987 and was part of the Drew Associates team that developed cinéma vérité / cinema direct in the early sixties,30 puts it, "What we were doing was finding an upcoming event in which some character would have a great stake. At that moment they would either win or lose. And it didn't matter whether they win or lose. What mattered was that they cared a whole lot about what they were doing" ( _Cinema Vérité_ ).
And what this caring "a whole lot" meant for a movie is that no matter what a character might do on camera during that make-or-break moment, viewers knew what she or he was really feeling. So if the character showed emotions, such as anxiety, happiness, or disappointment, those emotions could be counted on as being authentic; and if she did not show any, she would still be transparent, because the viewers knew that she was trying hard to conceal her feelings. In the words of Robert Drew, who directed the documentary _Primary_ (1960), featuring presidential candidates Hubert Humphrey and John F. Kennedy during the Wisconsin primary elections: "The idea of capturing human emotion spontaneously as it happens was the key idea that made _Primary_ work, that made all of our films work, and that is making cinéma vérité work today in many ways across the spectrum of television" ( _Cinema Vérité_ ). As historians of documentary film Jack C. Ellis and Betsy A. McLane put it:
> [In] _Primary_ , Humphrey and Kennedy were much more concerned with winning an election than with how they would appear on screen. . . . _Mooney v. Fowle_ (1961, also known as _Football_ ) builds up the climaxes with high school football game in Miami, Florida, between two rival teams. It concentrates on the players, coaches, immediate families—those most completely preoccupied with this contest. _The Chair_ (1962) centers on the efforts of a Chicago attorney, Donald Page Moore, to obtain a stay of execution for his client, Paul Crump, five days before it is scheduled to take place. _Jane_ (1962) concerns Jane Fonda in the production of a play, from the rehearsal period through the negative reviews following its Broadway opening and the decision to close it. (216)
So, when in _The Chair_ , "the attorney breaks into tears and expresses his incredulity after he receives a phone call from a stranger offering support for him in his efforts to save his client's life" (219), we cannot doubt that we see the man at his most transparent, that his body language provides direct access to his mind.
Yet doubt is never far enough given the cognitive underpinnings of the phenomenon that we are dealing with. We cannot escape for long the double view of the body: the promise of direct access is always fraught with the possibility of manipulation and deception. Once a culture becomes aware of a seemingly reliable representational context for embodied transparency in its midst, that context is rendered suspect. (We have seen it happening in the eighteenth-century sentimental novel: in _Clarissa_ , Lovelace carefully fakes "spontaneous" body language.)
Thus, on one hand, it was the credo of cinéma vérité that their "subjects would reveal what they really felt and were like when unself-consciously relaxed or deeply involved in some activity" (Ellis and McLane 217). On the other hand, however absorbed the subject may be by what she is doing, it is still possible that the presence of the camera influences her emotional responses, if even ever so slightly. As Jean Rouch, the director who originated the term cinéma vérité, saw it, "the camera acts as a stimulant. It causes people to think about themselves as they may not be used to doing and to express their feelings in ways they ordinarily would not" (Ellis and McLane 217).31 The double perspective of the body worms its way into "vérité."
So perhaps we should look at certain films made between the late 1960s and now as attempts by filmmakers to reclaim the documentary as a reliable context for embodied transparency.32 Among such attempts would be documentaries that depict people killed during rock concerts (e.g., _Gimme Shelter_ 1970), victims of war in Vietnam (e.g., _Hearts and Minds_ 1974), people dying of AIDS (e.g., _The Broadcast Tapes of Dr. Peter_ 1994), and people literally growing up on camera (Michael Apted's "Up Series" of 1970, 1977, 1984, 1991, 1998, and 2005). Such a perspective would be both cognitive (that is, acknowledging that we remain ever-seducible by the body's promise of direct access to the mind) and historicist (that is, following specific circumstances that led, but did not _have to_ lead, to the creation of each film).
Note how many of these films depict death: the ultimate moment of embodied transparency. Of course, with exception of certain unique circumstances, such as those created by the epidemic of AIDS in the 1980s and 1990s,33 the directors did not and could not set out expecting to film _that_ kind of transparency.34 It is interesting, then, that the 1970s saw a number of movies appearing to be documentaries shot in cinéma vérité style, such as _No Lies_ (1973), a "staged film about rape,"35 and _Rushes_ (1979), a staged film about suicide.36 Following in the wake of the original cinéma vérité of the 1960s, these fake documentaries indicated the genre's intuitive awareness that a strenuous extra effort might eventually be required to sustain its claims to direct access. And they were right to the extent to which in the years to come the real cinéma vérité did turn to death and other unfakeable physiological experiences, such as growing up.
Where would _The Office_ and other recent mock documentaries fit our hypothetical narrative of the twentieth-century documentary filmmakers' intuitive quest for reliable contexts of embodied transparency? They are the direct descendants of the fake cinéma vérité films of the 1970s. (Note that not all of those films dealt with injury and death. There were others, for example, ones in which the camera filmed everyday life of families and of women in the workplace.37) _The Office_ takes the original cinéma vérité promise of transparency as its absolute raison d'être. Whereas cinéma vérité _hoped_ to catch at least some moments of embodied transparency and looked for situations that were likely to produce such moments, a mock documentary _transforms_ every situation into an occasion for embodied transparency.
As one brief illustration of how this transformation happens, consider Gervais's commentary on the "talking heads" of _The Office_. He refers to the scenes in which the protagonists are interviewed individually and thus can presumably put on any attitude or personality:
> I really love the talking heads in the show. Because we shot it like a documentary, we couldn't do things people wouldn't do in front of the camera. They can't shut the door and take a line of coke. Or they can't blurt out things they're thinking.
>
> But, ironically, when they're by themselves and they're just being filmed, they're a little bit more honest. People do let their guard down because it's flattering. When a camera's pointed at someone, they think, "This is my chance, this is my platform. I can tell the world my all great philosophies on life." And of course, they open their mouth and they blow it and can't take it back.
Note what just happened here. The least likely moment for revealing one's true thoughts is turned into its opposite. It is one thing to be _caught_ on camera as Dawn is in the scene with David and the two job candidates; it is quite another to be invited into a separate room to be filmed during a formal interview. In the latter case, you have all the opportunities in the world to prepare well and to act out your better self. After all, many early cinéma vérité directors prided themselves on _never_ interviewing their subjects. Interviews were the mainstay of stodgy traditional documentaries—they spelled out performance. But in _The Office_ this context of performance par excellence becomes yet another context for transparency. The protagonists "blow it and can't take it back."
### 6. SUGGESTIONS FOR FURTHER INTERPRETATIONS
I wrote this essay because I wanted to understand why our cultural representations repeatedly invent new contexts for making characters betray their true feelings and why these moments of embodied transparency are generally short-lived and unstable. To think through these issues, I developed a four-level model, going from the broadest question to the most specific. I first asked how our worldview will change if we begin thinking about our culture as enmeshed with our hungry ToM (Level One). More specifically, I wondered how our cultural representations engage with our double view of the body as both a highly informative and at the same time quite unreliable source of information about the mind (Level Two). Even more specifically, I wanted to see how our novels, paintings, and moving images construct the contexts of forced transparency, when the double view of the body is temporarily transcended and the observable behavior serves as the reliable index to the underlying mental state (Level Three). And, most specifically, I wanted to know if certain features of these fictional moments of mental transparency are actually defined by the impossibility of reliable and lasting escape from the double perspective of the body, itself grounded in our cognitive evolutionary history (Level Four). Thus I suggested that artists and writers might actively look for pointedly short-lived social situations—such as the moment between the man's proposal and woman's answer (in proposal compositions) or a moment of acute social embarrassment (in mock documentaries)—to construct more _plausible_ contexts for complete mental access.
To what extent is this four-tier model applicable to further literary and cultural analysis? It seems to me that whereas my fourth question is geared toward a very specific set of problems, the first three can be used to approach a variety of cultural representations. Of particular interest here might be cognitive-historicist interpretations that place our flawed need to read bodies for states of mind within specific historical and socioeconomic environments. One may ask, for example (staying on the first level), what happens when a mind-reading species, such as ours, enters a capitalist economy. How do the laws of supply and demand operate when you have a mind that wants to process representations of other minds and a marketplace geared toward offering an increasingly broad range of such representations? A student of late seventeenth- and eighteenth-century literature and culture may speculate here about the proliferation of hybrid genres and publication outlets and ask how the economics of publishing drove the readers' cognitive appetites and vice versa. A student of contemporary culture may similarly think of how the rapid fragmentation of cultural production and consumption caters to and cultivates new mind-reading interfaces.
To quote Nick Gillespie, when "everywhere we look, the cultural marketplace is open and ready for business," when "in economic terms, the opportunity costs of both making and enjoying culture have dropped through the floor; [and when] it keeps getting cheaper and cheaper to produce and to consume culture under increasingly diverse circumstances," one result is "more and more of everything" (48–49). And "more of everything" means, in the words of Charles Paul Freund, that in a consumerist society, people can "grab the first opportunity to escape the traditionalist boundaries of selfhood" by experimenting "with different modes of self-presentation" (29).38 To put this in terms of second-level inquiry, people can now engage in endlessly diverse experimentation with our double view of the body as a privileged yet unreliable source of information about the mind.
Moreover (to continue with this example, but coming to the third level) new cultural forms and niches generate new attempts to carve zones of certainty in our perception of the body. Think of the Internet today—a powerful cultural institution that thrives because it offers us yet another way of figuring out what is happening in other people's minds. Yet if you ask yourself which feature of the Internet has been causing us the most anxiety since its inception, the answer will be the absence of the body behind the message. The "disembodied" nature of electronic communication leaves its users strikingly vulnerable to deception. As Peter Steiner's cartoon in _The New Yorker_ has it, "On the Internet, nobody knows you're a dog." The physical body thus gets implicitly re-valorized as the source of "true" information about our correspondent. If only we could see the actual man or woman typing in his or her message—if only for a second!—oh, we would surely know what he or she is really about!
As a matter of fact we would not. To be reminded just how informative the sighting of the actual body is, we have Web sites such as dontdatehimgirl.com, which features letters from women who have been deceived by flesh-and-blood men and now want to warn other women who may unknowingly come in contact with those cheaters. Ironically, the disembodied medium thus re-alerts readers to the unlimited lying potential of the body. Some aspects of this situation are captured by yet another _New Yorker_ cartoon, by Liza Donnelly, in which a street vendor with a laptop offers to passersby a service called "Vet your date." (Although, of course, there is no way of knowing if anonymous reports posted on dontdatehimgirl.com are actually written by women, just as there is no way to gage the trustworthiness of those posts.39) The double view of the body thus continues to shape our representations. Whereas we cannot predict the exact forms that the construction of embodied transparency will yet take on the Internet, we know that the arms race between rendering some aspects of observable behavior transparent and subverting that transparency will go on.
### NOTES
1 This essay also appears in the journal _Narrative_ 16 (2008): 56-92. Reproduced by permission from The Ohio State University Press.
2 For discussions of applications of research on ToM to literature, see Palmer, _Fictional Minds_ ; Vermeule, "God Novels"; Zunshine, _Why We Read Fiction_.
3 As Goldman puts it, in his account of simulation theory of mind-reading, "people routinely track the mental states of others in their immediate environment" (301).
4 I am referring here to the scene from Bryan Singe's _The Usual Suspects_ (1995), in which, we assume, Roger "Verbal" Kint uses the name of the cup manufacturer to corroborate his yarn.
5 See Goldman for a discussion of the relationship between mirror neurons and ToM.
6 See also Ramachandran for a discussion of anosognosia (38) and autism (119) as possibly associated with damage in the system of mirror neurons.
7 Unless, of course, our visual system is severely damaged.
8 And what you yourself are thinking, too. For discussion, see Palmer.
9 Compare to Baron-Cohen's description of Daniel Dennett's view of the Intentional Stance: "Dennett is not committed either way on the question of whether there really are such things as mental states inside the heads of organisms. We ascribe these simply because doing so allows us to treat other organisms as rational agents" ( _Mindblindness_ 24).
10 I concede that in a sci-fi movie, his jump may mean that there was a magnet planted in his body and he was pulled up by aliens who use such magnets to reel in earth-lings to their ship. Note two things, though. To counterbalance our immediate tendency to read a mental state into his behavior, I had to come up with a truly outlandish scenario. ToM-less explanations apparently require quite a bit of work. And, furthermore, my explanation is not really completely ToM-less. I did not manage to get rid of intentionality altogether. Only instead of ascribing an intention to the man, I ascribed it to the aliens. See if you can do any better.
11 Spolsky, _Satisfying Skepticism_ (7); "Darwin and Derrida" (52).
12 For a related discussion, see Spolsky's "Purposes Mistook."
13 Note that in this context I use the word _emotion_ interchangeably with _mental state_. For a discussion of the relationship between emotion and cognition, see Hogan, Keen, and Palmer.
14 For a discussion, see Zunshine, "Lying Bodies."
15 Compare to physical torture as another attempt to render the body the direct index of the mind. As Michaels puts it in his discussion of _American Psycho_ , "You can be confident that the girl screaming when you shoot her with a nail gun is not performing (in the sense of faking) her pain" (70). See also Scarry, _The Body in Pain_ (35–38).
16 Using James Phelan's categorization of unreliable narrators, we can say that Tyler underreports and underreads (219) Raymond's mental states.
17 Compare to Alan Richardson's analysis of Johanna Baillie's tragedy _Count Basil_ (1798), in which Victoria, the main heroine, "ends up resorting to all but nonstop simulation of emotion, allowing her genuine feelings to become manifest only obliquely, largely against her conscious wishes, and in the briefest of flashes" ("Facial Expression").
18 Compare to Cohn's discussion of E.T.A. Hoffmann's _Master Flea_ in _Transparent_ _Minds_. She writes that in Hoffmann's story, "the microscopic magician of the title gives to his human friend Peregrinus Tuss a tiny magic lens, that, when inserted in the pupil of his eye, enables him to peer through the skulls of all fellow human beings he encounters, and to discern their hidden thoughts. Peregrinus soon curses this 'indestructible glass' for giving him an intelligence that rightfully belongs only to the 'eternal being who sees through to man's innermost self because he rules it'" (3). Peregrinus thus forces "all fellow human beings" into a state of embodied transparency _which can last infinitely_ —an ethically indefensible situation that is resolved to the extent to which Peregrinus is rendered unhappy by his privileged access.
19 For a related argument, see Spolsky, "Elaborated Knowledge."
20 See Mullan (61).
21 See Zunshine, "Richardson's _Clarissa_."
22 _OED_ ; second online edition, 1989.
23 I am using the titles of paintings from Kern's lavishly illustrated study.
24 For discussion, see Kern (71).
25 For discussion, see Kern (93).
26 For discussion, see Kern (65). Note that I am purposely choosing proposal compositions of very different styles.
27 For discussion, see Zunshine, "Theory of Mind and Michael Fried's _Absorption and_ _Theatricality_."
28 For a useful broader perspective on ToM and cinema, see Per Persson's brilliant _Understanding Cinema_.
29 Here I use the terms cinéma vérité and direct cinema interchangeably, but there are important differences between the two. For an overview of these differences, see Ellis and McLane (216–18).
30 Quoted verbatim from Hope Ryden's official Web site, <http://www.hoperyden.com/disc.htm>, as accessed on August 18, 2007.
31 And the other side of this issue is that such revelatory moments were carefully staged and edited. As Frederick Wiseman, a pioneer of direct cinema, puts it, "It's all manipulation. Everything about that kind of movies is a distortion" ( _Cinema Vérité: Defining the Moment_ ).
32 Compare to Michael Fried's argument in _Absorption and Theatricality_ (61), and my analysis of it in "Theory of Mind and Michael Fried's _Absorption and Theatricality_."
33 _The Broadcast Tapes of Dr. Peter_ (1994) and _Silver Lake Life: The View from Here_ (1990) were made by filmmakers diagnosed with AIDS, who chronicled their fight with the disease until they died. For a discussion of these movies, see Ellis and McLane (284–87).
34 Albert and David Maysles and Charlotte Zwerin ( _Gimme Shelter_ ) could not know that a person would be killed on camera during a Rolling Stones performance. For a discussion of the history of that moment, see Ellis and McLane (290–91).
35 Ellis and McLane (236).
36 For a discussion, see Ellis and McLane (237).
37 For a discussion, see Ellis and McLane (235–36).
38 Compare to Dick Hebdidge's earlier argument about "the articulation of commodity consumption, personal identity and desire which characterizes life under hypercapitalism" ( _Hiding_ 168) and his yet earlier analysis of subcultures' movement through "commercial and cultural matrices" ( _Subculture_ 130).
39 For a discussion, see Alvarez and Jones.
### ACKNOWLEDGMENTS
I am deeply grateful to James Phelan for his thoughtful response to an earlier draft of this essay, to Stephen Kern for his detailed corrections and suggestions, and to Anna Laura Bennett for her superb editing skills.
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## Theory of the Murderous Mind: Understanding the Emotional Intensity of John Doyle's Interpretation of Sondheim's _Sweeney Todd_
DIANA CALDERAZZO
Through the experience of engaging in the theatrical event, audiences take part in a process of storytelling on a visceral level—one that appeals to sensory reception and often elicits evidence of emotional response in audience members who relate on some level to the contents of the characters' minds. This path from sensory reception to emotional response with regard to theatrical storytelling, from a cognitive perspective, is the topic of this article; I will explore its function in defining stated audience emotional reactions to a specific theatrical production—Stephen Sondheim's _Sweeney Todd_ , debuting on Broadway in 2005 with direction by John Doyle. In the fall 2007 issue of _The Sondheim Review_ , I explored emotional intensity in Doyle's production based in cognitive mirroring, conceptual blending, and the link between gesture and speech; here, I retain the cognitive approach but build a more in-depth analysis of mirroring and empathy. In addition, I offer an exploration of the aspects of storytelling and concepts of obsession and compulsion as these elements serve to link the minds of characters and audiences within the production. Finally, I will address the function of laughter as audience reaction to the macabre characters in Doyle's _Sweeney_.
Doyle's production of Sondheim's musical lends itself to such a study for several reasons. First, Sondheim composed the piece, which he wrote in 1979, with the intent of emphasizing its visceral appeal to its audiences. In 1980, the composer stated that "the theatre is the one place where you can create larger than life . . . " and melodrama, the stuff of _Sweeney Todd_ , is "high theatre" (Sondheim 4). Significantly, the story itself is based upon an ancient European legend that has served to inspire images of the macabre for nearly two hundred years: that of a mad barber who murders his clients with his razor only to have their bodies baked into meat pies and fed to a ravenous nineteenth-century London public. Sondheim has expressed his fascination with this legend, further asserting that "the true terror of melodrama comes from its revelations about the frightening power of what is inside human beings. . . . [That is] exactly what this show is" (6). _Sweeney Todd_ , then, is a prime manifestation of Sondheim's own dedication to this traditionally heightened form.
The fact that Sondheim proposed to appeal to his audiences on a basic human level through his vision of melodrama in _Sweeney Todd_ is appropriate, considering the nearly two-hundred-year history of the Todd legend and the popular allure of its interpretation on the melodramatic stage and in literature, visual art, and cinema. In his 1980 book on melodrama, Daniel Gerould carefully relates much of this history, enumerating and briefly describing seventeen artistic and literary interpretations of the London-based Todd legend (believed to have previously originated in France) between the years 1825 and 1970 (43-47).1 Many of these interpretations no doubt elicited further public intrigue and the suggestion of historical veracity through their eerie resemblance to the Jack the Ripper murders in Whitechapel, during which prostitutes in the poor areas of 1880s and 1890s London became the victims of an infamous string of throat slittings. The legend made its way to Broadway for the first time in 1924 (a version written by George Dibdin-Pitt), followed in 1979 by the debut of Sondheim's version, which received two revivals, in 1989 and 2005. Simultaneously, literary and popular culture has displayed sustained interest in the Todd legend, spurred by folklore historian Peter Haining's controversial 1997 book, _Sweeney Todd: The Real Story of the Demon Barber of Fleet Street_ , in which Haining claims to have obtained access to documents proving the existence of a very Todd-like murderous barber in London around the turn of the eighteenth century. This book and others like it,2 as well as the Broadway revivals of Sondheim's musical, sustained popular interest in the Todd legend and led to the much-anticipated and hailed 2007 American movie version of _Sweeney_ , based on Sondheim's musical and starring Johnny Depp.3
It was a 1970 version of the legend, written by British playwright and actor Christopher Bond, however, that most inspired Sondheim. Expressing his attraction to Bond's version over others he had experienced, Sondheim asserted that "Christopher Bond humanized all the characters and gave the story motivation which had never existed before in the earlier versions of _Sweeney Todd_. Yet while enlarging the human dimensions of the play, Bond remained true to the melodramatic tradition" (5). It was Bond's version, then, that offered the combination of the traditional, larger-than-life feel of the Grand Guignol and an emphasis on humanity ("the frightening power of what is inside human beings") that Sondheim admired and strived to feature in his 1979 musical.
Twenty-six years later, Doyle's production, it seems, fulfilled Sondheim's goal more effectively than previous productions, as Sondheim himself noted. "Of all the productions I've seen," the composer told Charles Isherwood of _The New York Times_ , "this is the one that comes closest to Grand Guignol, closest to what I originally wanted to do. I characterize all the major productions I've seen in terms of a single adjective. Hal's [Prince, 1979 on Broadway] was epic. Declan Donnellan's [1993 in London] production was exactly the reverse, it was very intimate. John's [Doyle], for me, is the most intense" (qtd. in Isherwood 1). The emotional intensity Sondheim observed and commended was also noted by critics. For example, Celia Wren, writing for _Commonweal_ , observed that in general "director Doyle ratchets up the emotional voltage" (18-19); while Elysa Gardner of _USA Today_ stated that the show "rel[ies] almost entirely on emotional expression and interpersonal contact . . ." (3E). Ben Brantley of the _New York Times_ described the characters as follows: "Empty-eyed and defeated-looking in their German Expressionist makeup and thrift-shop clothes, they nonetheless vibrate with the energetic anger of folks done wrong" (1). Robert Brustein ( _New Republic_ ) noted a similar parallel with the emotionally heightened genre of German Expressionism, stating that "Michael Cerveris's bald-pated, pasty-faced, hollow-eyed Sweeney [was] skulking, slinking, stalking, like the somnambulist in _The Cabinet of Doctor Caligari_ " (25). Other visceral adjectives found in the reviews described either the characters or the production as a whole as "smarmy," "mournful," "sinister," and "sulfurous," suggesting through their onomatopoeic nature the show's visceral appeal—the element that lent the production its "capacity to freeze your vertebrae" (25), in the words of Brustein. Given these observations, it is no wonder Sondheim applauded the "intensity" of Doyle's production.
In commending such intensity within a story that has for centuries fascinated its audiences through its eerie appeal, Sondheim emphasizes the importance of communicating to audience members on an emotional level. Referencing _Sweeney Todd_ , he states that "most people can understand and identify with any emotion; the writer simply must draw the audience into the feelings of the characters that he has created on the stage" (10). His interest in his spectators' emotional engagement, in fact, lies behind Sondheim's decision to subtitle the piece a "musical thriller," rather than a musical melodrama. Aware that the use of word "melodrama" would conjure audience expectations of a more traditional style, Sondheim stated, "If we had called _Sweeney Todd_ a musical melodrama, it would have unfailingly suggested making fun of the genre, with villains galloping across the stage, the heroine tied to the railroad tracks, and the audience cheering and clapping" (7). Though Sondheim expresses personal fascination with melodrama (the form arguably considered to feature the most exaggerated interpretation of humanity found on the stage), he argues for the maintenance of a presentation style that does not become so over-the-top that it loses its capacity to engage its audiences on more complex emotional levels. The retention of the "human dimensions" that Sondheim applauded in Christopher Bond's adaptation of the Todd legend is indeed critical, even within a piece that is "larger than life as a story and larger than life in technique" (Sondheim 5). His concept of the "musical thriller," then, serves to illustrate Sondheim's brand of melodrama, featuring many of the exaggerated elements of Grand Guignol while preserving the piece's ability to "draw in" audiences through engagement in genuine human emotions. To begin to understand how the suggestion of "the frightening power of what is inside human beings" onstage may elicit emotionally intense reactions in audience members of a musical as engaging as _Sweeney Todd_ , we turn to cognitive conceptualizations of audience engagement in thematic threads and presentational elements as well as music and lyrics. My ongoing research focuses on cognitive science's growing contributions to the study of music and lyrics as these contributions relate to the work of Sondheim, while the present analysis addresses cognitive audience reception of thematic and presentational elements within the context of Theory of Mind. I begin by referencing the concept of mirroring, a cognitive procedure often said to be linked to the development of Theory of Mind.4 In a recent short article in _Psychiatric Times_ , John Medina described in basic terms the tendency of the human brain to mimic, or mirror, functions occurring in other human brains to which it is exposed. This includes the mimicking of and response to affect, a process known as empathy, defined by Medina as "the capacity to comprehend and react to the distinct affective experiences of another person" (14). Based upon a number of studies conducted by neuroscientists, this mirroring concept and the resulting emotional contagion describes a process of receiving sensory images, such as those presented by actors on the stage; mimicking the brain functions suggested by those images, and manifesting that mimicry in empathetic comprehension and response—perhaps in the form of cathartic laughter or tears in theatrical audiences.5 Thus, mirroring appears to describe an essential aspect of the theatrical experience, and its resulting empathy may account for much of the emotional response we detect in audiences—an emotional response, which, as noted above, was apparently particularly "intense" among audiences of _Sweeney Todd_ in 2005.
In applying a cognitive approach, then, it will be necessary to first understand the process of the formation of emotions as described by cognitive neuroscientist Antonio Damasio. Damasio states that emotions develop within six basic categories (and their subsets)—happiness, sadness, fear, anger, surprise, and disgust—and are stimulated either internally by a modification in chemical patterns within the body, or externally by sensory perception that is visual, aural, topical, etc (35-81). For the purposes of analyzing emotional reaction to a theatrical event such as _Sweeney Todd_ , I will assume that the stimulants discussed here are external, received by the senses (predominantly the eyes and ears) and emanating from the staged performance. Damasio finds five characteristics common to all emotions, quoted as follows:
> (1) Emotions are complicated collections of chemical and neural responses. . . . Their role is to assist the organism in maintaining life.
>
> (2) Notwithstanding the reality that learning and culture alter the expression of emotions and give emotions new meanings, emotions are biologically determined processes, depending on innately set brain devices, laid down by a long evolutionary history.
>
> (3) The devices which produce emotions occupy a fairly restricted ensemble of subcortical regions. . . .
>
> (4) All the devices can be engaged automatically, without conscious deliberation. . . .
>
> (5) All emotions use the body as their theater . . . (51).
I will not expand upon numbers 1 and 3, which address the chemistry and regions of the brain from a solely scientific approach; however, number 2 is relevant to this analysis in its statement that emotions are "biologically determined." In other words, the human ability to emote is fairly universal, although the causes and behavioral manifestations of emotions vary among individuals. This may shed light on the cause of the multi-century popular interest in a tradition such as melodrama, and more specifically on the legendary nature of a narrative such as that of Sweeney Todd. Since emotions tend to formulate under the six simple headings listed above and appear to be natural human responses to stimuli presented, it is not surprising that the Sweeney legend has, as described above, resonated with audiences in similar manners throughout the past two hundred years and within multiple countries and cultures. Thus, on a simple cognitive level, the nature of emotion supports Sondheim's goal of appealing to the emotions as demonstrations of "the frightening power of what is inside human beings." Indeed, biologically, emotions are "inside human beings" and they form in response to the human experience.
Further, Damasio's fourth stipulation regarding emotion is relevant to this analysis because it posits that much of the brain activity that contributes to the formation of emotions may occur on a subconscious level. Emotions appear to have the ability to affect us cognitively long before we become aware of them and long before we form conscious reactions to them. According to Damasio, we become aware of emotions only after we begin to "feel" them: "Feeling an emotion is a simple matter. It consists of having mental images arising from the neural patterns which represent the changes in body and brain that make up an emotion. But knowing that we have that feeling, 'feeling' that feeling, occurs only _after_ we build the second-order representations necessary for core consciousness" (280). Subsequently, Damasio states that such awareness of our emotions depends not only upon the development of such representations leading to core consciousness; it also relies upon autobiographical memory, allowing us to experience emotions within the context of our own past experiences.6 This indicates that we often cannot help experiencing the emotions we experience, even though we may often judge ourselves afterward.7 Perhaps this relative unavoidability of formulating emotions in specific situations explains what is "frightening," in Sondheim's terms, regarding the power within human beings. It certainly suggests frightening possibilities in terms of the results of emotions often leading to obsessive thoughts and compulsive behavior, an observation that will become relevant as I further analyze _Sweeney Todd_.
Finally, Damasio's fifth characteristic of human emotion, addressing the body as emotional theater, provides the foundation for examples of bodily reflection of emotion, such as audience laughter, to be discussed later.
A second concept functioning on a basic cognitive level to heighten emotional intensity in terms of both Sondheim's piece and Doyle's interpretation is the concept of compulsion. Compulsion as it is represented on stage forms a basic-level concept addressed in some detail by Bruce McConachie, whose recent book, _Engaging Audiences_ , examines audience experience from a cognitive perspective. In applying specific cognitive concepts to the audience experience, McConachie quotes scientist Mark Johnson's concept of compulsion as follows:
> In his comments on the embodied nature and gestalt cognitive structure of compulsion, Mark Johnson notes, "Everyone knows the experience of being moved by external forces, such as wind, water, physical forces, and other people. When a crowd starts pushing, you are moved along a path you may not have chosen, by a force you seem unable to resist." (McConachie 201)
Compulsion, then, describes the type of repeated behavior that typically drives tragic heroes to their downfall. From Oedipus to Willy Loman, tragic heroes have succumbed to their own compulsive behavior resulting from pressure exerted by external forces; and Sweeney appears to follow this pattern exactly. When Sondheim's _Sweeney Todd_ begins, the audience sees a main character who is acted upon by external forces—his unjust imprisonment at the hands of Judge Turpin, the Judge's lecherous behavior toward Todd's wife, and the evident social imbalance created by Victorian London society's treatment of those less fortunate than others. When Todd narrowly misses an opportunity to kill the judge, irrationality takes hold of him, compelling him to attain revenge not only on Judge Turpin in particular, but on society as a whole, making it his goal to murder every person who possesses the misfortune to enter his shop. His compulsion, if it has not been irresistible until this point, certainly becomes so now. Exhibiting the blindness of Oedipus, and driven by a parallel determination to "set things right," he ultimately kills the Beggar Woman, who was his wife, as he discovers only afterward. Todd, like Oedipus and other tragic heroes throughout history, displays the type of compulsive behavior that drives him to his downfall.
Sondheim's vision for the show confirms this conceptual approach to understanding _Sweeney Todd_ :
> For me what the show is really about is obsession. I was using the story as a metaphor for any kind of obsession. Todd is a tragic hero in the classic sense that Oedipus is. He dies in the end because of a certain kind of fatal knowledge: he realizes what he has been doing. I find it terribly satisfying—much more so than any kind of accidental death, which often occurs in flimsy forms of melodrama. (qtd. in Zadan 245)
Sondheim, then, bases his melodrama on the concept of obsession, one that has appealed to human audiences since the time of the ancient Greeks. Obsession, the term Sondheim applies to the driving force behind _Sweeney Todd_ , describes a mental "preoccupation or feeling" ( _Merriam-Webster's Dictionary_ ), often leading to compulsion, or the need to act upon obsessive thoughts. Thus, Todd's compulsive behavior and accompanying obsessive thoughts form the concept that drives his own need for revenge as well as Sondheim's excitement in regarding the piece as good melodrama.
In Doyle's production, in which the characters played the orchestral instruments and thus did not leave the stage, audience members were never really allowed relief from Todd's presence and the compulsion associated with his mental world. As a result, reviewer Celia Wren credited the presence of "the ten dynamic performers on stage the entire time" (18-19) with heightening the emotional nature of the show; and Charlotte Headrick noted that audiences may likely find themselves identifying with Sweeney beyond the level of comfort: "Through all the blood and violence, we are on an electric and grueling journey with Sweeney, but we also know that going to the dark side may not bring us release" (478). In witnessing Sweeney's compulsion, it seems, audiences may have on some level entered the world of Todd's mind, feeling themselves compelled to join the deranged barber on his journey despite knowledge of imminent downfall.
In establishing this driving concept of obsession/compulsion in _Sweeney Todd_ , Sondheim relies on the assumption that spectators will identify with this concept, and he argues that they will find themselves emotionally engaged with Sweeney's desire for revenge. Indeed, we have seen that at least one reviewer found herself on a "journey" with Sweeney, even though she was consciously aware of the dark nature of such a journey. This process relies once again on the human cognitive tendency of the mind/brain to "mirror" and therefore emotionally empathize with the actors on stage. Indeed, when audiences can be potentially "drawn in" to the minds of the characters through emotional contagion on a subconscious level, as Damasio describes, a piece as visceral as _Sweeney Todd_ may be said to possess great affective potential; and both Sondheim and Doyle appear to have made choices designed to heighten this potential. In other words, a key goal of both artists seems to have been to "draw in" audience members on a subconscious level—to "catch" them unaware in the midst of this macabre London world where disconcerting human tendencies such as impulsiveness, vengeance, and, as we have seen, obsession/compulsion reign supreme.
The first line of the opening song, in fact, "Attend the tale of Sweeney Todd," serves to "draw in" the audience in a way that presupposes the allure of childhood storytelling. In addition, throughout the piece, as Sondheim scholar Craig Zadan notes, the composer stimulates audience cognitive interaction with the stage scenario by presenting the story of the Beggar Woman as a puzzle: "The Beggar Woman is in disguise," Sondheim explains. "A few, very alert people caught on right away, though, . . . because when the young wife appears and is raped, the minuet they're playing is the Beggar Woman's theme in a different guise" (qtd. in Zadan 252). Sondheim ingeniously communicates through the language of musical motif that the Beggar Woman and Sweeney's wife are the same character, drawing audiences in on a level meant to appeal to their sense of the allure of mystery. Sondheim offers other, more obvious clues to the Beggar Woman's identity as well, such as her indication early in the show that Todd appears familiar to her. In these ways, he calls upon his audience to participate, and although such participation obviously requires cognition on more than a simple subconscious level, it inevitably returns to the most basic level of emotional engagement culturally associated with storytelling, and the empathetic response likely to result when audiences are "drawn in" to the theatrical event.
In 2005, Doyle sought to further heighten this visceral style of communication between performer and spectator, utilizing a minimal set and focusing on the storytelling aspect of the piece. "It's more poetic than prescriptive. It's playful—almost told in a way that children would tell the story, using what was available to them" (qtd. in Isherwood 1), said Doyle. Sondheim agreed: "When an audience's imagination is engaged they enjoy it even more. . . . The theatre is a poetical medium . . . that's the fun of the theatre" (qtd. in Isherwood 1). Finally, producer Richard Frankel proudly noted what he viewed as the success of this poetic, storytelling approach based upon a nearly tangible tension within the audience: "I think it's great that at previews people are leaning forward in their seats hanging on every word. I stand in the back of the theatre and am delighted by the silence" (qtd. in Isherwood 1). Thus, an observed sense of collective audience suspense during the 2005 production as spectators watched and listened to the piece points to the existence of a heightened intimacy between performers and spectators—evidence of the performance's tendency to engage its audiences cognitively in ways that do not appear to require conscious awareness of engagement. Doyle thus creates the ideal setting for communicating affect in a way that illustrates the formation of emotion, as Damasio states, "without conscious deliberation."
As stated above, the fifth main characteristic of human emotion, according to Damasio, is the utilization of the body as theater for emotions. Damasio notes that the purpose of emotions is to maintain a healthy existence in the organism (either animal or human), and the result is an appropriate manifestation of the emotion within the body: "In an animal, for instance, the reaction may be to run or become immobile or to beat the hell out of the enemy or to engage in pleasurable behavior. In humans, the reactions are essentially the same . . ." (53-54). This type of reaction in response to spectators' theatrical experiences has of course been observed through the ages as what Aristotle termed catharsis, based in the emotions of pity and fear, while cognitively oriented studies such as McConachie's offer a more scientifically-based understanding of the truly beneficial nature of audience cathartic response. Although I cannot detail here the elaborate bodily process leading to physical manifestation of empathy and emotion in spectators, I would like to address the basic emotional precursors to the physical response of laughter, since Sondheim discusses laughter specifically with regard to _Sweeney Todd_. Sondheim has stated that "as Alfred Hitchcock made millions of dollars proving, there is a very thin line between melodrama and comedy, between being scared and laughing. An audience is more vulnerable to laughter—as we are in real life — not just in the theater, when it's most tense. If somebody is tense, you can tickle him; if he's relaxed, you can't" (12). Robert Latta confirms these statements from a cognitive perspective, explaining that in order for laughter to occur, a subject must exist in an "un-relaxed state," only to experience a "cognitive shift" from the un-relaxed state to a relaxed state, resulting, as Sondheim states, in the release of tension. This release of tension consequently leads to laughter (Latta 37-44). This formation and release of tension is similar to what Sondheim notes in spectators who laugh at the Beggar Woman. As Sondheim explains: "there is something creepy about the Beggar Woman, and by laughing at somebody or something, we attempt to ward it off. . . . This is the kind of laughter you can only create in melodrama where an audience will laugh to protect itself from being scared" (12). In this example, spectators' fear of the Beggar Woman and the resulting formation of tension result in the necessary release of such tension through laughter. Tension, fear, melodrama, and laughter—all resonate closely in Sondheim's _Sweeney Todd_ and all appear closely related as cognitive stimulants and reactions played out in the theater of the body.
In a parallel vein, Sondheim also posits that his characters in _Sweeney Todd_ possess a simplicity in their melodramatic nature that paradoxically allows them to communicate a passionate richness prompting them to become comic:
> In _Sweeney_ , Mrs. Lovett's venality can be treated in a comic way because that is what she is: a venal character. . . . She is defined by her practicality combined with her greed. . . . Todd is a man bent on revenge, he thinks of nothing else; that is his dimension. . . . When you have characters like that, you can get laughs quite easily. (12)
Sondheim refers to audience laughter that can result from these characters' exaggerated obsessions, such as the practicality Mrs. Lovett displays when she hits upon the master plan of processing Todd's victims into meat pies. Her actions are larger-than-life, and they evoke tension among spectators—tension that finds its release in laughter. Interestingly, spectators' ability to laugh specifically at this character finds confirmation in the writing of Robert R. Provine, whose book examines laughter from a cognitive perspective. In his description of laughter within the context of theatrical and operatic repertoire, Provine describes the following scenario:
> As the house lights dim and the curtains part, settle into your pricey front-center seat and get ready for a wild ride. Much operatic laughter occurs in a decidedly grim setting. . . . Laughter merges with cannibalism in Stephen Sondheim's _Sweeney Todd_ , where the invective Mrs. Lovett recycles the byproduct of Sweeney's homicidal tendencies in her meat pies ("A Little Priest"). (65-66)
Though Provine's example comprises a single sentence within his book, his selection of _Sweeney Todd_ illustrates the ease with which Sondheim's work is understood on a cognitive level; and it speaks to the grotesquely comic nature that Sondheim rightly believes to be inherent in these characters.
It is likely the tension and fear created by all that is creepy and abnormal, then, that explains John Lahr's ghoulish delight in observing:
> Patti Lupone is Sweeney's pragmatic and pathological helpmate, Mrs. Lovett. A gargoyle of toughness, with an angular haircut, red lips and nails, a short skirt, and fishnet knee socks, LuPone looks positively Weimar. . . . [She] is never better than when Mrs. Lovett hits on the idea of disposing of one of Sweeney's victims by getting others to swallow the evidence. "Think of it as thrift / As a gift / If you get my drift," she sings. (96)
This is merely one example out of many of the moments in Doyle's _Sweeney_ that combined the look of the characters with Sondheim's lyrics to create spectator engagement resulting in laughter as a form of tension release.
Doyle's _Sweeney Todd_ , then, as Sondheim has stated, successfully drew out the emotional intensity already implicated within the writing of Sondheim, as evidenced through the beginnings of a cognitive analysis of audience reception of the production. As audiences became drawn emotionally into Todd's mental world despite their understanding of its inherent creepiness, processes of mirroring and empathetic response allowed a nearly firsthand experience of basic concepts, such as compulsion, on a heightened level. Laughter, it seems, offered a form of tension release; but those who wrote about the _Sweeney_ experience did not soon forget its emotional intensity, and Sondheim and Doyle appear to have justly reawakened the legend that has gripped audiences for centuries.
### NOTES
1 Gerould's study specifically focuses on the Todd legend, and therefore leaves out earlier relatable British narratives such as that of the pie-eating cannibalism in Shakespeare's _Titus Andronicus_.
2 Related books include: (1) _The London Monster: A Sanguinary Tale_ , by Jan Bondeson (2000), detailing the history of Jack the Ripper's late eighteenth-century murderous precursor; (2) _The Wonderful and Surprising History of Sweeney Todd: The Life and Times of an Urban Legend_ , by Robert L. Mack (2007), discussing the Todd legend itself. Mack's book and a 2007 paperback edition of Haining's book, mentioned above, were both released in time to accompany the debut of Tim Burton's 2007 film version of _Sweeney_.
3 Three British films, debuting in 1936, 1928, and 1926 (the latter two are silent) and based on the Dibdin-Pitt version, preceded the American film.
4 See Roch-Levecq (2006), whose study of a group of congenitally blind children revealed a possible link between these children's underdeveloped empathetic mirroring abilities and their comparatively low scores on Theory of Mind tasks. Also, in her 2005 article, "Finding the Self in Mind: Vermeer and the Reflective Function" (in _Psychodynamic Practice_ 11.3, 255-68), Rosemary Rizq analyzes apparent connections between mothers' visible mirroring of their infants' emotional facial expressions and the infants' later development of Theory of Mind.
5 In _Engaging Audiences_ , Bruce McConachie draws this connection between mirroring and experiencing the theatrical event in more detail.
6 See Damasio pages 222-26 for descriptions of the role memory plays in an integrated model closely orienting previous descriptions of emotion with elements of consciousness and memory.
7 Patients and clients have been shown to exert some control over their emotional reactions through Aaron T. Beck's cognitive-behavioral therapy process, during which they learn to modify their belief systems in order to consciously create an environment in which more positive emotions and fewer negative emotions will form. However, this process is not spontaneous—it does not allow for the regulation or exchange of emotions as they are forming; instead it seeks through practice to establish mental outlooks conducive to the formation of positive emotions and less conducive to the formation of negative ones.
### WORKS CITED
Bondeson, Jan. _The London Monster: A Sanguinary Tale_. Philadelphia: University of Pennsylvania Press, 2000. Print.
Brantley, Ben. "Grand Guignol, Spare and Stark." _New York Times_ 4 Nov. 2005: Sec. E, P. T1, Col. 3: 1. Print.
Brustein, Robert. "Making the Familiar Strange." _New Republic_ 19 Dec. 2005: 25-27. Print.
Damasio, Antonio. _The Feeling of What Happens: Body and Emotion in the Making of Consciousness_. New York: Harcourt Brace and Co., 1999. Print.
Gardner, Elysa. "Great Musical Returns on a Much Smaller Scale." _USA Today_ 4 Nov. 2005: 3E. Print.
Gerould, Daniel, ed. "A Toddography: Sweeney Todd in Literature and the Performing and Visual Arts." _Melodrama_. New York: New York Literary Forum, 1980. Print. Haining, Peter. _Sweeney Todd: The Real Story of the Demon Barber of Fleet Street_. London: Anova, 2007. Print.
Headrick, Charlotte. "Performance Review of _Sweeney Todd_ ," ed. Sonja Arsham Kuftinec. _Theatre Journal_ 58 (2006): 477-78. Print.
Isherwood, Charles. "Cutting 'Sweeney Todd' to the Bone." _New York Times_ 30 Oct. 2005: Sec. 2, P. 1, Col. 3. Print.
Lahr, John. "Music Men." _New Yorker_ 81:36, 96-97. Print.
Latta, Robert L. _The Basic Humor Process: A Cognitive Shift Theory and a Case against Incongruity._ Berlin: Mouton de Gruyter, 1999. Print.
Mack, Robert L. _The Wonderful and Surprising History of Sweeney Todd: The Life and Times of an Urban Legend_. New York: Continuum, 2007. Print.
McConachie, Bruce. _Engaging Audiences: A Cognitive Approach to Spectating in the Theatre_. New York: Palgrave MacMillan, 2008. Print.
Medina, John. "Mirroring in the Human Brain." _Psychiatric Times_ 24:10 (2007): 14-17. Print.
Provine, Robert R. _Laughter: A Scientific Investigation_. New York: Viking, 2000. Print.
Sondheim, Stephen. "Larger than Life: Reflections on Melodrama and _Sweeney Todd_." _Melodrama_. Ed. Daniel Gerould. New York: New York Literary Forum, 1980. Print.
Wren, Celia. "'Rabbit Hole' and 'Sweeney Todd.'" _Commonweal_ 10 Mar. 2006: 18-19. Print.
Zadan, Craig. _Sondheim and Co._ Second Edition. New York: Harper and Row, 1989. Print.
## Distraction as Liveliness of Mind: A Cognitive Approach to Characterization in Jane Austen
NATALIE PHILLIPS
> "How could you begin?" said she. "I can comprehend your going on charmingly, when you had once made a beginning; but what could set you off in the first place?" "I cannot fix on the hour, or the spot, or the look, or the words, which laid the foundation. It is too long ago. I was in the middle before I knew that I _had_ begun."
>
> —Jane Austen, _Pride and Prejudice_
What should we make of Elizabeth Bennet's liveliness? At the end of _Pride and Prejudice_ , she asks Darcy "to account for his having ever fallen in love with her": "'Now be sincere,'" Elizabeth asks, "'did you admire me for my impertinence?'" Darcy replies: "'For the liveliness of your mind, I did'" (378). Frequently, Jane Austen uses three main words to describe a character; for Elizabeth, these are "lively," "playful," and "quick." Liveliness of mind and quickness of thought are, of course, Elizabeth's defining traits in the novel. However, as I will argue, the nuances of her "lively, playful disposition" only fully emerge when Elizabeth's habits of mind are compared with those of the other, minor characters by whom she's surrounded—specifically those within her family (51). It is, then, no accident that when Mr. Bennet introduces Elizabeth into the novel he sets her mind against her siblings': "'They have none of them much to recommend them,' replied he; 'they are all silly and ignorant like other girls; but Lizzy has something more of quickness than her sisters'" (45).
In this early scene, Austen cues the reader to pay close attention to Elizabeth's quickness of mind by having Mr. Bennet describe it explicitly. She also encourages the reader to understand the character in comparative terms: Elizabeth is noteworthy because she possesses a swiftness that Lydia, Mary, Kitty, and Jane lack. Throughout, the novel itself rewards such attention, for Austen strategically modulates her narrative so that implicit comparisons among characters can begin to convey—rather than merely describe—that quick liveliness that makes Elizabeth so vibrant. As Austen works to capture Elizabeth's cognitive agility and intellectual vivacity within what Blakey Vermeule calls the "fragile casing of narrative," however, she also develops a new structure for literary characterization using representations of attention—a paradigm in which distraction can build the fiction of a heroine's liveliness of mind (7). Darcy may say as much about Austen's narrative, then, as he does about love when he says: "'I cannot fix on the hour, or the spot, or the look, or the words, which laid the foundation . . . I was in the middle before I knew that I _had_ begun'" (378). Darcy cannot pin down his impression of Elizabeth's impulsive quickness to a single moment because Austen has so finely integrated distraction within the novel's texture; Elizabeth's vivacity—and the techniques that produce it—are embedded almost intangibly within the story and structure.
A number of critics have recognized the rich effects of Austen's structure of characterization without quite being able to identify their cause, to say _how_ they give Elizabeth such energy and life. Tony Tanner notes that Elizabeth has a "dimension of complexity, a questing awareness, [and] a mental range and depth" which makes her seem "trapped in a constricting web of . . . simple people" (126). Alex Woloch builds on this, suggesting that this depth is the effect of the "central tendency of the narrator" in _Pride and Prejudice_ "to closely link the description of an 'intricate' mind with a counterpoised description of a 'simple' one" (52). Understanding how attention structures the novel's characterization, however, gives such arguments new specificity and texture; for, as I will show, Austen carefully links intricate minds with distraction and simple minds with narrow focus—and it is the _comparative_ "reading" of these minds and attention spans on the reader's part that gives Elizabeth's questing awareness the illusion of range and depth.
_Pride and Prejudice_ often brings minor characters into focus so—and only so—we can compare the inferior quality and complexity of their concentration to Elizabeth's. Austen severely limits the complexity of minor characters' attention spans; against them, the intricacy of Elizabeth's mind emerges in relief, and she claims the psychological richness of a central character. This means that Austen's minor characters play anything but a minor role in this drama of attention and characterization. Mary Bennet, for example, is essential to the narrative development of Elizabeth's "quickness." Austen carefully deprives Mary's attention of complexity and sets it against Elizabeth's livelier distraction. As I argue, Austen's process for simulating her heroine's vivacity has two key components: first, she associates Mary's rigid attention with mental stupor and uses this to play up the "something more" of Elizabeth's "quickness"; next, she develops a well-crafted set-piece of distraction that seems able to model—and perhaps actually narrate—the heroine's vibrant mind in action, and places it at the novel's center. Using distraction to convey Elizabeth's creative vibrancy, the novel develops a structure for comparing characters' minds and attention spans—one where Mary's rigid focus defines by contrast the playful flexibility of Elizabeth's mind.
### DEEP INTERSUBJECTIVITY AND COMPARATIVE THEORY OF MIND FOR LITERATURE
Many of Austen's novels use such uneven comparisons to develop psychological intricacy for a main character—which gives her famous portrayals of deep intersubjectivity a complex twist. In his book, _I Know that You Know that I Know: Narrating Subjects from Moll Flanders to Marnie_ (2004), George Butte defines deep intersubjectivity as
> the web of partially interpenetrating consciousnesses that exists . . . when a self perceives the gestures, either of body or word, of another consciousness, and . . . perceive[s] in those gestures an awareness of his or her own . . . [O]ut of this conversation of symbolic behaviors emerges a web woven from elements of mutually exchanged consciousness. (28)
Lisa Zunshine's pioneering work uses Theory of Mind to discuss such moments, revealing that Austen's "layering of human consciousness" emerges, in part, from her ability to represent increasingly recursive levels of "multiply embedded states of mind" in prose ("Why Jane Austen Was Different" 278). The model of deep intersubjectivity has produced excellent readings of the layered scenes of observation in Austen's novels; however, when we bring it into the larger, messy world of a novel, we need new language. Phrases like "mutually-exchanged consciousness" begin to ring oddly false here, especially in narratological discussions of Austen's characterization. Why?
Butte and Zunshine can use deep intersubjectivity's language of openness and exchange so constructively, I think, because they tend to approach Austen's novels from a micro-level, often analyzing multiple short episodes or passages. When these same moments of multi-character observation are viewed from a macro-level, however, they seem to perform a radically different function. At the level of characterization, the idealized principles behind theories of deep intersubjectivity—its webs of interpenetrating minds, its mutuality of exchange, its reciprocal fashioning of multiple selves—give way to a much more imbalanced system of exchange. And, though many things are negotiated, narratologically speaking, through Austen's webs of fictional consciousness, such exchanges are rarely mutual.1 Austen creates networks of characters, I argue, not to model the mutuality and reciprocity of multiple subjectivities, but to produce competition for the scarce resources of narrative space and cognitive richness. Scenes of intersubjectivity (and mutual exchanges of observation) from her novels, ironically, support a distinctly anti-reciprocal, _intra_ subjective process of characterization.2
Analyzing the interplay between Mary and Elizabeth's attention to reading (and recognizing how vital it is to Austen's depiction of their mental states) thus underscores the need for what I call a _comparative Theory of Mind_ for discussions of literature, as compared to Theory of Mind proper. Theory of Mind (ToM), or "mind reading," is a model used in cognitive science to describe "our ability to explain people's behavior in terms of their thoughts, feelings, behaviors, and desires" (Zunshine 6). In this essay, I emphasize the importance of distinguishing between ToM and ToM for literature in one specific point: how we manage the presence of "extra people." When we attribute mental states to fictional characters, peripheral figures have an unusually strong influence on literary mind reading. When I walk into a supermarket, ToM engaged, and decide that the person in front of me looks angry and the person behind me bewildered, I may compare them implicitly, but the fact that they are in the same line will have very little to do with the precise mental states I attribute to them.3 In the novel, however, such a juxtaposition of characters can change everything.
As Zunshine points out, literature relies on "stimulat[ing] our Theory of Mind mechanism":
> On some level, then, works of fiction manage to "cheat" these mechanisms into "believing" that they are in the presence of . . . agents endowed with potential for a rich array of intentional stances.
>
> The very process of making sense of what we read seems to be grounded in our ability to invest the flimsy verbal constructions we generously call "characters" with a potential for a variety of thoughts, feelings, and desires and to look for the "cues" that would allow us to guess at their feeling and thus predict their action. ( _Why We Read Fiction_ 10)
When certain psychonarratological cues in _Pride and Prejudice_ encourage us to "mind read" Elizabeth as distracted, however, something more happens than "guessing at" Elizabeth's feeling. Such cues urge us to interpret the heroine's distraction as a basic sign of her cognitive complexity—the literary mark of a fictional mind endowed with a rich array of intentional stances, or one worth spending the time "reading."
However, our ability to invest characters like Elizabeth with an imaginary mind worthy of complex mentalizing may not only rely upon but arise from our awareness of other, "simpler" minds, specifically those of surrounding minor characters like Mary. This opens up a crucial difference between the ToM process for real and fictional minds. In the world, whether we believe people to be fascinating or dull, we still automatically assume that the people we see have minds, and that their mental states are worth "reading" for signs of potential threat or benefit. This is not so true of fiction, where the illusion of a character's mind (and a fictional mind with psychological richness) must be built, sentence by sentence. Moreover, unlike a person, a novel can be skimmed, or ignored, without much danger to us. Literature, in this sense, must not only develop narrative strategies that "'cheat' these mechanisms into 'believing' that they are in the presence of . . . [real] agents" (Zunshine, _Why We Read Fiction_ 10), they must also give enough realism to a fictional character so that we exercise our ToM extensively on his or her mind. Since narrative space is limited, underdeveloped minor characters that receive less description (and thus provide us with less ToM information to process) must vie with protagonists both for our attention and for our activation of literary ToM. This competition for cognitive richness may be essential to crafting the illusion of psychological depth for main characters in the first place—a narratological mechanism for fooling our ToM into thinking a character made up of typeface and sentences has a mind to read at all.
If so, how can we describe a character's potential for this rich array of intentional stances? The techniques that build this cognitive richness? In the pages that follow, I suggest that the analysis of attention not only offers us new models for understanding Austen's novels, but also reframes discussions of the fictional representation of human minds to better appreciate the intricate variety of thoughts (and habits of thinking) modeled in narrative fiction. Traditional models of characterization, often based on E. M. Forster's classic distinction between the "flat" and the "round" character, tend to rely on a generative but ultimately limited paradigm of psychological "depth" that requires us to measure cognitive richness and realism in terms of volume. Here is Forster: "[A] novel that is at all complex often requires _flat_ people as well as _round_ , and the outcome of their collisions parallels life more accurately" (108). Looking at how characters' inner lives and minds develop around collisions of attention as I do, however, opens this discussion to descriptors of character far more complex than the flat and the round. Analyzing characters' various modes and habits of focus moves us toward a range of characteristics for fictional minds that explode the simplicity of mental volume—qualities of mind like quickness, endurance, flexibility, capacity, intensity, divisibility, and fluidity. My reading of _Pride and Prejudice_ looks at how Elizabeth's "liveliness" inter-weaves three of these: flexibility, quickness, and fluidity.4
### CHANGING THEORIES OF ATTENTION AND DISTRACTION
Understanding why distraction, of all things, could work in Austen's novel to convey such mental liveliness, however, requires us to return to its historical moment. For most of the early modern period, distraction was associated with madness, sin, and error. Such interpretations emerged from religious (and thus moral) readings of concentration. Attention and distraction were seen as the product of effort, and consequently were linked to virtue and vice. The basic metaphor for attention was a straight line, with distraction as a divergent angle. Distraction was therefore a sinful mental digression—a wandering from the right way of godliness. Such ideas supported a single-object theory of concentration, where proper focus was seen as maintaining stasis, or fixing the attention—imagined as narrowing the concentration to a single point or path (and keeping it there).5 This is the basic logic behind the allegorical topography of John Bunyan's _The Pilgrim's Progress_ (1678): the sin of the pilgrim, Christian, is allowing himself to be diverted by characters like Mr. Worldly Wiseman, and thus straying from the narrow road of righteousness on his way to the "Celestial City."
In the late seventeenth and early eighteenth century, however, writers began to move away from such conspicuously moralistic descriptions of distraction to develop a more neutral, psychological framework for interpreting attention. By the 1740s, there were at least two competing philosophies of attention at play: a moral one, in which attention and distraction mark the line between virtue and vice; and a psychological one, in which attention and distraction are just two different modes, or states, of mind. Those using this newer psychological model believed that thoughts rarely stand still or follow a straight path, and created new definitions to accommodate the mind's natural movements. John Locke and David Hume, for example, emphasized that the narrowing stretch of intensely focalized attention was an essentially temporary mental posture, only one of the many states the mind shifts through on a daily basis.6 They were more interested in describing the spontaneous movements—and changing modes—of attention over time than in assessing its success or failure in sticking to a point. Accordingly, philosophers in this tradition began to describe distraction, not as a geometric angling off path, but as an interruption in attention's proper temporal rhythm, a moment (or set of moments) when our moving thoughts do not match up with our intended object of focus. Distraction, for them, was a kind of temporal disjunct, an inequality between the amount of time a task requires and the time the mind devotes to that task. Some began to suggest that this odd mental tempo could have benefits as well as costs—specifically, the ability to bring unusual ideas together, and thus spark creativity.
Denis Diderot articulates this shift in ideas about attention most explicitly, and he voiced the strongest challenge to the perception of distraction as pure vice:
> La distraction a sa source dans une excellente qualité de l'entendement, une extrème facilité dans les idées de se réveiller les unes les autres. C'est l'opposé de la stupidité, qui reste sur une même idée... Un bon esprit doit être capable de distractions, mais ne doit point être distrait.
"Distraction," Diderot argues in the _Encyclopédie_ (1754), "stems from an excellent quality of the understanding."7 It has its source in an "extreme facility" of mind—for distraction, he argues, "allows ideas to trigger, or awaken one another." Diderot believed distraction to be a vital state of mind for generating new thought. He also considered it the creative opposite of compulsive attention, which he described as the over-focused stupor of "remaining, unchanging, on one idea." According to Diderot, a "good mind" was spontaneous enough to accommodate distractions but stable enough to avoid going mad.
If Diderot's redefinition of distraction articulated the wider cultural reorganization around the concept of attention, Austen's _Pride and Prejudice_ is its narrative realization.8 The novel takes up the idea that distraction contains a productive kernel of creative energy (as well as the suggestion that excessive concentration could make one intellectually numb) and turns attention into a literary device for better simulating the complexity of Elizabeth's inner life. In this context, Elizabeth's capacity for distraction (and other characters' lack of it) is just as important as her oft-discussed powers of observation. _Pride and Prejudice_ sets up Elizabeth's distraction to signify her liveliness and spontaneity and silhouettes it against a backdrop of obnoxiously over-attentive minor characters. The most noteworthy of these is Elizabeth's sister, Mary. In Austen's careful (and satirical) hands, the girl's unshifting attentiveness becomes the antithesis of creativity.
### MARY BENNET: OR, THE PROBLEM OF HYPERFOCUS
Austen's Mary Bennet gives us one of the earliest and clearest descriptions of a character stuck in the heightened state of absorption scientists now call _hyperfocus_. Psychologists like Thomas Brown, who approach distraction from a neuro-cognitive perspective, describe this state of mind as follows:
> Hyperfocus is an intense form of mental concentration that focuses consciousness on a narrow subject. [Those who hyperfocus] are unable to stop focusing on one thing and redirect their focus to another when they need to. They end up "locking on" to some task, sight, or sound—and totally lose track of everything else. (33)
If the cognitive term "hyperfocus" is anachronistic, the concept it refers to is not. John Locke began looking at the problems of absorption in his 1688 _Essay on Human Understanding_ , in which he notes that intense contemplation threatens to shut out other thoughts. When the mind "fixes it self with so much earnestness on the Contemplation of _some_ Objects," Locke worries, it will "take no notice of the ordinary Impressions made then on the Senses" (228). Such over-attention, for both Locke and Brown, has distinct costs: a shutting down and a subsequent inability to notice the ordinary impressions of the surrounding world.
Austen experiments with this idea (and its narrative implications) in _Pride and Prejudice,_ where she makes Mary Bennet a figure of hyperfocus.9 For much of the book, the girl remains inflexibly locked in a state of close attention, taking "no notice of the ordinary Impressions made on [her] Senses." Mary focuses stubbornly on two main actions: reading books and playing piano. And, though Austen shows her absorption in these to be deep, earnest, and focused, it is also, importantly, static. Mary seems to attend to all tasks in precisely the same way—with unremitting, relentless concentration. The costs for her in terms of character development are intentionally high. When Elizabeth and Jane return home from Netherfield, "they found Mary, as usual, deep in the study of thorough bass and human nature" (95). The novel implies that Mary's focus has not shifted at all in the meantime: she is still reading as intently as she was when they left a few chapters before. The key is not just that Mary's still reading, but that she is still absorbed—deeply absorbed—in her study.
Fifty years prior, when more traditional religious ideas about attention clearly held sway, depicting a character's absorption in the study of morality could be imagined to reliably convey piety and goodness, as well as the psychological depth of that character's mind and interest. As Michael Fried notes in _Absorption and Theatricality,_ French painters of the 1760s intentionally sought to "capture" characters in visible states of heightened attention in their artworks so as
> to arrive in the end at the sort of emotionally charged, highly moralized, and dramatically unified situation that alone was [thought] capable of embodying with sufficient perspicuousness the absorptive states of suspension of activity and fixing of attention that that painter and critic alike regarded as paramount. (56)
However, for absorption to carry such cultural significance and emotional resonance, such moments of intense focus needed an additional quality: they needed to be temporary. With the short iterative "as usual" (and the tedious cycle of repetition it connotes), Austen drains Mary's absorption of the potential for such emotional and narrative charge.
The monotony of this "as usual" holds throughout the novel. Mary is only granted one mode of attention—hyperfocus. This stoic intensity limits her as character, conversationalist, and thinker; for the narrator implies that Mary's clockwork diligence robs her of mental agility. Inflexible-minded from too much study, she cannot even keep up, Austen suggests, with the basic temporal flow of conversations going on around her. We see this most clearly when Mr. Bennet teasingly asks Mary to comment on the etiquette rules for introducing a suitor like Bingley. She fails woefully, producing the following painful exchange: "'What say you, Mary?,' Mr. Bennet asked, 'for you are a young lady of deep reflection I know, and read great books, and make extracts.'" Here the narrator interrupts to note that though "Mary wished to say something very sensible," she "knew not how." "'While Mary is adjusting her ideas,'" Mr. Bennet interjects, tongue-in-cheek, "'let us return to Mr. Bingley'" (47). The joke relies on the sharp contrast between Mary's supposedly "deep reflections" and her shallow—in fact, absent—contribution to the conversation. Mary's habit of absorption seems to slow down her mind, producing such verbal stupor that the conversation moves on before she can adjust her ideas, or get her thoughts flowing so as to speak. Mary thus becomes a figure for the cognitive costs of excessive attention—inflexibility, rigidity, and a deep social awkwardness.
Since Mary cannot loosen her attention, she is also barred from character change. After Lydia's elopement, an event that fuels much character development for Elizabeth, Mary remains unmoved. Unable to cobble together a single new thought in the face of family disaster, Mary recycles undigested maxims she's picked up in her old readings. In this, her longest speech of the novel, she produces four rather frozen gems of morality, culled from her conduct books:
> "that loss of virtue in a female is irretrievable—that one false step involves her in endless ruin—that her reputation is no less brittle than it is beautiful,—and that she cannot be too much guarded in her behaviour towards the undeserving of the other sex." (298)
Mary's focus on morality (much like her blind concentration on piano performance at the Netherfield ball) remains disturbingly undisrupted by any attention to her family's reaction. The problem, as Zunshine might put it, is that Mary is attuned, not to the dynamic "presence, behavior, and emotional display" of people around her, but to an inert set of objects—her books, and to a static memory of the morals they contain ("Why Jane Austen Was Different" 277). Indeed, undismayed by (and perhaps completely oblivious to) Elizabeth's uplifted eyes, Mary continues "to console herself with such kind of moral extractions from the evil before them" (298).
This concentration on studying becomes the novel's "excuse" to keep Mary "indoors," not only excluded from the many walks in _Pride and Prejudice_ , but also from any kind of productive mental wandering.10 Set up only to study and be static, Austen's Mary can "never spare time" from reading, and thus can't indulge in a walk:
> Lydia's intention of walking to Meryton was not forgotten; _every sister except Mary_ agreed to go with her. . . .
>
> Mrs. Bennet was not in the habit of walking, _Mary could never spare time,_ but the remaining five set off together. (105, 364-65; emphasis added)
Indeed, when Lydia tries to tempt Mary to go outside by describing the joys of travel— coach rides, new ribbons, and cucumber sandwiches—Mary can only respond: "'Far be it from me, my dear sister, to depreciate such pleasures. . . . But I confess they would have no charms for _me_. I should infinitely prefer a book'" (238). No model for Austen's audience, the over-attentive Mary is instead a cautionary figure for the dangers of reading too much. The novel mocks Mary for her habit of reading to the exclusion of all else, and lauds Elizabeth for her ability to stop (and to digress in thought and action). In such a world, when Miss Bingley calls Elizabeth a "great reader" who takes " no pleasure in anything else," Mary's novelistic presence sharpens the insult. And, Austen's caricature of Mary both fuels the outrage (and raises the value) of Elizabeth's famous retort: "I am _not_ a great reader, and I have pleasure in many things" (74).
Mary's absorption in books is literalized in her lack of mobility in the novel, and her absorption limits her both in mental (and in narrative) space. Austen now uses Mary's motionlessness to reflect a still deeper inflexibility—that of mind and character. For example, when Elizabeth returns home from Pemberley, stunned and altered by the news of Lydia's decision to run off with Wickham, Jane hurries downstairs to meet her. Mary and Kitty, however, stay upstairs, both too focused on their daily routines to notice their older sister's entrance. The narrator suggests that this habitual self-absorption has left the two girls oddly unmoved by any reaction to Lydia's elopement:
> [They] had been too busily engaged in their separate apartments to make their appearance before. One came from her books, and the other from her toilette. The faces of both, however, were tolerably calm; and no change was visible in either. (297)
Their sister's tragedy, Austen implies, has not influenced them in any way. Their surface calm reflects the shallowness of their respective characters. Mary's calm derives from obliviousness; she is too busy in her books, it seems, to be bothered. A few lines later, Jane accidentally explains Mary's inertia—and why she has not asked Mary to help manage their mother's hypochondriac nerves. The others "'would have shared in every fatigue, I am sure,'" Jane tells Elizabeth, "'but I did not think it right . . . and Mary studies so much, that her hours of repose should not be broken in on'" (301). With her absorption in books seemingly unbroken by any distraction, the ever-attentive Mary is not only compressed but crystallized into a minor figure. She remains a tableau—not character but unshifting caricature—of excessive, and ultimately impoverished, attention.
### ELIZABETH BENNET: OR, THE BENEFITS OF DISTRACTION
Elizabeth, swift of thought and quick of tongue, has (distinctly unlike Mary) the capacity for distraction, and thus, I argue, the capacity for change. Playing on the notion that distraction reawakens ideas, Austen mixes in some absentmindedness with Elizabeth's normal attentiveness to establish her difference from flatter characters like Mary and to demonstrate her superior flexibility of mind. Though Elizabeth's powers of observation are unparalleled in the first half of the novel, in the second half, her attention is frequently interrupted—often while reading—and still more often by her conflicting emotions for Darcy. Austen places the most important scene of disrupted focus at the center of _Pride and Prejudice_ , when Elizabeth reads Darcy's letter. And it is here—at the novel's turning point, or _volta_ —that Austen uses Elizabeth's distraction to cultivate her most detailed, and realistic, descriptions of the heroine's mind.
Distraction, for Diderot, marked a creative mind in movement, though one perhaps not always quite moving at the right tempo. Michael Posner and Raja Parasuraman, two contemporary leaders in the field of attention, both emphasize this aspect of attention's cognitive complexity, and the importance of such movements in time. Each describes attention as a continuous, multilayered process—optimally, the ongoing process of focusing on the right things, at the right time, at the right speed—which allows us to adapt to the shifting conditions and demands of the surrounding world. Attention, for Parasuraman, "is not a single entity but the name given to a finite set of brain processes that can interact, mutually and with other brain processes, in the performance of different perceptual, cognitive, and motor tasks" (3). Posner's _The Cognitive Neuroscience of Attention_ explains attentional orienting "as the set of processes by which neural resources are deployed selectively . . . on the basis of changing motivation, expectation, or volition in order to optimize perception and action" (157). Brown, I think, best synchronizes these two theories for a discussion of distraction when he writes:
> The continuous process of attention involves focusing and shifting focus, regulating alertness, sustaining effort, and regulating the mind's processing speed and output . . . Attention, in this sense, is essentially a name for the integrated operation of the executive functions of the brain. Distraction marks the failure to activate and manage these various cognitive functions in the right way at the right time. (11, 21)
In one sense, distraction is still clearly a cognitive failure here—or at least a mark of cognitive inefficiency. The neurochemical networks associated with the executive functions have failed to direct and shift the mind's focus appropriately to concentrate on the "right" object or task at the "right" time. But distraction is not the opposite of attention, Brown makes clear; it simply marks a certain disruption, or pattern of cognitively clustered disruptions, in attention's normal process and rhythm. What Diderot (and I think Austen) emphasize is that such mistimings and disruptions can be oddly useful—at times, essential—for bringing new thoughts together spontaneously to catalyze innovation.11
Distraction, in this sense, marks not a deficient but an unpredictable, or sporadic, tempo of focus. The mind bounces loosely and haphazardly at a changing pace, moving between intense concentration and sharp interruption, delving into an idea and then snapping away from it. Austen spends a full chapter describing Elizabeth reading (and re-reading) Darcy's letter in this quick shifting mode of mind, and her description is replete with such spontaneous, playful leaps of attention. This chapter—and the distraction it contains—fleshes out the heroine's otherwise stock characteristic of liveliness by creating a space in which the narrative can model her mind's quickness, flexibility, and fluency, creating a more textured sense of her inner life.
For Elizabeth's reading there is none of Mary's frozen "as usual." Elizabeth's distracted reading habits thus open out into complexity.12 As her mind moves swiftly and erratically through Darcy's letter at ever-shifting paces, it begins to convey not only the heroine's impulse and emotion, but also the fluency of her ideas and her elasticity of mind. Mind, again, is literalized in motion. Unlike Mary, Elizabeth not only walks, but walks while she reads; and, in truth, reads much like she walks: "crossing field after field at a quick pace, jumping over stiles and springing over puddles with impatient activity" (70). Austen grants this same breathless quickness to Elizabeth's reading of Darcy's letter:
> She read, with an eagerness which hardly left her power of comprehension, and from impatience of knowing what the next sentence might bring, was incapable of attending to the sense of the one before her eyes. (223)
This energetic, distracted reading—a kind of mental jumping and springing through the text—as well as her strong yearning for the next line make Elizabeth unable to focus fully on the sentence before her.
At the end of this first speed-reading of Darcy's missive, Austen is careful to tell us, Elizabeth's mind is left so scattered that she "scarcely know[s] anything of the last page or two." She ends the letter still pacing restlessly in a "perturbed state of mind," with "thoughts that [can] rest on nothing" (223). Though such mental restlessness may keep Elizabeth from fully absorbing the meaning of Darcy's letter at first, her distraction produces what Mary's plodding mind cannot: a rapid burst of thoughts.13 Pausing only halfway through, Elizabeth "instantly" 1) establishes Darcy's belief of Jane's insensibility to Bingley to be false; 2) assesses the letter's style as "not penitent, but haughty"; and 3) firmly labels him "all pride and insolence" (223). Only thirty seconds later, Elizabeth begins reading the letter again; however, this second time, distracting thoughts and emotions now endlessly interrupt the swift flow of her reading. She must stop and re-read after almost each paragraph, and at times, after each sentence. This makes her reading seem full of motion, if also restless and haphazard. Moreover, these interruptions create narrative, and clearly, it is the back and forth within Elizabeth's mind, and not Darcy's letter, that Austen ultimately seeks to convey.
The narrator, in fact, lingers for an entire chapter over Elizabeth's distraction, using the space created to articulate the new thoughts it produces. Austen here seeks to capture Elizabeth's mind in process within the narrative form itself; the very structure of the novel begins to take on the tempo of Elizabeth's particular liveliness.14 The illusion of the heroine's cognitive intricacy and vibrancy of mind seems to emerge from within the chapter's montage of leaps, stops, and starts:
> The letter was unfolded _again_ . . . she _again_ began the mortifying perusal.
>
> But when she read, and re-read . . . _again_ was she forced to hesitate. She put down the letter.
>
> _Again_ she read on.
>
> After pausing on this point a considerable while, _she once more continued to read_ . . .
>
> From herself to Jane—from Jane to Bingley, her thoughts . . . soon brought to her recollection that Mr. Darcy's explanation _there_ had appeared very insufficient; _and she read it again_ . . . (223-25, 227; all but the penultimate emphasis added)
Again and again, her returns to Darcy's letter are interspersed with extensive descriptions of her distraction. Some interrupting new thought (often in free-and-indirect discourse) necessitates the "again" of a more attentive reading—and each new close reading again scatters her thoughts. The narrator shuttles between narrative input 1 (Elizabeth's stages of reading) and input 2 (the thoughts that make up her distraction). The more ragged texture of this distracted reading can connote Elizabeth's liveliness, in part, because the caricature of Mary as all-absorbed monotonous reader works as such a fit foil for her sister's energetic mind. Mary's strict mode of study does nothing better than show us what Elizabeth's style of reading is not.
A lively-minded heroine, the novel imagines, must not read in a neat and orderly fashion. Instead, Elizabeth's mind, like the narrative that describes it, refuses linear development. And as the novel layers distraction upon distraction, we seem to get a bundle of insights produced in short, erratic bursts of concentration. This makes the novel, like her mind—or, in fact, her mind, like the novel—dart in and out, back and forth, as it throws light on a variety of fresh ideas and emotions. This switching back and forth creates an eccentric narrative rhythm—one that plays out spontaneously against the steady, calm rhythm of Darcy's carefully written letter itself, embedded in full (and uninterrupted) just a chapter before. It is here, in this narrative simulation of distraction-punctuated thoughts, that I think Austen renders Elizabeth's liveliness of mind most subtly—for she invites her audience not only to "read" the heroine's distraction, but to enact it (and reflect upon it) as readers themselves.
Incorporating distraction within the novel's form, Austen not only finds a way to capture the richness of her heroine's spontaneity; she also turns distraction into a powerful catalyst for narrative movement and character development. For example, it is in one of the moments of free-and-indirect discourse that emerge in the distracted spaces between her re-readings that Elizabeth finally realizes: "Mr. Darcy's conduct,"
> which she had believed it impossible that any contrivance could so represent, as to render . . . less than infamous, was _capable of a turn_ which must make him entirely blameless throughout the whole. (224, my emphasis)
In the first half of the novel, of course, Elizabeth declares Darcy "'the last man in the world whom I could ever be prevailed on to marry'" (214). Only after her distracted reading of his letter does Elizabeth undergo "so material a change" that her "sentiments" become precisely the opposite (366). The very halts and leaps, slants and errors of the heroine's mind as she reads seem, in fact, to fuel the novel's crucial moment of double transformation: Elizabeth's acknowledgment of her emotions for Darcy and her profound recognition of her prejudices against him. _Pride and Prejudice_ imagines that distraction stimulates rapid bursts of thought in a strong mind; it is this production of new, if disordered, thoughts that makes Elizabeth's distraction the novel's vehicle for deep mental change.
Playing on this idea of distraction-as-transformation, the chapter ends with a complex scene that shows Elizabeth's creative mind adrift; distraction, as always, prompts in her an energetic, chaotic awakening of ideas:
> [Elizabeth] wander[ed] along the lane for two hours, giving way to every variety of thought; re-considering events, determining probabilities, and reconciling herself as well as she could, to a change so sudden and so important. (227-28)
Wandering, it seems, is what allows her mind to churn, both giving way to and producing "variety of thought." Elizabeth's final ability to let her mind roam affirms that she has true mobility of mind and is capable not only of rich thought and variety, but of character change both "sudden" and "important." Austen builds the eccentric rhythm of concentration—and its generative power—into the chapter itself.
In a letter to her sister, Cassandra, Jane Austen claims that Elizabeth's playfulness is in fact the novel's, and teasingly suggests that the insertion of some distraction would perfectly offset its liveliness:
> Upon the whole however I am quite . . . well satisfied enough.—The work is rather too light, & bright, & sparkling;—it wants shade;—it wants to be stretched out here & there with a long Chapter—of sense, if it could be had, if not, of solemn specious nonsense—about something unconnected with the story: an Essay on Writing, a critique on Walter Scott, or the history of Buonaparté—or anything that would form a contrast & bring the reader with increased delight to the playfulness & general Epigrammatism of the general style. (203)
Liveliness. Comparison. Distraction. The building blocks are all here: "the playfulness of the general style"; "anything that would form a contrast"; "a long chapter . . . about something unconnected with the story." It is only in these moments—when distraction's playful creativity becomes, in this sense, both its protagonist's main trait and the defining stylistic quality of the novel—that we are able, I think, to read Elizabeth's mind as that of a fully fleshed-out heroine. And if attention has become part of the novel's structure, as I suggest, the modality of distraction is not only vitally tied to Austen's strategies of characterization, but inextricable from the novel's deepest levels of meaning making.
In this sense, it is no surprise that _Pride and Prejudice_ encourages us to measure out the spontaneity and vitality not only of novels' styles, but of fictional minds by their ability to accommodate creative distraction. Though descriptions of attention and multilayered observations can provide both readerly challenges and rich narrative texture, they too are enfolded into Austen's comparative, competitive structure of characterization. Elizabeth is invested with the potential Zunshine speaks of, the rich array and variety of thoughts, feelings, and desires; Mary is not. Novels often encourage us to attribute more mental depth and ToM to characters (like Elizabeth) who are given the room to demonstrate mental variation than we do to characters (like Mary) who can only demonstrate one rigid habit of mind, or what Alan Palmer has called a "mind-rut." However, as my discussion of _Pride and Prejudice_ also shows, some characters activate readers' ToM more than—or even, perhaps, against—others, and this may need to occur if we are ever to gain a sense of our heroes and heroines as fully-realized psychological entities. Maneuvering the reader's selective attention and attribution, as orchestrated by the novelist, becomes an important and deliberate tactic for developing psychological richness in the closed, structured world of fiction.
If it is only against the unchanging absorption of simple characters like Mary that scenes of Elizabeth's distraction can build the illusion of her cognitive complexity, we need a comparative literary model of ToM to talk about Austen's novels—and perhaps about fiction in general. Using a comparative model to understand how ToM works when we read fiction lets us take the potential distribution of mentalizing into account in a novelistic zero-sum-game of character development. Though all characters may stimulate our ToM, some characters stimulate it more than others. Not all minds (or mind readings) are created equal in narrative fiction; it is only by depriving Mary of mental richness that Austen can give Elizabeth's novelistic mind such subtlety. And it is only by caricaturing Mary's absorption that Austen can use distraction as she does: as a measure of character, as an invitation to mind read "liveliness," and as a literary device for conveying the cognitive richness of an intricate mind.15
### NOTES
1 The major exception to this point, of course, is Austen's modeling of reciprocity for the central couples of her marriage plot. The chemistry, for example, between Elizabeth and Mr. Darcy, Emma and Mr. Knightley, or Anne and Mr. Wentworth all rely on this narrative principle of mutual exchange. These pairs, however, are still pairs of major characters; and they still claim subjectivity, cognitive complexity, and narrative space at the expense of minor characters like Mr. Collins, Miss Bates, or Mrs. Clay.
2 It is no accident, I think, that our favorite passages modeling intersubjectivity from Austen make main characters—Elizabeth (not Lydia or Mr. Hurst) or Anne Elliot (rather than Louisa or Miss King)—the subject of the sentence. In the famous example from _Persuasion,_ "It did not surprise, but it grieved Anne to observe that Elizabeth would not know [Wentworth]. She saw that he saw Elizabeth . . ." (117), there is an intricate hierarchy of observers and observation embedded within the passage's multiple recursivity—with Anne at the top, then Wentworth, then Elizabeth. Butte implicitly acknowledges this privileging of certain characters over others in deep intersubjectivity when he notes that Austen's "favorite observers" are often her heroines, "women . . . finely attuned to the network of perceptions that surround them" (110). And, it is Anne's finely honed skills for "observing observation, even observation of herself," according to Butte, that give this scene's deep intersubjectivity such power (112).
3 These inferences, presumably, are based on the separate sets of physical and verbal cues I have picked up unconsciously from each shopper, not on a comparison of the two based on their accidental position in the same line.
4 Fluidity of thought can be imagined as a combination of what cognitive scientists call ideational fluency, that is, the ability to "generate a certain _quantity_ of new and imaginative responses to a task" and what they call ideational flexibility, that is, the ability to produce novel responses in a wide range of _categories_ (Turner 190).
5 _Attention_ originally comes from the Latin _ad-tendere_ (to stretch toward), implying that the attentive mind consciously elongates and narrows itself in a fine, taut beam toward the thing upon which it wishes to focus. Traditional seventeenth- and early eighteenth-century definitions of attention conventionally describe attention as a voluntary mental stretch. Writers of this period often represented it as an intense gaze, where attention seems to reach along an invisible line from the eye to the object. Similarly, since distraction derives from _dis-trahere_ (Latin for to drag away, or apart), distraction originally referred to anything that dragged the mind from the direct line of attention. This idea of attention as line supported religious readings of concentration as a moral good, for if one could only be on the proper path of attention or off of it, the divine consequences of attention could seem equally clear.
6 Locke argued that there were a number of "various _Modes of thinking_ which the Mind may observe in itself" (227). He believed that "the Mind employs itself about [its present Ideas] with several Degrees of Attention," and, in a day, necessarily moves through "a great variety of degrees," ranging "between earnest Study & very near minding nothing at all" (227-28). "Men," argued Hume in his _Treatise_ , "are nothing but a bundle or collection of different perceptions, which succeed each other with an inconceivable rapidity; and are in a perpetual flux and movement . . . the thought has evidently a very irregular motion in running along its objects, and may leap from the heavens to the earth, from one end of the creation to the other, without any certain method or order. . . . The capacity of our mind is not infinite" (39, 92). By 1757, Hume imagined that the mental stretch required of aesthetic attention necessitated so much self-discipline that, at even the slightest interruption, the Mind, like a spring, would relax: "The least exterior hindrance to such small springs, or the least internal disorder, disturbs their motion," he claimed, "and confounds the operations of the whole machine" ("Taste" 139).
7 Translations in this paragraph are my own.
8 Associations of distraction with spontaneity, innovation, and creativity like those we see in Diderot's article gathered currency throughout the 1760s, 70s, and 80s, and such suggestions appeared in a number of Austen's favorite literary works— including Samuel Johnson's _Rasselas_ (1759), Laurence Sterne's _Tristram Shandy_ (1759-1767), and William Cowper's _The Task_ (1785). However, Austen may well have read Diderot herself. Tutored by her French cousin, Eliza de Feuillide, she "read French with facility" (Holbert 83, citing Leigh 24) and Diderot's _Encyclopédie_ was published widely in cheap editions (see Darnton's _The Business of Enlightenment_ ).
9 For a more detailed reading of Jane Austen's relationship to John Locke—specifically her "dramatization" of Locke's educational tracts in _Northanger Abbey,_ see Jocelyn Harris's _Jane Austen's Art of Memory,_ 1989, pp. 1-33.
10 For more on this point, see Alex Woloch's _The One vs. The Many._ I base my cognitive work on attention in _Pride and Prejudice_ on his brilliant analysis of Austen's structure of characterization.
11 Useful terms for discussing Austen's early association of distraction and creativity may be found in the modern work of cognitive psychologists interested in the conditions for increasing our ability to link seemingly disparate concepts into novel synthesis (Snyder 416). For such increased creativity, researchers like Alice M. Isen and Ellis Paul Torrance emphasize the importance of maintaining a defocused attention, thus having additional cognitive materials available for association, increasing the breadth and range of the elements seen as pertinent, and facilitating cognitive flexibility, all of which raise the chances that unusual, or diverse, cognitive elements will in fact become associated to produce new thought (Isen 1987; Torrance 1974).
12 The contrast Austen sets up here between Mary and Elizabeth's styles of reading resonates with the differentiation of general crystallized intelligence (gC) and general fluid intelligence (gF) in cognitive science. Crystallized intelligence refers to the acquisition and maintenance of culturally specific knowledge (Mary's memorization of specific lines from Francis Burney's _Evelina_ , or of Bach's musicological principles for analyzing figured bass). Fluid intelligence applies to the flexibility of mind needed to solve complex and changing problems (Elizabeth's unscripted process of interpreting Darcy's letter).
13 This depiction of Elizabeth's reading as made up of short bursts of attention punctuated by distraction, interestingly, seems to follow the rhythms of what we now call working memory. In his article, "Cognitive Science and the History of Reading," Andrew Elfenbein defines _working memory_ as the "mental faculty allowing readers to store, process and manipulate recently read textual input through such operations as comparison with other parts of the text, retrieval of relevant background knowledge, and the creation of forward and backward inferences" (487). As he argues, individual reading span is determined by a reader's "ability to control attention" so that he or she can "maintain information in an active, quickly retrievable state" (Engle 20). For an investigation of how working memory serves as a predictor of reading span, and how reading span tests figure into assessments of general fluid intelligence and ideational fluency, see Engle et al., 1999.
14 This experimental representation of mind in _Pride and Prejudice_ foreshadows her later stylistic innovations in _Persuasion_. As A. Walton Litz claims, _Persuasion_ moves "away from the Johnsonian norm," developing a "rapid and nervous syntax designed to imitate the bombardment of impressions upon the mind" (228-29). In "Of Heartache and Head Injury: Reading Minds in _Persuasion_ ," Alan Richardson says further that the stylistic innovations critics have ascribed to the novel result from its prominently including "the gaps and disruptions in the represented flux of consciousness" (11).
15 I am grateful to John Bender, Denise Gigante, Blakey Vermeule, Lauren Caldwell, and the anonymous readers from the _Theory of Mind and Literature_ volume for their abundant insights and assistance in the preparation of this essay.
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Chicago: University of Chicago Press, 1980. Print.
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Hume, David. "Of the Standard of Taste." _Selected Essays_. Ed. Stephen Copley and Andrew Edgar. New York: Oxford University Press, 1993. Print.
\---. _Treatise on Human Nature_. Ed. L. A. Selby Bigge. Oxford: Clarendon Press, 1978. Print.
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\---. _Why We Read Fiction: Theory of Mind and the Novel._ Columbus: The Ohio State University Press, 2006. Print.
## Sancho Panza's Theory of Mind
HOWARD MANCING
### SANCHO PANZA'S DILEMMA
Sancho Panza faces a dilemma. He had previously served as squire to the knight-errant Don Quixote de la Mancha on a chivalric quest (this was the story of the first part of Cervantes's novel, published in 1605), and after a brief interval of just over one month, the two of them have again set out in search of adventure and fame (and this comprises the second and final part of the novel, published in 1615). But rather than wander aimlessly, as before, this time they head directly for the village of El Toboso, supposedly the home of the incomparably beautiful lady named Dulcinea del Toboso. Right now they are located outside the village, and Don Quixote has ordered Sancho to go to Dulcinea's palace and deliver a message of the knight's love and devotion, tell her that he requests an audience, and then bring back her response. The trouble is that Sancho knows full well that Dulcinea does not exist. How is he to carry out his master's orders? How do you deliver a message to, and return with a response from, a figment of someone's imagination?
There is in this scene a subtle and complex interplay between fictional minds as each character employs his Theory of Mind in various ways to achieve his private agenda. Before he sends his squire off on his errand, Don Quixote prepares him for the mission by laying out a probable scenario. His instructions are for Sancho to remember all the details of how Dulcinea receives him: Does her color change as the message is delivered? Does she show agitation upon hearing his name? Does she shift her weight from one foot to another, repeat her answer two or three times, change emotional state from harsh to loving? Does she subconsciously reveal her agitation by smoothing her hair (even though it is not disarranged)? Pay close attention, he says, and report back all these details because
> si tú me los relatares como ellos fueron, sacaré yo lo que ella tiene escondido en lo secreto de su corazón acerca de lo que al fecho de mis amores toca; que has de saber, Sancho, si no lo sabes, que entre los amantes, las acciones y movimientos exteriores que muestran, cuando de sus amores se trata, son certísimos correos que traen las nuevas de lo que allá en lo interior del alma pasa. (104)1
>
> [if you relate them to me just as they occurred, I shall interpret what she keeps hidden in the secret places of her heart in response to the fact of my love; for you must know, Sancho, if you do not know it already, that with lovers, the external actions and movements, revealed when the topic of their love arises, are reliable messengers bringing the news of what transpires deep in their souls.] (514)2
Note here that Don Quixote's Theory of Mind is based less on personal experience in life than on his readings in the romances of chivalry where such scenes between lovers frequently take place. He is providing Sancho with what Roger Shank and Robert Ableson call a "script": a prototypical scenario for a frequently-encountered and highly-structured scene. All Sancho has to do is come back after a while and report: Yes, as you said, when I delivered your message, she blushed; she became agitated when I mentioned your name; she nervously shifted her weight from one foot to the other; and she reached up her hand to smooth her perfect hair. But the only thing Don Quixote does not provide is the lady's answer: will she agree to a meeting with her lover?
Sancho rides off, leaving his master in a state of mind that the narrator reports to the reader thus: "Don Quijote se quedó a caballo, descansando sobre los estribos y sobre el arrimo de su lanza, leno de tristes y confusas imaginaciones" (104) ["Don Quixote remained on horseback, resting in the stirrups and leaning on his lance, full of melancholy and confused imaginings" (514)]. Meanwhile, Sancho rides away "no menos confuso y pensativo" (104) ["no less confused and thoughtful than his master" (514)]. Once he is out of sight, Sancho stops, dismounts, and sits at the foot of a tree to contemplate his situation, to work to clarify his confusion. Speaking out loud, he queries himself about what he is doing (and I am paraphrasing here):
> Well, brother Sancho, where are you going? —To look for a princess.
>
> Where? —In the village of El Toboso.
>
> Why are you doing this? —For the sake of the famous knight Don Quixote.
>
> Fine, and do you know where this princess lives? —My master says it is a royal palace.
>
> Have you ever seen her? —No, and neither has my master.
Sancho ends this self interview with the conclusion that "¡El diablo, el diablo me ha metido a mí en esto; que otro no!" (106) ["the devil, the devil and nobody else has gotten me into this!" (515)].
But here Sancho shifts gears and brings into play not only his Theory of Mind, but his "Machiavellian intelligence," a term coined by Richard Byrne and Andrew Whiten to describe that practical, pragmatic ability to act in one's own best interests, often in a way that involves deceit of others. He rejects Don Quixote's script and writes his own. First, he acknowledges that his master is crazy (and he adds that he must be just as crazy, since he follows and serves him). Furthermore, he reminds himself, Don Quixote often transforms one thing into another—windmills into giants, flocks of sheep into warring armies—so he, Sancho, ought to be able to deceive him by turning the tables on him and making the transformation for him. So, thinks Sancho:
> [N]o será muy difícil hacerle creer que una labradora, la primera que me topare por aquí, es la señora Dulcinea; y cuando él no lo crea juraré yo, y si él jurare, tornaré yo a jurar, y si porfiare, porfiaré yo más, y de manera que tengo de tener la mía siempre sobre el hito, venga lo que viniere. . . . [y] quizá pensará, como yo imagino, que algún mal encantador de estos que él dice que le quieren mal la habrá mudado la figura por hacerle mal y daño. (106)
>
> [It won't be very hard to make him believe that a peasant girl, the first one I run into here, is the lady Dulcinea; and if he doesn't believe it, I'll swear it's true; and if he swears it isn't, I'll swear again that it is; and if he insists, I'll insist more; and so I'll always have the last word, no matter what. . . . [and] maybe he'll believe, which is what I think will happen, that one of those evil enchanters he says are his enemies changed her appearance to hurt him and do him harm.] (516)
In fact, the scene plays out exactly as Sancho has scripted it. Just as he returns to where Don Quixote is waiting, three peasant women are coming from El Toboso, riding on three donkeys. Sancho tells Don Quixote that Dulcinea and two of her damsels are coming from the city to meet with him now. Quixote's first reaction is one of disbelief (perhaps because he never really expected to see his nonexistent lady). He says, "¡Santo Dios! ¿Qué es lo que dices, Sancho amigo? Mira no me engañes, ni quieras con falsas alegrías alegrar mis verdaderas tristezas" (107) ["Holy God! What are you saying, Sancho my friend? Do not deceive me, or try to lighten my true sorrows with false joys" (516)]. Sancho maliciously responds with his best Machiavellian deceptiveness, "¿Qué sacaría yo de engañar a vuesa merced?" (107) ["What good would it do me to deceive your grace?" (516)]. When Sancho points to the three peasants and calls them beautiful ladies riding richly adorned palfreys, Don Quixote is confused: "Yo no veo, Sancho, sino a tres labradoras sobre tres borricos" (108) ["Sancho, I do not see anything except three peasant girls on three donkeys" (517)]. Sancho pretends to be stunned and insists that they are beautiful ladies on snow-white palfreys, and all Don Quixote can do is say that he sees nothing but three women riding on some kind of donkeys, adding, "a lo menos, a mí tales me _parecen_ " (108; emphasis added) ["at least, that is what they _seem_ to be" (517; emphasis added)].
At this point, Sancho knows that his deception has been successful. When Don Quixote admits that they seem to be peasants on donkeys, he opens the door to the possibility that there could be another explanation of what they actually are. I will not go into further detail about how the scene plays out.3 Don Quixote winds up talking with what he now believes is an enchanted Dulcinea, and much of the remainder of the novel deals with the restoration of his beloved lady to her original form. This is the single most significant episode in the second part of the novel, and it all hinges on the characters' Theory of Mind. If one does not read _Don Quixote_ as a constantly evolving, ever more intricate series of scenes involving thoughtful fictional minds interacting with each other in the widest possible variety of ways, both theorizing about how other minds work and simulating the feelings of other minds, the novel simply cannot be understood.
Perhaps the most difficult thing Sancho has had to learn in his service to a knight-errant is how to deal with two kinds of people he cannot even see. The first of these is an evil enchanter (later there are multiple enchanters) who, out of envy, constantly attempts to thwart the knight's achievements. But, even more importantly, he learns that all knights-errant are in love with a beautiful lady, and Don Quixote loves the incomparably beautiful princess Dulcinea, who, furthermore, lives in the nearby village of El Toboso. Throughout the first part of the novel Sancho becomes increasingly familiar with the chivalric mind of Don Quixote and learns how to function in relation to it.4 By the time we get to the scene I have described in some detail, Sancho has mastered the art of manipulating not only his master, but invisible enchanters and imaginary women as well.
The brilliant combination of Theory of Mind and Machiavellian intelligence employed by Sancho during this episode, how he is increasingly able to take control in his relationship with Don Quixote, and the way in which the events that play out in this scene become the focal point for the remainder of the novel, are some of the subjects that could be explored at length. Sancho's mind at work in episodes such as those involving the duke and duchess who entertain him and Don Quixote for their amusement, his brilliance as governor of the island of Barataria, the clever ruse he invents to "disenchant" Dulcinea, and his genuine grief when a defeated and disillusioned Don Quixote returns home and dies, provide ample material for study. Sancho Panza's emergence as the greatest mind reader of a world full of expert mind readers should be considered one of the best examples ever of a Theory of Mind at work.
What I would like to do in the limited space remaining is to place Sancho's Theory of Mind in context. Recapping: Sancho _knows_ [1] that Don Quixote _believes_ [2] that evil enchanters _want_ [3] to foil his _desire_ [4] to gain fame as a knight errant; therefore, he (Sancho) is _sure_ [5] that he can make Don Quixote _believe_ [6] that a peasant woman is Dulcinea.5 This is as mature and modern an example of the narrative presentation of the fictional mind as I have ever read. It suggests that the way our minds function to understand the minds of other people has not changed in very many years, nor has the narrative presentation of these minds. Yet it has recently been proposed that such an ability to present complex, multileveled mind reading abilities simply did not exist at the time Cervantes was writing.
### _T HEORY OF MIND IN CERVANTES_
George Butte is the author of a book bearing the wonderful title _I Know that You Know that I Know_ (2004). Butte discusses what he refers to as "deep intersubjectivity": "the ways human subjects narrate and get themselves narrated" (vii). In effect, deep intersubjectivity is nothing but a sophisticated Theory of Mind. Butte never uses the term Theory of Mind and displays no knowledge of this concept as it has been studied in primatology and cognitive and developmental psychology; rather, he works within what he calls a "poststructuralist phenomenology." Butte begins his book by citing an interesting scene of recognition and understanding from Jane Austen's _Persuasion_. His comments on this scene deserve being cited at length:
> This scene in _Persuasion_ , about the observation of observations, gives voice to a profound new way of shaping narrative that takes coherent form, at least in English literature, in the early nineteenth century. When Anne Elliot watches Wentworth and Elizabeth negotiating complex force fields of memory and protocol, the enabling strategy of her story is a new layering of human consciousness, or a new representation of those subjectivities as layered in a specific way. Deep intersubjectivity has made its appearance in storytelling in modern culture, and it has altered our sense of self and community and the discourses that construct and reflect them. To conceptualize credibly this sea change in narrative practices requires a different kind of reading of consciousness, a reformed phenomenological reading, by means of a poststructuralist phenomenology, or at least a phenomenology attuned to poststructuralist questions that also attends to the situating of reader and text in a gendered world of class, race, and political power. (4)
This is quite a claim: that the narrative tools needed for representing "deep intersubjectivity" were not developed until about the time of, and perhaps specifically by, Jane Austen.
Lisa Zunshine has been the major theorist of Theory of Mind and literature, primarily in her book _Why We Read Fiction: Theory of Mind and the Novel_ (2006), a work that is essential to any consideration of the subject. In a recent essay titled "Why Jane Austen Was Different, and Why We May Need Cognitive Science to See It," Zunshine first praises Butte's theory and then demonstrates that in fact his deep intersubjectivity is precisely the concept of Theory of Mind as studied in evolutionary psychology and cognitive psychology, thus providing a much stronger theoretical background for the idea than Butte's so-called "postmodern phenomenology." Next, Zunshine extends the technique studied by Butte back into English sentimental fiction of the eighteenth century and, further still, to one example from the English theater of the late seventeenth century. We can presume that both Butte and Zunshine, like virtually all other professors of English literature, subscribe to the classic theory of Ian Watt (1957) that the novel "rose" for the first time in England in the eighteenth century. Perhaps because of such an assumption, it seems not to occur to them to look much further back than the Enlightenment period or to other countries in search of examples of Theory of Mind (or deep intersubjectivity).6
But the scene I have described in _Don Quixote_ seems to suggest that the narrative skill required for the multi-leveled presentation of fictional minds existed and was practiced much earlier, at least in Spain. Furthermore, my claim that such presentation of fictional minds is common in Cervantes's novel calls Butte's assertion into question in a very fundamental way. If Cervantes was presenting fictional minds with every bit as much subtlety in 1605 and 1615 as Jane Austen was in her novel published posthumously in 1818, then it becomes more difficult to make Butte's claim for Austen's originality.
But perhaps it is possible to set forth Cervantes as a lone genius, two centuries ahead of his time in his psychology and his narrative technique. Even Butte recognizes that "elements of deep intersubjectivity occur in . . . Cervantes" (37-38). If this were the case, then Butte's argument might still hold for the most part. Unfortunately for Butte's thesis, however, Cervantes was not alone in the fictional presentation of a sophisticated Theory of Mind in fiction. Let us briefly consider the example of the anonymous 1554 picaresque novel _Lazarillo de Tormes_ , the story of a street urchin who rises to become town crier of the city of Toledo.
By the end of the short novel, Lázaro, the protagonist and first-person narrator, is married to the servant and mistress of his patron, an archpriest who tells him that everything is perfectly legitimate and that he should pay no attention to what people say but look to his own honor, that is, to what is in his best material interest:
> Lázaro de Tormes, quien ha de mirar a dichos de mala lenguas nunca medrará; digo esto porque no me maravillaría alguno, viendo entrar en mi casa a tu mujer y salir della. Ella entra muy a tu honra y suya. Y esto te lo prometo. Por tanto, no mires a lo que pueden decir, sino a lo que te toca: digo a tu provecho. (133)
>
> [Lázaro de Tormes, anyone who pays attention to what evil tongues have to say will never get ahead in the world. I say this because I wouldn't be at all surprised if someone who sees your wife enter and leave my house would say such a thing. She enters very much to your honor, and hers. And I promise you this. Therefore, pay no attention to what people might say, but to what matters to you: I mean to what is to your advantage.] (translation mine)
Although Lázaro swears that his wife is as virtuous as any woman in Toledo, the reader is left to assume that he is a consenting cuckold, a willing participant in his wife's (and the archpriest's) lies and their illicit sexual affair. He forbids his friends to talk about the matter and ends the book by proclaiming that at that time he was at the height of all good fortune.
But here is Lázaro's dilemma (not, in a way, unlike Sancho's described at the beginning of this essay): a high-ranking church official (identified only as _Vuestra Merced_ , "Your Grace") has written to him requesting information about the affair. What to tell him? Lázaro _knows_ what is going on. The archpriest and his wife _know_ that he _knows_ , and he _knows_ that they _know_ it. But the archpriest, because he _understands_ Lázaro's situation, _realizes_ that Lázaro doesn't _dare_ say or do anything about it because he is _certain_ that the archpriest would _withdraw_ his support and ruin Lázaro's materially comfortable life. The inquiring church official must have heard the rumors about the affair, and he _knows_ (or _assumes_ ) that Lázaro is _aware_ of the situation and can, if he _wants_ to, _tell_ him the full story about the matter. Your Grace _knows_ (or at least _suspects_ ) what is going on. Lázaro _knows_ that Your Grace _knows_ , and he _knows_ that Your Grace _knows_ that he _knows_. And they both _know_ what the archpriest and his lover _know_. Furthermore, the reader _knows_ all of this and fully _understands_ the implications of this web of knowledge and lies. And, best of all, the anonymous author _knows_ that the reader _knows_ all this.
Or if 1554 is not early enough, then we can go back to 1499 and the publication of Fernando de Rojas's novel in dialogue titled _La Celestina_.7 The young nobleman Calisto loves the beautiful Melibea but believes that she disdains him. In fact, she secretly loves him but pretends not to. But the old bawd, sorceress, and procuress Celestina knows that she has to deceive Melibea's mother (who knows Celestina's reputation and is suspicious of her) in order to talk with Melibea and get her to reveal her true feelings, so that she, Celestina, can manage a secret love affair between the two youths. The entire work is filled with beliefs, desires, secrets, lies, manipulations, levels of intentionality, Theory of Mind, and Machiavellian intelligence, all as complex and compelling as anything found in any novel written by Jane Austen or anyone else.
In the long passage previously cited from George Butte's book, the author claimed that Austen's deep intersubjectivity was "a profound new way of shaping narrative that takes coherent form, at least in English literature, in the early nineteenth century" (4). I believe that such a statement must be qualified in two important ways: First, it certainly was not anything new in Spanish literature, where, as the examples I have briefly cited, show that it was common in the Renaissance. Second, if Butte is right that there was nothing comparable in the English Renaissance, then apparently he believes that writers like William Shakespeare were not capable of as profound an understanding of human nature as Jane Austen was two centuries later. One does not need to be a Shakespeare scholar to recognize that Butte is guilty of a serious misunderstanding of literature, history, and human psychology.8
A failure to make use of human biology and evolution sometimes leads humanists into truly embarrassing errors. George Butte's myopic view that pre-nineteenth-century storytellers were not able to convey a sense of an advanced form of human subjectivity is an example of such an error.9 Not only has the human mind functioned as he describes it since at least the early modern period, as has been illustrated by the examples I described from Spanish literature, but it has been basically unchanged throughout all of human recorded history and well before that. Daniel Levitin reminds us that, according to best estimates, "it takes a minimum of fifty thousand years for an adaptation to show up in the human genome. This is called evolutionary lag—the time lag between when an adaptation first appears in a small proportion of individuals and when it becomes widely distributed in the population" (250). _Homo sapiens_ had virtually the same brain and the same mental capabilities fifty millennia ago that our species has today.10 To claim that Jane Austen was doing something unprecedented in the history of literature is not convincing. Just ask Sancho Panza.
### NOTES
1 This and all subsequent citations from _Don Quijote_ are from Volume II of the revised Cátedra edition by John J. Allen.
2 This and all subsequent citations from _Don Quixote_ in English are from the translation by Edith Grossman.
3 The classic study of this scene is Erich Auerbach's chapter on "The Enchanted Dulcinea." I have previously studied the comic role of the terms used to describe the three animals in "Dulcinea's Ass"; see also Johnson's response to this article.
4 Roger Schank, an authority in artificial intelligence and human storytelling, points out that our "[e]xplanations of human behavior are always grounded in the beliefs of the person we are trying to understand" (63). It is because Sancho has spent so much time with Don Quixote that he has come to know better than anyone else— perhaps even better than Don Quixote himself—how his master the knight-errant thinks and what he believes about the world.
5 This illustrates what Daniel Dennett calls multiple levels of intentionality. Dennett illustrates the concept of intentionality thus: "I suspect [1] that you wonder [2] whether I realize [3] how hard it is for you to be sure [4] that you understand [5] whether I mean to be saying [6] that you can recognize [7] that I can believe [8] you want me to explain [9] that most of us can keep track of only about five or six orders [of intentionality], under the best of circumstances" (1988, 185-86). As evolutionary psychologist Robin Dunbar has suggested in a recent book on human evolution (46), humans can rarely follow a complicated multi-level expression of intentionality beyond about five levels, and Sancho is working at the upper limit in this scene with Don Quixote.
6 Zunshine's claim is not as extravagant as Butte's, but she, too, suggests that a very sophisticated Theory of Mind is essentially a modern cognitive skill. This is seen, for example, when she doubts that it would have been possible to depict a Theory of Mind as sophisticated as what we see since the eighteenth century in a pre-literate work like _Beowulf_ (2006, 73-75). It is worth noting, too, that Merlin Donald, whose brilliant work on the evolution of the modern mind is essential reading, falls into the same trap as Butte and Zunshine when he asserts that "[w]ith a few notable exceptions, literature did not plumb consciousness in depth until the last two hundred years or so" (78). My point throughout this essay is that a profound understanding of human consciousness and Theory of Mind has been a characteristic of literature for more than four centuries.
7 Earlier I cited Ian Watt's theory of the "rise of the novel," the idea that what we consider the modern novel appeared for the first time in eighteenth-century English literature. This is the view that is almost universally accepted in Anglo-American critical circles. But a more convincing alternate view is that of M. M. Bakhtin in the essays of _The Dialogic Imagination_ and other writings that the novel "emerged" in the Renaissance and was best exemplified in the fictions of Rabelais and Cervantes. It seems clear to me that, by any standard, _Lazarillo de Tormes_ "qualifies" as a novel and that works like _Celestina_ probably do also. Although I have referred to _Celestina_ as a novel in dialogue, others consider it a humanistic comedy; similarly, Rojas's work is considered to be the last great exemplar of Medieval Spanish literature while others consider it one of the first masterpieces of the Renaissance. It is one of the great complex, fascinating, original, influential works of European literature.
8 One example of a perceptive non-literary scholar is Robin Dunbar, who, in his previously-cited book on human evolution (161-62), cites two examples of fourth-and fifth-orders of intentionality in Shakespeare.
9 To be as clear as possible: Lisa Zunshine does not by any means share Butte's apparent lack of familiarity with biology and evolution. In fact, she has studied with the evolutionary psychologists Leda Cosmides and John Tooby, prominently cites their work, and acknowledges the embodied nature of human cognition.
10 Michael Tomasello has proposed that there has really only been one significant adaptation in the evolution of human beings, one that because of its overarching importance and inclusive ability to be manifest in numerous ways is the basis for all other human cognitive skills: "understanding others as intentional (or mental) agents (like the self)" (15)—essentially, that is, having a Theory of Mind. Gilles Fauconnier and Mark Turner have proposed that what they call "double-scope" blending—a concept intimately related to Theory of Mind—became a characteristic of the human species some 50,000 years ago, that is, about the time of the emergence of _homo sapiens_ (183-87; see also the later essay by Turner). Mind reading—together with the imaginative taking of cognitive elements from one input space and projecting them onto another space (conceptual blending)—has been a characteristic of our species much longer than Butte or Zunshine would have it.
### ACKNOWLEDGMENTS
I am thankful to my colleagues and friends Catherine Conner, Charles Ganelin, Paula Leverage, Rich Schweickert, and Jen William for their perceptive and helpful comments and suggestions after reading an earlier draft of this essay.
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\---. _Don Quixote de la Mancha_. Trans. Edith Grossman. New York: Ecco, 2003. Print.
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Turner, Mark. "Double-Scope Stories." _Narrative Theory and the Cognitive Sciences_. Ed. David Herman. Stanford, CA: CSLI Publications, 2003. 117-42. Print.
Watt, Ian. _The Rise of the Novel: Studies in Defoe, Richardson and Fielding_. Berkeley: University of California Press, 1957. Print.
Zunshine, Lisa. "Why Jane Austen Was Different, and Why We May Need Cognitive Science to See It." _Style_ 41.3 (2007): 275-99. Print.
\---. _Why We Read Fiction: Theory of Mind and the Novel_. Columbus: The Ohio State University Press, 2006. Print.
## Is Perceval Autistic? Theory of Mind in the _Conte del Graal_
PAULA LEVERAGE
The character of Perceval as a knight of the Grail quest appears for the first time in the twelfth century in Chrétien de Troyes's _Conte del Graal_ , and thereafter in various incarnations throughout history to the films of the twentieth and twenty-first centuries.1
_Le Conte del Graal_ is the last of five romances attributed to Chrétien de Troyes. It was written in the 1180s for Chrétien's patron, Philippe d'Alsace, Count of Flanders, and at 9,234 verses, it is unfinished.2 In France the story was taken up by four authors who wrote what are commonly referred to as the _Continuations_.3
Perceval across the literary and filmic representations of his character is variously described as a "bumbling hero" (Wise 52), "naïve, charming and slightly absurd" (Lacy, Ashe, and Mancoff 344), "socially inept" (McCullough 48), and a "country bumpkin" (Eckhardt 205). Chrétien's romance itself has been called "a story of the blunders of a teenage knight" (Wise 48). In Monty Python's _Holy Grail_ Perceval has devolved into Piglet, played by the actress Avril Stewart.4 This modern and critical reception of the character of Perceval as a less than perfect knightly model is already evident in the selection of material from Chrétien's romance, which is reworked in later medieval narratives, and in the choice of scenes illustrated in the manuscript tradition of the romance (Busby, "The Illustrated Manuscripts" 359-60). Beate Schmolke-Hasselmann has shown that it is material relating to Perceval's "niceté" (naïveté) that is reworked, while Keith Busby observes that almost all of the miniatures in the illustrated _Perceval_ manuscripts depict scenes that belong to the canon of reworked material (360).5 Given the apparent bias in the reception of the character of Perceval, which is ostensibly stable by 1250 (ibid. 360), it is not surprising that, generally speaking, the characterization of the Perceval of Chrétien's romance is simplified and loses interest within the tradition, until his position as the principal knight of the Grail Quest is finally usurped by Galahad.6
Putting aside centuries of insistence on Perceval's _niceté_ , and returning to the first literary incarnation of the character, I re-examine here Chrétien de Troyes's presentation of the character, which, I argue, has been misrepresented to the extent of distorting modern critical discourse about the romance as a whole. My principal interest is in the representation of Perceval's mind in Chrétien de Troyes's twelfth-century romance _Le Conte del Graal_ , which is not simply naïve, comic, bumbling, or effeminate, but appears to be struggling with a very specific cognitive deficit. At least one modern interpretation of the Perceval character approaches a similar reading. Robert Wilson in his production of _Parzival_ cast the autistic actor, Christopher Knowles, as Parzival.7 In spite of this casting decision, critical insistence on a naïve, foolish Perceval continued to trump any possible suggestion of cognitive difference in the character, as is clear from John Rockwell's review in the _New York Times_ (May 8, 1988), which suggests that this casting decision was made "to portray a Parsifal closer to the ideal of a pure fool than any seen or heard before in opera or play."
Chrétien de Troyes, however, is known for his fine psychological portrayals of his characters and their relations, and to that extent we could compare him today to novelists such as David Lodge and Ian McEwan, who consciously explore the workings of their characters' minds.8 Regardless of how he, or his contemporaries, might have labeled Perceval's behavior, in _Le Conte del Graal_ Chrétien de Troyes appears to be exploring what we now call Theory of Mind (ToM). Recognizing Perceval's apparently deficient ToM has important consequences for reading _Le Conte del Graal_ and understanding the Grail / Perceval tradition as a whole, as well as for literary ToM studies, specifically in the following three areas of enquiry: a) the influence of the reception of Chrétien's romance on the subsequent tradition; b) the relationship between Perceval and the other characters of the romance; c) transhistorical analysis of cognition and how it is represented.
Starting with the question of the reception of Chrétien's romance, it is clear, as we have seen above, that subsequent literary and filmic treatments of Chrétien's character have extrapolated from the atypical behavior of Perceval to present him as a country bumpkin, a naïf, or an effeminate character. In many works, most notably Mallory's late fifteenth-century _Morte d'Arthur_ , under the influence of the Vulgate Cycle, Perceval becomes a secondary character to Galahad, whose purity replaces, or is an extension of, the naïveté of Perceval. Under the considerable influence of works such as the _Morte_ , it is easy to read back into Chrétien's romance characterizations, which mask the original depiction of character. By reading Chrétien's text and recognizing ToM in action, it is possible to return to a characterization of Perceval that is not filtered through the common clichés so readily associated with him. Secondly, by considering ToM in connection with Perceval, it becomes clear that the romance also explores the ToM of other characters, who are not usually discussed in criticism of the romance. Thus ToM emerges as a central interest of the romance and demystifies some of the problematics of its textual interpretation. Thirdly, in the context of much current discussion about ToM in literature, theorists have raised the issue of the importance of looking at the fictional mind in a historical perspective, and this single example of a twelfth-century fictional mind begins to extend the discussion of ToM and literature beyond the nineteenth- and twentieth-century novels, which have been its focus in recent books.9
ToM is mind reading, or the attribution of mental states and intentions to others that affords us the possibility of adapting our behavior in response, and it has been argued that this evolved ability supports characteristically human activities, such as the use of irony and sarcasm, as well as more basic communicative functions (Baron-Cohen, _Mindblindness_ 132).10 While some researchers would go so far as to suggest that ToM is what makes us human (O'Connell 5), others have argued that TOM activity can be observed in non-human animals. However, it is difficult to ascertain in non-linguistic situations to what extent observed actions in animals are driven by mind reading rather than stimuli.11 Two groups of individuals notably lack function in mind reading to varying degrees: most notably autistic individuals, and young children who do not begin to develop ToM until around the age of four or five.12 Simon Baron-Cohen describes autism as "mindblindness," which emphasizes quite graphically the relationship between ToM and autism ("Autism: A Specific Cognitive Disorder of Mindblindness"). Sanjida O'Connell lists three core deficits in autistic people: 1) they cannot communicate; 2) they cannot imagine (and so show no pretend play); 3) they are unable to deal with people socially ( _Mindreading_ 12). These three deficits must, of course, be assessed on a relative scale, since there are varying degrees of autism. Asperger's syndrome, for example, is often termed "high functioning autism."
Several scenes in Chrétien's _Le Conte du Graal_ suggest that Perceval is deficient in reading minds, and perhaps autistic. Naturally, Chrétien de Troyes would not have thought of Perceval either in terms of autism or ToM, since these are modern terms. The term "autism" was coined independently in 1943 and 1944 by Leo Kanner and Hans Asperger, respectively (O'Connell 11), while the term "Theory of Mind" dates from David Premack and Guy Woodruff's 1978 study of chimpanzees, which investigated whether these animals could attribute states of mind to others and then use this information to predict the others' actions. However, in the past, individuals known as idiot savants, noble savages, or holy or "blessed'' fools were probably autistic (O'Connell 9-11). The so-called Wild Boy of Aveyron is one such "noble savage" from the late eighteenth century who was discovered at the age of twelve in a forest in France. He could not talk and displayed behavioral characteristics that Uta Frith has identified as being characteristic of autism. Frith further points out that autistic people are relatively well equipped to survive in the wild, since they can better tolerate extremes of temperature, pain, hunger, and discomfort ( _Autism and Asperger Syndrome_ 95). While these are examples of historical individuals, it is plausible that Perceval is modeled on an autistic individual known to Chrétien de Troyes. However, this hypothesis is not necessary in order to recognize that Chrétien is exploring ToM in his romance.
_Le Conte de Graal_ assigns three kinds of explanations to Perceval's atypical behavior: a) physical; b) social; and c) racial. The first is reflected through the characters who meet Perceval and who describe him in terms that imply that he is cognitively challenged. For example, he is described at various junctures as "sos" (mindless) (v. 200); "fols" (mad) (v. 688); "niches" (simple) (v. 689); "muiaus" (mute) (v. 1863).
The social explanation, which derives from the circumstances of Perceval's upbringing, is very prominent in scholarly analyses of the romance and it is frequently associated with psychoanalytical readings of Perceval's relationship with his mother.13 He has been raised in isolation by his mother after the death of his father and two brothers in combat. Devastated by this loss, his mother has raised Perceval in the country, far from any contact with knightly culture in the hope that she can prevent him from becoming a knight. Thus when he leaves his mother's home, he is ignorant of the world beyond the one he knows. The author remarks that at the time of this traumatic family event, Perceval is two years old and still nursing. In his mother's account to him of the loss of his father and brothers in childhood, she states:
> Et vos, qui petis estiiez,
>
> .II. molt biax freres aviiez;
>
> Petis estïez, alaitans,
>
> Peu aviiez plus de .ii. ans. (vv. 455-8)
>
> [And you, who were little,
>
> You had two very handsome brothers;
>
> You were little and still at the breast,
>
> You were barely older than two years.]
Chrétien seems to be highlighting the tragedy of such a young child losing his father and siblings to heighten the pathos of the passage. A psychoanalytic reading might point to the traumatic impact on Perceval of this loss, and the consequences for his subsequent development. In the context of reading Perceval through an approach that focuses on ToM, it is relevant to observe that autism is often recognized in children of around the age of two through behavioral patterns.14 Secondly, the age difference between Perceval and his brothers suggests that he was born to older parents, and studies have recently associated a higher risk of autism with advanced parental age (Croen, Fireman, and Grether 338-39).15
The third explanation within the romance of Perceval's atypical behavior relies on a racial explanation, which is tantamount to saying that he is odd because he is Welsh. This explanation surfaces in a conversation between two knights after they have encountered Perceval near his home at the beginning of the romance. The knight who has engaged Perceval in conversation gives the following assessment of Perceval:
> "Il ne set pas totes les lois,"
>
> Fait li sire, "se Diex m'amant,
>
> C'a rien nule que li demant
>
> Ne me respont il onques a droit,
>
> Ains demande de quanqu'il voit
>
> Coment a non e c'on en fait."(vv. 236-41)
>
> ["He doesn't know the rules of conduct,"
>
> Said the lord, "for the love of God,
>
> Because he does not ever reply to anything
>
> I ask him properly, but
>
> He asks the name of everything he sees
>
> And what it's used for."]
To this, his knight companion offers the following explanation:
> "Sire, sachiez tot entresait
>
> Que Galois sont tot par nature
>
> Plus fol que bestes en pasture;
>
> Cist est ausi come une beste." (vv. 236-45)
>
> ["Lord, know straight up front
>
> That the Welsh are all, by nature,
>
> More stupid than a beast in pasture;
>
> This one is just like a beast."]
These three explanations (stupidity or naïveté, education, and race) spill over into scholarship on the _Conte del Graal_ , with the exception of the racial explanation, which is re-interpreted as cultural difference. While all of these perspectives on Perceval's character and behavior have some validity, they do not, either individually, or accumulatively, account fully for Perceval's actions across the romance. This has resulted in readings that explain awkwardly individual scenes in relation to the work as a whole, which might be more straightforwardly interpreted by recognizing Perceval's difficulties with ToM. Before returning to this point, it is important to establish which scenes suggest that Perceval's ToM is deficient. Two scenes especially illustrate Perceval's difficulties in understanding: the scene in which Perceval encounters knights for the very first time (vv. 69-342), and the narration of his experience with the damsel in tent (vv. 635-781). The first scene also includes a brief perspective on the ToM of the knights, as one of their group attributes a state of mind to Perceval, in stark contrast to Perceval, who is apparently incapable of such mind reading.
After the prologue, the romance opens with a description of Perceval riding across his mother's land. He hears the discordant sound of knights in the forest and thinks that these must be the devils his mother had told him about. Then he sees the knights emerging from the forest with their armor shining in the sun and jumps to the immediate conclusion that these are angels, since his mother has told him that angels are the most beautiful creatures of all. He falls to his knees in prayer, whereupon the principal knight of the group, attributing an emotional state to Perceval, advises the others to stay back because a boy has fallen to the ground in fear ("A terre est de paor cheüs" v. 161), and he may die of fear if they approach. The knights approach him since they want to ask him if he has seen five knights and three maidens pass this way. The oddity of the conversation that then ensues between him and one of the knights exemplifies Perceval's social awkwardness and complete lack of communicative skills, which lead the knights to attribute his foolishness to his identity as a Welshman, as we have seen above. Perceval pays no heed to the conversation. He does not answer the knight's questions, and instead, referring to the knight's equipment repeatedly asks him questions of the type: "Ce que est et de coi vos sert?" "What's that? What's it for?" (v. 214).
This inability to engage in conversation resurfaces when his mother, realizing that her son now has knowledge of the knightly culture from which she has successfully shielded him, decides to tell him about his knightly heritage and how his brothers and father were knights and were killed in combat.16 She also describes her emotional state then and now. However, Perceval is detached, he does not attribute emotion to his mother (i.e., he does not read her mind), and he does not respond appropriately in the context of the conversation. He simply interrupts and demands food:
> Li vallés entent molt petit
>
> A che que sa mere li dist.
>
> "A mangier, fait il," me donez (vv. 489-91)
>
> [The young man is hardly listening at all
>
> To what his mother is saying to him.
>
> "Give me something to eat," he says]
This indifference, or inability to attribute a mental state to his mother, is re-emphasized in the scene in which Perceval leaves his home and mother in search of knighthood:
> Quant li vallés fu eslongiez
>
> Le get d'une pierre menue,
>
> Si se regarde et voit cheüe
>
> Sa mere al pié del pont arriere,
>
> Et jut pasmee en tel maniere
>
> Com s'ele fust cheüe morte. (vv. 620-5)
>
> [When the young man was at a stone's
>
> throw distance
>
> he looked and saw his mother back there
>
> at the foot of the bridge
>
> and she was lying in a faint in such a way
>
> that it was as if she had fallen down dead.]
The use of the phrases "en tel maniere" (v. 624 "in such a way") and "com s'ele" (v. 625 "it was as if") preserve a necessary ambiguity that is exploited at the end of the narrative when we learn that Perceval's mother had not just fainted, but had died precisely at this moment and that he is subsequently considered to be guilty of her death because of the emotional pain his departure had caused her to suffer. However, even in circumstances in which his mother is not dead, the total lack of empathy is disturbing. The subsequent verses describe how Perceval whips his horse with a switch and speeds away.
Before he leaves in search of Arthur's court, Perceval's mother gives him advice about how to treat women, how to deal with men he meets, and how to pray. She is clearly concerned about her son, and indicates an awareness of his difficulties in social situations:
> Ce que vos ne feïstes onques,
>
> Ne autrui nel veïstes faire,
>
> Coment en sarez a chief traire? (vv. 518-20)
>
> [How will you know how to accomplish
>
> That which you have never done,
>
> Nor seen anyone else do?]
Perceval's mother's concern highlights the basic principles of what is known as a simulation ToM, which holds that ordinary people discern the mental states of others by trying to replicate or emulate them. This is in distinction to theory theory, which proposes that people construct or are endowed with a naïve psychological theory that guides their assignment of mental states, or the rationality theory, which suggests that people map others' thoughts by assuming that all follow the same postulate of rationality.17 The mother is ostensibly referring to knightly conduct, and how her son will ever know how to be a knight when he has never been a knight, nor seen anyone act as a knight. However, her question goes to the very heart of Perceval's problem with ToM. He cannot ever truly fit into the knightly culture because his low level of function in mind reading does not allow him to emulate and replicate the mental states of others by which process he might better understand how to engage with the community to which he aspires. It is thus significant that throughout most of the romance Perceval is wandering alone through the country. He cannot assign a mental state or intention to another because he has difficulty in emulating the mental states of others. Thus to paraphrase his mother, how can he be a knight if he cannot simulate something he has never been and thus understand others by trying on their mental states? Perceval's inability to read minds properly is illustrated in the following scene in which he tries to put into practice what he has misunderstood his mother to mean.
When Perceval comes upon a woman in a tent (vv. 635-781), he tries to follow the advice his mother has given him before his departure on how to treat women (vv. 532-56). The advice and Perceval's literal application of the advice is repeatedly underlined throughout the scene by Perceval's references to his mother: for example, "Si com ma mere le m'aprist" "Just as my mother taught me (v. 683); "que ma mere le m'enseigna" "as my mother taught me" (v. 695). The relationship between the mother's advice and how Perceval applies this advice to real situations illustrates beautifully the role of ToM in communication. As Simon Baron-Cohen, basing his statement on the theorists Grice, Sperber, Wilson, and Austin, explains:
> . . . the key thing we do as we search for the meaning of the words is imagine what the speaker's communicative intention might be. That is, we ask ourselves "What does he mean?" Here the word "mean" essentially boils down to "intend me to understand."( _Mindblindness_ 27)
Perceval takes his mother's advice extremely literally, to the point of terrifying a young woman whom he kisses and from whose finger he forcibly takes a ring. While a rape does not occur, the use of phrases such as "volsist ele ou non" "whether she wanted or not" (v. 708) or the phrase "a force" "by force" (v. 718, v. 720) suggests a violent scene in which the woman is stretched out beneath Perceval and kissed forcibly.18 Then while the woman is crying and wringing her hands (vv. 756-64), he makes a hearty meal of the food and wine in the tent.
Clearly Perceval has not read his mother's mind properly, and has not discerned what she intended him to understand, but he has followed her advice to the letter and kissed the woman he encountered, without going further, and he has taken her ring as a token. Perceval's literalness is a hallmark of autism, and it results from an inability to represent the minds of others, or in other words, from a deficient ToM. Temple Grandin, who is probably the most famous person with Asperger's syndrome, has described in her autobiography how she used to believe that other people were telepathic because they magically understood the meaning behind words and the intentions of others ( _Thinking in Pictures_ ). Not only has Perceval not understood his mother's intention in the advice she gives him, but he is completely oblivious to the mental state of the woman he finds and assaults in the tent.
These scenes suggest persuasively that Perceval's skills in ToM resemble those of an autistic individual, but beyond problems with ToM, there are of course other indicators of autistic spectrum disorder, which should also be considered. Analysis of Perceval's ToM in the scenes reviewed thus far has already yielded several of the diagnostic criteria for autism: a) he is unaware of others' feelings; b) he appears not to hear at times; c) he repeats words or phrases verbatim but does not understand how to use them; d) he cannot easily start a conversation or keep one going; and e) he is unusually sensitive to light, sound, and touch (the thunderous sound of the knights' arrival and the sun reflecting on their armor have such an impact on the young boy that he believes them to be heavenly).19 Other signs of autism in Perceval, which will emerge from subsequent analysis of scenes in which ToM is important, include: a) a propensity to retreat into a world of one's own; and b) constant movement (in the sense that he never stays in one place for long).
One of the less frequently cited characteristics of autistic spectrum disorder is a difference in visual processing, which is often manifest in a superior grasp of engineering, mathematical, or geometric problems:
> Increasing evidence show an atypical visual processing style in autism. This includes the processing of details in priority, in opposition to controls for whom the global context is most salient. This particular processing style in autism has been named weak central coherence (eg. Frith 1989) and it has been suggested that this capacity to focus on local elements may bring about superior performance in various domains, notably in visual tasks. (Rondan and Deruelle 197)
Autistic individuals will engage in local rather than global processing of facial profiles (Rondan and Deruelle 198). In other words, rather than analyzing a whole impression of a face, it is more likely that they will perform a feature by feature analysis. Perceval demonstrates this tendency toward weak central coherence in response to questions about where he has been and where he is going. Instead of giving a general, functional description, Perceval focuses somewhat obsessively on details.
In the first example, Blanchefleur has just asked Perceval from whence he has traveled that day. Perceval's reply is striking in its emphasis on the architectural structure of the castle where he has stayed, and its lack of generalization either in terms of location, name, or description that would help Blanchefleur identify it:
> Damoiselle, fait il, je gui
>
> Chiez un preudome a un chastel
>
> Ou j'oi hostel et bon et bel;
>
> S'i a cinc tors fors et ellites,
>
> Une grant et quatre petites;
>
> Si sai tote l'oevre assomer,
>
> Mais le chastel ne sai nomer;
>
> Et si sai bien que li preudom
>
> Gornemans de Gorhaut a non. (vv. 1884-92)
>
> [Lady, he says, I stayed
>
> With a nobleman in a castle
>
> Where I received great hospitality;
>
> It has five strong, tall towers,
>
> One big one and four small ones;
>
> And I can reckon the whole edifice,
>
> But I can't name the castle;
>
> And I know that the nobleman's
>
> Name is Gornemant of Gorhaut.]
In the second example, Perceval is again trying to describe the location of a castle, but this time with the explicit intention of giving directions to a knight whom he has defeated in combat and whom he is dispatching as a prisoner. Once again the description is more local than global, focusing on detail to such an extent that the narrator comments that a mason could not have described the castle more effectively:
> Et lors li dist cil que il aille
>
> A un chastel, chiez un preudome;
>
> Del preudome le non li nome.
>
> En tot le monde n'a maçon
>
> Qui mix devisast la façon
>
> Del chastel qu'il li devisa;
>
> L'eve et le pont molt li prisa,
>
> Et les torneles et la tour
>
> Et les murs fors qui sont entor,
>
> Tant que cil ot tres bien et set
>
> Que en liu ou on plus le het
>
> Ke velt envoier en prison. (vv. 2292-2303)
>
> [And then he told him that he should go to
>
> A castle, the home of a nobleman
>
> And he gave him the name of the nobleman.
>
> There is not a mason anywhere in the world
>
> Who could have outlined better the aspect of the castle
>
> He described to him.
>
> He praised the water and the bridge
>
> And the small towers, and the tower
>
> And the strong surrounding walls,
>
> To such an extent that that man heard and knew very well
>
> That he wanted to send him as a prisoner
>
> To the place where he was most hated.]
This geometrically oriented perspective is rendered exceptionally well by the set of Eric Rohmer's film, which eschews realism for stark shapes and contrasts. While Rohmer is not as explicit as Robert Wilson in making an explicit association with autism, his sets and framing render eerily perspectives of a weak central coherence.
I return now to the issue I raised previously about how the three explanations (physical, social, and racial) for Perceval's odd behavior, given by the text, and adopted to a greater or lesser degree in criticism, do not offer a satisfactory global reading of the romance. The logic of the social explanation demands that we recognize a development in Perceval, and this is precisely what we find in criticism. For example, the romance has been described as "the story of his education" in which the naïve youngster of the _enfances_ or childhood episodes of the romance matures ( _The Arthurian Handbook_ 344). It is generally accepted that Perceval develops during the course of the romance and that he makes progress in knighthood, love, and religion, in this order.20 Penny Simons who sees Perceval as "naturally curious and disposed to ask questions" (9) attributes his problems to his education:
> Part, at least, of Perceval's difficulty arises from the fact that his education has been sadly deficient. A deficient education will produce a deficient human being as skills do not come naturally, no matter how great natural aptitudes. (8)
We can argue for a degree of development in Perceval. He does leave behind his rustic life in isolation with his mother and is more or less successfully initiated into knightly culture. To some extent this perception is the consequence of the social explanation of Perceval's behavior. In other words, he has to be seen to develop, once he leaves the social isolation that is seemingly responsible for his behavior, and enters society. However, by the middle of the extant romance, he is still exhibiting strange behavior, illustrated clearly in three of the most famous, and frequently analyzed scenes of the romance, which I discuss here in the following order: a) Perceval's contemplation of the three drops of blood on the snow (vv. 4162-236); b) Perceval's encounter with the pilgrims on Good Friday (vv. 6217-314); and c) Perceval's failure to ask the relevant, necessary questions about the Grail procession in the castle of the Fisher King (vv. 3191-311).
In the episode known as the scene of the three drops of blood on the snow, Perceval suddenly becomes obsessively withdrawn. One morning as Perceval is out riding in a snow-covered landscape, he sees a falcon attack one of a flock of wild geese. From the wounded goose's neck three drops of blood fall to the snow and spread out, and upon seeing this, Perceval is reminded of the white and rose complexion of his beloved, Blanchefleur. This causes him to fall into a trance, which the text describes using the Old French verb "s'oblier" (v. 4202), which is translated as "to be carried away" or "to become distracted" but which literally means "to forget oneself." He spends all morning gazing at the drops of blood, and the squires who find him run to tell the king that there is a knight outside dozing on his horse. When Sagremor rides up to Perceval and demands that he come to the king, we are told that Perceval does not move, and appears not to hear him. Perceval does not return to awareness until Sagremor, after challenging him, is charging toward him tilting his lance. Perceval's reverie over the three drops of blood on the snow is usually interpreted as symbolic of his initiation into the world of love, and of chivalry motivated by love (e.g., Poirion 155; Henri Rey-Flaud 23), but when it is considered in the context of the scenes discussed above, it appears to belong to an emerging pattern of behavior.21 While the progression interpretation would suggest that Perceval's behavior improves as the romance proceeds, and he learns more about the world and knightly culture, in fact his behavior in the scene of the three drops of blood on the snow is markedly more troublesome. In earlier scenes, he is isolated because he is unable to engage in meaningful social exchanges, since he has difficulties in discerning the mental states and intentions of others. Here Perceval is completely withdrawn and absorbed with his own thoughts. This is where the social explanation of his atypical behavior in the opening scenes fails, since in spite of his acculturation, his behavior remains atypical, while an interpretation based on cognitive deficit remains valid for both the earlier and later scenes.
A second example of atypical behavior toward the end of the romance involves Perceval's encounter with a group of pilgrims on Good Friday (vv. 6217-314). Once again the narrator emphasizes a mental decline, and once again Perceval is withdrawn and indifferent. He has lost his memory, he has been wandering for five years, and he doesn't know which day it is. This scene is often juxtaposed to the scene in which Perceval first encounters the knights and believes them to be angels, and underscores a consistency in cognitive deficiency that is first evident in the parallel opening scenes of _Le Conte del Graal_. If Perceval has acquired knowledge between these scenes, it is clear that in the later scenes he is still exhibiting behavior suggestive of a cognitive deficiency.
The most famous scene of Chrétien's romance, and the one that inaugurates the tradition of the Grail quest in literary tradition, is also one of the most informative in an analysis of the behavioral characteristics of a possibly autistic Perceval. In addition to demonstrating further that Perceval's naïveté of the opening scenes is significantly more complex than a temporary lack of _savoir-faire_ in the knightly culture, reading this passage in view of the scenes discussed above that suggest cognitive deficiency in Perceval casts new light on an obscure and enigmatic episode of the romance.
At the Grail Castle of the Fisher King, Perceval first glimpses the bleeding lance and grail that pass before him repeatedly in procession. The central issue in this part of the narrative is that Perceval fails to ask the key question about the grail procession, which he must ask if the Fisher King is to be healed. There is a considerable body of scholarship on this issue of the unasked question.22 Ann McCullough has recently argued that the foregrounding of the question in a ceremonial context brings to mind the ritual questions which are part of the Passover seder: for example, "Why is this night different from other nights?" These questions are posed by the youngest present at the seder, and thus McCullough draws a comparison between the purportedly naïve Perceval and children present at a seder (51-2). Another frequently cited explanation, and one that appears to be supported by the text is that Perceval does not ask the question because he remembers advice given to him by a patron named Gornemant earlier in the text that he should not talk too much (McCullough 53-4). Yet another explanation within the romance itself, and cited often in criticism, is that Perceval's failure to ask the question is a punishment imposed upon him in retribution for his mother's death (McCullough 55). This explanation is given both by the Demoiselle Hideuse whom Perceval meets shortly after the Grail episode, and the hermit who hears Perceval's confession after he meets the pilgrims on Good Friday (vv. 6392-8). Explaining his failure to ask the questions, the Demoiselle Hideuse says:
> Por lo pechié, ce saiches tu,
>
> De ta mere t'est avenue,
>
> Qui est morte de doel de toi. (vv. 3593-5)
>
> [Know that this happened to you
>
> Because of the sin committed against your mother,
>
> Who died sorrowing for you.]
Jean Frappier's insightful observation that Perceval notices the brightness of the Grail as it passes before him, but yet fails to ask the question that would reflect consciousness of the Fisher King's suffering anticipates Perceval's deficient capabilities in mind reading before the concept of ToM had been named and recognized ( _Le Roman breton_ 54-5).
The correlation between Gornemant's advice and Perceval's silence, and between his mother's death and his silence, is not as straightforward as the criticism on the romance, or the romance itself, may suggest. In light of the evidence presented here, relating to Perceval's difficulties with ToM and his autistic characteristics, it is possible that Perceval does not ask the question simply because he is heeding advice, but rather because he is taking the advice too literally, in the same way in which he took his mother's advice about how to treat women too literally. He is not struck silent in retribution for his mother's death, but rather the lack of ability to read his mother's gestures and her mental state is the same cognitive deficiency that prevents him from engaging in the social exchange of question and answer, which is so dependent on ToM. Williams, Donley, and Keller have conducted research into question-asking in autistic children, and specifically how to teach autistic children to ask questions about hidden objects. They state:
> Most children with autism fail to engage in typical social interactions. For example, often they are not skilled in asking questions (Charlop & Milstein, 1989). According to Charlop and Milstein, asking questions must often be explicitly taught. Researchers have demonstrated recently that behavioral techniques are effective in teaching children with autism to ask questions. For example, Taylor and Harris (1995) taught young children to ask "What's that?" when presented novel pictures in a classroom, and then when encountering new objects on a walk in the school building. Similarly, Koegel, Camarata, Valdez-Menchaca, and Koegel (1998) taught children to ask "What's that?" in training and nontraining settings with novel items as reinforcers. (627)
In light of this study Perceval's failure to ask the question essential to the Fisher King can be understood as yet another instance of a general cognitive deficiency that is so marked in the protagonist throughout the romance.
While we may not be prepared to go as far as identifying an autistic individual in Chrétien de Troyes's Perceval, the analysis of these episodes of the romance, well known by both medieval and modern audiences, suggest an interest on the part of the author in what we now call ToM. One of the advantages of considering Perceval as autistic is that this description is relevant to many scenes that in the past have been interpreted separately, using in each case quite different models to explain Perceval's behavior. In approaching Chrétien de Troyes's story of Perceval from the perspective of ToM, we highlight the importance in this romance of communication between the characters, and, more significantly, the failure of that communication.
### NOTES
1 The tradition is rich and too complex to adduce here completely. I refer throughout this article to various versions of the Perceval legend where it is relevant to do so. For an overview of medieval texts relating to the Grail legend, see the comparative table in Mahoney's _The Grail: A Casebook_ (101), and the narrative comparison in the introduction of this volume (1-78); for an overview of texts relating specifically to Perceval, see the essays in Groos and Lacy's _Perceval = Parzival: A Casebook_ ; for a more general, but comprehensive, tabulated chronology of Arthurian literature throughout the centuries, see _The Arthurian Handbook_ (eds. Lacy, Ashe, and Mancoff), xviii-xxxv. Examples of modern films that feature the character of Perceval are as follows: _Monty Python and the Holy Grail_ (dirs. Terry Gilliam and Terry Jones, 1975); _Perceval le Gallois_ (dir. Eric Rohmer, 1978); _Excalibur_ (dir. John Boorman, 1981); _The Natural_ (dir. Barry Levinson, 1984), an adaptation of Bernard Malamud's novel of the same name, which models a gifted baseball player who plays for the New York Knights (managed by Pop Fisher) on Perceval; and a new film by Christophe Mavroudis, _Perceval_ , which is based, like Rohmer's film, on the text of Chrétien's romance, is currently in production. I refer throughout to William Roach's edition of Chrétien's romance. Translations from the Old French are my own.
2 The date of the romance has been the subject of much discussion. While the romance was dated to 1177-81 and 1182-83 by Lejeune and Fourrier, respectively, during the 1950s, more recently Luttrell (30-32) and Diverres (97) have proposed the later dates of 1189-90 and 1188-91, associating the romance with the Third Crusade. In his 1990 edition, Charles Méla dates the poem to 1181-85.
3 There are four _Continuations_ of which the Second, signed "Gauchier de Donaing" (c. 1200), the Third by Manessier (c. 1230), and the Fourth by Gerbert de Montreuil (c. 1230) continue the story of Perceval, while the First (c. 1200) follows the story of Gauvain (Lacy, Ashe, and Mancoff,76-7).
4 Piglet accompanies Galahad to Castle Anthrax where in a clever comic take on Perceval's failure to ask the question that will save the Fisher King, Piglet declares "There's no grail here" to Galahad's "I have seen it! I have seen it!"
5 "An zweiter Stelle stehen Motive aus dem _Perceval_ : hier jedoch scheint es, als sei nur den Anfangsteil, nämlich Percevals _niceté_ , sein Abschied von der Mutter, die Begegnung mit den Rittern und sein erster Besuch am Artushof, als bleibende Eerinnerung in das literarische Bewusstsein des Publikums arthurischer Versromane eingegangen" (161).
6 Although Galahad is foreshadowed in the _Perlesvaus_ (1200-10), he is not named until the _Queste del Saint Graal_ (1215-35) of the Vulgate Cycle. Thereafter he becomes the prominent knight of the Grail quest, significantly in Sir Thomas Malory's fifteenth-century _Le Morte d'Arthur_. Galahad is the only knight to succeed in the quest for the Holy Grail.
7 I am grateful to Fritz Breithaupt (Indiana University) who told me about Wilson's casting of an autistic actor in the role of Perceval during the course of the Theory of Mind and Literature Conference at Purdue (November 2007). It is intriguing to ask if Eric Rohmer's version of Chrétien's romance also suggests autistic tendencies in the principal character. I return to this below in my discussion of weak central coherence.
8 Much of this reputation rests on the author's nuanced, extended interior monologues, which, for some, earn him the merit of being designated as the first French novelist. Robert Anacker, for example, dubs him "The First French Psychological Novelist." Jody Enders has analyzed memory and the psychology of the interior monologue in another of Chrétien's romances entitled _Cligés_. Robert A. Johnson, a Jungian analyst in private practice, has published a popular book which interprets Perceval and his search for the Grail as the story of a developmental masculine psychology. Barbara Nelson Sargent-Baur takes a more cognitive approach in her article on vision and cognition in the _Conte del Graal_.
9 For example, the books of Alan Richardson, Alan Palmer, and Lisa Zunshine.
10 Alvin I. Goldman uses the term "mentalizing" as an alternative to mind reading, ostensibly since it points to the process of representing others' mental states in our own minds: "Mentalizing may be the root of our elaborate social nature. Would there be language and discourse without mentalizing? Would the exquisitely coordinated enterprises of cultural life, the structures of love, politics, and games, be what they are without participants' attending to the mental states of others? Would there be a human sense of morality without an understanding of what others experience, of what their lives are or might be like?" (Goldman 3).
11 For conflicting conclusions about the ToM of primates, see B. Hare, J. Call, and M. Tomasello, "Do chimpanzees know what conspecifics know and do not know?" and D. J. Povinelli, K. E. Nelson, and S. T. Boysen, "Inferences about guessing and knowing by chimpanzees ( _Pan troglodytes_ )," _Journal of Comparative Psychology_ 104 (1990): 203-10.
12 The development of ToM manifests itself at around the same time as pretend play, which requires the ability to model the epistemic states of others and distinguish these from one own's epistemic state (Baron-Cohen, _Mindblindness_ 56).
13 Ann McCullough states: "Perceval's complete lack of social skills is a result of the strange education his mother gave him." (48). See also Claude Luttrell's "The Upbringing of Perceval Heroes."
14 While current research is trying to develop screening tools to detect autism in children younger than two, since improved outcomes are associated with early intervention (cf. the research of Rebecca Landa of the Kennedy Krieger Institute), in a survey reported in _Neurology_ in 2000, most parents recognized a problem by the time their child was eighteen months old and sought medical assistance by the age of 2, although the average age of official diagnosis was 6 years (Filipek, Accardo, Ashwal et al. 469).
15 Boys could enter service as a squire at a young age, but most would not be dubbed until adolescence.
16 "Del doel del fil morut li pere." "The father died of grief for his sons" (v. 481), but he has already sustained a serious injury from combat.
17 Cf. P. Carruthers and P. Smith, _Theories of Theory of Mind_. On simulation theory especially, see Gallese and Goldman.
18 Evelyn Birge Vitz, "Rereading Rape" (19). She argues that Chrétien is falsely accused of setting up scenes of rape to provide opportunities for knights to rescue damsels, and notes that there are no scenes of rape in any of his romances (6).
19 The authoritative source for the signs and symptoms of autism is the American Psychiatric Association's _Diagnostic and Statistical Manual of Mental Disorders_ , fourth edition, Washington DC: American Psychiatric Publishing, Inc., 1995. The signs and symptoms of autism are also widely published in generally available pediatric pamphlets, articles in the popular press, and Internet sites.
20 Alexandre Micha, "Le Perceval de Chrétien de Troyes (roman éducatif)"; Jean Frappier writes of "les apprentissages de Perceval....L'initiation à la chevalerie est au savoir-vivre est suivie d'une initiation aux délices sentimentales de l'amour courtois, puis d'une initiation plus difficile et plus lente à la vie spirituelle" ( _Chrétien de Troyes et le mythe du Graal_ 68).
21 A notable exception is Peggy McCracken's reading of the scene through its corollary in the _Perlesvaus_. Following Poirion's insight that the scene should be read in conjunction with the scene in which Perceval encounters the lady in the tent, which we have discussed above, McCracken suggests reading the scene as symbolic of "a myth of social order that is grounded on the sacrifice of women" (168).
22 For example: Carolyn Whitson, "Why Does the Lance Bleed? Whom Does the Grail Serve? Unasked Questions from a Working-Class Education"; Anna-Marie Ferguson, "Percivale and the Grail: Always Ask the Foolish Question"; Michel Stanesco, "Le Secret du Graal et la voie interrogative"; T. J. Cherian, "To Ask or Not to Ask the Question: East-West Encounters in the Perceval Legend"; Anton Janko, "Parzivals Fragen"; Harry F. Williams, "The Unasked Questions in the _Conte del Graal_."
### ACKNOWLEDGMENTS
I would like to thank my colleagues Howard Mancing, Rich Schweickert, and Jen William for reading an earlier draft of this paper and offering perceptive commentary and suggestions.
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## Whose Mind's Eye? Free Indirect Discourse and the Covert Narrator in Marlene Streeruwitz's _Nachwelt_
JENNIFER MARSTON WILLIAM
_I can see it in my mind's eye_. Whether in reference to a fully imagined or a remembered image, this common expression reveals how the mind's workings are often conceptualized. In this simplified view of cognition, the mind becomes analogous to an optical system with a separate "eye" mechanism that processes experience and makes imagination and recollection possible. This metaphor implies a uniquely subjective standpoint; a mind's-eye view can be described, but not directly shared. This essay explores the metaphor of the mind's eye1 in relation to the phenomenon of free indirect discourse. By way of example I consult Marlene Streeruwitz's _Nachwelt_ , a novel that relies heavily on free indirect discourse (FID)2 as a means of conveying its protagonist's thoughts and emotions. Starting with the premise of Ann Banfield's revolutionary theory of "unspeakable sentences," with its emphasis on the voicelessness of FID utterances, I argue for a further reconceptualization of FID by examining it in light of the mind's eye metaphor. This approach leads to a more productive reading of the narrative dynamics of Streeruwitz's novel in particular, and to a greater understanding in general of how a character's thoughts are conveyed through language in a third-person text. The following demonstrates how deictic shifts (which I will analogize as the focal shift from one mind's eye to another) in a third-person narrative need not indicate the narrator's Theory of Mind (ToM) or the characters' thoughts as filtered through the narrator, as conventional and intuitive conceptions of FID would imply. Rather, FID points toward the author's temporary silencing of the narrator, providing the reader with more direct glimpses into the protagonist's mind. The reader is thereby given an opportunity to practice mind reading without the narrator's involvement.
Although contemporary narratologists have offered numerous definitions for FID, the technique is best illustrated through specific instances from literature. Since Gustave Flaubert was one of the earliest authors who relied on FID, a passage from his novel _Madame Bovary_ serves as an appropriate example (FID is italicized):
> Ce qui l'exaspérait, c'est que Charles n'avait pas l'air de se douter de son supplice. La conviction où il était de la rendre heureuse lui semblait une insulte imbécile, et sa sécurité là-dessus de l'ingratitude. _Pour qui donc était-elle sage? N'était-il pas, lui, l'obstacle à toute félicité, la cause de toute misere, et comme l'ardillon pointu de cette courroie complexe qui la bouclait de tous côtés?_ (145)
>
> [What exasperated her was that Charles did not seem to be aware of her torment. His conviction that he was making her happy looked to her a stupid insult, and his self-assurance on this point sheer ingratitude. _For whom, then, was she being virtuous? Was it not for him, the obstacle to all happiness, the cause of all misery, and, as it were, the sharp clasp of that complex strap that buckled her in all sides?_ ] (77)
The first two statements of this passage are clearly related by the narrator, who is reporting on the emotions of the desperate Emma Bovary. There is an apparent shift in the two questions that follow, however; while conceivably these queries could stem from the narrator who directs them toward the reader, it is much more likely that the questions depict Emma's own state of mind—these are the questions she is asking herself as she reflects on the marriage she finds so constraining. Flaubert uses FID here to temporarily diverge from the narrator's report, taking us directly "into the head" of Emma instead.
Not surprisingly, given its frequent ambiguities, many scholars have issued caveats about relying on FID as a useful category for analysis of literary discourse. In her valuable work _The Fictions of Language and the Languages of Fiction_ , for example, Monika Fludernik comprehensively outlines problems with the tripartite model of viewing narrative communication as consisting of the two poles of direct and indirect discourse, with FID situated somewhere between the two. She also reviews and extends the criticisms from Banfield and others of Roy Pascal's celebrated theory of a "dual voice" that allegedly combines the voices of narrator and literary characters (see Fludernik 322 ff.).3 In his path-breaking book _Fictional Minds_ , Alan Palmer asserts that speech categories like FID—which, as he points out, is often misidentified—are not especially useful in analyzing presentations of fictional thought, and he notes that "the high regard in which free indirect discourse is held can lead theorists into inaccuracy" (61). Such warnings are wise; the zeal to neatly classify complex linguistic structures can hinder rather than further narrative analyses. At the same time, we need a useful framework for better understanding and discussing FID, which holds an important key to the fictional minds created by a text's author. I propose that the mind's-eye metaphor4 helps illuminate the nuances and the very nature of FID, providing us in turn with greater insight into the disparate but interconnected Theories of Minds of readers, writers, and fictional figures.
A review of some commonly asserted characteristics of FID makes the connections between it and ToM immediately clear. Readers practice mind reading when engaged with a third-person narrative;5 an utterance in FID then provides them with some confirmation or denial of what they could previously only infer about characters' thoughts,
feelings, and motivations. FID, often less formal in style than that of the surrounding text, may be realized as a casual observation, an exclamation, or as a character's self-interrogation. The appearance of a question such as "What had she gotten herself into?" in a third-person narrative6 is often thought to reduce, if only temporarily, the mental proximity between the narrator and character in question, as well as between the reader and that character. FID need not signify a direct or indirect quotation, an important point to which I will return shortly, and it remains in the third person. But upon encountering FID, the reader senses having been granted a glimpse into the character's inner world. FID thus allows readers to shift their gaze from the external standpoint that the third person usually promotes to a more internal one.7 This viewpoint approaches but does not quite reach the first-person perspective, in part because the pronouns used tend to remain in the third person. Generally speaking, FID is employed to convey characters' thoughts that remain unuttered. As readers we are not quite "in the head" of the character—as we might believe to be while reading a first-person narration—we do not have enough access to the character's mental space to lay claim to a mind's eye view, although it might seem as if the narrator does have that privilege. However, conceding that the narrator retreats from narrating altogether at such times alleviates this kind of suspicion and allows us a greater appreciation of FID as the linguistic interpretation of the character's thoughts. This verbalization is not that of a mind-reading narrator, but simply that of the text's author. I propose that in considerations of third-person FID, we follow the lead of Ann Banfield in her radical divergence from the notion of an omnipresent and ever-narrating narrator:
> Represented speech and thought is neither an interpretation of the reported speech or thought which implies an evaluating speaker, nor a direct imitation or presentation of the quoted speaker's voice. Instead, the speech or thought of the SELF represented retains all its expressivity without suggesting that its grammatical form was that uttered by an original speaker, whether aloud or silently. (Banfield 108)8
In other words, we must remember that the representation of consciousness effected through FID is not the same as simply turning a direct utterance into an indirect one. One could perhaps argue for FID as the expression of the narrator's reading of the main character's mind. Yet it seems that an author relying on her third-person narrator to this extent generally does so in the direct reportive form of "She thought . . . ," or through thought report.9 The author's decision to use FID instead of these forms indicates that s/he relieves the narrator of any reportive tasks at those moments, thereby shifting the narrator to a covert position within the narrative world, as the following analysis demonstrates.
Streeruwitz's novel _Nachwelt_ , published in 1999, provides fertile ground for exploring the nuances of FID. A highly autobiographical novel, the narrative perspective toggles between straight third person and third-person FID. Streeruwitz's preference for a third-person over a first-person narrator reinforces the distinction between her narrator's voice and her own, despite the many details in the novel that overlap with the author's biography. _Nachwelt_ documents ten days in the life of Austrian dramaturge Margarethe Doblinger, who travels to Los Angeles to conduct research and interviews for a biography of the émigré sculptor Anna Mahler, the daughter of composers Gustav Mahler and Alma Mahler-Werfel. Instead of concentrating on this project, Margarethe becomes absorbed with her personal life, particularly relationship troubles and various health concerns. The account of the protagonist's trip begins on 1 March 1990, but the novel pays only minimal attention to this significant historical period at the Cold War's end, as the preoccupied protagonist skims newspaper articles or listens halfheartedly to newscasts about the watershed events happening in Europe. The mentions of these news items are but one component of the novel's multilayered narration, which also includes Margarethe's personal stories, conveyed primarily via FID, as well as details about Anna Mahler's life, as reported in first-person voice by sources in varying degrees of proximity to the artist. Margarethe originally plans to incorporate the fragmented stories from these primary sources into Mahler's biography, but she quickly grows weary and wary of her role as a mediator between past and present, ultimately casting the project aside as futile and mercenary. Meanwhile, she is distracted by, and at times obsessed with, the complexities of her personal life, especially a deteriorating relationship with her partner back in Austria.
Author Marlene Streeruwitz has quite a bit in common with her protagonist Margarethe Doblinger, even beyond their hometown of Vienna and the first three letters of their forenames. Like Streeruwitz, a dramatist with a number of successful plays brought to the stage over the past fifteen years, Margarethe feels at home in the realm of theater. But also like Streeruwitz, her intellectual and creative pursuits are not confined to the theatrical. Before writing _Nachwelt_ , Streeruwitz had also attempted to write Anna Mahler's biography, for which she embarked on a research trip to Los Angeles, where the artist had resided after fleeing Europe during the Nazi period. But like her protagonist, Streeruwitz also abandoned this endeavor after realizing the impossibility of condensing a woman's life into biographical form. Margarethe's less than sympathetic depiction prompts questions about the possible connections between these autobiographical aspects and the author's decisions in regard to narrative perspective. It would have been feasible, and arguably easier, for Streeruwitz to portray the innermost thoughts of Margarethe through a first-person voice. Yet the author opted instead for a style that underscores the division between author and protagonist, while providing considerable insight into the mental world of the latter.
By virtue of its frequent and at times muddled deictic shifts, _Nachwelt_ demonstrates the potential of FID to both convey a literary character's thoughts and ToM as conceived by the author, and to complicate clear-cut distinctions between the various minds of a fictional world (e.g., narrator versus character whose story is narrated). The novel's opening invokes immediate uncertainty regarding the narrative perspective. Because the first section is headed with the boldfaced date of Thursday, 1 March 1990, the reader's reasonable expectation might be that a first-person journal entry is to follow. The first several sentences provide no evidence to the contrary: "Im Haus war es warm. Roch nach Krankheit. Desinfektionsmittel und Urin. Stechend. Und süß" (7) ["In the house it was warm. Smelled like illness. Disinfectant and urine. Pungent. And sweet"].10 The particular subjectivity of the senses evoked in the descriptive opening statements, namely the very proximal senses of touch and smell, suggests that these are Margarethe's own perceptions, and not those of the third-person narrator. Indeed, these opening impressions need not be understood as presented by the narrator, who does not seem to appear until the sixth sentence, "Manon führte sie durch Zimmer" ["Manon led her through rooms"]. Here it becomes clear that the narrator is not of the first-person variety, as Margarethe is introduced to the reader via the third-person feminine object pronoun _sie_. In retrospect, the novel's first five sentences appear to have been written in FID through Margarethe's mind's eye, and not until the sixth sentence does the deictic center shift to that of the narrator.
Streeruwitz conveys Margarethe's impressions of Los Angeles in a similarly direct fashion through her tendency toward strings of short utterances. We are provided with a list of sites passed by Margarethe as she drives around in unfamiliar territory:
> Sie fuhr Venice Boulevard bis Sepulveda. Holzhäuser. Einfamilienhäuser. Tank-stellen. Burger Kings. Bestattungsinstitute. Kirchen. Motels. Holzhäuser. Keine Bäume. Die Büsche staubig und grau. Kaum Menschen unterwegs. Autos. Aber nur wenige. Die Sonne hinter einem Wolkenschleier. Das Licht blendete. Sie fuhr kurz auf Pico. (128)
>
> [She drove Venice Boulevard to Sepulveda. Wooden houses. Single-family houses. Gas stations. Burger Kings. Undertakers. Churches. Motels. Wooden houses. No trees. The bushes dusty and gray. Hardly any people on the way. Cars. But only few. The sun behind a veil of clouds. The light dazzled. She drove quickly onto Pico.]
The reader gleans some insight here into the state of Margarethe's mind, which at this point is not formulating concrete thoughts as much as it is accumulating impressions. The nearly complete lack of verbs in this passage enhances the immediacy of the listed images, and makes them appear to the reader not as narrative descriptions, but as flashes of the protagonist's consciousness. While we receive no explicit commentary on her reactions to the succession of buildings and desolate surroundings, we sense implicitly Margarethe's feelings of alienation as she is immersed in this new environment. Whether this type of stream-of-consciousness can be classified strictly as FID is up for debate, but more significant to the present discussion is that the third-person narrator seems to disappear in between the two framing statements ("She drove . . ."). A narrator is not necessary (and perhaps would be unable) to mediate these observations from Margarethe's mind's eye; indeed, the role of the narrator in this novel is reduced to providing fairly bland statements of actions—there are no instances of narrative thought report, as Streeruwitz relies only on FID to convey the protagonist's ruminations.
The deictic shifts between the third-person narrator and protagonist Margarethe are even more pronounced in those statements that would be more readily classified as FID. The following passage is representative in its revelation of Margarethe's thought processes as she procrastinates on her research project and anxiously attempts to pass the time in her Los Angeles apartment:
> Und warum saß sie allein. Hier. Warum war er nun wirklich nicht mitgekommen. War sie schon wieder dabei, sich von ihm etwas einreden zu lassen. Sie hätte ihm die Telefonnummer nicht geben sollen. Zwei Wochen nicht anrufen. Nicht reden können. Aber das hätte sie wieder nicht fertiggebracht. Es war nach neun Uhr. Was sollte sie hier tun. Konnte man wirklich nicht hinaus. Sie konnte doch nicht jeden Abend nach Einbruch der Dunkelheit hier herumsitzen. (17)
>
> [And why was she sitting alone. Here. Why had he actually not come along. Was she already prepared to be persuaded by him. She should not have given him the telephone number. Two weeks not calling. Not able to talk. But she would not have been able to manage that either. It was after nine o'clock. What should she do here. Could one really not get out. She could not just sit around here every evening after nightfall.]
Such long passages of FID permeate the novel, interspersed with third-person narrative reports comprised largely of trivial details about mundane activities. Streeruwitz stresses the protagonist's temporal and spatial position through repeated deictic references, such as two instances of _wieder_ (again), and three instances of _hier_ (here).11 As
Fludernik notes, "Deictic expressions (including their syntactic equivalents)—in so far as they are _shifters_ or indicative of a SELF's point of view—are intrinsically 'subjective'"; they often "refer to a SELF's referential deictic centre" (431). Yet Fludernik claims the language itself, even when presented in FID, still marks "the language of the current speaker or text" (432). In _Nachwelt_ , the language remains in third person, but the mind's eye represented is often clearly that of Margarethe. Thus it makes little sense to speak of a "third-person narrator" at those points containing FID, as it is clearly not the case that her thoughts are being filtered through the narrator's mind's eye.
Through FID characterized by deixis, and through other linguistic means,12 Streeruwitz shifts the deictic center frequently back to that of the protagonist's self. In the sample passage above, the statement-question "What should she do here" does not imply that the character necessarily said or thought the exact words, "What should I do here"; FID is, in such a case, an approximation of a thought process that need not have a direct verbal equivalent. It seems then that what is communicated via FID has nothing to do with the omniscient narrator's perspective on the character's thoughts.13 Rather, it signifies the author's attempt to translate thought, or at times ToM, into language. We need not assume the assertion of a narrative voice at junctures of FID; as Banfield demonstrates in her work, "Subjectivity is not dependent on the communicative act, even if it is _shown_ through language. And if it is not subordinated to the communicative function, then language can contain speakerless sentences" (70). A focus on the question of whose mind's eye is represented at a given narrative moment allows for a clear dissociation of FID and a narrative voice.
At times the reader of _Nachwelt_ must determine whether an utterance is intended as FID or third-person narrative, accentuating the different challenges involved in reading fictional minds as opposed to mind reading with an actual person. When presented with a question of character motivation, such as "And why was she sitting alone. Here" from the above passage, we rely on context and the text's particular narrative patterns to tell us whether it reflects the character's self-questioning, or a critical stance of the narrator that has nothing to do with the character's own thoughts. Given the premise that thought is generally not directly translatable into the written word (i.e., in all likelihood Margarethe was not thinking verbatim "Why am I sitting alone?"), FID becomes a convenient means of inviting us to read the literary figure's mind. As readers we receive the essence of the thought, but we are still required to fill in the gaps in regard to impetus and emotion behind it (e.g., Margarethe's loneliness and her apparent ambivalence toward going out). During this interim, the narrator becomes completely covert,14 even when the third-person pronoun remains as a reminder that this is not a first-person account, and the reader becomes privy to a more direct view of the fictional mind.
This determination of narrative perspective—that is, of knowing on whose mind's eye to focus at a given time and of sensing when the narrator is actually covert—might be easier in _Nachwelt_ than in many other third-person texts, due to the narrator's consistent lack of involvement in reporting Margarethe's thought processes throughout. Geoffrey Leech and Michael Short have pointed out the difficulties of distinguishing the "mind-styles"15 of narrator and literary characters, since "novelistic descriptions, having different discourses embedded inside one another often 'flow,' in a manner difficult to pin down, from say, a narratorial description, to a description where the narrator and the character's perspective are fused and back again" (Short 32). This kind of flow strikes me as a problem only if scholars of narrative choose to constitute it as one. That is, regardless of how we choose to label the discourse at a given textual moment, the fact remains that a fictional mind's eye guides our reading gaze at that moment, and the shifts between the mind's eye of the third-person narrator and that of characters are hardly unexpected by the mind-reading reader. Seeing through a literary mind's eye involves much more than looking at a narrative situation from another visual vantage point, as the metaphorical mind's eye encompasses the subjectivity of a character's prejudices, predispositions, and previous experiences. Deictic references like those found in the sample passage from _Nachwelt_ above are thus significant not only because they signify a particular spatial and temporal position occupied by the character, but because they remind us that the fictional mind dealing with relative concepts such as _here_ / _there_ and _now_ / _then_ also carries with it the implication of a different set of experiences from those of the narrator—and perhaps from those of the author.
As I suggested earlier, the prevalence of FID in this novel as well as its similarities to a personal diary beg the question of why Streeruwitz did not simply compose it in the first person. There are a few possible reasons for this stylistic choice. First, Streeruwitz has remarked in interviews and essays on her desire to find a new, anti-patriarchal, and anti-hierarchical language.16 Arguably, a unique literary style that ultimately privileges neither the narrator's nor the protagonist's mind would mark a start in that direction. Second, because her novel is so autobiographical, a logical presumption would be that Streeruwitz had hoped for her readers to view through _her_ mind's eye some of her experiences from Los Angeles while working on the Anna Mahler research project.17 At the same time, she might not be eager for readers to equate this often pathetic character Margarethe18 with herself—a fallacy more easily committed when faced with a first-person narration. In any case, the pervasiveness of FID does not tell the whole story about what makes Streeruwitz's novel work so well despite its concentration on the protagonist's mundane activities and procrastination tactics. Frequent shifts of the deictic center from narrator to protagonist allow Streeruwitz to provide the mind-reading reader with considerable access to Margarethe's thoughts and mental representations that remain unmediated by the narrator. The reader's mind-reading abilities are thus engaged by the lack of narrative hints regarding the motivations and emotions behind these thoughts.
The conclusions reached through this analysis of FID in _Nachwelt_ can be extrapolated to considerations of third-person narratives in general. First, insofar as the mind'seye view in represented thought is purely that of the literary character in question, FID implies the temporary silencing of the third-person narrator's voice. Sentences written in the style of FID are best regarded as "speakerless," as Banfield has argued, and the subjectivity inherent in the mind's eye metaphor helps to clarify both why this is the case and why it matters for literary studies. Second, accepting the possibility that even an overt narrator becomes covert in such sentences enables us to more comfortably view FID as a verbal approximation of characters' nonverbal thoughts and motivations; these expressions can be viewed as guideposts for the mind-reading reader. Finally, in analyzing literary works with strong autobiographical elements, a mind'seye view can illuminate the stylistic choices made by the author in the process of fictionalizing her own story.
### NOTES
1 See Lakoff and Turner, _More than Cool Reason_ , 151. See also Barnden's chapter in _Two Sciences of Mind_ , especially 317 ff., for a summary of the linguistic-philosophical debates surrounding the cognitive metaphors of COGNIZING AS SEEING and MIND AS PHYSICAL SPACE.
2 There are at least a dozen variations on the term describing this narrative phenomenon, and related ones, as Alan Palmer lists in his book _Fictional Minds_ (55). Besides "free indirect discourse" and "narrated monologue," the French and German terms _style indirect libre_ and _erlebte Rede_ are often used and left untranslated. While the phenomenon may have different implications depending on the language in which it is expressed, and debates abound among literary theorists regarding exact definitions, the general concept of a non-reportive style remains constant. I use the common and easily abbreviated term "FID" here, although I prefer Ann Banfield's more descriptive but more cumbersome term "represented speech and thought" (12). See Fludernik, especially chapter 2, for the history behind the terminology and details on previous studies of the phenomenon.
3 The dual voice concept, put forth by Pascal in 1977, is still endorsed by many narratologists in one form or another. For instance, Palmer describes "Free Indirect Thought" as a form that "combines the subjectivity and language of the character, as in direct thought, with the presentation of the narrator, as in thought report" (54).
4 Although in common usage "the mind's eye" tends to refer to visual mental imagery, here I expand the metaphor to describe an individual's perceptions of the external and internal world (using all the senses as well as ToM) and the recollection and retelling of those perceptions through narrative. It differs from "perspective" in that a mind's-eye view is wholly individual, whereas a perspective can be shared by two people (or multiple literary figures, or a narrator and literary characters).
5 Cf. Zunshine, _Why We Read Fiction_ , for a comprehensive analysis of this phenomenon.
6 FID is found in first and second-person narratives as well (see Fludernik 85), but here I will concentrate on the particular deictic shifts that result from the switch between third-person, omniscient narration and third-person FID representing a character's thoughts.
7 At the same time, the often abrupt appearance of FID marks an interruption in the reading process that may produce an alienating effect. The absorbed reader becomes momentarily conscious of the text's fictionality, and of the fact that s/he is reading and practicing mind reading with fictional characters rather than with real people.
8 Among other scholars, Palmer does not agree with Banfield's "dogmatic antiverbal approach"; he asserts that "sometimes thought is highly verbalized and can accurately be described as inner speech; sometimes it is not and so cannot" (65). Palmer's criticism is reasonable but not especially relevant for my purposes here, as I concentrate on the deictic shift that occurs with FID and renders the narrative voice silent, regardless of whether a character's thoughts are represented verbatim or in an approximated fashion. The main point I am borrowing from Banfield is that a sentence containing represented thought can be speakerless and need not be understood as conveyed by a narrator.
9 Also known as "psychonarration," thought report is a technique employed to "touch on states of consciousness or feeling that the character may be _unaware_ of (in which case this implies a narrator's external description) or of which s/he may only be dimly cognizant, and certainly not in the terms of description supplied by the narrative" (Fludernik 297).
10 Translations from _Nachwelt_ are my own.
11 The German adverb _hinaus_ in this passage is also notably deictic, as is its opposite _heraus_ ; both can be translated into English as "out," but the directional prefixes _hin_ \- (to) and _her_ \- (from) determine the spatial reference point.
12 For example, extreme syntactic reductionism and fragmentation that reflect Margarethe's distracted and frazzled thought patterns.
13 The complexities behind the concept of the narrator's mind warrant careful consideration as well, but begin to go beyond the scope of this essay. See Fludernik, 58 ff., for a summary of some debates regarding the essence of narrators in fictional texts.
14 Although "covert" and "overt" are common narratological terms to describe the level of active involvement from a narrator within a text, I am using the term "covert" here to describe the narrator specifically at the juncture of FID. While the narrator of _Nachwelt_ happens to be only minimally involved and thus largely covert throughout the novel, I am suggesting that all third-person narrators, even otherwise overt ones, become temporarily covert through FID. My own use of "covert" to describe the narrator's position during instances of FID also alludes loosely to Catherine Emmott's application of the term to literary characters within a primed contextual frame that are not being specifically referenced in a sentence, but which the reader still assumes to be present (cf. Emmott 132). To say that the narrator is "absent" altogether at such times seems too extreme, since the reader is hardly surprised when the narrator takes over again.
15 The term "mind-style" was coined by Roger Fowler in his book _Linguistics and the_ _Novel_ (1977) to describe the phenomenon of how worldviews are projected by the language of a text; the concept was developed further by Leech and Short in their work _Style in Fiction_. Mind-style characterizes "the reader's interpretation of certain consistent linguistic choices in novels as reflecting the way characters . . . perceive the world" (Short 31).
16 "Die bekannten Sprachen müssen verlassen werden. Der Subjekt/Objekt-Grammatik der Rücken gekehrt werden. Langsam. Schrittchenweise. Vorsichtig. Lang-weilig vorsichtig" ["The familiar languages must be left behind. Back turned on the subject-object grammar. Slowly. Tiny step by tiny step. Carefully. Slowly carefully"] ( _Hexenreden_ 26).
17 This dynamic recalls Zunshine's discussion of the mind-reading writer who engages the mind-reading reader out of a kind of "cognitive necessity" (160).
18 Britta Kallin, who has written an otherwise astute analysis of the novel, would likely disagree with my negative assessment of Margarethe. Kallin labels the protagonist "a self-confident woman," "a successful agent," and "a woman who pursues professional goals and can take care of herself" (341).
### ACKNOWLEDGMENTS
I would like to express my gratitude to Atsushi Fukada, Paula Leverage, Howard Mancing, and Richard Schweickert for their insightful comments on previous drafts of this essay.
### WORKS CITED
Banfield, Ann. _Unspeakable Sentences. Narration and Representation in the Language of Fiction_. Boston: Routledge and Kegan Paul, 1982. Print.
Barnden, John A. "Consciousness and Common-sense Metaphors of Mind." In _Two Sciences of Mind: Readings in Cognitive Science and Consciousness_. Eds. Seán Ó Nualláin, Paul McKevitt, and Eoghan Mac Aogáin. Amsterdam: John Benjamins, 1997. 311-39. Print.
Emmott, Catherine. _Narrative Comprehension. A Discourse Perspective_. Oxford: Clarendon Press, 1997. Print.
Flaubert, Gustave. _Madame Bovary._ New York: Dell, 1964. Print.
\---. _Madame Bovary. Backgrounds and Sources, Essays in Criticism_. Ed. and trans. Paul de Man. New York: Norton, 1965. Print.
Fludernik, Monika. _The Fictions of Language and the Languages of Fiction. The Linguistic Representation of Speech and Consciousness_. London and New York: Routledge, 1993. Print.
Fowler, Roger. _Linguistics and the Novel_. London: Methuen, 1977. Print.
Kallin, Britta. "Marlene Streeruwitz's Novel _Nachwelt_ as Postmodern Feminist Biography." _The German Quarterly_ 78.3 (Summer 2005): 337-56. Print.
Lakoff, George, and Mark Turner. _More than Cool Reason_. _A Field Guide to Poetic Metaphor_. Chicago: University of Chicago Press, 1989. Print.
Leech, Geoffrey N., and Michael H. Short. _Style in Fiction: A Linguistic Introduction to English Fictional Prose_. London: Longman, 1981. Print.
Palmer, Alan. _Fictional Minds_. Lincoln, NE: University of Nebraska Press, 2004. Print.
Pascal, Roy. _The Dual Voice. Free Indirect Speech and Its Functioning in the Nineteenth-Century European Novel_. Totowa, NJ: Rowman and Littlefield, 1977. Print.
Short, Michael H. "Mind-Style." _The English European Messenger_ 1.3 (Autumn 1992): 31-34. Print.
Streeruwitz, Marlene. _Hexenreden_. Eds. Gisela von Wysocki and Birgit Vanderbeke. Göttingen: Wallstein, 1999. Print.
\---. _Nachwelt. Ein Reisebericht._ Frankfurt am Main: Fischer, 1999. Print.
Zunshine, Lisa. _Why We Read Fiction_ : _Theory of Mind and the Novel_. Columbus: The Ohio State University Press, 2006. Print.
## Attractors, Trajectors, and Agents in Racine's "Récit de Théramène"
ALLEN G. WOOD
Racine's tragedy _Phèdre_ (1677) is one of the most famous, most produced, and most analyzed texts of seventeenth-century French classicism. Generations of French school-children have memorized and recited passages from the play, especially the extremely important and emotionally-charged account of the death of the young hero Hippolyte, as told by his mentor and guardian Théramène. Occurring in the fifth act, the narrative _récit_ is a long (almost one hundred lines), static, and nearly monologic interlude in the otherwise dialogic, interactive drama. It has often been characterized as a _morceau de bravoure_ , a virtuoso speech, similar in cultural stature and popularity to Hamlet's soliloquy or Cyrano's description of his nose. Théramène's _récit_ relates the events of the young prince Hippolyte, who, falsely accused of an incestuous love for his stepmother Phèdre, is banished by his father Thésée. Upon leaving the city of Trézène, Hippolyte and his followers are attacked by a sea monster, sent by Neptune in response to Thésée's hasty prayer. Hippolyte fights the creature, which, wounded, startles his horses. The prince is thrown from his chariot, becomes entangled in the reins, and is dragged to his death.
The passage is crucial in the performance of the play, and it has received much attention from a wide range of literary critics. Leo Spitzer analyzed the philological stylistics of the text as primarily ornamental rhetoric, and Philip Butler considered it a baroque element in a classical play. Psychological critics such as Charles Mauron have examined the passage for signs of repressed desire, the return of the repressed, and signs of the maternal. Other critics have looked to thematic elements to emphasize historical, mythological, theological, and intertextual aspects. Yet one approach that should also be utilized is that of cognitive poetics, which is rich in possibilities for explaining the ways in which both the characters themselves, as well as spectators of the play, process the information in the passage, make sense of an event that is to some extent "senseless," and react emotionally to it.
It is indeed a passage that moves characters and audiences. Aristotle proposes a cathartic purpose for tragedy in his _On_ _Poetics_ , that it should cause emotions of terror and pity for the audience, and this is echoed by Racine in the "Préface" to _Phèdre_ when he states that Hippolyte possesses all the characteristics of a hero "qui sont propres à exciter la compassion et la terreur" (745) [which are appropriate to arouse pity and terror]. Appearing in the penultimate scene of the play, the _récit_ is the focal point for these emotions.
Théramène's long speech is different from the surrounding text because it is a monologic narrative; the action of the play is suspended as the just recently transpired death is recounted. Yet this unique situation is not without irony, since most of the action in the play consists of talk, whereas this concentrated talk contains the most (physical) action in the play, as Hippolyte fights for his life, and loses. His death is a physical, real consequence to a series of previous situations and speech acts that make his death, and the _récit de_ _Théramène_ , a summary of the preceding play. In the midst of the theatrical spectacle, in which the audience sees and hears the characters interact (in movement and words), the _récit de Théramène_ appeals to the spectators' imagination, to a mental playing out of the events related. The mental images are stronger and more effective than any staging of the events, which could not credibly handle the dimensions or extraordinary nature of the action. What could devolve into the laughable on stage is instead horrific in verbal imagery.
In the beginning of the fifth act, Thésée has already begun to doubt his hasty judgment of his son's guilt based on the false accusation from Phèdre's nurse. Aricie confirms Hippolyte's claim that he was really in love with her, not Phèdre, and the nurse, Oenone, is reported to have committed suicide. The condemnation of his son, which Thésée claimed was based on "témoins certains, irréprochables" (1441) [reliable, irreproachable witnesses], quickly loses credibility, as the witnesses become "peu fidèles" (1485) [hardly trustworthy]. Thésée realizes he was perhaps wrong about his son, wants to recall him from exile, and hopes Neptune did not carry out his request for vengeance. It is at this moment that Théramène enters to give his account.
Théramène cannot know that the king has begun to doubt his erroneous belief that his son is guilty of an incestuous desire for Phèdre. His account of Hippolyte's death must stress both the heroism and the innocence of the prince, in order to change Thésée's mind. The _récit_ needs to function as an UpDater in modern mind-reading theory:
> Though we have no idea how the process of belief updating works, it is obvious that it _does_ work and that it generally happens swiftly, reasonably, accurately, and largely unconsciously. So there must be a cognitive mechanism (or a cluster of them) that subserves this process. We will call this mechanism the _UpDater_. (Nichols and Stich 30)
Although Théramène has just gone through the catharsis of the experience, is still shocked, and drained of all emotions except for a few tears, he must frame his account to be effective. He can accomplish this task since he is a master rhetorician, and a teller of tales of the king's former exploits against mythic creatures (74).
The king sees that his son's guardian, who has been entrusted with the care of the prince since his early childhood: "Je te l'ai confié dès l'âge le plus tendre" (1489) [I have entrusted him to you since his tenderest youth]. He has returned to the palace unexpectedly, and is crying. Théramène immediately responds that the prince has died: "Hippolyte n'est plus" (1492) [Hippolyte is no more]. This removes any narrative suspense from the _récit_ , since we are aware of the final outcome from the beginning. As spectators we process the information, not to construct a meaningful conclusion that will occur sequentially and inductively, but for the details that support an already-given conclusion. The narrative is thus ultimately circular, in that when it arrives at the recounting of Hippolyte's death, it fulfills the news that was initially announced. And Théramène knows the mind of the king, and constructs his account accordingly, thereby illustrating the following phenomenon:
> Indeed, to initiate conversation at all, speakers make judgments about the other person's willingness and ability to participate in the desired interaction. These mental models about the hearer's presumed belief and intention states must shift constantly as conversation proceeds, as new information is exchanged. (Barker and Givón 224-25)
The _récit_ is informative, but it is also performative rhetoric, in that it persuades the characters and audience. The passage is exculpatory, in that it answers Thésée's accusation "Qu'as-tu fait de mon fils?" (1488) [What did you do to my son?], as Théramène makes clear that it was a monster, sent from the sea, that led to both Hippolyte's heroic act, and ultimately his death. On the surface, the language is objective and dispassionate, although a strong unstated or understated affective purpose is clear. One way of examining the highly organized details in the _récit_ comes from Gestalt theory, and strategies for distinguishing prominence and pertinence of various elements. Théramène does not allow his account to appear frantic or disjointed, despite his need to discuss multiple subjects engaged in various actions simultaneously. Théramène's task is to tell a credible tale about an incredible event.
Recently many literary texts have been the object of insightful studies using principles of cognitive theory. Peter Stockwell notes that "[p]erceiving a figure and ground in a . . . field involves _selection for attention_ " (15). Visually this may involve greater size, unity, or detail of an item that attracts our attention more than others.
> A literary text uses stylistic patterns to focus attention on a particular feature, within the textual space. . . . attention will only be maintained by a constant renewal of the stylistic interest, by a constant process of renewing the figure and ground relationship. . . . literature is literally a _distraction_ that pulls attention away from one element onto the newly presented element. I will call these objects or devices **attractors** in this context. (Stockwell 18)
An examination of the attractors in the _récit de Théramène_ reveals a dynamically shifting pattern of figures and grounds, found in a sequence of changing nouns (usually subjects) and pronouns that refer to those entities with a main role in the event. As Russell Tomlin notes, an item occurring in the foreground "is characterized as information which is more central or salient or important to the development of the discourse theme" (89).
Théramène's narrative begins with the well-known "A peine nous sortions des portes de Trézène" (1498) [Scarcely had we left the gates of Troezen], where the vaguely undetermined "nous" [we] refers to Théramène himself (as he initially places himself into the account), to Hippolyte, and his followers. But the focus does not remain on this "nous," as it becomes the ground against which the prince, unmistakably referred to by the pronoun "il" [he], is described in the next eight lines. Hippolyte is given vertical and social prominence, since he is "sur son char" [on his chariot], and both his guards and horses imitate the somber mood of the unjustly exiled prince making his way along the shore. In fact, his horses are so much under his control that Hippolyte lets their reins fly freely. This detail is pushed to the background at this point, but later both horses and reins will figure prominently in the action.
While visual images formed the initial, cognitive figure of Hippolyte, presented in the "silence" (1500) maintained by the prince and his guards, a loud, unexpected auditory interruption prepares for the next attractor. We hear it before we see it (or rather they heard it before they saw it), and it is not clear where to look or what will be seen. A frightening roar ("un effroyable cri" 1507) comes from the water, and a formidable voice responds from deep within the earth. The backgrounded characters are briefly recalled; the men, the previous "us," were afraid—"notre sang s'est glacé" (1511) [our blood was frozen]—and the horses' hair stood on end. All look to the sea, which rises vertically as a "montagne humide" (1514) [wet mountain], but the object itself is obscured by the water in which it is still submerged. The huge mass comes to shore as a wave and finally it is revealed to be a "monstre furieux" (1516) [furious monster]. Its horns and body are described in the following five lines (1517-21), with possessive pronouns referring back clearly to the newest figure, the water born half-bull, half-dragon. The next five lines (1522-26) indicate the effect the beast has on various natural elements (earth, air, and water) and how "chacun" (1525) [each one] sought refuge.
But the "chacun" was inaccurate, since one person alone dared to confront the dangerous beast. Our attention is focused again on Hippolyte, who stopped his horses, grabbed his javelins, and threw one at the monster. What follows is a confusing and fast succession of these three prominent figures (prince, monster, and horses) at the heart of the _récit_. The monster is hit and comes to land at the feet of the horses (1530-33), where it covers them with flame causing them to take off in a panic (1534-41). The chariot breaks apart, and Hippolyte is caught up in the reins (1542-44). It is at this point that the narrator interrupts himself, for although he had presented a rather rhetorically-composed, dispassionate account, the image of his pupil and prince caught in the reins makes him pause and excuse himself before his king:
> Excusez ma douleur. Cette image cruelle
>
> Sera pour moi de pleurs une source éternelle.
>
> J'ai vu, Seigneur, j'ai vu votre malheureux fils
>
> Traîné par les chevaux que sa main a nourris. (1545-48)
>
> [Forgive my grief. This cruel image
>
> Will be for me an eternal source of tears.
>
> I've seen, my lord, I've seen your unfortunate son
>
> Dragged by the very horses his hand has fed.]
And while the horses continued to flee, the young prince suffered such injuries that his body was soon little more than a mass of wounds. References to the plural pronouns of the horses rapidly alternate with singular, third-person references to the dying prince.
When the horses finally stop, Théramène interjects for the first time in the _récit_ his own first-person narrative of his actions. Up to this point his account had been objective, told primarily in the third person, as he had formed part of the background, the group that had accompanied the main figure, Hippolyte. Théramène had been part of the initial "nous" (we) that set out from the gates of the city, then one of the "chacun" (each one) who had fled for safety from the monster. But as his prince lay dying, Théramène is foregrounded when he states that "I run there, sobbing" (1555) with the guards and others following him (a detail reminiscent of their following Hippolyte in the opening line). The guardian then relates how "I reach him, call him" (1559), using a historical present that, when combined with the first-person narrative, creates a strong sense of immediacy and nearness.
The prince has just enough life left in him to rally one final moment to attract our attention. He extends his hand, opens a dying eye, and speaks. His final action is to speak. In the six and one-half lines that are reported verbatim, Hippolyte's dying request is that his father, once he learns of his son's innocence, treat Aricie with kindness. The focus of attention, the speaking prince, utters his final words and immediately becomes no more than a thing, "triste objet" (1569) [lamentable object], in Théramène's arms.
The effect upon the king is conveyed at this point by his interrupting the narrative, and it is clear that his previous anger against his son, and accusation of fault by Théramène, have changed to grief and awareness of his own culpability. The _récit_ was performatively effective, as the king laments that his son was a "cher espoir que je me suis ravi" (1571) [a dear hope that I have taken from myself] and this will be a source of considerable "regrets" (1573).
Théramène concludes his _récit_ with an account of the final figure, that of Aricie. Her gestures are at first similar to those of the guardian—she arrived at the spot, approaches her beloved, and has difficulty recognizing him in his current state—although Hippolyte is now dead. But unlike him, she accuses the gods, faints at Hippolyte's feet, and is brought back to life by her servant. At this point Théramène leaves the scene to return to the palace with his horrible news.
The _récit de Théramène_ is a narrative recounting the death of Hippolyte, and indeed the prince is the primary figure, the one who grammatically and semantically attracts the most attention throughout the passage. He is kept in the foreground to increase the grief felt by his father and the audience. The two other main figures are the sea monster, and collectively his horses, and the central portion of the narrative contains a dynamic interchange, as our attention switches rapidly among the three. It is significant to note, however, that once the monster and the horses have fulfilled their primary function (of scaring the horses, of dragging Hippolyte), they do not just fall into the background, but they disappear, from the narration and our consideration. It is not clear if the monster died of the wound that the prince inflicted, making Hippolyte triumphant in his final hour, or if, being after all a supernatural phenomenon sent by Neptune, he simply vanished once the horses were frightened. And the horses, once they finally stop of their own accord—"At last their mad, wild galloping abates" (1553)—are simply not mentioned again. As it were, in their absence, the new figures of Théramène himself, and then Aricie, become secondary figures, called upon to interact with the dying, and react to the dead, young hero.
In the beginning of the _récit_ Hippolyte presents a strong, positive figure, magnificent even in exile, who is young, noble, and handsome. Because of his many admirable qualities, he attracts much attention. The prince is serene, almost immobile in his composure, as his horses first carry him away from the city. When he is called to action, his courage and skill are singular when he wounds the monster. But from the moment he falls, entangled in his horses' reins, he is broken, losing figural integrity as he is both literally and symbolically disfigured. He disintegrates from the pronominal whole of "Il" in "Il veut les rappeler" (1549) [He wants to recall them] to the next line with "son corps" (1550) [his body] that so soon becomes "un corps défiguré" (1568) [a disfigured body], in other words a "Triste objet" (1569) [lamentable object]. The dictates of French classicism required Racine to describe Hippolyte's death in a decorous, unshocking manner, and so the extent of his injuries are only vaguely evoked, as when Aricie finds his body unrecognizable: "Elle voit Hippolyte, et le demande encore" (1582) [She sees Hippolyte, and still calls for him]. Previous versions of the play, however, especially those of Seneca and the French baroque poet Garnier (1573), were not bound by such niceties, and in fact often reveled in gory details. As Hippolyte was dragged along, details abound concerning his dismemberment, an eye dangling here, a leg thrown there. There is certainly not enough of him left to say anything when he is finally found by his men, who start picking up the scattered pieces to bring back to king Thésée. Hippolyte is no longer physically, literally attractive, nor is he a narrative attractor except in the memory of his wholeness.
Another structuring device in the _récit de Théramène_ that can be analyzed using cognitive theory involves figures that are foregrounded as trajectors. Whereas a figure can be presented in stasis, a trajector attracts attention by its motion. Leonard Talmy indicates that "[t]he Figure is a moving or conceptually movable entity whose site, path or orientation is conceived as a variable, the particular value of which is the relevant issue" (19). Stockwell comments on specific terms involved in such movement: "the moving figure can be seen to follow a **path**. . . . Within the image schema, though, the element that is the figure is called the **trajector** and the element it has a grounded relationship with is called the **landmark** " (16).The opening line of the _récit_ , "A peine nous sortions des portes de Trézène" (1498) [Scarcely had we left the gates of Troezen], reveals these very elements, with the landmark being the gates of the city Troezen, from which the "we" (Hippolyte, Théramène, and his followers) issued forth, the point of origin or terminus a quo. The path they were following, "le chemin de Mycènes" (1501) [the road to Mycenes], is also clearly specified, forming a second landmark, the _terminus ad quem_ , at which they never arrive.
From out in the middle of the sea, "du fond des flots" (1513) [from the depths of the waves], first arises a terrible cry, then "sur le dos de la plaine liquide" [on the back of the wet plain] arises an enormous wave. The watery landmark is perhaps not precise, but clearly denotes Neptune's realm. The mass is not static, however, as it moves for the shore: "L'onde approche, se brise, et vomit à nos yeux, / Parmi des flots d'écume, un monstre furieux" (1515-16) [The wave approached, broke and vomited before our eyes / From among the frothy waves a raging monster]. The initial trajectory of Hippolyte's travel along the shore to Mycenes is intersected by the perpendicular path of the monster, forming a deadly crossroads.
Michel Serres observes that in Greek mythology and literature, the topos of the crossroads serves as a place of crisis and decision. At the chiasmus occurs a clash of disparate intents and fates, and it is often a place of death and revenge. Oedipus encounters unknowingly his father at a crossroads, and "it is a place so catastrophic and so confined that he must kill his father" (47). Ubersfeld speaks of the crossroads as an "impossible place where the horses of the forest have their deadly encounter with the monsters of the wave" (204).
At the approach of the monster, all of Hippolyte's men leave the road, but their flight is scattered, lacking figural unity, and limited in narrative scope—"tout fuit" (1525) [all fled]—as they seek safety in a nearby temple. Even nature abhors the monster, since the wave that brought it in rolls back, frightened. Only Hippolyte moves toward the monster, as he throws a spear, in a line that foregrounds for the first time in the _récit_ his name, his lineage, and his bravery: "Hippolyte lui seul, digne fils d'un héros" [Alone, Hippolyte himself, worthy son of a hero] (1527).
After the injured monster falls dying at the feet of the horses and frightens them, they take off. Whereas the previous trajectories indicated a clear teleological landmark for completion (Hippolyte headed to Mycènes, the monster bearing down on the shore and Hippolyte, all the attendants fleeing to the temple, the wounded monster approaching the horses), the frightened horses are directionless, without any guidance from their master Hippolyte. And as they run, they drag him along, in a trajectory that causes his death. The horses' flight is a deviation from his path; he is thrown off course and into the wilderness that will kill him. Théramène and the other men are able to find Hippolyte by following the bloody trail that he leaves:
> De son généreux sang la trace nous conduit:
>
> Les rochers en sont teints; les ronces dégouttantes
>
> Portent de ses cheveux les dépouilles sanglantes. (1556-58)
>
> [The trail of his generous blood led us:
>
> The rocks were colored by it, the dripping brambles
>
> Wear the bloody remains of his hair.]
The less decorous, earlier versions of the tragedy portray an even more gruesome path, littered with the prince's body parts. Racine's _récit de Théramène_ ends with Aricie also following the path to her beloved fiancé, and the guardian making his return to the palace. This is done in order to fulfill the promise of the dying prince who begs him to have the king protect Aricie, as well as inform, convince, and move Thésée. A circular motion occurs that takes the narration, and our imaginations, out to the scene of death and back again to the palace.
The main figures in Théramène's account are brought into focus for our cognitive processing by their nominal positions grammatically and semantically as attractors; they move along a trajectory, and they are agents of action. Craig Hamilton explains the distinction between agents and patients:
> To profile objects conceptually first as figures and then as trajectors linguistically entails marking some objects as _agents_ and others as _patients_ ; things that are more active in relation to less active things, or things that act versus things that are acted upon. (58)
At a grammatical level, the concept is often tied with subject or object positions of transitive action verbs.
The first section of the _récit de Théramène_ is more concerned with motion than with other actions. Hippolyte and his group leave the city (he does not even have to guide his horses); the sea monster appears, comes on shore, and seems threatening. The prince figures most prominently as an agent when he seizes his javelins and throws one at the beast. Wounded, the monster is weak (it rolls over) and not capable of much more action than exhaling some flames at the horses. But this is enough to frighten the horses, or perhaps there was another agent in this. Théramène admits at this single juncture a different voice into his narration, the vague "on" [somebody] of other witnesses to the event: "On dit qu'on a vu même, en ce désordre affreux, / Un Dieu qui d'aiguillons pressait leur flanc poudreux" (1539-40) [Some even say they saw, in this frightful disorder / A god who pricked with spurs their dusty flanks]. Whatever the cause, the horses ran, and Hippolyte could not control them. This is the first instance of powerlessness, and the transition from the prince as agent to that of patient. He falls, is entangled in the reins, and the still fleeing horses drag him to his death. It is an accident, the horses did not intend to kill him, and so they are unwittingly, perhaps even unknowingly, agents of his death. The rest of the _récit_ involves the reactions and consequences of this chain of agents (or rather of instruments used by an agent)— the horses drag Hippolyte to his death, but they were acted upon by the monster, sent by Neptune, at the request of Thésée. It was the king, Hippolyte's father, who was the originating agent and cause of the prince's death, although chance, the accidental fall, complicates a strict causal chain.
An analysis of the _récit de Théramène_ using principles of gestalt perception and cognitive poetics, namely attractors, trajectories, and agents reveals multiple ways in which Hippolyte is foregrounded (which play out in the pattern of nouns, pronouns, and verbs), and the dynamic interplay of and mind reading between the other main figures (the monster, the horses, even Théramène himself, who is both narrator and actor). And it reveals a particularly pertinent line of inquiry, as we see the intersecting trajectories of Hippolyte and the sea monster. It was a fateful and fatal encounter, one that changed Hippolyte's initial and intended path, as he effortlessly guided his horses to exile in Mycènes and beyond, to a path of destruction, as he was dragged by his horses, resulting in death by dismemberment, and a lack of figural unity. It is a tale that arouses terror for both the king and the audience, and not just terror at the horrendous aspects of the monster, but at the injustice of both human and divine persecution of an innocent prince. The cold clarity of Théramène's focused account brings about a complete change in Thésée, who has not only recognized his own role in his son's death, but reconciles himself with his former enemy Aricie as the play closes.
### WORKS CITED
Aristotle. _On Poetics_. Trans. Seth Bernardete and Michael Davis. South Bend: St. Augustine's Press, 2002. Print.
Barker, Marjorie, and T. Givón. "Representation of the Interlocutor's Mind during Conversation." _Other Minds: How Humans Bridge the Divide between Self and_ _Others_. Eds. Bertram F. Malle and Sara D. Hodges. New York: Guilford Press, 2005. 223-38. Print.
Butler, Philip. _Classicisme et baroque dans l'oeuvre de Racine_. Paris: Nizet, 1959. Print.
Hamilton, Craig. "A Cognitive Grammar of 'Hospital Barge' by Wilfred Owen." _Cognitive Poetics in Practice_. Eds. Joanna Gavins and Gerard Steen. London: Routledge, 2003. 55-66. Print.
Mauron, Charles. _L'inconscient dans l'oeuvre et la vie de Racine_. Paris: Gap, Éditions Ophrys, 1957. Print.
Nichols, Shaun, and Stephen P. Stich. _Mindreading: An Integrated Account of Pretence, Self-Awareness, and Understanding Other Minds_. Oxford: Clarendon, 2003. Print.
Racine, Jean. _Phèdre_. _Œuvres completes_. Vol. 1. Ed. Raymond Picard. Paris: Gallimard, 1950. 735-803. Print.
Serres, Michel. "Language and Space: From Oedipus to Zola." _Hermes: Literature_ , _Science, Philosophy_. Eds. Josué V. Harari and David Bell. Baltimore: Johns Hopkins University Press, 1982. 39-53. Print.
Spitzer, Leo. "The Récit de Théramène." _Linguistics and Literary History: Essays in_ _Stylistics_. Princeton: Princeton University Press, 1948. 87-134. Print.
Stockwell, Peter. _Cognitive Poetics: An Introduction_. London: Routledge, 2002. Print.
Talmy, Leonard. "Concept Structuring Systems in Language." _The New Psychology of Language_. Vol. 2. Ed. Michael Tomasello. Mahwah, NJ: Lawrence Erlbaum Associates, 1998. 15-46. Print.
Tomlin, Russell. "Foreground-Background Information and Syntactic Subordination." _Text_ 5.1-2 (1985): 85-122. Print.
Ubersfeld, Anne. "The Space of Phèdre." _Poetics Today_ 2.3 (Spring 1981): 201-10. Print.
## The Importance of Deixis and Attributive Style for the Study of Theory of Mind: The Example of William Faulkner's Disturbed Characters
INEKE BOCKTING
### INTRODUCTION
Teaching William Faulkner's _The Sound and the Fury_ to English major students in Holland, Norway, and France, I have always been asked why Faulkner made the story so difficult to read, the choice of words so bizarre, the syntax so convoluted, and the situation in time and space so extremely complex. To this question I have come to answer: because he wants to create compassion for the characters. He wants you to feel as deserted as they do—that is why I avoid explaining anything about the book beforehand—he wants you to be as desperate, as lost as they are. I might have said: he is testing your Theory of Mind and soliciting it to the fullest, expanding it beyond its everyday use.
I use the term Theory of Mind, here, not just as the ability of individuals to perceive other people to "know, want, feel, or believe things" that may be different from what they themselves know, want, feel, and believe, per Simon Baron-Cohen (37-46), but as an actual capacity to understand what it is that they "know, want, feel or believe"1— what Lisa Zunshine calls "our mind-reading ability" (47)—and, going further, to even experience it in their place. This is why I prefer to talk about the capacity to step into the shoes of someone else, seeing the world from his or her point of view. This taking "the other fellow's point of view," to use the words of the linguist Charles Fillmore in his _Santa Cruz Lectures on Deixis_ (44)—or viewpoint shift—is not an unusual phenomenon at all. Douglas Hofstadter, in his majestic work _Gödel, Escher, Bach: An Eternal Golden Braid_ , expresses it as follows: "I can fire up my subsystem for a good friend and virtually feel myself in his shoes, running through thoughts which he might have, activating symbols in sequences which reflect his thinking patterns more accurately than my own" (386). Faulkner's text is just more challenging in this respect than what our everyday experience teaches us.
The fact that Faulkner's style is so extraordinary, of course, makes any kind of linguistic study of his work worthwhile, but it is a real gold mine for an analysis of the ways in which Theory of Mind is realized linguistically in the text. I will look here at the linguistic nitty-gritty of its artistic creation, both in terms of what is presented and in terms of how that is done, starting, therefore, with a discussion of the textual markers of viewpoint shift, or _deictic projection_ (Lyons 579), and subsequently analyzing the various forms the attribution of mental activity to a subject may take, discussing both the different mental verbs (including perception verbs) and their complements.
Cognitive linguistics, of course, has taught us that reality is not simply out there to be discovered, named, and talked about, but rather a construction in the individual mind consisting, basically, of three elements: _universal reality_ (that which we all share because we are humans; that which is "hardwired" into our brains); _social reality_ (those preexisting social relations we are born into); and _personal_ _reality_ (that which our own psychological makeup and our own personal experiences add to the previous two). An interactive mixture of these three elements, both determined by and determining our position in the world, makes up our individual point of view. Language offers different types of linguistic terms that will encode this individual point of view into the language we use: the so-called _deictics_. Indeed, deictics normally point to things in the world from the position of the language user. This position is called the _deictic center_ (Levinson 664), the point to which all deictics are "anchored."2 If we say, then, that language is egocentric in nature, it means that, normally speaking, all phenomena in the world are related to the position of the language user. To illustrate this, I will briefly review the different classes of deictics.
Temporal deictics function to relate the time of an event to the time of the deictic center, that is to say, to the speech event, the major ones being the adverb ( _now_ , indicating simultaneity, _then_ anteriority, and _soon_ posteriority of the event),3 and the system of the tenses (present tense indicating simultaneity, past tense anteriority, and future tense posteriority of the event). Tense, in fact, shows the profound way in which, at least in our Western culture, human beings are embedded in time, and it is hard to avoid them, every well-formed sentence in English containing at least one of its elements. (One might cite Shakespeare's "To be or not to be . . ." as a counter-example, were it not that the next sentence brings us back into time, with its "whether t' _is_ best . . ."; emphasis added.)
Spatial deictics function to relate the position in space of a phenomenon to the position of the deictic center and include the common demonstratives _this / these_ and _that / those_ , and the adverbs _here_ and _there_ , which indicate physical closeness to or farness from the deictic center. The spatial deictics may also indicate emotional stance. By using the deictic _this_ , for instance, a speaker may not only evoke spatial proximity, but also emotional nearness, which John Lyons calls "emphatic deixis" (677). Certain motion verbs, in addition, have built-in deictic aspects, such as _come_ and _go_ and _bring_ and _take_ , indicating directions or destinations toward and away from the deictic center.4 That temporal deixis and spatial deixis are closely related, finally, is clear, and this shows itself for instance when Faulkner makes one of his characters say: "You've been running a long time not to've got any further off than meal time" ( _Sound and Fury_ 81).
Social deixis concerns kinship terms, which include a relation of superiority to the deictic center (such as _parent_ , _mother_ , and _father_ , _grandmother_ and _grandfather_ , _aunt_ and _uncle_ ), of inferiority to the deictic center (such as _child_ , _daughter_ , and _son_ , _niece_ and _nephew_ ), or equality with the deictic center (such as _sister_ and _brother_ , and _cousin_ ), as well as hierarchical terms, again including superiority, inferiority, or equality with regard to the deictic center (for instance _boss_ , _pupil_ , and _colleague_ , respectively).
Personal deixis, finally, concerns the set of personal pronouns, of which the first person pronouns _I_ and _we_ (as well as their derivatives _me_ , _us_ , _mine_ , and _ours_ ) create and sustain the deictic center itself (a personal one, or one shared with another person), the pronoun _you_ (as well as _your_ and _yours_ ) relate the deictic center to a speech partner, or speech partners, and the pronouns _he_ , _she_ , _it_ , _and they_ (and _his_ , _her_ , _its_ , _their_ , and _theirs_ ) posit a subject or subjects as outsider or outsiders with regard to the language event. Indeed, in the present tense, the s of the third-person singular can be seen as a visible and sonorous marker of absence from the speech event (Joly and O'Kelly 415-16).
The pronoun _I_ obviously occupies a special position, as it is the one through which the deictic center establishes itself. At birth, the young individual—as psychologists agree—is still undifferentiated, not yet occupying any subject position at all. Then follows the mirror stage, which the psychoanalyst Jacques Lacan saw as an essential stage in human development, when the child starts to recognize itself as a separate individual. Subsequently, as the child starts to speak, the deictic center—that is, the encoding of his or her personal position as a language user—starts to fall into place. However, with their unstable reference—that is why they are sometimes called "shifters"—deictics are notoriously difficult to learn how to use. Young children will therefore use the proper name—as a sort of "proto-deictic"—until they have mastered this difficulty, and require others to do so well.
We may turn to the first chapter of William Faulkner's novel _The Sound and the Fury_ now, where Caddy Compson, returning from her day at school, sees her brother holding on to the fence around their house, looking distressed, and says to him: "Did you come to meet Caddy. . . . What is it. What are you trying to tell Caddy" (7).
By using her proper name rather than the pronoun _me_ to refer to herself, Caddy can be seen to take the point of view of her brother; indeed, she has "stepped into his shoes" and uses a proto-deictic, as he would (as a matter of fact, she might have used two proto-deictics and have said "What is Benjy trying to tell Caddy," but presumably her brother has enough information if he just understands "telling Caddy" or else he is already in the process of learning the deictic system, being able to accommodate one deictic). In any case, Caddy's Theory of Mind is in place. She is able to project herself into the point of view of somebody else, her brother. In linguistic terms, she is able to use _deictic_ _projection_.
A more everyday, yet instructive, example of deictic projection is where we find Mr. Compson—Benjy's and Caddy's father—using the noun phrase "your grandfather" in a discussion with his eldest son, Quentin. The social deictic that performs the deictic projection—that is, which shows the workings of Mr. Compson's Theory of Mind—is the term _grandfather_. Indeed, it is the term that Quentin himself would use to refer to this relative, not, of course, his father. Without the deictic projection, the term used by Mr. Compson would have been _father_. What is important to notice, however, is that the deictic projection is not complete. While the social deictic _grandfather_ , indeed, shows the deictic projection, the personal deictic _your_ remains anchored to the deictic center, that is, to the position of the speaker, Mr Compson. After all, it is the term that Mr. Compson uses to refer to his son, not the term Quentin would use to refer to himself. If his deictic projection had been total, Mr. Compson would have said "my grandfather," which would have created total confusion.
This simple, everyday example shows, in fact, that deictic projection must always remain anchored, otherwise chaos will occur. That is to say, for Theory of Mind to be successful, one must be able to perform deictic projection while at the same time holding steady one's own deictic center. This anchoring function is essential. Apart from the social and artistic conventions of role-play and acting in the theater or the movies, which are beyond the scope of this paper, this anchoring can in spoken language be indicated by changes in speed, rhythm, and tone of voice and in written language be taken up by graphological signs—the comma, the colon, and quotation marks—as well as by deictics and by the system of attribution.
Faulkner's novels form a gold mine for showing what happens if deictic projection is disturbed, that is, if it is either absent or unanchored. From the way his sister talks to him, we can deduce that Benjy is not yet able to perform deictic projection; that is, his Theory of Mind is not yet in place: he cannot yet see the world from Caddy's point of view. This observation will be confirmed in the rest of the chapter, for instance where he has drunk from the champagne at his sister's wedding. Remembering how his brother Quentin had to lead him, Benjy's text reads: "The ground kept sloping up. . . . Quentin held my arm and we went towards the barn. Then the barn wasn't there and we had to wait until it came back" (23).
By using the deictic _we_ , Benjy creates a combined deictic center for himself and his brother, thus showing that because he is dizzy—seeing everything turn before his eyes—he believes that his brother is too. This would not be surprising, indeed, in a child of three. But Benjy is thirty-three, an adult autist rather than a child. The grown man he is shows that he is unable to perform deictic projection or, in Baron-Cohen's terms, that he is incapable of perceiving other people to feel things that are different from what he himself feels, which is seen as one of the diagnostic symptoms of autism. Indeed, in their work _The Language of Psychosis,_ Bent Rosenbaum and Harly Sonne argue that in contrast to other types of special needs children, the problems of language acquisition that autistic children show are especially in the area of deictics (52).5
The other example I want to present here is that of Benjy's older brother, Quentin. In contrast to Benjy's text, in which deictic projection is limited or perhaps not realized at all, it is easy to see that in Quentin's text it has run wild. Indeed, we can recognize that Quentin's viewpoint shifts are largely unanchored, missing both lexical anchors—the pronouns that are supposed to remain anchored to the deictic center—and grapho-logical ones—the comma, the colon, and the quotation marks. Here is a sample from the first page of the chapter narrated by him:
> When the shadow of the sash appeared on the curtains it was between seven and eight o'clock and then I was in time again, hearing the watch. It was Grandfather's and when Father gave it to me he said I give you the mausoleum of all hope and desire; it is rather excruciatingly apt that you will use it to gain the reducto absurdum of all human experience which can fit your individual needs no better than it fitted his or his father's. I give it to you not that you may remember time, but that you might forget it now and then for a moment and not spend all your breath trying to conquer it. (86)
In addition to the absence of lexical and graphological anchors, his text shows very few attributive clauses, and when they occur they often sound like afterthoughts, added at the last minute, such as in: "It's always the idle habits you acquire which you will regret. Father said that. That Christ was not crucified: he was worn away by a minute clicking of little wheels. That had no sister" (87). The result is that Quentin, for more than half a page at a time sometimes, takes over the viewpoint of his father without any markers at all. We can say that, in contrast to Benjy's lack of Theory of Mind, Quentin's has become so expansive—so unanchored—that he can no longer distinguish what his father knows, wants, believes, or feels from what he does himself.
Critics have diagnosed Quentin as a psychotic, his language as schizophrenic,6 and although the usefulness of such labels is debatable, of course, it is true that the collapse of the deictic system that Quentin's text shows has been described as a symptom of schizophrenia. As Rosenbaum and Sonne put it, in schizophrenic language the "otherwise relatively firm boundary between internal and external, between self and surroundings" becomes "penetrable" (105), until finally the speech of many different persons becomes merged in totally unanchored deictic projection, as Quentin's text shows it in the following passage:
> _Three times. Days. Aren't you even going to open it_ marriage of their daughter Candace _that liquor teaches you to confuse the means with the end_ I am. Drink. I was not. Let us sell Benjy's pasture so that Quentin may go to Harvard and I may knock my bones together and together. (200)
As Theory of Mind thus runs rampant, operations upon language become more and more central, as a last attempt to hold together some sort of personal viewpoint.7 In Quentin's text one can see clearly how thinking becomes, in Rosenbaum and Sonne's words, "metonymically substitutive" (64), with concepts following each other not because they form part of an integrated conceptualization of the world, but as a result of "phonetic and semantic similarities and contiguities." Such "sliding," as it is called, may as if by accident land on the crucial point, which is clear in the following passage:
> _the first car in town a girl Girl that's what Jason couldn't bear the smell of gasoline making him sick then got madder than ever because a girl Girl had no sister but Benjamin Benjamin the child of my sorrowful if I'd just had a mother so I could say Mother Mother_. (197)
The text finally presents the total collapse of Quentin's deictic center when the personal deictic "I" is written as a lower case "i" in a memory of a direct confrontation with his father. Quentin, here, "confesses" the incest he has not committed:8
> . . . and i i wasnt lying i wasnt lying and he you wanted to sublimate a piece of natural human folly into a horror and then exorcize it with truth and i it was to isolate her out of the loud world so that it would have to flee us of necessity and then the sound of it would be as though it had never been and he did you try to make her do it and i i was afraid to i was afraid she might . . . (175-76)
From the study of schizophrenic language it is known that with the "fragmentation of the imaginary body"—that is, when the "I" loses its "body-bound identity," as Rosenbaum and Sonne put it (105)—certain body parts take over, thus becoming its acting and speaking counterparts, as where Quentin remembers confronting his sister's lover: ". . . then I heard myself saying Ill give you until sundown to leave town. . . . My mouth said it I didn't say it at all" (183).
We see here that Quentin uses an attributive clause— _I heard myself saying_ —to attribute mental activity to himself at an earlier time. In other words, Quentin steps into the shoes of his younger self, the past tense of the verb— _heard_ —providing the anchor to his deictic center while the pronoun "I" with its reflexive _myself_ and the progressive form of the verb _say_ perform the deictic projection.
Even if this passage happens to be anchored somewhat more firmly than some of the passages we have seen before, it has an unsettling quality. One reason is the splitting in "behaver and observer" that the deictic projection produces, which specialists of schizophrenia have recognized as characteristic of the affliction (Bernheim and Lewine 56), the other is the syntactic form of the attributive clause itself. This brings me to the second part of my discussion, the importance of the attributive clause for the study of Theory of Mind.
If Theory of Mind concerns the attributing of mental activity to a subject—whether this goes all the way to "virtually feel[ing] [oneself] in his shoes," as Hofstadter puts it, or stays within the more distanced realm of "mind reading"—it seems useful to me to study the specific mental activity verb that is used when this attributing of mental activity is represented in the text—the verb _hear_ in this case—as well as the form of the complement (the clause _myself saying . . ._ ). By using _not_ a "that-clause," such as _I heard that I was saying . . ._ (in which Quentin would have attributed to his younger self a conclusion based on prior existing knowledge), _not_ a "how-clause," such as _I heard how I was saying_ (which would have signified his awareness of a process going on inside him), and _not_ a "to-infinitive," such as _I heard myself to be saying . . ._ (which would have expressed his assessment of himself based on immediately available knowledge), but instead a so-called "small-clause." Quentin, as narrator or "attributor," expresses the mental experience of his younger self in a pure and intellectually unprocessed way.
As defined by the linguist Frederike van der Leek, the term _small-clause_ refers to the complement of a mental activity verb—in this case the verb _hear_ —which, if it is verbal, is non-finite and not accompanied by the infinite marker _to_ (1-28), such as the clause _myself saying_ used here. This small-clause complement of a mental activity verb, rather than conveying the assessment of a situation (as does the to-infinitive), the awareness of a process (as does the how-clause), or the conveying a conclusion (as does the that-clause), refrains from making any reference to a truth-judgment on the part of the experiencing character, conveying his or her experience "raw," as it were.9 As a matter of fact, the grammatical form of the small-clause actually prohibits the addition of an epistemic evaluation, this becoming clear if we use a verb that includes such an evaluation, such as "I _knew_ myself saying," or "I _concluded_ myself saying," which are clearly ungrammatical.
Because the small-clause conveys phenomena as experienced rather than epistemically—that is, in terms of what is considered true of the world—such a construction lends itself especially well to the attributing of states of altered consciousness to an experiencer, such as reveries, dreams, psychological stress, drugs-induced conditions, hallucinations, and mental deficiency, when truth-judgment is avoided, deferred, disturbed, altered, impaired, or plainly impossible. Faulkner's stories and novels, with their concentration on deviant states of mind, contain many other beautiful examples, for instance _Light in_ _August_ , where Joe Christmas's depersonalized behavior is described: "He heard his voice say . . . and he watched his hand swing" (261); as well as in the story "Pantaloon in Black," where the young man Rider has to bury his wife after only six months of marriage: "He stood in the worn, faded, clean overalls which Mannie herself had washed only a week ago and heard the first clod strike the pine box" ( _Go Down, Moses_ 131).
In each case the mental activity verb _heard_ occurs in combination with a small-clause. To take the last example, instead of using attributive clauses such as _He heard that the first clod was striking the pine box_ , so as to give Rider's conclusion; or _He heard how the first clod was striking the pine box_ , so as to convey Rider's awareness of a process witnessed; or else _He heard the first clod to be striking the pine box_ , to indicate Rider's assessment of the situation before his eyes, the narrator uses his Theory of Mind to present the character's unmediated, "raw" experience.
Rider's text, however, is not totally devoid of other types of attribution. One finds, in fact, a small number of that-clauses. Consider the following sentence, in which the external narrator ascribes mental activity to the character Rider:
> Then he could stop needing to invent to himself reasons for his breathing, until after a while _he began to believe he had forgot_ about breathing since now he could not hear it himself above the steady thunder of the rolling logs; whereupon as soon as _he found himself believing he had forgotten it_ , _he knew that he had not_ . . . (140-41; emphasis added)
At first glance only one that-clause is obvious: "he knew that he had not." In this construction the narrator, in order to enter Rider's state of mind, combines the truth-judgment on his—Rider's—part, which the that-clause conveys, with a choice of the attributive verb _knew._ This attributive verb, _know_ , belongs to a group called factives, normally presenting the type of mental activity that a subject engages in, and at the same time indicating narratorial consent; that is to say, the narrator vouches for the truth of the matrix subject's observation. That this is so becomes clear when one makes the consent explicit. We can see easily that the sentence "he knew that he had not forgotten his breathing, and he was right" is redundant, while conversely "he knew that he had not forgotten his breathing, but he was wrong" is obviously paradoxical. Thus it is clear that speaker consent is part of the intrinsic meaning of the verb _to know_.
Two other that-clauses can be found in the passage:
a) he began to believe [that] he had forgot about breathing;
b) he found himself believing [that] he had forgotten it.
Obviously, the intellectual mediation attributed to Rider, because of the ellipsis of _that_ , is of a more covert type here, and what is more, it is combined with the attributive verb _believe_ , which refrains from any truth-judgment on the part of the narrator. This again becomes clear if the implicit is made explicit: "he believed that he had forgotten it and he was right" and "he believed that he had forgotten it but he was wrong" are both possible. This proves that no truth-judgment is part of the intrinsic meaning of _to believe_. The second that-clause, furthermore, is embedded within the small-clause, "he found himself believing," which casts the whole experience as a "raw" sensation that the reader shares with Rider. This effect is still further enhanced by the depersonalizing reflexive "he found himself."
The passage reveals as further treasures for the study of Theory of Mind its temporal organization. Changing from the adverb _then_ in the first clause, to _now_ in the third, the text shows well the temporal flexibility, or as Cohn calls it the "elasticity," of "psychonarration" (38).10 Rider's experience, thus, is captured again through deictic projection, as the narrator, through the usage of _now_ and further on of terms such as "six months ago" instead of "six months before" (134), projects himself into the time perspective of the experiencer, that is to say, steps into the shoes of this young black laborer gone crazy with grief over the death of his young wife and takes us, his readers, with him.
### CONCLUSION
I hope that I have been able to show that for the study of Theory of Mind in literature it is worthwhile to descend into the linguistic nitty-gritty of the text, first of all because linguistically speaking, Theory of Mind is a matter of deixis. As we have seen, the stepping into the shoes of another, and presenting the world from the point of view of that other, necessitates both the anchoring and the shifting of deictics. Secondly, Faulkner's texts have shown, I hope, how it can be worthwhile to study closely the specific mental verbs (and that includes perception verbs) through which a narrator, using his Theory of Mind, attributes mental activity to an experiencer, including himself at a younger age. In addition, we have seen that the specific attributive verb chosen may or may not include a truth-judgment on the part of this narrator himself, while the form of the complement of the attributive clause itself indicates the degree of intellectual awareness of the experiencer, going from a totally conscious conclusion to an intellectually unmediated, "raw" perception.11 It is not just a matter of having a Theory of Mind or not; it is a matter of how, when, and to what degree. One day, hopefully, brain imaging will be subtle enough to show us these differences on film.
### NOTES
1 See also Douglas Frye and Chris Moore.
2 See also Duchan et al., who bring together a series of essays concerned with "deictic shift theory," the various ways in which "readers and writers of narratives sometimes imagine themselves to be in a world that is not literally present" (14).
3 In the case of the "if . . . then" conditional, Barbara Dancygier and Eve Sweetser argue, the deictic _then_ creates a mental space of temporal dimensions (144-46).
4 As a matter of fact, the motion verb _come_ is more complex, as it may stipulate a movement toward the deictic center (as in _The mail has finally come_ ), toward an addressee (as in _I don't mind coming to get you_ ), or toward the home-base of either (as in _Why did you come all the way to my house?; I could have come to yours_ ).
5 See also Harold I. Kaplan and Benjamin J. Sadock 557. For a discussion of Benjy's autism, see Sarah McLaughlin 34-40.
6 Joseph Blotner calls him Benjy's "psychotic older brother (213). Bruce Kawin speaks of a young man "who is not just chaotic and allusive, but frankly psychotic" (17). John Irwin argues that Quentin's narration shows the "bipolarity typical of both compulsion neurosis and schizophrenia" (29). The literary critic and psychiatrist Jay Martin believes he is best qualified as a male hysteric (personal communication).
7 See Julius Laffal 19; Sigmund Freud 203; and Rosenbaum and Sonne 64.
8 Faulkner explained this omission of capitals as follows: "because Quentin is a dying man, he is already out of life, and those things that were important in life don't mean anything to him any more." In Frederick L. Gwynn and Joseph Blotner 18.
9 A beautiful example of the small-clause is found in Henry James's short story "The Beast in the Jungle": "He saw the Jungle of his life and saw the lurking Beast; then, while he looked, perceived it, as by a stir of the air, rise, huge and hideous, for the leap that was to settle him." In Morton Dauwen Zabel, ed., 327-83. (See also James Phelan's interesting discussion of the story "The Beast in the Jungle" in _Reading People, Reading Plots_ , 61ff.). But the narrator might have allowed for some form of epistemic evaluation of the experience on the part of the matrix subject by choosing the _to-infinitive_ ("he perceived it _to be rising_ "), the _how-clause_ ("he perceived _how it rose_ "), or the _that-clause_ ("he perceived _that it rose_ ").
10 This form of narration often moves seamlessly into what Cohn calls "narrated monologue" (38), also known as free indirect discourse.
11 The approach developed here is, of course, not limited to William Faulkner's works. Indeed, deictic projection and attributive style are worthwhile subjects of study in connection with Theory of Mind for many other literary works. In Flannery O'Connor's short story "Parker's Back," for instance, the main character, the non-believer Parker, crashes his truck into a tree, causing what amounts to a religious vision: "All at once he saw the tree reaching out to grasp him. A ferocious thud propelled him into the air, and he heard himself yelling in an unbelievably loud voice, 'GOD ABOVE!'" (520). The small-clauses "he saw the tree reaching" and "he heard himself yelling" give the experience "raw," here, showing that neither Parker nor the narrating voice—this because of the truth-neutral attributive verbs _saw_ and _heard_ —questions his experience. In the second small-clause, the depersonalizing reflexive adds to the supernatural effect of the passage. In this context the exclamation "GOD ABOVE!" can no longer simply be seen as a case of blasphemy. For the third time in the story, Parker ends up on his back—he had fallen twice before—as the tractor crashes "upside down into the tree" and he sees his shoes "quickly being eaten by the fire" (another small-clause). Yet this is the first time Parker reaches some intellectual awareness. Indeed, the small-clauses, here, are followed by the attributive that-clause: "He only knew that there had been a great change in his life, a leap forward into a worse unknown, and that there was nothing he could do about it" (521). By making use of the that-clause and the factual attributive verb _know_ , the attributive clause "he knew that there had been a great leap forward" in this passage shows at once Parker's intellectual grasp—he is able to form a conclusion on the basis of his experience—and the narrator's consent. This conclusion is paired with a pure and simple explanation that cannot be contested, and that is strongly reminiscent of Christ's words on the cross: "it was for all intents accomplished" (521). (John 19:28 reads, "After this, Jesus knowing that all things were now accomplished, that the scripture might be fulfilled, saith, I thirst." One notices, here, the that-clause "knowing that," as well as the past participle "accomplished," which are also essential in Parker's awareness. In verse 30, moreover, Jesus says: "It is finished"; the Greek verb form is _tetelestai_ in both cases). From the "raw" sentiment of being brought out of himself and carried away, conveyed by the small-clauses "he saw the tree reaching out" and "he heard himself yelling," Parker has moved into the intellectual awareness—he "knew"—that he must let himself be moved toward the Christ on the cross.
### WORKS CITED
Baron-Cohen, Simon, Alan M. Leslie, and Uta Frith. "Does the Autistic Child have a 'Theory of Mind'?" _Cognition_ 21 (1985): 37-46. Print.
Bernheim, Kayla F., and Richard R. J. Lewine. _Schizophrenia: Symptoms, Causes, Treatments_. New York: Norton, 1979. Print.
Blotner, Joseph. _Faulkner: A Biography._ New York: Random House, 1984. Print.
Cohn, Dorrit. _Transparent Minds: Narrative Modes for Presenting Consciousness in Fiction._ Princeton: Princeton University Press, 1983. Print.
Dancygier, Barbara, and Eve Sweetser. _Mental Spaces in Grammar._ Cambridge: Cambridge University Press, 2005. Print.
Duchan, Judith F., Gail A. Bruder, and Lynne E. Hewitt, eds. _Deixis in Narrative_. Hillsdale, NJ: Laurence Erlbaum, 1995. Print.
Faulkner, William. _The Sound and the Fury_. New York: Vintage, 1987 [1929]. Print.
\---. _Light in August_. New York: Vintage, 1987 [1932]. Print.
\---. _Go Down, Moses_. New York: Vintage, 1987 [1942]. Print.
Fillmore, Charles. _Santa Cruz Lectures on Deixis._ Berkeley: The University of California Press, 1971. Print.
Freud, Sigmund. _The Standard Edition of the Complete Works_. Vol. 14. London: The Hogarth Press, 1978. Print.
Frye, Douglas, and Chris Moore. _Children's Theories of Mind: Mental States and Social Understanding._ Hillsdale, NJ: Lawrence Erlbaum, 1991. Print.
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## Alternative Theory of Mind for Artificial Brains: A Logical Approach to Interpreting Alien Minds
ORLEY K. MARRON
We have come to regard Theory of Mind (ToM) as a natural process through which readers interpret and comprehend characters in fiction. Authors such as Jane Austen or Virginia Woolf, writing books about human characters like Emma or Mrs. Dalloway, expect their readers to infer characters' motivations and feelings by assuming a similarity to the readers' own thinking and feeling bodies. The authors often construct their characters as human beings with ToM skills similar to the reader. However, writers may also undermine readers' and characters' assumptions that they correctly perceive others' intentions, by deliberately depicting mistakes in their construction of other people's minds—or _misconstruction—_ thereby creating complex webs of misunderstanding and misinterpretation between people of different cultures or even those growing up in the same home (as in Shakespeare's _Othello_ or _A Midsummer Night's Dream_ ). My interest in this essay is with authors who go so far as to construct test-cases of Theory of Mind by creating interactions between characters that are clearly human and characters that are artificial constructs, harboring alien, uncharted minds. Robots, golems, moving statues, speaking pictures, and dynamic chess pieces—objects visually and cognitively marked as different from humans and thus as more than normally unpredictable, may be presented in narratives as reliable resources for absolute "truth" and reflectors of the human soul, or, at other times, as completely opaque. By describing the interaction between human and nonhuman bodies, and the clash of cultural and biological expectations with alien neuron systems, the texts question and probe the limits of humanity and human expectations, and force the reader to rethink and reconsider reactions that seem "natural." Fantastical worlds and artificial minds provide a perfect laboratory for examining how human characters contend with, adapt to, and overcome—or fail to overcome—obstacles to their own survival and success. Such worlds allow for "extreme case analysis" and provide readers with optional cognitive "playgrounds" for their own thinking and feeling brains.
My examples here are taken from Isaac Asimov and Charles Lutwidge Dodgson (also known as Lewis Carroll), in their engagements with science fiction and fantasy. These authors probe the question of how can humans, when confronted with an alien mode of thinking or behavior, understand the reasoning inside such brains? Stanislaw Lem in _Solaris_ maintains that such feats are impossible: there are entities too great and too obscure for human minds to understand or even to approach (namely the ocean).1 Humans hardly understand their own hidden desires and motives, their obscure biology—how can they comprehend others? But authors like Asimov and Dodgson take the approach that even a partial success is worthwhile, and they optimistically create human characters that learn not to rely on intuitive human ToM skills, but rather to deploy a "Theory of Artificial Minds" (ToAM) in order to succeed or even survive in complex fictional environments. Their protagonists acquire ToAM skills as players acquire a game's rules and strategy, assuming the method in which logicians and scientists approach a problem: they study the environment and its constraints, they observe alien creatures' behaviors, collect as much empirical and background (or historical) information as possible, and then they deduce a set of premises, or game rules. These rules are used in turn to predict the creatures' responses, and must be modified according to the success or failure of their projections. Of course in a simple game scenario there is an underlying assumption that the alien behavior will be consistent, but occasionally inconsistency is also taken into account, and the human characters must live with uncertainty and develop heuristics to reduce this instability. By endowing their characters with an approach that is similar to Dodgson's description of formal logic— that is, take whatever "rules" or premises you observe (whether they seem true or false in the "real world") and follow them to their logical conclusion2—the authors reduce the emotional and even intuitive aspects of character to character interactions. The human character has the propensity for reacting emotionally, but it manages to control its spontaneous reaction (which may be very negative) and to use inference and logic instead to comprehend the alien one. A nonemotional reaction, however, does not mean inhumane. Frequently the protagonists that are most logical and least emotional, like Powell in _I, Robot,_ also prove to be the most open and accepting of alternative ways of thinking. The highly popular character Spock in the Gene Roddenberry _Star Trek_ series refrains from emotional reactions and uses logical inference to understand the needs of other species; he has become a model of respect for diversity and difference in both the television series and _Star Trek_ books. Alice in Dodgson's books for children is just a child, easily moved to tears or anger, but she manages often to control her emotional behavior and apply logical thinking in order to interact with and learn from alien beings. Dodgson and Asimov's human protagonists mediate for the reader a rational access to alien minds and thus encourage readers to adjust to new ways of thinking and alternative perspectives.
Isaac Asimov's _I, Robot_ , published in 1950, was groundbreaking in its inventiveness. Asimov introduced the robot with a positronic brain, the forefather of Data in _Star Trek: The Next Generation_. Science fiction differs from fantasy, explains Larry Niven,3 in that (at the time it is written) you cannot prove that it cannot happen: the premises upon which the sci-fi worlds are constructed must seem plausible and believable within the current scientific community (readers often argue about the details and authors like Niven relate to their concerns, as with Niven's _Ring World_ series). Fantastical worlds, however, while they need to be consistent within their own narrative schema, do not have to seem scientifically plausible (like Terry Pratchett's utterly flat Discworld, which rides on the back of four elephants standing on the back of Great A'Tuin the turtle). Asimov's artificial, human-constructed, positronic robot brain is consistent with a science fiction approach, and a hierarchy of three basic rules built into its logic prescribes the robot's behavior (or "motivation"). A robot may not injure a human being or, through inaction, allow a human being to come to harm; it must obey orders given it by human beings except where such orders would conflict with the first law; and it must protect its own existence as long as such protection does not conflict with the first or second law. However, how these rules are construed by the robot's electronic mind is often surprising, and the human characters face the problem of what happens "when one is face to face with an inscrutable positronic brain, which the slide-rule geniuses say should work thus and so. Except that it doesn't" (56). In "Runaround," Isaac Asimov's story of a drunken robot, two human robot specialists, Mike Donovan and Greg Powell, find themselves in dire need of a robot's services, but the robot, named Speedy, does not function as planned. Rather than bringing back selenium from a certain pool on Mercury's surface, the robot circles the pool again and again, swaying drunkenly and quoting lines from Gilbert and Sullivan. The narrator describes the desperation of the two men: unless they understand why Speedy behaves as it does, and motivate it to act differently, they will burn to death within a few hours. Having a Theory of Artificial Mind becomes here a matter of life or death: thus the story provides for the reader an extreme, nearly bare-bones testing ground of human versus machine thought. Asimov's narrative is constructed as a conundrum or dilemma similar to the detective story. As in detective stories, there is a riddle or problem that must be resolved, and the basic "rules of the game" are already known ahead of time: what are the given physical realities, and what are the possible motivations for the culprit's behavior. Using all the relevant facts, human characters combine Theory of Mind skills with deductive reasoning to construct a solution. In _I, Robot_ , the protagonists know that the three laws of robotics determine Speedy's behavior (rather than love or power or money found in human detective stories), and they must use the same rules to devise a solution to their problem. Lisa Zunshine suggests that detective stories exercise readers' metarepresentational ability, or the ability to keep track of sources of representations (who said what, when), to store such data under advisement and to reevaluate their truth-value once more information becomes available. Detective stories provide pleasure in the cognitive training of this ability (123). In addition, however, these "whodunits" provide the reader with the enormous satisfaction of solving riddles and bringing open questions to a neat and logical conclusion—a response shared by mathematicians, scientists, crossword puzzle solvers and novel readers alike. While Asimov's stories exercise readers' metarepresentational skills to some extent, they emphasize more the gratification of riddle resolution—provided by detective like scientists and robopsychologists with unusual ToM skills.
In "Runaround," Asimov's human characters model two distinct approaches to artificial behavior interpretation. The narrator employs language patterns, action, and thought descriptions to stress the difference. Mike Donovan represents the average human point of view: he reacts very emotionally to the stressful situation, and becomes frustrated and angry, attributing human agency to the robot's mind. Donovan construes the robot's behavior as deliberately antagonistic and menacing. When he calls Speedy _drunk_ , he suggests the negative connotations of tales of self-induced inebriation (e.g., the biblical Noah and Lot), and when he applies the word _thinks_ he implies that Speedy is an independent, free-willed thinker. He thus attributes a malevolent intention to Speedy's refusal to obey orders. By suggesting that Speedy is playing a game Donovan conjures an intentionally irresponsible, even teasing, and aggravating attitude: "'To me, he's drunk,' stated Donovan, emphatically, 'and all I know is that he _thinks_ we're playing games. And we're not. It's a matter of life and very gruesome death'" (43). Greg Powell, on the other hand, is more rational and inclined to proceed logically, and he does not easily fall back on intuitive human ToM. Powell refuses to treat Speedy as a human, nor does he assume deliberate insubordination. Refuting the negative implication of self-induced inebriation, he replies, "Speedy isn't drunk—not in the human sense, because he's a robot, and robots don't get drunk. However, there's _something_ wrong with him which is the robotic equivalent of drunkenness" (43).
When readers first read Donovan's allegations, they may construct a certain view of Speedy's agency and motivation. However, once Powell reacts, Donovan's view will come under advisement. Powell's approach asks the reader to revise an earlier opinion of Donovan's highly emotional message. He refutes Donovan's humanization of Speedy's intentions, by objectifying Speedy, saying that: "A robot's only a robot. Once we find out what's wrong with him, we can fix it and go on" (43). As do investigators in detective stories, Powell attempts to reconstruct the scenario before Speedy "went bad," so as to imagine what could have brought about Speedy's bizarre behavior. Powell thus acts for the reader as a mediator to the artificial mind. He analyzes the environment around the pool, pointing out that while Speedy is perfectly adapted to normal Mercurian environment, the particular region they are in is abnormal and particularly toxic to the Robot's iron body ("There's our clue"). He then requires Donovan to reiterate exactly what he asked of Speedy and realizes that Donovan failed to emphasize the urgency of the request. Powell, using the basic laws of robotics upon which Speedy's positronic brain is constructed, analyzes the miscommunication and the electronic conflict that occurred in the machine brain. Speedy's Rule 3 (saving himself) was strengthened as a result of his costly construction, given that he is "expensive as a battleship." Although obeying Rule 2—human commands—is in general of higher priority than Rule 3, the orders given by Donovan were given casually and without special emphasis, so the Rule 2 potential was rather weak. Powell concludes that the conflict between potentials is what drove Speedy to the pool and away from it, causing him to circle it; the point where the potentials strike an equilibrium is the distance from the pool that Speedy maintains. This explains the drunken attitude, says Powell:
> And that, by the way, is what makes him drunk. At potential equilibrium, half the positronic paths of his brain are out of kilter. I'm not a robot specialist, but that seems obvious. Probably he's lost control of just those parts of his voluntary mechanism that a human drunk has. (46)
Thus using inference and logic, Powell shows the reader how a specific combination of decision states, or algorithmic possibilities, caused a conflict of interest that confused the machine brain. The lack of urgency in Donovan's command, given a low priority level by the machine brain, is what triggered the conflict with Speedy's unusually high priority instruction to protect his expensive iron body. Powell alerts the reader to a more rational representation of the particular reality of machine "thought"—a type of short-circuiting of metal neurons. This human character thus acts like a detective, psychologist, and computer analyst in one. He probes the source of conflict that could cause a robot to appear malevolent and confrontational, without making the mistake of thinking that the robot has an autonomous, self-governed inner self to which his actions are clues. Powell's deductions prove correct and they allow him to devise a solution that brings Speedy out of his confused state and back to normal functioning, saving the lives of the two humans. The scientist puts himself at risk (effectively triggering a response to a No. 1 Rule), overriding Speedy's circuit malfunction with a higher priority command that forces Speedy into positive action. Once Speedy returns to normal and realizes how poorly he performed, he apologizes profusely, thereby disproving Donovan's insinuation of deliberate robot insubordination and reinforcing Powell's approach.
Many of the stories in _I, Robot_ repeat this pattern of strange robot behavior followed by detective-like logical analysis, or ToAM, that explains it. The author thus encourages readers to adopt analytic and deductive thinking when dealing with initially incomprehensible brains. Zunshine suggests that high intensity detective stories pay a price for their demanding cognitive exercise. Such stories must be more focused and emphasize a certain plotline above all others. They do not, for example, give equal time to romance and mystery, since that may create competition with the "metarepresentational framing required to process the detecting elements of the story" (143, 147). Emphasizing aspects of romance or other genres may weaken the mystery plot by compromising readers' ability to keep track of characters' states of minds, especially when they are under advisement. Asimov's stories, too, focus on specific dilemmas, and they rarely create multiple plot lines or complex intentionality levels. Rather, they concentrate on the "artificial mind" problem at hand. Readers need to concentrate on the state of mind of the people trying to understand the machines, the current artificial world rules, and the possible options for robot thought processes. Readers (along with human characters) are working within a type of symbolic logic framework, in which each new premise that the author introduces is used in constructing or testing the next stage.
An exception to this focus on logic is Asimov's "Liar!," a story in which the schema is reversed: instead of humans trying to read the minds of robots, a mind-reading robot attempts to exercise theory of human minds. This strange creation has on the one hand direct access to human thoughts (literally "reading them"), but on the other it has no human judgment or evaluative skills nor any human intuition and common sense. The robot's behavior is structured upon the three rules embedded into its logic and its positronic brain, the first of which is not to hurt people, and the second of which is to obey orders. His interpretation of these commands is wish-fulfillment: when he perceives a certain desire or wish within the human characters, for example a romantic affection for someone or an ambition to take over another's position, the robot attempts to fulfill this desire through stories, or lies. Because the robot lacks the ability to generate embedded narratives of the minds he reads, he cannot imagine the full story of the humans' reactions, and thus he cannot predict the consequences of any actions he takes based on his privileged knowledge. Disregarding the misery that will result in feeding humans on fantasies, he misleads the human characters with invalid "facts," promising them what they cannot have. They, in turn, assume that a machine cannot lie, they fail to verify his input, and act foolishly on false assumptions. When he tells the robopsychologist Susan Calvin that her affections for one of the scientists are reciprocated by that person, even though the man in fact loves another woman, Susan acts on his information and makes futile overtures. The robot does not understand relative magnitudes of human pain, and that the shame of being openly rejected and losing one's pride may be far worse than living with unrequited love. The tenuous balance between the fear of being made to look like a fool in contrast to obtaining one's desire is not apparent to a robot brain. "Liar!" thus provides a test case depicting what happens when one's mind is bound by certain fixed rules, and is privileged with special knowledge, but does not have access to human cultural and biological requirements and expectations. While the robot fails to develop a correct theory of human mind, Susan does figure out how his artificial mind works. She realizes that her assumption that a machine is unbiased and therefore provides true facts was wrong, and that all her coworkers fell for the same illusion, eager to have their fantasies fulfilled. Incensed by being made to look a pathetic fool, she destroys the robot by charging him with causing pain to humans and thus driving him mad.
About a century before Asimov, Charles Lutwidge Dodgson also created worlds full of artificial beings. Dodgson was a professor of mathematics and loved math and logic, observing a unique type of truth and beauty in these subjects. He believed everyone should be trained in logic theory and mathematics, as it provided enjoyment and an excellent way to develop the mind. Using various visual aids he developed teaching methods fit for different audiences, including those not expected to study logic in Victorian England—young children, high school girls, and women teachers. In his first volume of _Symbolic Logic_ , which appeared in February 1896, he addressed schoolteachers in a separately printed prospectus, where he points symbolic logic out as a healthy mental recreation in the education of young people between twelve and twenty:
> I claim, for Symbolic Logic, a very high place among recreations that have the nature of games or puzzles; and I believe that any one, who will really try to understand it, will find it more interesting and more absorbing than most of the games or puzzles yet invented. (45)
Dodgson claimed that people harbor three false ideas about the study of logic: "that it is too hard for average intellects, that it is dry and uninteresting, and that it is useless." He rebutted all three, and concluded that this fascinating subject will be "of service to the young" and should be taken up in high schools and by families.4
Similar to Asimov's construction of robotic mind-riddles for youth and adults, Dodgson invented for children fascinating games, puzzles, and riddles that engaged their attention and encouraged them to solve problems and to experience the satisfaction of coming up with clever solutions. I suggest that Dodgson's Alice books themselves ( _Alice in Wonderland_ and _Through the Looking Glass_ ), the first originally written for Alice Liddell,5 were a powerful means of expressing to thousands of children and adults, across generations, the beauty and wit and power of logic and math. In these books, Dodgson creates a new model protagonist who acts out his ideas, a heroic paradigm that boldly seeks new and unknown regions—not just physical, but also of the mind.6 Dodgson's Alice models a logician's approach to alternative ways of thinking and to the understanding of alien minds.
Dodgson's imaginative worlds, created for a young audience, are different from Asimov's more serious, scientific-sounding constructs, and they do not claim to be possible technologically. The inhabitants are purely fantastical, a combination of mythological creatures, characters out of fables, political cartoons, and _humanobs_ (humanlike, man-made animated objects). While the stories seem like a wild collage of children's memories and scenes from pretend games, they display Dodgson's remarkable acuity as a teacher for children. He uses objects and concepts that are familiar to many Victorian children (e.g., poetry, puns, and visual art) to introduce new ideas, and thus he provides a type of learning lab for the character Alice—and for the children who read her story. Unlike Powell and Donovan, Alice does not come equipped with a predetermined set of premises through which she can interpret alien behaviors. There are no "three laws of Wonderland creatures." Like a child in a bewildering and alien adult world, she is unprepared and must undergo a complex process of learning to interpolate world rules and acquire meaning. Many of the _Wonderland_ or _Looking Glass_ world creatures with which Alice interacts, especially the playing cards and chess game pieces, are comparable to elements of formulas and sets that Dodgson presented in his mathematic lectures and game theories. They have values and attributes that can be used to categorize them as elements of sets, and they are bound by specific game or world principles. Pawns, for example, can only move in specific directions, and in the _Looking Glass_ world creatures must sometimes walk backward in order to move forward. When Alice approaches them with her Victorian child's conceptions, she often finds that they cannot be understood simply through her own embodied ToM. Her critical, reasoning mind is the bridge that enables her to close the gap between human expectations and alien modes of thought, allowing her to construct a ToAM. Alice develops her ToAM gradually, by using—as Powell did—empirical information, inference, and deduction. At every new encounter she gains more knowledge. As she compiles a set of rules of behavior by which the strange creatures abide, she improves her ability to predict their reactions, and she devises adaptive strategies to deal with them and to maneuver toward her goals. Dodgson, through the narrator character (a type of "fond uncle"), constructs Alice as a curious, independent, and humorous person with deductive powers, as well as frustrations and concerns—someone with whom children can identify and whose learning methods they may imitate. Alice's learning process thus becomes a model through which Dodgson promotes logic and inference as tools for dealing with new ideas and unconventional minds.
How does Dodgson convey this process to the reader? Like Asimov, he uses his protagonist to mediate the access to alien minds. Such mediation may not be necessary when alien creatures exhibit humanlike emotions and behaviors, such as fear or anguish or anger. The narrator uses what Alan Palmer terms a thought-action (or action-disposition) type of description—for example, the gardener Five "had been anxiously looking across the garden," and when the Queen appears "the three gardeners instantly threw themselves flat upon their faces" (Palmer 108). These behaviors are easily interpreted by both Alice and readers, and evoke immediate responses: Alice pities the frightened gardeners when they are threatened with decapitation, and she hides them in a flowerpot. But behaviors that are remarkably bizarre and cannot be understood intuitively, such as physical reactions based on prediction of the future or disembodied responses, require mediation and cognitive analysis. For example, many of the characters seem to ignore physical risk to themselves and to others, and they misinterpret each other's (and Alice's) needs. The Red Chess Queen, after an exhausting race, offers Alice dry biscuits to "quench her thirst" rather than water. The footman nearly gets his nose smashed, but utterly ignores the event.7 And the Duchess rocks her wailing baby under a barrage of flying pots and pans, insensitive to the blows and callously ignoring the danger to the baby's well-being—in contrast with Alice, who (in keeping with her humane character) becomes greatly anxious ("jumping up and down in an agony of terror") for the poor child's welfare (84). In addition, the physical world rules of the fantastical worlds shape the physical reactions and assumptions of the inhabitants. They must run extremely fast to stay in the same place, or move backward to advance. In the _Looking Glass_ world where future events are known in advance (that is—creatures "live backwards"), the characters frequently act on the knowledge of a future or potential event rather than in response to a direct physical cause. Cakes must be served before they are cut. People are punished before they are found guilty, and injuries are tended to before they occur. The white chess Queen cries in pain—anticipating a pinprick from her brooch, but remains smiling when the event actually takes place, and refrains from screaming again having "done all the screaming already." The Queen thus displays a disembodied emotional response to a physical event occurring on her body, as if she is insensitive to pain resulting from the prick.8 The earthly empirical concept of direct "cause and event," with which Alice is familiar, becomes distorted by the effect of "living backwards."9 At this more complex level of behavior, Dodgson's artificial creatures seem quite inscrutable, and Alice's ToM inference falls short of predicting their reactions or guiding her as to how she should best respond.
The narrator has several ways of revealing the internal workings of alien minds to readers. He can give authorial commentary, or expose the alien mind to the reader directly, or use the protagonist, Alice, to figure out the ToAM for the reader. As it turns out, like many other science fiction and fantasy narratives including Asimov's robot stories, the narrator creates an asymmetric mode of mind exposure. Unlike Alice's broadly exposed contemplations and embedded narratives, the alien characters' minds are opaque: the narrator does not expose them for viewing and there are no descriptions of their thought processes, nor of their embedded narratives. That is, there are no unequivocal authoritative statements such as "the Gardener felt sure that his end was near . . ." or "he hoped that Alice would save his life, despite her youth." Nor does the text describe the cards' mental plan to escape their plight, and the Queen's attempt to prevent them. This mental opaqueness dehumanizes the artificial characters, and keeps the reader from "seeing through" their eyes and from adopting their point of view, maneuvering the reader to focus on Alice and look through her perspective. Alan Palmer describes a similar narrative technique in Evelyn Waugh's _Vile Bodies_ , where the narrator dehumanizes even human characters by hiding their thoughts and embedded narratives from the readers.10 It is Alice's analytical and probing deliberations, revealed to the reader, that allow, as do Powell's inferences, access to the characters.
But because Alice cannot rely on intuitive (or previously developed) understanding and intuition, she needs to adopt a more formal analytical approach. Alice needs to collect empirical data by observing the characters' conduct (including their reaction time and cause-and-effect schema), by listening to their input and by repeatedly questioning them. She needs to acquire new terms and language skills to better comprehend their world and to understand their obscure language and its implications (just as logicians and mathematicians require a definition of terms). Alice must then extrapolate new world rules from the creatures' speech and body motions, test these rules by experimentation, draw conclusions, and finally apply them—both to social situations and to physical maneuvers en route to her goal.11 Thus fantastical worlds such as these encourage in Alice the exercising of a logician's type of ToAM. Since Alice's thoughts _are_ described by the narrator, the reader becomes privy to the deductive processes, inferences, and conclusions she generates as she constructs a schema of new world rules and behaviors.
One example of the process of deliberation and new schema construction is the remarkable question and answer dialogue between the White Queen and Alice. In this discussion Alice studies the idea of futuristic causality, a phenomenon that explains alien creatures' disembodied behavior as well as their ability to conceive "impossible things." In a world where potential events are seen as if they already occurred, empirical data is not immediately necessary for something to be proven as true and believable, but only the _expectation_ or potential that it will occur. Thus, explains the White Queen, the King's Messenger is already sitting in prison, being punished, "and the trial doesn't even begin till next Wednesday, and of course the crime comes last of all." Alice expresses her confusion about such a system of causal rules: what if the event does not occur—that is—it becomes a failed alternative? The Queen insists that it may be so much the better, but Alice has a hard time accepting this alien notion, and especially its ethical implications. Not only will there be effect without cause, but conventions of justice based on the concept of punishment only of the guilty may be altogether eradicated. Alice's encounter with alien world rules thus makes her aware of cultural conventions she (and readers) have taken for granted:
> "Suppose he never commits the crime?" said Alice.
>
> "That would be all the better, wouldn't it?" the Queen said . . .
>
> Alice felt there was no denying _that_. "Of course it would be all the better," she said: "but it wouldn't be all the better his being punished."
>
> "You're wrong there, at any rate," said the Queen. "Were you ever punished?"
>
> "Only for faults," said Alice.
>
> "And you were all the better for it, I know!" the Queen said triumphantly.
>
> "Yes, but then I had done the things I was punished for," said Alice: "that makes all the difference."
>
> "But if you _hadn't_ done them," the Queen said, "that would have been better still; better, and better, and better!" . . .
>
> Alice was just beginning to say "There's a mistake somewhere—", when the Queen began screaming . . . (248)
Alice's analytical acquisition of a theory of alien minds seems to be highly conscious and goal-oriented, perhaps similar to the descriptions of autistic savants' attempts at learning to read human faces and intentions.12 It seems to differ from the automatic or nonconscious manner in which normal human children acquire ToM skills. However, it may be that the basic cognitive / neurological structures underlying both automatic and self-aware ToM acquisition are alike or even common to both. The manner of acquisition of ToM skills by children is still debated by researchers, a crucial question being of the relative weights of innate capabilities versus social training. Some researchers emphasize a more modular and "innate" approach, while others, such as Jay L. Garfield, Candida C. Petterson, and Tricia Perry, argue that the acquisition of ToM is "dependent as well on social and linguistic accomplishment, and that it is modular in only a weak sense" (505). Garfield et al. present a developmental model that is "social and ecological as opposed to being individualistic." I suggest that for any model of automatic ToM acquisition in normal children, underlying that capability is a basic biological capacity for rule acquisition, or positive-negative value acquisition, (perhaps using neurological constructs such as those as found in conditioning described by Larry Squire and Eric Kandel, 57). That is, infant brains, on the basis of social and environmental interaction, must be able to construct certain cause and effect rules—albeit automatically and not in a language form. For example—"Given milk, the pain in my stomach disappears," or "the light is turned on, someone will come to me," and eventually "if I hit someone, they will feel pain, shout at me, or hit me back." Children's ability to read visual body cues and to comprehend others' intentions may develop automatically (or without awareness) through social interaction, but it relies on a neurological logical-comparative mechanism, which enables the child, unconsciously, to test inferences and to discern and remember causal rules. I believe this comparative mechanism comes into play, and becomes more noticeable to the interpreting person when an obscure or alien mind crops up, and the interpreter finds that he or she has to consciously extract information and generate new causal rules to comprehend that mind. Authors like Dodgson and Asimov, a logician and a scientist who use analytical thinking in their work, externalize and expose the logical process in their narratives by depicting the interaction of human and alien characters. They seem to suggest humans should recognize the limitations of their normal human intuition and, when confronted with seemingly alien modes of behavior, they should adopt a more formal logical approach. Like their fictional protagonists, readers should observe others, collect empirical data and extrapolate new world—or cultural—rules. This approach can encourage readers to adopt a more flexible, and perhaps sympathetic, approach to different modes of thinking. Many science fiction and fantasy writers (including _Star Trek_ producer Gene Roddenberry), believed that their narratives could thus encourage a more pluralistic society.13
### NOTES
1 Stanislaw Lem, 1921-2006, was a science fiction writer whose works explore philosophical themes such as the impossibility of mutual communication and understanding. The official Lem site is _Solaris_ , ed. Stanislaw and Tomasz Lem 2000-2007 http://www.lem.pl/.
2 Morton N. Cohen, _Lewis Carroll, a Biography_ (New York: Alfred A. Knopf, 1995) 446.
In 1886 Dodgson wrote _The Game of Logic_ , in which he explains that it does not matter what the beginning premises are, as long as they are followed logically to the end:
> I don't guarantee the Premises to be facts . . . It isn't of the slightest consequence to us, as Logicians, whether our Premises are true or false: all we have to make out is whether they lead logically to the Conclusion, so that, if they were true, it would be true also.
3 The prolific science fiction writer Larry Niven has written over sixty books, among them the _Ring World_ series, and has created the unique alien the Pierson's Puppeteer; personal interview, March 18, 2008, Bar Ilan University Creative Writing Visitor Lecture.
4 In his introduction to his _Symbolic Logic_ book, Charles Dodgson (a.k.a. Lewis Carroll) writes:
> this is, I believe, the very first attempt . . . that has been made to popularize this fascinating subject . . . But if it should prove, as I hope it may, to be of real service to the young, and to be taken up, in High Schools and in private families, as a valuable addition to their stock of healthful mental recreation, such a result would more than repay ten times the labor that I have expended on it. (47)
5 Alice Liddell was one of Dodgson's favorite child-friends. Morton Cohen in the biography of Dodgson describes the famous trip that took place on July 4, 1862, in which Charles Dodgson, Robinson Duckworth (Fellow of Trinity College), and three Liddell sisters set off for Godstow for one of their picnics. On the way, Charles told Alice her story, and Alice petitioned that the tale might be written out for her, and _Alice's Adventures Underground_ (later "in Wonderland") was created (89).
6 For discussion of the new paradigm in Dodgson's books, see Orley Marron, "Fictions Beyond Fiction: A Cognitive-Literary Study of Human-like Animated Objects in the Fiction of Lewis Carroll and Nathaniel Hawthorne," diss., Bar-Ilan University, 2008.
7 The footman in _Alice's Adventures in Wonderland_ does not react as a normal human would to a near catastrophe:
> At this moment the door of the house opened, and a large plate came skimming out, straight at the Footman's head: it just grazed his nose, and broke to pieces against one of the trees behind him. "—or next day, maybe," the Footman continued in the same tone, exactly as if nothing had happened. (81)
8 In _Through the Looking Glass_ (hereafter _TLG_ ), the White Queen displays anticipatory responses before she is injured, crying out loudly, and then explaining to the worried Alice that "I haven't pricked it yet, . . . but I soon shall—oh, oh, oh!" (249).
9 The essence of "living backwards" as explained by the White Queen in _TLG_ 247 is that "one's memory works both ways." Alice explains that her memory works "only one way" and that she "can't remember things before they happen."
> "It's a poor sort of memory that only works backwards," the Queen remarked.
>
> "What sort of things do you remember best?" Alice ventured to ask.
>
> "Oh, things that happened the week after next," the Queen replied in a careless tone. (247)
10 Palmer discusses presentation of consciousness in _Vile Bodies_ by Evelyn Waugh:
> The key to the fictional minds in Vile Bodies is that there is a good deal of intermental thinking but very little evidence of doubly embedded narratives. There is very little indication of the existence of one character in the mind of another...The lack of doubly embedded narratives demonstrates some very solipsistic states of mind . . . [it] contributes substantially to the callous and unfeeling quality of the novel. (233-34)
11 For example, by conversing with the talking flowers in _TLG_ , Alice learns to walk backward in order to move forward toward the Queen.
12 Autistic savant Temple Grandin wrote the book _Animals in Translation: Using the Mysteries of Autism to Decode Animal Behavior_. Temple attributes her success as a humane livestock facility designer to her ability to recall detail, which is a characteristic of her visual memory. She has become aware of details to which animals are particularly sensitive, giving her insight into their needs, and allowing her to design humane animal-handling equipment. Her site is http://www.grandin.com/.
13 Gene Roddenberry's _Star Trek_ scripts promoted many forms of interracial relations, exemplified by the first interracial kiss broadcast on television. Actress Nichelle Nichols, who played Lieutenant Uhura, discussed the importance of this symbolic event as well as Martin Luther King's conviction that her role in the series provided an immensely important non-stereotypical model for African-Americans. She was presented on an equal basis, on a level of dignity and authority, and with the highest of qualifications. Interview and text can be found in the "Trek Nation" news site, "Nichols Talks First Inter-Racial Kiss," _Trek_ , ed. Christian Hohne Sparborth, 5 Sept. 2001, http://www.trektoday.com/news/050901_05.shtml.
### WORKS CITED
Asimov, Isaac. _I, Robot._ New York: Bantam Dell Books, 2004. Print.
Bartley, William Warren. "Introduction." _Lewis Carroll's Symbolic Logic, Part I & Part II_. 1896. By Lewis Carroll. Comp. and ed. William W. Bartley. New York: Potter, 1977. Print.
Carroll, Lewis. _The Annotated Alice: Alice's Adventures in Wonderland & Through the Looking Glass Today._ Comp. and ed. Martin Gardner. London: Penguin Books, 1970. Print.
\---. _Lewis Carroll's Symbolic Logic, Part I & Part II_. 1896. Ed. William Warren Bartley, III. New York: Potter, 1977. Print.
Cohen, Morton N. _Lewis Carroll, a Biography_. New York: Alfred A. Knopf, 1995. Print.
Dodgson, Charles Lutwidge. _The Game of Logic._ London: McMillan & Company, 1887. Print.
Gardner, Martin. "Introduction and Notes." _The Annotated Alice: Alice's Adventures in Wonderland & Through the Looking Glass_. By Lewis Carroll. London: Penguin Books, 1970. Print.
Garfield, Jay L., Candida C. Peterson, and Tricia Perry. "Social Cognition, Language Acquisition and the Development of Theory of Mind." _Mind and Language_ 16.5 (2001): 494-541. Print.
Hambly, Barbara. _Ishmael (Star Trek No. 23)_. New York: Pocket Books, 1985. Print.
Palmer, Alan. _Fictional Minds_. Lincoln: University of Nebraska Press, 2004. Print.
Squire, Larry R., and Eric R. Kandel. _Memory: From Mind to Molecules_. New York: Henry Holt and Company, 2003. Print.
Waugh, Evelyn. _Vile Bodies._ Boston: Little, Brown and Company, 1930. Print.
Zunshine, Lisa. _Why We Read Fiction: Theory of Mind and the Novel_. Columbus: The Ohio State University Press, 2006. Print.
## Reading Phantom Minds: Marie Darrieussecq's _Naissance des fantômes_ and Ghosts' Body Language
MIKKO KESKINEN
> l'absence de mon mari à mes côtés me désigne plus que jamais comme un corps étranger.
>
> —Marie Darrieussecq, _Naissance des fantômes_
>
> the absence of my husband at my side marks me more than ever as a foreign body.
>
> —Marie Darrieussecq, _My Phantom Husband_
Currently, Theory of Mind could be dubbed Practice of Body without losing much of its functional content. Actual people, real readers, and fictional characters are all accustomed to interpreting human body language as indicative of thoughts and feelings, that is, of otherwise hidden mental structures. This move from body to mind usually happens so automatically that to call it a _Theory_ of Mind is something of a misnomer.1 But the theory in Theory of Mind becomes more justifiable than the practice in Practice of Body, when the default reading cannot take place. This failure in the mind reading ability may be due to a number of reasons. The person occupying the position of the mind reader may be suffering from autism or Asperger's syndrome, making the subject incapable of such immediate inference. Or, on the other side of the process, there may not be a body present to infer a mind from.
In this essay, I address the latter case by analyzing Marie Darrieussecq's 1999 novel _Naissance des fantômes_ ( _My Phantom Husband_ ).2 In that novel, the female narrator's husband suddenly disappears, and possibly dies, leaving her in a state that develops from the initial disquietude to numbness and eventually to hallucinations (or, alternatively, to witnessing actual visitations).3 The wife's mind reading activities necessarily—and radically—differ from those usually performed by fictional characters. Since her husband is not corporeally present, she cannot interpret his body language and behavior as indicative of his mental structure or state. Instead, she must rely on her memories and the residual markers of his past behavior.
Not surprisingly, her very first mind reading activities relate to the possible reasons for his disappearance, to his corporeal absence. The speculation on the workings of his mind or on his mind-set in general is made explicit on the novel's opening page:
> Mon mari a-t-il disparu parce que, se soir-là, après des années de négligence de ma part, excédé, fatigué par sa journée de travail, il en a eu subitement assez de devoir, jour après jour, redescendre nos cinq étages en quête de pain? (11)
>
> [Has my husband disappeared because that evening, after enduring years of my negligence, exasperated, worn out by his day at work, he was suddenly fed up with having to go back down our five flights of stairs, day after day, in search of bread?] (1)
The investigators provide some more scenarios, based on the statistical data of two hundred missing persons a day, ranging from the possibility of his going on a vacation with a beautiful blonde to that of a suicide. Soon, however, it becomes evident that there is no cunning plan, no scheming mind behind the mysterious disappearance. The husband is probably as dull and unimaginative as the mundane dreams ("d'une quotidienneté consternante" [45]; "distressingly ordinary" [35]) he has of his business or of shopping at the local bakery. Significantly, it is his wife ("beaucoup ennuyée" [42]; "extremely bored" [32]) who makes up Freudian interpretations of his dreams. He is, indeed, in the real estate business, with the main emphasis on the first word, whereas she is busy being unreasonable or even surreal. Narrationally, this means that she can seamlessly move, in free direct discourse, from reporting the investigators' utterances to her own speculative details of forensic precision, and back again:
> d'autres encore passaient la frontière et il ètait sage, là, de renoncer à tout espoir, d'autres enfin se jetaient dans la mer, échouaient tout gonflés sur les plages, les yeux et la langue mangés par les crevettes, des colonies de bigorneaux dans le rectum, il valait mieux s'épargner ce genre de retrouvailles. (12)
>
> [while others cross the border, in which case it's wisest to renounce all hope, and still others throw themselves into the sea, washing up bloated on the beaches, eyes and tongue eaten by shrimp, bodily crevices colonized by sea snails, it's best to spare yourself this type of reunion.] (2)
Immediately after her rambling digression into the corporeal decay, she resumes the investigators' crucial question: "Les enquêteurs m'ont demandé si mon mari était dépressif" (12); "The police asked me if my husband was prone to depression" (2). That question could justifiably be posed to herself as well.4
Her husband turns out to be, in the police investigation, curiously ambiguous and semi-transparent even before his disappearance. The real estate clients he meets on the day of his mysterious vanishing fail to remember his facial features (12 / 2), and the baker does not recall his customary visit to the corner _boulangerie_ (21 / 12). Objective evidence also testifies to his vacillating physical appearance. Trying to get a "un sens de la réalité" (49) ("sense of reality" [40]), the narrator looks at her wedding photo but notices that her husband's image is (or has become) vague: "La photo avait bougé. Elle était devenue floue [. . .]. C'était la photo de sa disparition" (49–50); "The picture had moved. It had blurred [. . .]. It was the photograph of his disappearance" (40–41). This man without qualities (if not without qualia) thus resembles a ghost or a zombie even before his possible death.
Although the husband is both physically unassuming and mentally unimaginative, the narrator has, at three o'clock in the morning, an illumination that he had framed his disappearance for a purpose:
> mon mari n'avait mis en scène le scénario de sa disparition que pour me rendre à la conscience de cet amour, à seule fin, lorsqu'il aurait eu la prevue (à son tour) de mon désespoir, de revenir en conquérant recevoir les multiples gifles que je ne manquerais pas de lui administrer, quel abruti, après m'avoir livrée combien de temps aux épouvantes du veuvage. (35)
>
> [my husband had stage-managed the scenario of his disappearance only to bring me to an awareness of my love, in the sole aim of making a triumphal return as soon he (in his turn) wielded the proof of my despair; to receive the multiple blows I would not fail to administer, what a jerk, after having abandoned me for how long to the terrors of widowhood.] (26)
This plan, cunning as it may seem, only generates a third- or fourth-order intentionality: she thinks or believes that her husband wants to make her aware of her love, giving him the reason to return to receive his punishment. The final phrase starting with "de revenir" is not likely part of the husband's plan but, rather, in its detailedness, her wishful thinking. So is the whole speculation, as she herself admits in the next paragraph when she does some source-monitoring (in disguise of self-scrutiny) fifteen minutes after her illumination: her husband is "un homme dépourvu d'imagination, et, de plus, extrêmement gentil" (35); "a man devoid of imagination, and in addition extremely kind" (26). The whole elaborate scheme derives from one source: her mind. She is the ultimate (and only) mastermind behind the theory that seeks to solve the mystery by casting it as a love plot or test of love. The explanation she provides is therefore essentially susceptible. She seems to be an unreliable narrator in the sense that she misreports, misreads, and misregards, which forces the audience to discard her "words and reconstruct an alternative" (Phelan 219) to get the things right. On the other hand, that she admits she was wrong in this particular case makes her tend toward reliability. She is systematically neither. Just as a narrator can be "unreliable in different ways at different points" (Phelan 52), s/he can also fluctuate between reliability and unreliability in the course of narration (which makes the audience's role all the more challenging). This kind of narrator, by being alternately or in degree reliable and unreliable, is always to be doubted (Phelan 53), which makes him or her, basically, unreliable until found truthful at a particular point. Hence, not even the narrator's unreliability is always reliable.
In Darrieussecq's case, the situation is even more complex than that. First, the nature of the phenomena that she perceives and reports defy accurate narrating beyond reasonable doubt. The very ontology of a ghost is questionable, as is the epistemology of its human perception. One could dub the novel's ghost an "unreliable character," were it not an unnecessary term for a persistently reticent agent.5 The narrator could not, even if she wished, be totally reliable in connection with ghosts. Second, even when giving an unreliable account of the world around her, she performatively provides a reliable or at least truthful-seeming representation of her inner world, the workings of her own mind.
### READING GHOSTS' BODY LANGUAGE
The first complication, mentioned above, relates to the novel's main concern, made explicit in its French title, the birth (and consequent existence) of ghosts. Ghost, in the meaning of "the soul of a deceased person, spoken of as appearing in a visible form, or otherwise manifesting its presence, to the living" ("Ghost," def. 8a) is a matter, albeit an immaterial one, of belief, not a verifiable fact. It is symptomatic that we ask if someone believes in ghost _s_ , in the plural, rather than in _a_ ghost, in the singular. The ghosts' mode of existence problematizes not only reliability, but also the very idea of Theory of Mind or mind reading as constructs or activities based on corporeal data.
The ghosts do not have bodies (of their own) to base a theory or a reading on. Ghost, in the meaning of the "immaterial part of man as distinct from the body or material part" ("Ghost," def. 3a), seems, by definition, bodiless. The ghost does not have a proper body, even doubly so. When still in the living body of its host, the immaterial part (the soul or spirit) of a human is not called a ghost. Ghost as the "soul of a deceased person" appears "in a visible form" or otherwise manifests its presence ("Ghost," def. 8a). The visible form of a ghost is the body of another, an artifactual or prosthetic body, or an ambiguous entity like fog (Derrida 7, 126, 194).6 A corollary of this state of affairs is that visuality or, more accurately, physical resemblance is not the ghost's primary attribute. Not only does a ghost's appearance differ from what used to be its original bodily host, the ghost is also unidentical with its _present_ physical appearance in the sense that it is traditionally believed to lack a mirror-image. Therefore a ghost will not recognize itself in a mirror, but _others_ will recognize it as one by this very token (Derrida 156). Nevertheless, the very distinction of the ghost as mind, spirit, or soul from the body demarcates its invisible contours by negation. A ghost is what body or matter is not. Without what frames it, a ghost would be not only imperceptible, but also unspeakable.
The incarnation of spirit in _some_ body other than the one it inhabited while still alive, indeed gives birth to a ghost (cf. Derrida 127). A ghost is, thus, engendered posthumously, but the most important aspects of its mind derive, to create a preposterous neologism, from the subject's "pre-posthumous" phase.7 Hence, perhaps, the _birth_ of phantoms in Darrieussecq. The _ghost's spirit_ , conceived of as preceding incarnation, would be an impossibility, for the possessive case contradicts, in spectral logic, the state of not being possessed. What all this means to Practice of Body (also known as Theory of Mind) is that the ghost's body language is, by definition, a second language, a language articulated or, rather, gesticulated in and by a foreign body.
In _Naissance des fantômes_ , ghosts appear in the narrator's life as metaphors and representations before there are any manifestations proper. She has been afraid of ghosts since her childhood and has shown exceptional talent in dreaming up their qualities in dreadful detail (43 / 33). She is afraid of ghosts, not because of their snatched, horrendous bodies, but because of their evil minds. She does have a Theory of Posthumous Mind based on a Practice of Invaded Body. Ghosts are, in her reading of their semivaporous corporeality, malicious and scheming, stalking innocent humans with "des griffes, des dents, des ventouses" (43); "claws, teeth, suckers" (34). She has a number of ghostbusting means, the most important of which is her husband and his steady breathing at night (31). With her husband gone, she is exposed to the shadows of the night and left with her old, less efficient ghost-repelling tricks: thinking of her own madness ("folie" [43]), light reading (even gory horror, but not fantasy or flat realism), repeated turning on of lights, and staying awake all night long (43–44 / 33–34).
In a dreadful twist commonly found in modern ghost stories, spectrality draws so near that it verges on incorporating the narrator: she momentarily acquires visual attributes of the very entities she fears. When she spreads, for cleansing purposes, green paste on her face, it makes her look, besides an alien and creature of the depths, like "un fantôme vieux modèle" (37); "like an outdated model of ghost" (28). She is, thus, a veritable mirror-image of a ghost, of an entity that allegedly has none. Reflective absence is conventionally a token of a ghost's presence. That she can see her face in the mirror indicates that she only resembles, and not actually is, a ghost.
To her surprise and awe, she witnesses a manifestation in broad daylight, which deviates from the usual midnight appearance of the classic specters she is accustomed to dreading.8 Significantly, the narrator does not explicitly call the apparition a "ghost" but a "shadow" ( _ombre_ ). It could be read as referring to another kind of (supernatural or unnatural) entity besides a phantom, but, in the context of literary ghosts, it rather reads as a euphemism reminiscent of some staged apparitions in Marlowe and Shakespeare (cf. Greenblatt 156, 170). The odd presence is a "densification de l'espace" ("densification of space") and "un épaississement de l'air" ("thickening of [. . .] air," a three-dimensional "colonne de l'air dans l'air" ("column of air in the air") and, although an "ombre" ("shadow"), "si claire, si fluctuante dans la lumière" ("so bright, so fluctuating in the light") (54–55 / 46). She also tries to make contact with this strange phenomenon, but it does not react to her gestures ("il n'y avait pas d'effet de miroir" [55]).9 Astonishingly, the entity is not completely immaterial: when the narrator steps inside it, she feels "une pression, une prise" (55); "a pressure, a hold" (46) In any case, the narrator seems to presume that the phenomenon is a living entity, perhaps even having a mind, but she does not regard it, unlike she does the imagined nocturnal ghosts, as malicious or frightening.
Some thirty pages later, the thickening and densification of space is specified as that of a "solidification du vide" (86); an emptiness or "solidification of void" (77). That empty space is, predictably, created by the disappearance of her husband. It is not exactly an absence, a vacuous space formerly occupied by him, but an invisible entity substituting for that space ("Le vide s'était fait à l'endroit même qu'il occupait" [86]).10 The difference is significant, since it implies that her husband, or his soul or mind, has returned, as the aptly named revenants customarily do, incarnated in another, albeit indiscernible, "body." This state of affairs reminds one of Derrida's formulations in _Specters of Marx_ : "there is never any becoming-specter of the spirit without at least an appearance of flesh, in a space of invisible visibility, like the dis-appearance of an apparition" (126). Again, the only thing that the narrator reads in this fluctuating texture of a body is that it relates to the _existence_ of a substituting intending subjectivity, specifically to that of her husband, but without any modalities.
Soon the narrator does, however, arrive at a conclusion concerning the reading of phantom minds, on both general and specific levels. In general, she concludes, "[m]ême nommés, touchés ou traversés, les fantômes ne perdent ni en puissance ni en indulgence" (96); "[e]ven when named, touched or crossed through, ghosts lose none of their power or indulgence" (87). And, in particular, "[m]on mari, copiant les morts, allait me faire signe et me rendre à l'éxistence" (96); "[m]y husband, copying the dead, was going to give me a sign and return me to existence [. . .]" (88). Neither of these readings is based on actual bodily information. The first is a generalizing statement, a deductive _idée reçue_ of sorts. The second is speculation disguised as logical necessity:
> Mon mari était forcément quelque part, gazeux peut-être, à la limite de sortir de l'univers, mais quelque part forcément, penché sur les bords (ce qu'il faut bien supposer de bords) et me regardant; comme les morts dont les vivants savent qu'il sont encore là [. . .]. (96)
>
> [My husband was necessarily somewhere, vaporous perhaps and on the verge of exiting the universe, but necessarily somewhere, leaning over the edge (whatever edge we must suppose there is) and watching me; like the dead whose living know they are still there [. . .].] (88)
The postulated, gas-like body becomes visible a few pages later at a supermarket where "[l]e bac fumait et dans cette fumée se devinait un corps gazeux, informe et flottant, doué pourtant d'une sorte de volonté [. . .]" (109–10); "[t]he freezer gave off a vapor and in that vapor a gaseous body could be distinguished, formless and floating, though endowed with a kind of will [. . .]" (101). The materialization is not only visual but also haptic: "je tendais la main et cela me caressait délicatement les doigts, me pressait amoureusement la paume" (110); "I stretched out a hand and it caressed my fingers delicately, amorously pressed my palm" (102). In both manifestations, then, the almost incorporeal body is readable or at least indicative of a will or emotion.
The next phase in the gradual materialization of the phantom (or shadow, or whatever it should be called, for it is not unequivocal that the supernatural entity is the same one on each occasion) is the predictable manifestation of the husband. At her mother's farewell party toward the end of the novel, the narrator and some other guests witness the husband's semitransparent figure emanating before their very eyes. The narrator-wife reads his mind more specifically than that of the supermarket specter: "mon mari sembla hésiter, vacillant sur le seuil" (144) ("my husband seemed to hesitate, trembling on the threshold" [136]); "[m]on mari, malgré son teint lumineux, avait l'air anxieux et comme délavé" (145) ("[m]y husband, despite his luminous hue, looked anxious and washed-out" [138]). When she tries to touch him, he disappears, leaving behind only some residual light on her fingers and eyes (149 / 140).
The novel ends with the narrator returning home and finding her husband there, looking exactly the same as the day he disappeared. She reads his body language and infers from it even more complicated intentions than at the party: "il semblait attendre quelque chose, comme si c'était à moi de l'accueillir, de lui demander, que sais-je, s'il avait bon voyage" (157); "he seemed to be waiting for something, as if it were up to me to welcome him, to ask him, what do I know, if he had had a nice trip" (149–50). He embraces her with his "mains peu matérielles" ("barely material hands") thus without exactly touching her but charging her with a particular energy, and disappears (161–62 / 153–54). This also puts an end to her kind of supernatural (and natural, in the sense of faunal) mind reading activities:
> je sais seulement avoir cessé, à ce moment-là, de me demander si mon mari (si les chats, les oiseaux, les poissons et les mouches aux yeux à facettes) sentait et voyait tout de même ce que moi je sentais et voyais. (162)
>
> [at that moment I stopped wondering whether my husband (or cats, birds, fish and flies with their many-faceted eyes) felt and saw what I felt and saw.] (154)
This last sentence of the novel may be read as a statement implying that all the apparitions she has recounted so far do in fact relate to her as a feeling and seeing narrator, and therefore are not necessarily reliable accounts of actual happenings in the story-world, but, rather, projections of her own mind.
As is commonplace in ghost stories, either ghosts actually exist in _Naissance des fantômes_ 's storyworld or the narrator only believes she sees them (possibly because she is mad). In the latter case, it is the narrator's and not the supernatural entity's mind that is foregrounded, and the recursiveness easily becomes self-referential: the mind represents itself even when ostensibly representing others' minds (or any constituents of the storyworld). The "architectural truth" (cf. Cosmides and Tooby 60–61) of the narrator's metarepresentations is possibly that of a haunted house.
The epigraph taken from _Alice's Adventures in Wonderland_ by Lewis Carroll relates to the function and traumatic nature of the apparitions in Darrieussecq's novel. Trying to find a way to escape a possible beheading, Alice notices a "curious appearance in the air" that soon turns out be a grin; she says to herself, "It's the Cheshire-Cat: now I shall have somebody to talk to" (Carroll 112). Darrieussecq's autodiegetic narrator can be seen as endeavoring to cope with the shock of her husband's possible death by dreaming up or conjuring an apparitional addressee or, rather, silent companion (for she does not address her words to the entity, nor does it, or he, ever verbally respond to her). In any case, curiosity and airiness connect the two entities at the initial stage. The Cheshire Cat also ties in with the narrator's own spectral qualities. Listening to her friend's relentless speech, the narrator produces a veritable _risus sardonicus_ on her face:
> Je m'enforçai à un sourire que j'espérais aimant, attentif, et le plus convaincant possible, mais qui dut ressenbler, à mon corps défendant, à une manifestation physique aussi étrange que les coquetteries du chat du Cheshire [. . .]. (78–79)
>
> [I forced myself to give a smile that I hoped was loving, attentive, and as convincing as possible, but that must (against my will) have seemed a physical manifestation as strange as the coquetries of the Cheshire cat [. . .].] (69)
This feline metamorphosis is all the more prominent as it comes immediately after the extended section in which the narrator is metaphorically cast as a dog (75–76 / 66–67).
### MINDWORKS
_Naissance des fantômes_ contains not only heightened descriptions of (possibly) supernatural entities and their relation to posthumous minds, but also minute accounts of natural processes inside the brain and nervous system. The workings of the mind are hence depicted in quite a literal sense of the term. Narrationally, the posthumous and the neural are equally paranormal in the sense that they are invisible to the naked eye and only available to the mind's eye. The narrator repeatedly zooms in on the molecular level of the mind but also brings metaphoric language (drawn mainly from the source of mechanical domain) into play. She, as it were, reverses her narrational strategy of projecting her mind to the outside world by introjecting external realms to thematize the structure and processes of the mind.
When telephoning her mother-in-law in the novel's second chapter, the narrator describes what she hears happening:
> Je n'ai plus entendu qu'une respiration raidie, la pulsation presque palpable d'influx cérébraux, la forme que peut prendre, entre deux personnes en contact dans le noir, un effort ou un vacillement de la mémoire. (27)
>
> [I heard nothing but a tense breathing, the almost palpable impulse of every nerve cell in the brain, the form that contact between two people in the dark can take, a straining or flickering of memory.] (17)
Elsewhere, she continues the mechanical or machinic metaphor and conceives of her neurons as "rouillé" (92); "rusting" (84). She also takes a cue from the moving sea foam she sees during a walk in conceptualizing her mind:
> J'imaginais ainsi mon cerveau, soumis à la pression et vaporisé, toutes les connexions de mon cerveau peu à peu défaites, brumeuses et floues [. . .], et ma pensée se vaporisait à son tour [. . .]. (68–69)
>
> [That was how I imagined my brain, subjected to great pressure and vaporized, all its connections undone little by little, foggy and slack [. . .], and my mind was vaporized in turn [. . .].] (59)
In addition, she repeatedly refers to the molecules of thought, thinking, and body (94, 98, 125, 140–41 / 86, 89, 116, 132–33).
The narrator even gives lively accounts of how the animal mind functions. These takes on the animals' inner worlds are hypothetical, since she presumably cannot enter their "consciousness" or the organs where it possibly resides. She is not, in other words, omniscient in any reliable way as regards these strata of the storyworld. The narrator makes both normal and paranormal contact with a sea lion on the beach:
> de toutes mes forces j'ai lancé un caillou, pile entre les deux yeux que l'abruti a ouverts un peu plus grands, l'information a cheminé à travers ses synapses bardées de lard, deux neurones ont réussi à se détacher de leur graisse pour inaugurer leur roulement à billes [. . .]. (67)
>
> [I threw a pebble with everything I had and got him [the dominant male sea lion] right between his two large eyes, which the brute opened a little wider, the information made its way along his lard-encased synapses, two neurons succeeded in detaching themselves from their bed of fat to set the marbles rolling [. . .].] (57–58)
Another animal mind is described through the narrator's metaphoric metamorphosis into a dog. Listening to her friend Jacqueline, the narrator finds herself, dog-like, only reacting to a familiar string of sounds:
> Je restais béate devant cet étonnant phénomène, et comme les chiens, dont on dit que dans le discours le plus assertif de leur maître ils n'entendent, salivants, que la répétition martelée de leur nom [. . .], j'étais assise sur mon derrière, la bouche ouverte et le poil lissé, une queue presque tangible battant dans mon dos au rythme bien marqué des paroles de Jacqueline [. . .]. (75–76)
>
> [I sat and gaped at this astonishing phenomenon like a dog that supposedly understands, as it listens, drooling, to its master's most vehement speeches, only the hammering repetition of its name [. . .]; I was sitting on my hindquarters, mouth open and panting, fur smoothed down, an almost tangible tail thumping behind me to the heavy beat of Jacqueline's words [. . .].] (66)
The extended canine metaphor generates a "dog schema" for reading the poor meta-morphosed creature's (that is, her own) mind: "j'écoutais fascinée, salivant toujours davantage, attendant quoi, un sucre, la promenade, qu'on me flatte du plat de la main" (77); "I was listening [. . .] in fascination, still drooling copiously and waiting—for what? a sugar cube, my walk, someone to pat me with an outstretched hand?" (67). There is a connection here to the "spectral schema" that the narrator applies, at the beginning of the novel, to the phantoms' or shadows' minds. The received knowledge or _idées reçues_ is triggered almost unwittingly, but might as well turn out to be Barthesian _doxa_ , false knowledge. Whatever their truth value, these schemata are versions of the narrator's paranormal and supernatural Theories of Mind.
### ELECTRONIC MEDIA AND SPIRITUALISTIC MEDIUMS
The narrator is not content with speculating on minds or with trying to reach them by means of extended metaphors. She also has recourse to several methods of making supernatural contact. Most of these attempts appear as ordinary media use on the surface, but the novel clearly taps many occult streaks discernible in media history (for spiritualism and electronic telecommunication, see Sconce, and Keskinen 93–119).
The first medium in her search for her disappeared husband is the telephone. She tries to call him at his real estate agency, but the only response she gets is a sonic verification that he is not present (or not answering the phone): "J'entendais de très loin les pointillés de l'absence de mon mari" (22–23); "I heard the stipplings of my husband's absence from very far away" (13). She also calls her friend Jacqueline, her mother-inlaw, and, finally, the police. What is common to all these calls is that she does not primarily want to reach the person or instance at the other end of the line but to make contact with her husband, or at least to learn that he is alright. In this sense, she uses the telephonic medium in a spiritualistic fashion, since she cannot be certain whether he is dead or alive, and whether his well-being is posthumous or not.
In the midst of her telephone calls she turns on the television, which seems more like an automatic gesture than a conscious reach for information. Soon, however, the program flow begins to relate, in her hypothetical mind, to her husband: "il me semblait qu'à tout moment le présentateur prendrait une mine grave pour annoncer la disparition de mon mari [. . .]" (18); "any moment now, I thought, the announcer would put on a serious face to announce the disappearance of my husband," who would have ended up in a variety of freak accidents (9). After calling the police, she turns on the TV again, this time "très fort" ("very loud"), and the debate on "choses concrètes" ("hard facts") between two electoral candidates sounds strangely soothing to her (30/20). The medium itself does not, however, remain concrete or stable: "L'image bleue de la télé semblait surgir hors de son cardre et dégoutter en l'air, suspendue dans l'espace comme un rideau de douche" (31); "The blue image on the TV seemed to overrun its frame and precipitate in the air, suspended in space like a shower curtain" (21). Analogously, she senses that he is, as a bodily person, gone forever. What the overrunning and precipitating image resembles is ectoplasm oozing from a spiritualistic medium's mouth. It can refer to another kind of hope, to that of a paranormal encounter or contact with the absent husband. Later the TV screen sheds green light (originating from a documentary about the Amazon) (32 / 24), and the narrator's emotions sympathize with the vistas represented in the documentary, ranging from the low angle (panic and anguish) to the high (relief and newly kindled hope):
> D'un coup, comme un hélicoptère s'élevait droit par-delà les arbres, dévoilant le trajet du fleuve et la pelote de ses affluentes, d'un coup comme je reprenais mon souffle dans le ciel grand ouvert je dis que [. . .] il était évident et obligatoire que je dormirais avec mon mari [. . .]. (36)
>
> [Suddenly, as a helicopter rose straight into the sky over the trees, revealing the path of the river and the tangled mass of its tributaries, suddenly, as I began to breathe again in the wide open sky, I told myself that [. . .] I would be sleeping with my husband [. . .].] (27)
In her next phase of supernatural or at least paranormal media use, she is even more formally mediatic at the cost of the message or program content: "Je changeais de chaîne sans regarder, j'essayais de sentir sa présence quelque part, dans les rues, dans la ville, sur la planète" (38); "I changed the channels without watching, I was trying to feel his presence somewhere, in the streets, in the city, on the planet" (29). She explicitly conceives of her channel-hopping as technologically assisted "telepathy"—both in the literal and metaphorical senses of the term: "je voulais croire qu'en atteignant la bonne longueur d'onde j'allais pouvoir retrouver mon mari par la seule vibration de mes antennes [. . .]" (38); "I wanted to believe that if I achieved the correct wavelength I would be able to find my husband again by mere vibration of my antennae [. . .]" (29).
The narrator's attempts at making contact with her husband turn out to be, in spite of their apparent resemblance to detecting electronic voice phenomena (EVP), paranormal rather than supernatural (for EVP, see Sconce 85–91). She tries to reach her _living_ husband on the spiritual or mental level, not to reach him in the afterlife. In a fit of depression she comes to the following conclusion: "si je ne parvenais pas à joindre mon mari [. . .], c'est qu'il était mort" (38); "I could not succeed in reaching my husband [. . .] because he was dead" (29). There is, however, a spiritualistic streak in her paranormal activities. This is especially pertinent to her use of the computer. In her search for her husband, she repeatedly goes to his real estate agency, sits at his desk, and uses his computer.11 The computer provides a window on a world beyond the normal perception. She finds out that her husband's business goes on as usual, with the commissions bringing in money, although he is only virtually, not bodily, present in the transactions:
> autant d'opérations qui n'avaient sans doute de référence qu'informatique, mais indiquaient très réellement, sous les modifications spectrales des pixels, que peut-être, quelque part, mon mari (ou quelque esprit tutélaire) continuait de penser à moi. (114)
>
> [undoubtedly none of these operations had any reference outside of cyberspace, but they indicated in a very real way, beneath the spectral modifications of the pixels, that perhaps, somewhere, my husband (or some tutelary spirit) went on thinking about me.] (104)
Again, she reads the mind of her husband in the absence of his body, on the basis of unsubstantial evidence.
Sitting in his chair and participating in "ce qui avait été son atmosphère" (113–14) ("in what had been his atmosphere" [104]), she virtually becomes him, or at least assumes a number of his roles. She makes appointments in his name, visits the Web sites he had been to, and engages in online chats with strangers, possibly using his alias name. She, in a way, becomes a medium who channels her husband's gestures and words at electronic séances of sorts. The computer also figures as a writing implement, and thus functions in a more traditional manner, in her paranormal or supernatural pursuits. She starts writing the story of her husband's disappearance and her wait, that is, the narrative that constitutes the novel _Naissance des fantômes_. She writes about her writing: "frappant sur le clavier j'essayais ainsi [. . .] de me clarifier peut-être les idées [. . .]" (114); "I tried, tapping on the keyboard, [. . .] perhaps to clarify my thoughts [. . .]" (104–05). The rapping or knocking12 and the clairvoyant vision implied in the diction of her account of writing can be read as indicative of the spiritualistic or supernatural facet of her project. Indeed, the computer keyboard functions, formally, as an Ouija board, with the standardized QWERTY system making automatic-seeming (supernatural or paranormal) contact via writing possible. In practice, however, the act of writing marks separation rather than connection. The narrator finds the "writing to the moment" principle, routinely encountered in epistolary and diary fiction, to be impossible:
> je n'écrivais pas aussi vite que je vivais, et pourtant cette vie était lente. Et dans le retard accumulé, qui semblait mimer la disjonction temporelle dont je souffrais parfois si violemment [. . .], dans ce retard je reconnaissais le temps que je vivais mieux que si j'avais pu l'écrire sans écart. (115)
>
> [I didn't write as quickly as I was living, yet this life of mine was slow. And in the accumulating lag, which seemed to mimic the temporal disjunctions from which I sometimes suffered so violently [. . .], I recognized the time I was living through better than if I had been able to write it without the lag.] (105)
Thus, she makes contact with her own life rather than with her husband's possible afterlife. In fact, the very act of narrating, which could function as a connecting channel, generates a hiatus that becomes an emblem of the couple's temporal and ontological separation:
> le retard que prenait sur ma vie le récit de cette disparition, ou plutôt de ses effets en négatif, ce retard était celui que je me sentais prendre sur mon fantôme de mari, parti plus loin que moi dans des espaces qui m'échappaient. (116)
>
> [the growing lag between the story of this disappearance, or rather of its effects in the negative, and my life was the same one I felt widening between myself and my phantom husband, gone much farther than I into the spaces that eluded me.] (106–07)
The contact with herself, or with her own life, that the narrator's act of storytelling creates by widening the lag and gap between the two strata of happening, is also of a supernatural nature. During the process, the narrator herself acquires spectral attributes:
> Mais s'il ne restait de moi qu'une coque vide, ce que j'avais été se dissolvait dans l'atmosphère pour participer presque harmonieusement à la réalité de l'absence et du vide, d'une façon désormais plus gazeuse qu'immobile. (116)
>
> [What remained of me was only an empty shell, what I once had been dissolved in the atmosphere to participate almost harmoniously in the reality of absence and void, in a way that became more ethereal than immobile.] (106)
"Gas-like" or "vaporous" are the usual attributes attached to the "shadow" or phantom in the novel, and this choice of epithet connects the narrator to these semi-immaterial entities. She may be one of the ghosts raised that the plural in the novel's French title suggests. When she becomes semi-immaterial or incorporeal, she must perform her mind reading activities (aimed at herself) without a discernible body and articulate body language. In this case, too, her Theory of Phantom Mind equals Practice of Absent Body.
### READING WRITTEN MINDS: FROM THE UNNATURAL TO THE SUPERNATURAL
But perhaps reading the phantom minds of absent or dead bodies is not, after all, as curious an activity in fiction as I have suggested. I could turn the spiritualistic table and state that it is, in fact, the unmarked feature of reading literary language also known as narrative fiction.
This primordial supernaturalness or at least paranormality of narrative fiction can be crystallized in one word, albeit with a double meaning: character. Firstly, characters, in the sense of personages, only exist in writing, as inscribed representations of their bodies and minds. Both are absent as actual or bodily entities. In most cases, that absence is of an ontological nature, since fictional characters usually lack worldly referents. Fictional characters are thus, in an important sense, "posthumous" from the outset. They are dead because they never were alive, or, rather, they arise after their primordial death and give the impression of being living people. To emulate Lisa Zunshine, the problem of _why_ we read fiction turns into _how_ we _can_ read fiction (and claim that we read characters' minds).
Secondly, character, in the sense of a graphic sign, or a letter of the alphabet, is a synecdoche of language. Literary language can be thought of as a subset of natural or ordinary language, a heightened or second-order version of the everyday linguistic mode. In this sense, literary language is "unnatural." On the other hand, it is debatable whether "natural language" is natural at all. De Saussure and the majority of modern linguists would claim that language is an arbitrary and conventional system developed for communicative purposes, without any relation to "nature."13 In this constellation, the language of fiction appears even more unnatural or abnormal.
Consequently, reading becomes not only an unnatural, but a supernatural activity. All fiction conjures absent characters, gives them life and voice, preserving, in the residue of writing, what would not be otherwise present either on account of death or primordial nonexistence. Ghost stories perform these literary conjuring tricks explicitly, but the house of fiction is always, implicitly, haunted. The nature of fiction as make-belief is thus profoundly supernatural, and the reader's willing suspension of disbelief amounts to belief in literature's ghost world. By the same token, many of the acts performed by narration are not only unnatural, but paranormal. The narrator's telepathic or prophetic omniscience and superhuman mind reading capabilities are just the most obvious examples of literary paranormality.14
In addition to talking about natural narratives (Fludernik) or unnatural narration (Richardson), we could also mind the supernatural aspects of (reading) narrative fiction.15 If " _on one level_ our evolved cognitive architecture indeed does not fully distinguish between real and fictional people" (Zunshine, _Why We Read Fiction_ 19), the same could hold true of living and dead persons when they are not present. And if "fiction engages, teases, and pushes to its tentative limits our mind-reading capacity" (Zunshine, _Why We Read Fiction_ 4), one of the boundaries tested is the ultimate one, the one between life and death, between shared presence and irrecoverable absence. In this sense, the cognitive reward of reading fiction can turn out to be of an extreme kind.
### NOTES
1 Summing up the theory-theory branch of ToM, Alan Palmer writes, "No one suggests that what people use is a self-conscious and fully worked theory. All agree that it is used intuitively and non-consciously" (143). Presumably the same could be said of the simulation theory in ToM as well. For a discussion of the seeming facility of mind reading, see also Zunshine, _Why We Read Fiction_ 3–4, 13–16. In her recent article, Lisa Zunshine emphasizes that, in actual practice, " _theory_ in Theory of Mind and _reading_ in mind-reading" do not imply intentional and conscious attribution of mind ("Theory of Mind" 67; emphasis in original). For a metapsychological critique of the "theory" in ToM, see Costall and Leudar.
2 Marie Darrieussecq (1969-) is an internationally renowned experimental author of French Basque origin. She has written eight novels, a collection of short stories, a play, and an unpublished dissertation on autofiction in French literature. Darrieussecq's first novel, _Truismes_ (1996; _Pig Tales_ ), turned into an international success, which enabled her to leave her teaching position at the University of Lille and to become a full time writer of fiction. _Naissance des fantômes_ (1999; _My Phantom Husband_ ) is her second novel. I quote _Naissance des fantômes_ and Esther Allen's translation of it parenthetically in the text.
3 Strange disappearances are not uncommon in folklore and literature, and the explanations given to these mysteries often tend toward the paranormal. Two examples of the disappearance novel may suffice to demonstrate the dynamic within the subgenre. Tim O'Brien's _In the Lake of the Woods_ (1994) reverses Darrieussecq's basic situation by making a wife vanish and her husband search for her, or at least for a reason for her disappearance. Although the supernatural is not explicitly suggested as a solution to the mystery, ghosts do figure in the husband's Vietnam War secrets. In Georges Perec's lipogrammatic novel _La Disparition_ (1969; _A Void_ / _A Vanishing_ ), a character's disappearance takes place doubly in the sense that both a fictional personage, Anton Voyl, and the letter E are entirely absent from it. In Perec's _Les Revenentes_ (1972; _The Exeter Text_ ), the ghostly E returns _en masse_ , since it is the only vowel used in the whole novella.
4 The possible death of her husband is not the only case of demise in her immediate family. She has had several miscarriages during her seven-year marriage, and one of her pregnancies even lasted almost six months. The dead fetuses, the ghosts of her unborn babies, or at least the idea of pregnancy, can be seen as haunting her, albeit in a subdued way, in brief references to real or pseudo pregnancies (52, 54, 106 / 43, 45, 97) and to metaphorically or literally dead children (70, 130 / 61, 121). Her unsuccessful work of mourning could be interpreted as having turned into melancholia, resulting in apparitions in the classic Freudian manner (cf. Rickels 16). _Naissance des fantômes_ features themes also found in Darrieussecq's other works, and posthumous presence of sorts is a veritable leitmotif in her fiction. The disappearance of family members recurs in _Le Mal de mer_ (1999; _Undercurrents_ / _Breathing Underwater_ ), and the persistent grief over a dead child is focused on in both _Bref séjour chez les vivants_ (2001; _A Brief Stay with the Living_ ) and _Tom est mort_ (2007; _Tom is Dead_ ). _Le Bébé_ (2002; _The Baby_ ) shows that a living child, too, can haunt his parents. For critical essays on _Naissance des fantômes_ , see Robson; Horvath; Narjoux; and Jordan.
5 In addition to being ontologically fragile, a ghost can, when articulating, be as unreliable as any other narrator; cf. the lying ghost in Akira Kurosawa's 1951 film _Rashomon_ (Richardson 3, 94).
6 In this essay I mainly utilize Derrida's concise account of the lore and study of ghosts. Derrida's conception of specters is paradoxically ahistorical or at least synthetic, but its heuristic force lies elsewhere. Re-evaluating ontology in the tradition of continental philosophy, Derrida coins the neologism _hauntology_ , which comprises both ontology and haunting: "This logic of haunting would not be merely larger and more powerful than an ontology or thinking of Being [. . .]. It would harbor within itself, but like circumscribed places or particular effects, eschatology and teleology themselves" (10). In Thomas Mical's reading, hauntology is "an intellectual conjuring trick where the spectral after-image of the thought of ontology becomes its prior foundation—by situating hauntology as prior to ontology [. . .], as the unexpected chain-rattling return of repressed difference within any sweeping static ontological claims." I find hauntology productive in reading Darrieussecq's novel whose central concern pertains to the absent husband's ontological status. For contextualizing readings of ghosts in literature, see Greenblatt; Wolfreys; Dickerson; and Kolmar and Carpenter, eds.
7 In popular tradition, ghosts are earthbound spirits who will haunt the living "until a wrong done to them or by them has been set right" (Gillie 540). In Rickels's reading of Freud's _Totem and Taboo_ , "the hostility the dead, while still alive, had aroused in the still living is attributed to the phantoms of the deceased who now persecute the survivors" (16). The ghost's motivation or mind-set, whether actual or attributed, hence originates from the realm of the living.
8 Midnight is traditionally the specters' most common showtime, although some ghosts are also believed to make an appearance in the day (Greenblatt 41, 114–15, 274n77). Darrieussecq's novel follows a medieval and Renaissance pattern in the sense that "[t]he ghost generally appears shortly after death, while the memory of the deceased [. . .] is still fresh" in the mind of the loved one (Greenblatt 41, 291n45).
9 In Allen's translation, this sentence reads somewhat vaguely "it didn't move" (46); Helen Stevenson renders it more literally as "there was no mirror effect" (45). The mirror is important in at least three ways in the scene. The lack of the mirror effect, that is, of reflection, momentarily makes the narrator resemble a ghost in front of a looking glass. Second, that she can step inside and through this airy "mirror" puts her in the position of Lewis Carroll's Alice; this intertextual connection is further reinforced by the Darrieussecq novel's Cheshire Cat epigraph. Thirdly, the mirror effect relates, in the context of cognitive science, to mirror neurons and simulation theory (cf. Goldman); the shadow's reluctance to imitate the narrator's gestures may point to its autism, not being a primate, or fundamental nonexistence.
10 In Allen's wording, "Void had sprung up in the place he once filled" (77), and in Stevenson's, "The emptiness had appeared in the very place where he'd once been" (77). The ghost's invisibility equals perfect transparency. In this sense, invisibility is not a matter of "embodied transparency," that is, "perfect access to other people's minds via their observable behavior" (Zunshine, "Theory of Mind" 72), because there is no discernible body to observe. Nor is it exactly a matter of a "transparent mind" in Dorrit Cohn's sense; the ghost's consciousness is not psychonarrated but merely rendered as translucent as its "body."
11 Blond and curious, the narrator acts, in this scene, like a cyber-age Goldilocks. Another juvenile blonde, Carroll's Alice, and via her the fantastic in children's literature, are evoked throughout the novel. One example will suffice to demonstrate this economy of simultaneous allusion to supernatural technology and transgressive figures. Maniacally cleaning her apartment, the narrator has recourse to a homonymy of the mysterious oceanic element where spirit beings were once believed to travel, the ether (cf. Sconce 63–66): "I sprinkled all the windows with ether [. . .] and rubbed hard until they were spotless, you could pass right through them" (91).
12 There are also references to posthumous rapping, both imagined and (presumably) real, by skeletons or spirits in the novel (96, 139 / 88, 131).
13 In contrast with Saussurian arbitrariness, some contemporary linguists emphasize "natural factors affecting and motivating linguistic forms and functions" (Fludernik 18). Mark Turner's take on the naturalness / arbitrariness debate "naturalizes" what is usually regarded as the second-order language in both schools' literature. Turner claims that story, parable, and metaphor precede language and are basic to thought. In his model, language follows from mental capacities that are, primordially, literary.
14 In cognitive psychology, the concept of "mind-reading has nothing to do with plain old telepathy" (Zunshine, _Why We Read Fiction_ 6), but in criticism, it may have, as in Nicholas Royle's thematization of literary omniscience.
15 If Monika Fludernik is able to call her theory of natural narratives, which she defines on the basis of "the _cognitive frames_ by means of which texts are interpreted," _natural narratology_ (12, 312–15), perhaps my tentative coinage _supernatural narratology_ could have, at least in a jocular frame of mind, some heuristic value.
### WORKS CITED
Carroll, Lewis. _Alice's Adventures in Wonderland_ and _Through the Looking-Glass_. Harmondsworth: Penguin, 1966. Print.
Cohn, Dorrit. _Transparent Minds: Narrative Modes for Presenting Consciousness in Fiction_. Princeton: Princeton University Press, 1978. Print.
Cosmides, Leda, and John Tooby. "Considering the Source: The Evolution of Adaptations for Decoupling and Metarepresentations." _Metarepresentations: A Multidisciplinary Perspective_. Ed. Dan Sperber. New York: Oxford University Press, 2000. 53–116. Print.
Costall, Alan, and Ivan Leudar. "Where Is the 'Theory' in Theory of Mind?" _Theory & Psychology_ 14.5 (2004): 623–46. Print.
Darrieussecq, Marie. _Naissance des fantômes_. 1998. Paris: Gallimard, 1999. Print.
\---. _My Phantom Husband_. Trans. Esther Allen. New York: The New Press, 1999. Print.
\---. _My Phantom Husband_. Trans. Helen Stevenson. London: Faber and Faber, 1999. Print.
Derrida, Jacques. _Specters of Marx: The State of the Debt, the Work of Mourning, and the New International_. Trans. Peggy Kamuff. New York: Routledge, 1994. Print.
Dickerson, Vanessa D. _Victorian Ghosts in the Noontide: Women Writers and the Supernatural_. Columbia: University of Missouri Press, 1996. Print.
Fludernik, Monika. _Towards a 'Natural' Narratology_. London: Routledge, 1996. Print.
"Ghost." _The Oxford English Dictionary Online_. 2nd ed. 28 Feb. 2008. Web. http://www.oed.com.
Gillie, Christopher. _Longman Companion to English Literature_. London: Longman, 1972. Print.
Goldman, Alvin I. _Simulating Minds: The Philosophy, Psychology, and Neuroscience of Mindreading_. New York: Oxford University Press, 2006. Print.
Greenblatt, Stephen. _Hamlet in Purgatory_. Princeton: Princeton University Press, 2002. Print.
Horvath, Christina. "Le fantastique contemporain: un fantastique au feminin." _Iris_ 24 (2002–2003): 171–80. Print.
Jordan, Shirley. "Saying the Unsayable: Identities in Crisis in the Early Novels of Marie Darrieussecq." _Women's Writing in Contemporary France: New Writers, New Literatures in the 1990s_. Ed. Gill Rye and Michael Worton. Manchester: Manchester University Press, 2002. 142–53. Print.
Keskinen, Mikko. _Audio Book: Essays on Sound Technologies in Narrative Fiction_. Lanham: Lexington Books, 2008. Print.
Kolmar, Wendy, and Lynette Carpenter, eds. _Haunting the House of Fiction: Feminist Perspectives on Ghost Stories by American Women_. Knoxville: University of Tennessee Press, 1991. Print.
Mical, Thomas. "Introduction: Hauntology, or Spectral Space." _Perforations_ 29 (2007). 28. Feb. 2008. Web. http://www.pd.org/~chea/Perforations/perf29/p29–intro.pdf. Narjoux, Cécilie. "Marie Darrieussecq et 'l'entre-deux-mondes' ou le fantastique à l'oeuvre." _Iris_ 24 (2002–2003): 233–47. Print.
Palmer, Alan. _Fictional Minds_. Lincoln: University of Nebraska Press, 2006. Print.
Phelan, James. _Living to Tell about It: A Rhetoric and Ethics of Character Narration_. Ithaca: Cornell University Press, 2005. Print.
Richardson, Brian. _Unnatural Voices: Extreme Narration in Modern and Contemporary Fiction_. Columbus: The Ohio State University Press, 2006. Print.
Rickels, Laurence A. _Aberrations of Mourning: Writing on German Crypts_. Detroit: Wayne State University Press, 1988. Print.
Robson, Kathryn. "Virtual Reality: The Subject of Loss in Marie Darrieussecq's _Naissance des fantômes_ and Régine Detambel's _La Chambre d'écho_." _Australian Journal of French Studies_ 41.1 (2004): 3–15. Print.
Sconce, Jeffrey. _Haunted Media: Electronic Presence from Telegraphy to Television_. Durham: Duke University Press, 2000. Print.
Turner, Mark. _The Literary Mind_. Oxford: Oxford University Press, 1996. Print.
Wolfreys, Julian. _Victorian Hauntings: Spectrality, Gothic, the Uncanny and Literature_. London: Macmillan, 2001. Print.
Zunshine, Lisa. "Theory of Mind and Fictions of Embodied Transparency." _Narrative_ 16.1 (2008): 65–92. Print.
\---. _Why We Read Fiction: Theory of Mind and the Novel_. Columbus: The Ohio State University Press, 2006. Print.
## Theory of Mind and Metamorphoses in Dreams: _Dr. Jekyll and Mr. Hyde_ , and _The Metamorphosis_
RICHARD SCHWEICKERT AND ZHUANGZHUANG XI
### INTRODUCTION
An author has an opportunity to adopt a special point of view in a story when one character metamorphoses into another. In the famous opening of _The Metamorphosis_ , Franz Kafka immediately immerses the reader into the mind of the central character:
> When Gregor Samsa woke up one morning from unsettling dreams, he found himself changed in his bed into a monstrous vermin. He was lying on his back as hard as armor plate, and when he lifted his head a little, he saw his vaulted brown belly, sectioned by arch-shaped ribs . . . His many legs, pitifully thin compared with the size of the rest of him, were waving helplessly before his eyes. "What's happened to me?" he thought. It was no dream. (Kafka 3)
The entire story, except the epilogue, is told from Gregor's point of view: "Magically, the story adopts the victim's perspective, which produces both brief comic moments and a lasting pathos" (Adler 82).
However, a metamorphosis does not always prompt an author to attend to an accompanying mental change. In _Dr. Jekyll and Mr. Hyde_ , Robert Louis Stevenson describes Mr. Hyde transforming into Dr. Jekyll as seen by another person, Dr. Lanyon:
> He put the glass to his lips and drank at one gulp. A cry followed: he reeled, he staggered . . . there came, I thought, a change—he seemed to swell—his face became suddenly black . . . I leaped back, my mind submerged in terror . . . What he told me in the next hour, I cannot bring my mind to set on paper. (Stevenson 75)
The opportunity to tell the reader what Dr. Jekyll experienced is declined. Only the final chapter, the narrative of Dr. Jekyll, is written from the metamorphosed character's perspective, and there, as Joyce Carol Oates says, "It is significant that the narrator of Jekyll's confession speaks of both Jekyll and Hyde as if from the outside" (xv).
A character attributing perceptions, thoughts, feelings, and so on to another character is said to be using Theory of Mind (ToM). Recent work by Alan Palmer and Lisa Zunshine pays close attention to mental activity of characters, and of readers and authors as well, as a way of understanding the function of literature and of enriching the experience of reading.
Authors Stevenson and Kafka were both influenced by their dreams. In "A Chapter on Dreams," Stevenson says the source for the above scene in _Dr. Jekyll and Mr. Hyde_ was a dream (more precisely, a dreamlike-state, see Karen Goodwin). "For two days I went about racking my brains for a plot . . . on the second night I dreamed the scene at the window, and a scene afterwards split in two, in which Hyde . . . took the powder and underwent the change . . . " (Stevenson 103). For Kafka, dreams do not seem to be a direct source of material, but he reports his dreams in his diaries and letters, and says in the _Diaries_ , "My talent for portraying my dreamlike inner life has thrust all other matters into the background . . ." (qtd. in Corngold xvii).
Metamorphoses of one character or animal into another occur in roughly 1% of dreams (Hall and Van De Castle 168). In a study of metamorphoses in a particular dreamer, G. William Domhoff found the most common type to be human-to-human, but there are also human-to-animal, object-to-object, and so on (131). Studies by David Kahn and J. Allan Hobson and by Kahn, Edward Pace-Schott, and Hobson show that mind reading and other ToM activities frequently occur in dreams. With a metamorphosis of characters, where there was one mind, now there is another. Are mental states of transformed characters attended to in dreams, or do they pass unnoticed? Is one or the other of our two metamorphosis stories more like a dream in this respect?
We can pursue the questions with dream reports from a large collection on a Web site by Adam Schneider and Domhoff at the University of California at Santa Cruz. We find there reports of inner states of a metamorphosed character, for example:
> (Hall / VdC Norms: Female: #0288)
>
> I, as my maiden aunt, was standing in the front yard of my home in the country. I was not myself, but my aunt. A car drove by in which were my sister (my mother, my aunt's sister) and her husband. I did not determine whether her son and daughter were in the car. I felt very sad that I had no husband while she did. I envied her. Immediately, I changed back to myself, although the setting remained the same . . . (Schneider and Domhoff)
In another report the idea of minds transforming when bodies transform seems to be understood in the midst of dreaming:
> (Barb Sanders: #1985, 11 / 15 / 91)
>
> . . . Now Troi and I are in bed together cuddling. . . . Troi and I then exchangebodies so I can experience what it is like to be free of my own patterns of thought and feeling I've built up since birth, and to experience her way of feeling her own feelings. It's a delightful experience. Then I see her push a table over and hit her ankle and limp with pain. I go and kneel in front of her and try to be sympathetic. "Does it hurt?" I ask. She grimaces and says, "I'm fine." (Schneider and Domhoff)
The transformation is brief, and soon the dreamer has to ask about the pain when Troi hits her ankle.
On the other hand, there are also reports indicating that a metamorphosis does not automatically call attention to a possible change in mental experience. A good example is in the following part of a dream report of a woman in her thirties:
> (Merri: #032 6 / 12 / 1999):
>
> . . . There was a family and I was the little boy and on Saturdays we had to line up and get a big plastic cup and have it filled with sulfuric acid. Then we had to stick our hand in the acid (my right hand) and scrub—hard. I had to scrub the 4 sides of the light post. It was trapezoid in shape. . . . There was an underwater cave murder. There was a pocket of air in the underwater cave and 3 guys were there. One was the culprit, one was the victim, and one was the survivor. The victim knew the end was coming and he became very calm. . . .(Schneider and Domhoff)
The woman adult dreamer has the form of a male child, but the hand of this child in sulfuric acid produces no comment about inner feelings. A little later, however, the dreamer knows somehow that a different character, the victim, knows the end is near. In this dream report, a ToM event occurs, but quite unrelated to the metamorphosis.
Ignoring a metamorphosis is a little like the phenomenon of change blindness. A rather large change in a scene can pass unobserved unless the observer is attending directly to it, as Daniel J. Simons and Daniel T. Levin have shown. In one of their demonstrations, a male experimenter asks directions from a woman on a street. While they are talking, a person carrying a door blocks the woman's view for a moment, and, changing places with the experimenter, replaces him in the conversation. The woman continues the conversation, not noticing one person has been replaced by another. This suggests that some metamorphoses in dreams are overlooked. A metamorphosis must have been noticed in a dream for the dream report to mention it, but the consequence that a mind was transformed sometimes passes unremarked.
The examples show that mental states of metamorphosed characters are noted in some dreams but not in others, so _The Metamorphosis_ , which attends to such states, and _Dr. Jekyll and Mr. Hyde_ , which ignores them, are each faithful to dreams in this respect, albeit to different dreams. The examples do not suffice to determine whether ToM is typical or not when a metamorphosis occurs in a dream, and we take up this issue now with a sample of dream reports. A frequent and reasonable objection to using material from dream reports is that one must assume the reports are accurate accounts of what occurred in the dreams. Unfortunately, there is no way to test the accuracy of dream reports. Some indirect evidence for accuracy is summarized in Domhoff (40). At this time, if we want to study dream content we have no choice but to study dream reports, with the working assumption that they are accurate.
### OBSERVATIONS
Description of the study here is necessarily brief; more details are in the paper by Richard Schweickert and Zhuangzhuang Xi. All available dream reports in English from individuals on the Schneider and Domhoff Web site were searched for words and phrases indicating a metamorphosis. Phrases such as "change," "transform," and "turn into" were included, in addition to "metamorphosis" (see also Domhoff 132). Dream reports containing such phrases were read and selected for further study if a sentient or robotic entity was involved in a metamorphosis. A report was included if one of the metamorphosed entities was an object, for example, a chair turning into a cat, but not if all metamorphosed entities were objects. Reports were also selected for further study if a group, for example, a flock of sheep, was involved in a metamorphosis, but only dream reports of metamorphoses involving single entities will be discussed here.
More dream reports are available from some dreamers than others, so the number of dream reports in the sample from each dreamer is not the same. To prevent the sample from being dominated by dreamers with a large number of reports, no more than ten dream reports with a metamorphosis were selected from any one dreamer.
The Web site provides a procedure for selecting reports from a dreamer at random, with upper and lower bounds for the word length of the selected reports. Using this procedure, for each of the dream reports with a metamorphosis a control dream report from the same dreamer was selected at random, within plus or minus twelve words of the length of the corresponding report with a metamorphosis. A control report was replaced if it happened itself to contain a metamorphosis. Dream reports were put in a random order and printed with dreamer's names and word-lengths removed.
One coder (the first author) listed the characters, animals, sentient and robotic entities, and groups of such entities in each dream report. The coding system is a modified version of that of Calvin S. Hall and Robert Van De Castle; see also Schweickert.
Two other coders (one of whom is the second author) coded ToM events in the dream reports. The coding system was developed from ideas of Zunshine, of Kahn and Hobson, and, most directly, from a coding system for children's literature by Kimberly Wright Cassidy, Lorraine V. Ball, Mary T. Rourke, Rebecca Stetson Werner, Nohr Feeny, June Y. Chu, Donna J. Lutz, and Alexis Perkins. Briefly, each coder looked for phrases referring to internal states of entities. Internal states include what an entity is perceiving, feeling, believing, knowing, thinking, wondering, remembering, wanting, or intending. Each coder also looked for phrases referring to higher order internal states that themselves refer to internal states, for example, "Mary believed Harry was sorry." Results for higher order ToM events will not be discussed here.
Instructions were to "Underline with one line a word or short phrase identifying the entity whose internal state is referred to," and "Circle a word or short phrase mentioning the internal state." For example, if the dream report said "Mary was sad," the coder was to underline Mary and circle sad. A word or phrase mentioning a higher order mental state was to be circled twice, and the entity with this higher order state was to be double underlined. Coders were not informed that half the dream reports contained a metamorphosis and half did not, but the instructions said "If one entity transforms into another, consider the entities before and after the transformation as different."
### RESULTS
Each ToM event identified by a coder attributes a mental state to a particular character or other entity. The average over the two coders of the number of ToM events attributed to each entity was calculated. There are 65 dream reports from 21 dreamers.
### _M ETAMORPHOSIS OF DREAMER_
In 10 of the 65 dream reports with a metamorphosis, the dreamer was metamorphosed. For example, a dream report of the adult woman Alta says, "I have turned into a male policeman."
In these 10 dreams with a metamorphosis of the dreamer, 6.6 ToM events per dream are attributed to the dreamer in one form or another, on average. In the 10 corresponding control dreams, 7.0 ToM events per dream are attributed to the dreamer, on average. These rates are about equal; in other words, the frequency of ToM events referring to a form of the dreamer is about the same, whether or not the dreamer is metamorphosed.
Because each dreamer did not contribute the same number of dream reports, averages were calculated in the following way. The number of ToM events per dream was calculated for each dreamer. Then these numbers were averaged over the dreamers; hence each dreamer contributes one number to the average. All averages reported here were calculated this way.
In the dreams in which the dreamer is metamorphosed there were 21 dreamer forms (sometimes a dream report begins with the dreamer already transformed, and sometimes a dreamer undergoes more than one metamorphosis, so the number of dreamer forms is not exactly twice the number of dream reports). In dreams with a metamorphosis of the dreamer there are fewer ToM events per dreamer form, 4.8 on average, than in dreams without a metamorphosis of the dreamer, 8.1 on average. This happens, of course, because there are more dreamer forms when the dreamer meta-morphoses. To summarize so far, the dreamer, in whatever form, refers to his or her (or its) mental states the same number of times per dream, on average, whether or not the dreamer metamorphoses.
### _M ETAMORPHOSIS NOT OF DREAMER_
In 55 dream reports the dreamer did not undergo a metamorphosis, but at least one other entity did. For such dreams, there was an average of 2.3 ToM events per dream. This is quite close to the average of 2.4 ToM events per dream for the corresponding 55 control dream reports without a metamorphosis. (Because we have already discussed the dreamer, the dreamer is excluded from these calculations.) For entities other than the dreamer, we reach the same conclusion as for the dreamer. The frequency of ToM events in dreams is the same for nondreamer entities, on the average, whether there is a metamorphosis or not.
When an animate entity in one form changes into another form, followed perhaps by other changes in form, let us call the set of forms a _line_. When the dreamer changes from one form to another, the number of characters and animals is increased, but there is a single underlying line. Over all dreams with metamorphoses, the average number of forms is 6.2. Over all control dreams this average is smaller, 4.5. But, over all dreams with metamorphoses, the average number of lines is 4.9, about equal to that of control dreams, 4.5. A metamorphosis does not increase the number of lines. Putting this together with our result that the number of ToM events in a dream is the same for dreams with and without a metamorphosis, we conclude that the number of ToM events on a line is the same, whether or not there are metamorphoses along the line.
To summarize, one might expect the dreamer's attention to be drawn to an entity undergoing a metamorphosis, so the number of ToM events attributed to such entities would be larger than for other entities. However, the number of ToM events per dream is the same, whether or not there is a metamorphosis. Contrary to our initial intuition, the average number of ToM events per entity for metamorphosed entities is less than or equal to that for nonmetamorphosed entities. This is puzzling, perhaps, until we consider that a metamorphosis increases the number of entities, and thus decreases the average. It is as though the dreamer attends to mental states at a certain rate (not necessarily periodically). The focus of attention depends on who is present, but the rate does not. A metamorphosis does not call the dreamer's attention to the mental states involved, even if it is the dreamer who is metamorphosed.
### DISCUSSION
In dreams mental states of a character are referred to, or not, quite independently of whether the character has undergone a metamorphosis. Stevenson and Kafka drew on different dream experiences and produced two different types of metamorphosis story. _The Metamorphosis_ , one type, presents the feelings and thoughts of a metamorphosed character. _Dr. Jekyll and Mr. Hyde_ , the other type, largely ignores the inner life of meta-morphosed characters.
What does ignoring or attending do for the story? The story of _Dr. Jekyll and Mr. Hyde_ is better served by ignoring the minds of Jekyll and Hyde. In our day "Jekyll and Hyde" is a well-worn phrase, but Stevenson intended the metamorphosis to be a surprise. A contemporary review says "Mr. Stevenson evolves the idea of his story from the world that is unseen, enveloping everything in weird mystery, till at last it pleases him to give us the password" (137). For this to succeed, thoughts of Jekyll and Hyde cannot be revealed, of course. Moreover, because the description of the mind of Hyde is omitted, when Hyde is described as unlikable or evil, the reader fills in details, perhaps more luridly than the author would have dared to.
On the other hand, attention to the mind of the metamorphosed character provides opportunities for comparisons, as in _The Golden Ass_ , when Lucius says, "That wonderful magic of yours has equipped you with an ass's shape and an ass's hard life, but not his thick skin" (Apuleius 107). Opportunities also arise for irony, as in _The Metamorphosis_ when Gregor thinks "'This getting up so early,' . . . 'makes one a complete idiot. Human beings have to have their sleep'" (Kafka 4).
What might dreams have done for the authors? It is well known that Stevenson wrote _Dr. Jekyll and Mr. Hyde_ in great haste. He refers to dream images as the sources for Jekyll and Hyde. According to Ernest Hartman, many dreams have a central image that is vivid, emotion laden, and memorable (28). If, as seems to be the case, Stevenson held a dream image in mind while he wrote about the scene the image portrayed, he would feel the emotional impact the image had on him, and would know that a good description of the image would suffice for an impact on the reader. Mental states not accompanying the image need not be invented. The image guides what to write about and what not to write about.
Dreams also may have influenced the description of inner states of metamorphosed characters, when they are described. Not all mental states can be produced by the dreaming brain, because of biological limitations. A dreamer can experience flying because views of the ground from high up, feelings of wind, and so on are available in the dreamer's memory and imagination. But consider this dream report of Merri:
> (Merri: #224 1 / 04 / 2000):
>
> . . . I did a few lines of coke with a girl on the floor. She had a whole lot of coke in a clear plastic round cylinder, the size of a film canister but bigger circumference. We left there. They wanted me to see some other artists' work at this school I had gotten a scholarship to. . . .
No change of inner state is reported after the dreamer takes coke, perhaps because the dreaming brain cannot produce the experience of being high.
Let us take a closer look at one of the few passages in _Dr. Jekyll and Mr. Hyde_ where the mental experiences of the main character are described. In Henry Jekyll's full statement of the case Jekyll says:
> . . . I . . . woke the next day in bed with odd sensations. It was in vain I looked about me; in vain I saw the decent furniture and tall proportions of my room in the square . . . something kept insisting that I was not where I was . . . I began lazily to inquire into the elements of this illusion . . . in one of my more wakeful moments, my eyes fell upon my hand. . . . It was the hand of Edward Hyde. (82)
It is curious that Hyde does not notice he is Hyde through his mental state. His inner state hints he is not where he thinks he is, but it does not hint he is not who he thinks he is. He learns who he is by observing the hair on his hand. Hyde is said to be the very embodiment of evil; surely this must be manifest in his own mind. But what exactly could the writer describe of this? The author can conceive of a mind of pure evil, but he cannot produce the experience in his imagination or dreams.
Jekyll learns on waking that he has transformed into Hyde by looking at his hand, just as Gregor learns he has been transformed into a vermin by looking at his legs. No inner feeling informs either character that his mind has undergone a transformation. Kafka does not present the mind of a vermin—what words could he write? The voice is third person narrative. As dream reports are naturally in first person, the voice emphasizes that this is no dream. Kafka presents, as a dream would, a body changed to vermin, with a mind continuing on, half-asleep, but mostly human. Kafka's talent for describing his dreamlike inner life is shown in maintaining a steady pitiful reasoning through page after page of struggle with an absurd reality.
One might wonder whether in Kafka's dream reports, metamorphoses are accompanied by descriptions of inner states. In the following dream, the last known of his life, Kafka dreams he is transformed into the woman Milena:
> Last night I dreamt about you. What happened in detail I can hardly remember, all I know is that we kept merging into one another. I was you, you were me. Finally, you somehow caught fire. Remembering that one extinguishes fire with clothing, I took an old coat and beat you with it. But again the transmutations began and it went so far that you were no longer even there, instead it was I who was on fire and it was also I who beat the fire with the coat. But the beating didn't help and it only confirmed my old fear that such things can't extinguish a fire. In the meantime, however, the fire brigade arrived and somehow you were saved. But you were different from before, spectral, as though drawn with chalk against the dark, and you fell, lifeless or perhaps having fainted from joy at having been saved, into my arms. But here too the uncertainty of transmutability entered, perhaps it was I who fell into someone's arms. (qtd. in Hall and Lind 124)
The dreaming brain of a man cannot produce the experience of being a woman, so little is said about the inner experience of the dreamer as a woman. There is no report of sensation when her body is on fire, and at the moment of a hint of joy, her inner life is impossible because she has died, or, perhaps, has fainted.
Often we think of dreams as a source of wild ideas. But sometimes in a dream there is an equanimity disproportionate to a strange situation, which, if all is carefully described, can produce a curious effect in a story. The inner nature of Mr. Hyde remains a mystery even after his identity with Dr. Jekyll is revealed. For juxtaposition of the fantastic with the ordinary in _The Metamorphosis_ , dreams are a source for the ordinary as well as the fantastic.
### ACKNOWLEDGMENTS
We thank Sonya Basaran for help with coding, and G. William Domhoff, Paula Leverage, Howard Mancing, and Jennifer Marston William for helpful discussion.
### WORKS CITED
Aldler, Jeremy. _Franz Kafka_. New York: Overlook Press, 2001. Print.
Apuleius. _The Golden Ass_. Trans. E. J. Kenney. London: Penguin Books, 1998. Print. Cassidy, Kimberly Wright, Lorraine V. Ball, Mary T. Rourke, Rebecca Stetson Werner, Norh Feeny, June Y. Chu, Donna J. Lutz, and Alexis Perkins. "Theory of Mind Concepts in Children's Literature." _Applied Psycholinguistics_ 19 (1998): 463-70. Print.
Corngold, Stanley. "Introduction." _The Metamorphosis_. By Franz Kafka. New York: Bantam Books. Ed. and trans. Stanley Corngold. New York: Bantam Books, 1986. xi-xxii. Print.
Domhoff, G. William. _The Scientific Study of Dreams_. Washington, DC: American Psychological Association, 2003. Print.
Goodwin, Karen. "Drug Took Stevenson Face to Face with Hyde." _Sunday Times_ 20 Mar. 2005. Web. http://www.timesonline.co.uk/tol/newspapers/sunday_times/scotland/article432698.ece.
Hall, Calvin S., and Richard E. Lind. _Dreams, Life, and Literature: A Study of Franz Kafka_. Chapel Hill: University of North Carolina Press, 1970. Print.
\---, and Robert L. Van de Castle. _The Content Analysis of Dreams_. New York: Appleton-Century-Crofts, 1966. Print.
Hartman, Ernest. _Dreams and Nightmares: The Origin and Meaning of Dreams_. Cambridge, MA: Perseus, 1998. Print.
Kafka, Franz. _The Metamorphosis_. Ed. and trans. Stanley Corngold. New York: Bantam Books, 1986. Print.
Kahn, David, and J. Allan Hobson. "Theory of Mind and Dreaming: Awareness of Thoughts and Feelings of Others in Dreams." _Dreaming_ 15 (2005): 48-57. Print.
\---, Edward Pace-Schott, and J. Allan Hobson. "Emotion and Cognition: Feelings and Character Identification in Dreaming." _Consciousness and Cognition_ 11 (2002): 34-50. Print.
Oates, Joyce Carol. Foreword. _The Strange Case of Dr Jekyll and Mr Hyde_. By Robert Louis Stevenson. Lincoln: University of Nebraska Press, 1990. ix-xviii. Print.
Palmer, Alan. _Fictional Minds_. Lincoln: University of Nebraska Press, 2004. Print.
Rev. of _The Strange Case of Dr Jekyll and Mr Hyde_ , by Robert Louis Stevenson. _The Times_ 25 January 1886. In _The Strange Case of Dr Jekyll and Mr Hyde_. Ed. Martin A. Danahay. Peterborough, Ontario, CA: Broadview Literary Texts, 2000. 136-38. Print.
Schneider, Adam, and G. William Domhoff. _DreamBank_. Department of Psychology, University of California Santa Cruz. 16 December 2006. Web. http://www.dream-bank.net/.
Schweickert, Richard. "Properties of the Organization of Memory for People: Evidence from Dream Reports." _Psychonomic Bulletin & Review_ 14 (2007): 270-76. Print.
\---, and Xi, Zhuangzhuang. "Metamorphosed Characters in Dreams: Constraints of Conceptual Structure and Amount of Theory of Mind." _Cognitive Science_ 34 (2010): 665-84. Simons, Daniel J., and Daniel T. Levin. "Failure to Detect Changes to People during a Real-World Interaction." _Psychonomic Bulletin & Review_ 5 (1998): 644-49. Print.
Stevenson, Robert Louis. "A Chapter on Dreams." _The Strange Case of Dr Jekyll and Mr Hyde_. Ed. Martin A. Danahay. Peterborough, Ontario, CA: Broadview Literary Texts, 2000. 93-104. Print.
\---. _The Strange Case of Dr Jekyll and Mr Hyde_. Ed. Martin A. Danahay. Peterborough, Ontario, CA: Broadview Literary Texts, 2000. 29-92. Print.
Zunshine, Lisa. _Why We Read Fiction: Theory of Mind and the Novel_. Columbus: The Ohio State University Press, 2006. Print.
## Mother / Daughter Mind Reading and Ghostly Intervention in Toni Morrison's _Beloved_
KLARINA PRIBORKIN
Toni Morrison's _Beloved_ explores a triangular relationship between a mother (Sethe) traumatized by the horrors of slavery, her youngest daughter (Denver), and the ghostly figure of her deceased daughter (referred to as Beloved). The circumstances of Beloved's death as well as the mother's past experiences as a slave are the "unspoken secrets" that define the mother / daughter relationship in the novel. Growing up in a terrifying atmosphere, with the ghost of her dead sister haunting the house, Denver is unaware of the true reasons for the haunting. When she finds out that her mother killed Beloved to save her from slavery, she also notices the community's hostility toward her family. Not only the community, but Sethe's family and children, too, continuously misinterpret her intentions. Adopting the environment's interpretation of her mother's deed, Denver alienates herself emotionally, creating a demonic image of her mother. While Sethe understands the murder of her daughter as an act of love intended for that child's protection, Denver, similarly to other members of her family and community, believes that something horrible in her mother "makes it all right to kill her own" (Beloved 205).
This paper will examine three incidents, two early in the narrative and one toward the end, in which Denver's Theory of Mind (ToM) seems at first to fail, but finally succeeds in functioning. The change, I claim, is enabled by Beloved—the ghost of the murdered daughter—who facilitates Denver's acquisition of mind-reading skills. Beloved's mediation—the mediation of a ghost—is the narrative device that allows Denver's development of ToM, and its consequent application to her mother's stories. By encouraging Denver to visualize the details of her mother's narrative, Beloved compels her to change perspectives, recreating what Sethe must have thought, felt, and experienced as a slave mother. Such cognitive exercises, as well as continuous paralinguistic interactions with Beloved, gradually improve Denver's mind-reading skills. When Denver learns to retell her mother's story from her mother's perspective, rather than from her own, she learns to imagine Sethe's mental and emotional states. This allows her to fill in the gaps in the narrative, retell her mother's story, and eventually to succeed in interpreting her mother's mental states empathically. I will first discuss the reasons for Denver's difficulty in developing efficient ToM and then show the cognitive processes she undergoes as she learns to reinterpret her mother's states of mind.
The atmosphere of secrecy produced by Sethe's inability and unwillingness to share her traumatic memories with her daughter not only impedes mother / daughter communication, but also hampers Denver's cognitive and personal development. In "Mother-Child Reminiscing and Children's Understanding of Mind," Elaine Reese and Emily Sutcliffe suggest that mothers who discuss mental and emotional states and memories with their children advance their children's understanding of mind. These children relate richer and more elaborate autobiographical memories (18-19). Tragically, Sethe's failure to discuss the horrors of slavery and her own traumatic past with her daughter prevents Denver from developing the skills necessary for successful mind reading.
According to Nicolas Abraham and Maria Torok's theory of trauma, "secret discourse" or "cryptonymy" (the practice of word hiding) is symptomatic of traumatized individuals who experience difficulty talking about their past. Abraham and Torok claim that family secrets are transmitted without words and it is the offspring's suspicion that something has been left unsaid that impels the younger generation to listen for clues and hints in their parents' speech.1 This is precisely what happens in _Beloved_. Although Denver subconsciously knows that her mother killed her sister, she does not know why she did it or what led her to do it. Since Denver cannot discuss her assumptions regarding the murder with her mother, Sethe cannot correct her daughter's mistaken inferences regarding her original intentions.
Not only has no one ever corrected Denver's mistaken understanding of her mother's mind, but Denver's brothers and grandmother, Sethe's mother-in-law, and other members of the community2 also misinterpret Sethe's intentions. The boys tell Denver "die-witch!" stories that teach her how to kill her mother "if [she] ever needed to" protect herself" (205). Although Denver loves her mother, she also fears her. In Denver's words: "I love my mother but I know she killed one of her own daughters, and tender as she is with me, I'm scared of her because of it. She missed killing my brothers and they knew it" (205). When the brothers are old enough, they escape the haunted house without confronting their mother or talking with her about the murder.3 Denver, too, is secretly indulging the fantasy of leaving her mother; she is waiting for her father, who apparently failed to escape the slaveholders, believing that he will redeem her from her mother's potential violence.4 The children's inability to communicate with their mother is reinforced by their grandmother, who had been bought out of slavery by her son, and with whom Sethe and her children live on the outskirts of Cincinnati after they escape from the South.
Although Denver initially learns about the murder from her grandmother's cryptic narratives, Baby Suggs's stories conceal much more than they reveal. Denver remembers her grandmother telling her that she need not fear the ghost: "she said the ghost was after Ma'am and her, too, for not doing anything to stop it. But it would never hurt me. I just had to watch out for it because it was a greedy ghost and needed a lot of love, which was only natural, considering . . ." (209).5 Baby Suggs hints at but does not explain the murder to Denver. Indeed, she adds an additional layer of secrecy to the atmosphere of horror that already exists in the house. Instead of reducing Denver's fear of her mother by explaining to her the sociocultural context of the murder, Baby Suggs transfers the unspeakable secret to the next generation, reinforcing the tradition of reticence and secrecy.
Denver, thus, participates in a community of interpreters or rather misinterpreters of Sethe's intentions.6 Failing to react empathically to Sethe's tragedy or to try to understand her intentions, Denver's brothers, the grandmother, and the community of Cincinnati condemn Sethe's violence. They establish a cognitive framework of reference that Denver instinctively assimilates as her own. The environment, thus, shapes Denver's interpretation of her mother's violent act, leading the daughter to develop a demonic image of her mother.
In "Intermental Thought in the Novel," Alan Palmer defines intermental units as "groups [such as large organizations, as well as families, couples and friends] that regularly employ intermental thinking" (429). Palmer explains that this kind of thinking "is joint, group, shared, or collective, as opposed to intramental, individual, or private thought. It is also known as socially distributed, situated, or extended cognition, and as intersubjectivity" (427). Since all adult members of Sethe's community share the traumatic experience of slavery, they usually succeed in understanding each other's unspoken traumas and cryptic references to the past. This intermental communication is necessary for what Palmer refers to as an "intermental mind: these are intermental units that are so well defined and long lasting, and where so much successful intermental thought takes place, that they can be considered as group minds" (430). By attributing a common mind to a group of seemingly separate individuals, it is possible to analyze their social conventions in contrast to a single, private mind that may choose to rebel against a society.
Another vivid example of an intermental-group thought in the novel has to do with racial discrimination. According to the slaveholders, "slaves not _supposed to_ have pleasurable feelings on their own; their bodies _not supposed_ to be like that, but _they have to_ have as many children as they can to _please whoever owned them_ " ( _Beloved_ 209, emphasis mine). The words I marked belong to the white's "group mind" that imposed its conceptions on the black community. The white people communicate this racist message of enslavement to the blacks, making them incorporate it as an integral part of their own culture. In _Beloved_ , Baby Suggs tries to instill the opposite message, based on self-love and self-respect, thus attempting to reconstruct the black community and create a new group mind. As the spiritual leader of her community, Baby Suggs communicates with her people intermentally: "After situating herself on a huge flat-sided rock, Baby Suggs bowed her head and prayed silently. _The company_ watched her from the trees. _They knew she was ready when she put her stick down_ " (87, emphasis mine). The communal mind reading constructed at the spiritual site creates an unspoken understanding between Baby Suggs and her community. Baby Suggs's blessing and motivating speech that comes right after the ritual signifies intermental thinking, implying complete understanding between the leader and the group.
Binding the different characters in the novel into one intermental communal unit, Morrison exposes her readers to the cognitive processes of remembering and forgetting that the black community undergoes coming to terms with its past. According to Garciela Moreira-Slepoy, Toni Morrison's _Beloved_ provides an alternative to "the historical monolithic accounts dealing with slavery" (3) by integrating the scattered personal testimonies into a comprehensive account of these characters' past. These "series of local or private narratives . . . need to be told and retold once and again in order to reconstruct and come to terms with the past in a liberating way" (3). As Christopher Powers suggests, "Morrison creates characters whose consciousness of time is in fact spatialized, because it is part of her innovation as a writer to show how people experience memory synchronically" (34). Unlike Moreira-Slepoy, Powers does not believe that Morrison's main purpose is to undo "master narratives," rather, he claims that Morrison "spatializes time" and "synchronizes narrative" in order to "simulate subjective experience" of self-construction (33-34).7 While Morrison's focus on individual narratives indeed illuminates the characters' personal development, it is the connection between these stories that makes an authentic recovery of the past possible. When the individual voices in _Beloved_ are joined to tell a story, their personal testimonies confirm and legitimize each other, establishing a communal voice that reconstructs the white narratives about slavery.
According to Palmer a mind can "consist of more than one brain" (432). I suggest that the communal group mind constructed through Baby Suggs's rituals, cooperative remembering, and storytelling can recover truths that remain inaccessible to an individual mind. When several minds work together they form an intermental unit that consists "of individual minds pooling their resources and producing better, or at least different, results" (430). It is therefore through the cooperation between the members of the community that the individual self can access and come to terms with the communal past that shapes individual experience.
Although this group mind in _Beloved_ is initially determined to keep its past at a safe distance, Beloved's murder brings the horrors of slavery back to the people's lives. They attempt to repress it by ostracizing Sethe and her family. When Sethe challenges the cultural and moral presuppositions about the nature of motherhood and slavery, she becomes a rebel and an outsider in both white and black societies. By comparing Sethe's interpretation of her own beliefs and behavior to the ways white and black communities perceive them, it is possible to explain the workings of these distinct yet interrelated groups. Sethe challenges the social codes of the white community, insisting on her natural rights as a mother in spite of her supposedly inferior race and social status.8 Furthermore, she challenges the black community's moral standards. Sethe's community does not justify infanticide in the face of slavery, yet she rebels against this cultural or intermental construction. Sethe reinterprets her act of violence as an act of love and protection. Yet since this view does not correlate with the others' social codes, they create the intermental construct based on meaningful silence that everybody in the community understands. This silence defines Sethe as "the other," a murderer who broke a basic moral law.
When Sethe saw the slaveholder, called Schoolteacher, approaching her home to take her and her children back to the farm where she had been a slave before escaping, she rushes with her children to the shed and slashes the throat of her "crawling already!" baby daughter. She intends to kill all her children and herself, but her mother-in-law stops her. Stunned by her act, the white men conclude that Sethe is far too rebellious, and perhaps even mad, and therefore cannot be enslaved again. The people of the community, equally shocked, observe Sethe as she is taken to prison by the sheriff. Yet Sethe
> walked past _them_ in _their_ silence and _hers_. She climbed into a cart, her profile knife-clean against a cheery blue sky. A profile that shocked _them_ with its clarity. Was her head a bit too high? Her back a little too straight? Probably. Otherwise the singing would have begun at once . . . some cape of sound would have quickly been wrapped around her, like arms to hold and steady her on the way. (152, emphasis mine)
This compelling mind reading interchange between Sethe and her community is relayed by the third-person narrator who makes assumptions regarding both Sethe's and the community's mental states. The double mind reading, between the narrator and the characters, as well as between Sethe and her community, allows readers to participate in the creation of an intermental or group mind that observes and judges Sethe's behavior.
Since the readers are provided with Sethe's observable actions, such as holding her head too high or her back too straight, they understand, and perhaps participate in, the community's misinterpretation of Sethe's intentions. Showing no apparent or observable remorse through her body language, Sethe intentionally or unintentionally conveys her contempt and disrespect for the community, which simultaneously rejects her in return. Sethe later says that she could never explain why she killed her daughter "for anybody who had to ask. If they didn't get it right off—she could never explain" (163).9 Sethe refuses to communicate her motives to anyone but her reincarnated daughter Beloved, who, she believes, is the only one that can understand her motives without verbal explanation.10 This lack of linguistic or paralinguistic input makes Sethe rather opaque to her community and her family, leading them to draw incorrect assumptions regarding her mental states.11
In order to become conscious of the family secret, children of traumatized parents must decode the cryptic language of the adults. This is traumatic, especially when the ultimate revelation of the secret comes from a person that does not belong to the family. Although the atmosphere of secrecy oppresses Denver for years, it is only when she is directly asked about her mother's deed that she realizes her secret. Misreading not only her mother's, but also her peers' paralinguistic signals, Denver does not notice that her classmates often make "excuses and alter their pace not to walk with her" (102). By communicating their contempt to Sethe and her family through body language, the children subtly reject the outsiders, thus participating in the construction of the communal group mind. When one of her classmates confronts Denver about her sister's murder, she realizes her failure to read both her mother's and other people's minds. One of her fellow students asks Denver: "didn't your mother get locked away for murder? Wasn't you in there with her when she went?" Denver does not laugh the question off, because "something leap[s] up in her, when he asked **;** it was a thing that had been lying there all along" (102). The question brings Denver's subconscious knowledge about the murder into her everyday reality, thus allowing the daughter to start attributing meaning to the hints she has heard all her life.
Denver's realization that she has failed to understand her mother's and her community's intentions leads to her mistrust of language and communication. This is symbolized by her psychosomatic deafness and self-imposed isolation from social contact. She stops attending school and secludes herself in the house, concentrating on the ghostly presence of her dead sister. Although Denver's hearing impairment symbolizes her detachment from the outside world, it brings her closer to familial secrets and the ghost. Denver is capable of playing with the ghost without using verbal language: "that's how come me and Beloved could play together. Not talking. On the porch. By the creek. In the secret house" (206).
Assuming that Beloved's ghostly figure is not real, but rather belongs to the realm of Denver's imagination, the ghost can be thought of as Denver's imaginary friend. According to developmental psychologists, interacting with imaginary friends plays a positive role in a child's development both cognitively and emotionally, allowing children to practice managing social situations and handling conflict in a safe context. Cognitively it helps them to deal with abstract symbols and to start thinking about their own identity.12 Interacting with Beloved is indeed essential for Denver's cognitive development. Their silent communication, within the context of Denver's deafness, provides an essential learning experience; it trains her to read other people's faces and to "learn how to figure out what people were thinking, so [she] did not need to hear what they said" (206).
While the cognitive psychologists suggest that development of mind-reading abilities depends both on successful language acquisition and social interaction (Garfield, Peterson, and Perry 494), in _Beloved_ it is precisely Denver's detachment from society and language that allow her to sharpen her mind-reading abilities. Even if Denver did acquire some basic mind-reading skills as a child, they proved insufficient for decoding her traumatized mother or the community's paralinguistic signals. Denver learns the hard way that what people say does not necessarily correlate with what they think. However, although Denver rejects verbal communication, she hones her ToM skills to an exceptional degree. Mind reading becomes Denver's survival strategy; it allows the daughter to interpret her mother's facial expressions, watching out for murderous thoughts that she believes may be crossing her mother's mind.
Although the ghost teaches Denver to decode people's faces, it does not teach her to interpret her mother's intentions on a more complex level. Still fearing that her mother might kill her, Denver does not ask Sethe to tell her about the murder. She chooses to repress the story, thus protecting herself from additional traumatic revelations. As the novel progresses, however, the ghost of Sethe's murdered daughter establishes her physical presence in the house. This opens a new phase in Denver's relationship with Beloved. If prior to Beloved's reincarnation the sisters used to communicate in silence, Beloved's embodied presence initiates verbal interaction. The ghostly figure influences Denver's patterns of speech and storytelling, thus transforming the youngest daughter's relationship with her mother.
Prior to Beloved's intervention, the only story Denver is willing to hear from her mother is the story of her own birth. This can be interpreted as her refusal to accept Sethe as a separate individual with a personal history. Denver acknowledges only those parts of Sethe's past that are relevant to her. Since Sethe's function in Denver's life is defined mostly through her role as a mother, the daughter does not perceive her as a separate individual and therefore blocks those parts of her mother's story that define her as such.
In _Reproduction of Mothering_ , Nancy Chodorow claims that since mothers and daughters are of the same sex, their separation and individuation processes are more problematic and complicated than those of mothers and sons. While boys tend to define themselves in opposition to their mothers, girls experience themselves as contingent with them. For a girl there is no "absolute change of love object" and even her "libidinal turning to her father is not at the expense of, or a substitute for, her attachment for her mother" (125). Thus mothers and daughters maintain elements of their primary relationship throughout their lifetimes, and their ego boundaries remain unstable because they develop a relational, rather than an individualistic, sense of self.13
As Jessica Benjamin argues in _The_ _Bonds of Love_ , a healthy relationship between a mother and child is made possible through intersubjective connection and mutual recognition rather than through individuation and separation.14 Benjamin's intersubjective view suggests that "we actually have a need to recognize the other as a separate person who is like us yet distinct. This means that the child has a need to see his mother . . . as an independent subject" (23). Thus, paradoxically, it is only by acknowledging each other as individuals that the mother and daughter's identities can develop in relationship. In _Feminism and Psychoanalytic Theory_ , Chodorow shows that although mother / daughter relationships are extremely close across cultures, in matrifocal and traditional societies these relationships are less problematic. This is because "the people surrounding a mother while a child is growing up become mediators between mother and daughter, by providing a daughter with alternative modes for personal identification and objects of attachment, which contribute to her differentiation from her mother" (62).
The intersubjective communication and subsequent mutual recognition is hampered in Sethe and Denver's relationship, since the mother and daughter are ostracized by the community throughout most of Denver's childhood. As a result, Denver has very little social interaction with potential mediating figures in the community. Therefore, it is through Beloved's mediation that Denver is finally exposed to alternative modes of social interaction. Beloved's verbal intervention in Denver's storytelling practices facilitates the daughter's retelling of her birth as her mother's, and not only her own history. This rhetorical exercise develops Denver's creativity and independence, enabling her to imagine her mother's perspective as separate from her own, thus undermining the unhealthy patterns of dependence between the mother and daughter.
In "Explaining the Emergence of Autobiographical Memory in Early Childhood," Katherine Nelson suggests that by learning to tell stories about themselves children develop their autobiographical memory and thus construct their personal identity.15 In order to learn the skill of self-narration children must learn to "tell a coherent story, to tell the truth, get the facts right," and so on; "In doing all this, the child must become reasonably adept at taking the perspective of another, and of viewing events from a somewhat detached, meta-representational distance" (377). When Beloved teaches Denver to tell the story of her birth from her mother's perspective, she both evokes the daughter's empathy toward her mother and initiates the processes of the daughter's self-construction.16 Paradoxically, by telling her mother's story, Denver creates the necessary distance between Sethe and herself to recognize her own and her mother's individuality. As Paul Eakin suggests in _How Our Lives Become Stories_ , "mothers and daughters are so intimately bound in the process of identity formation that to tell the story of the one is necessarily to tell the story of the other" (179). By facilitating this relational storytelling, Beloved helps Denver to assert her individuality in relationship, rather than in opposition to her mother.
Moreover, by interacting with Beloved, Denver learns to read minds. This extensive exercise of her ToM enables Denver to apply her cognitive skills to her mother's as well as to other people's minds. At first Denver learns to understand the meaning behind Beloved's gazes:
> Beloved seldom looked right at her, or when she did, Denver could tell that her own face was just the place those eyes stopped, while the mind behind it walked on. But sometimes at moments Denver could neither anticipate nor create— Beloved rested cheek on knuckles and looked at Denver with attention. (118)
Denver differentiates between Beloved's absentminded and meaningful gazes by employing the embodied ability of mind reading that, as Lisa Zunshine suggests, allows her to "ascribe to a person a certain mental state on the basis of her observable action" (6). Beloved thus uses paralinguistic communication in order to engage in an intersubjective relationship with Denver: "it was lovely. Not to be stared at, not seen, but being pulled into view by the interested, uncritical eyes of the other. Having her hair examined as a part of her self, not as material or a style" (118). Denver enjoys being a subject rather than an object of observation. Beloved's recognition allows her to recognize herself as an independent individual in relationship, and later to apply this experience of self-assertion to her relationship with Sethe.
The changes in the narrative patterns between the two versions of Denver's retelling of her birth, which I now consider, reflect the transformations in Denver's perception of her mother. The first time Denver tells the story of her own birth, she basically repeats her mother's words instead of engaging in a creative retelling. Since Denver is not experienced in the techniques of self-narration, she is not accustomed to using her own narrative voice or to creating imaginative strategies in order to tell her own story. Thus she first renders the story through her mother's narrative voice, using reported speech: "she had good hands, she said. The whitegirl, she said, and then little arms but good hands. She saw that right away, she said" (76). Beloved, however, is not satisfied by Denver's abrupt narration and therefore provokes more elaborate storytelling by "asking questions about the color of things and their size" (77). Since Sethe has not assisted her daughter in the development of remembering and retrieving processes, Denver's storytelling techniques remain rather scant. Sethe's inability to discuss her past hampers not only Denver's cognitive abilities, but also the development of her autobiographical self-perception. Thus Beloved's assistance in the creation and shaping of life stories is essential for Denver's self-construction.
According to Reese and Sutcliffe, cooperative reflection on the past also enables parents to train children to shift perspectives. This serves as a powerful context for understanding minds (36). Indeed, when Denver, with Beloved's help, learns to perceive her birth story from her mother's perspective, she also becomes aware of her mother's mental and emotional states. Beloved facilitates Denver's storytelling by encouraging her sister to generate the visual details of Sethe's story. By joining in imaginative constructions, the two sisters come to share their mother's past, thus creating a mutual system of references that binds them both cognitively and emotionally into a new interpretive community.
In order to imagine her mother's thoughts and emotions when running away from the slaveholders, Denver has to change the narrative's perspective. This is conducted through the sisters' cooperation, since as a dyad, Denver and Beloved possess more interpretive authority to add details and develop the emotional framework of Sethe's story. Moreover, as mentioned above, Denver lacks the necessary interpretive background to engage in a complicated rhetorical enterprise. Although Beloved's guidance is necessary for Denver's cognitive and emotional development, the reincarnated sister does not overpower the narrative process.
On the contrary, Beloved refrains from imposing her own voice on Denver's narration. Instead, she facilitates a dialogic, intersubjective narration in which Denver and Beloved have equal parts. According to Mikhail Bakhtin, the intersection between voices, consciousnesses, or worldviews does not necessarily imply one voice overpowering another, but rather a "dialogic concordance of unmerged twos or multiples" (289). Bakhtin's concept of dialogism suggests simultaneous interdependence and autonomy in relation to the other. Since language, according to Bakhtin, is fundamentally social, the process of our self-construction17 depends on interaction with others. This view corresponds with Jessica Benjamin's concept of intersubjectivity that maintains that "the individual grows in and through the relationship to other subjects" which occurs through "mutual recognition" (20, 16). Both Bakhtin and Benjamin emphasize the possibility of individual development in relationship; interdependence need not negate personal freedom.
Indeed, in _Beloved_ , constructive narration becomes possible as a duet rather than as a monologue. By serving as Denver's audience, Beloved silently participates in the narration and encourages the storytelling process by creating an atmosphere of mutual support: "Denver spoke, Beloved listened and the two did the best they could in order to create what really happened" (78). By asking Denver to narrate the story: "'Tell me,' Beloved said. 'Tell me how Sethe made you in the boat,'" (76) Beloved undermines the patterns of silence and secrecy in which Denver has been growing up. She asks Denver to narrate that which has been indirectly relayed to her by her mother, brothers, grandmother, and the community at different times and under different circumstances. Denver is asked to transform the implicit into explicit: "[Denver] swallowed twice to prepare for the telling, to construct out of the strings she had heard all her life a net to hold Beloved" (76).
Beloved's rhetorical guidance helps Denver to identify with Sethe to such a level that she can fill in the missing traumatic details Sethe was unable to narrate. By helping Denver to describe events she has not actually witnessed, Beloved impels her sister to imagine; that is "to _see_ what she was saying and not just to hear it" (77, emphasis mine). At first, Denver recreates the mother's story from the perspective of the observer. However, as she goes on, she identifies with her mother so much that she starts not only seeing but also feeling the story through Beloved: "feeling how it must have felt to her mother. Seeing how it must have looked" (78). The change in perspective develops Denver's mind-reading ability, enabling her to experience the story from her mother's (i.e., the participant's) perspective. While in Sethe's storytelling about her own mother Beloved helps to transform visual, field memories into narrative memories, with Denver, Beloved initiates an opposite process, transforming the remembered story into visual images, which enhances Denver's identification with her mother.
According to John A. Robinson, "remembering can be organized through various perspectives or points of view. Subjectively, the rememberer may reexperience the event as a participant or a spectator." The participant's perspective enables imaginative participation in the events as if experienced firsthand. The spectator's point of view, on the contrary, registers "the self engaged in the event as an observer would," that is, without particular emotional involvement (206). Robinson further claims that flashbacks associated with post-traumatic stress disorder and other traumas tend to occur as participant memories (207). This may suggest that the participant's perspective is more emotionally loaded than a spectator's point of view. Since Denver's initial imagination of her mother's story occurred merely from a spectator's perspective, it blunted her emotional involvement in the story. Beloved's intervention allows Denver to develop an additional perspective. Although she maintains the initial observer's stance to a certain degree, after shifting her point of view she can also participate in the story as if she was actually present when the events took place.
However, even though Denver becomes emotionally involved in the story, she is not as traumatized by the events as her mother. This is because she now possesses a double vision as both an observer and a participant; she can both identify with her mother and maintain her own separate self. The novel's narrative strategy brings this out. Morrison narrates Denver's second retelling in the third person, rather than in the first, which is more immediately associated with personal testimonies. The third person voice provides the necessary distance for Denver's narration of her mother's story: "there is this nineteen-year-old slave girl—a year older than herself—walking through the dark woods to get to her children who are far away. She is tired, scared maybe, and maybe even lost" (77-78). On the one hand, the observer's perspective is maintained when Denver describes her mother, as if observing her from a distance. Yet, on the other hand, the daughter can relate to her mother's situation and narrate her emotions. By mentioning that her mother was her age when running away, the daughter brings her mother's past closer to her own present and can therefore understand how exhausted and scared her mother had been.18
When Sethe acknowledges Beloved as her daughter, she tries to amend their relationship. Beloved, however, refuses to forgive; instead, she attempts to avenge her own murder by killing Sethe. She makes Sethe quit her job and play with Denver and herself. Then, she drives Denver away from their play. Constantly asking for food and sweets, Beloved almost makes Sethe and Denver starve to death. Observing this deteriorating situation, Denver finally succeeds in drawing the right conclusions regarding the Sethe-Beloved relationship: she understood that Sethe "was trying to make up for the handsaw; Beloved was making her pay for it." Denver, however, also realizes "there would never be an end to that, and seeing her mother diminished, shamed and infuriated her. Yet she knew that Sethe's greatest fear was the same one Denver had in the beginning—that Beloved might leave" (251). Denver understands her mother's state of mind when comparing it to her own fear. By recognizing her own emotional connection and similarity to her mother, she establishes an intersubjective bond that allows her to read her mother's mind.
Beloved's intervention, despite its negative impact on Sethe's mental and physical health, promotes changes in Denver's assumptions. At first, Denver watches Sethe and Beloved's interaction alert for "any sign that Beloved was in danger . . . her eye was on her mother, for a signal that the thing that was in her was out, and she would kill again. But it was Beloved who made demands" (240); "The job [Denver] started out with, protecting Beloved from Sethe, changed to protecting her mother from Beloved. Now it was obvious that her mother could die" (243). Denver thus takes the initiative into her hands and turns to the black community of Cincinnati for help.
Interestingly, the same community that ostracizes Sethe for her violence at the beginning of the novel embraces her at the end of the novel, exorcising Beloved through vocal collaboration. The community once again creates an intermental unit that shares a common goal and a common mind. When the black women gather to banish Beloved, they use no words: "in the beginning there were no words. In the beginning was the sound, and they all knew what that sound sounded like" (259). The women's collective memory of the right sound allows them to access the primary, prelinguistic experience that precedes speech in order to accept Sethe back into their community.19 The women's collaboration during the exorcizing ritual reestablishes the intermental bond between the members of the community: "building voice upon voice," the women search for the right key, and when it is found, they produce "a wave of sound wide enough to sound deep water and knock the pods off chestnut trees" (261).
The community's silence, immediately after the murder juxtaposes with this later vocal attempt to help Sethe overcome her horrible past. In fact, after the women succeed in banishing Beloved physically, they begin the cognitive process of mental exorcising or—as Morrison puts it—"disremembering."
> Disremembered and unaccounted for, she [Beloved] cannot be lost because _no one_ is looking for her, and even if _they_ were, how can _they_ call her if _they_ don't know her name? . . . _They_ forgot her like a bad dream. After _they_ made up their tales, shaped and decorated them, _those_ that saw her that day on the porch, quickly and deliberately forgot her. . . . Occasionally, however, the rustle of a skirt hushes when _they_ wake, and the knuckles brushing a cheek in sleep seem to belong to the sleeper. Sometimes the photograph of a close friend or relative – _looked at_ too long—shifts, and something more _familiar_ than the dear face itself moves there. _They_ can touch it if _they_ like, but don't, because _they_ know things will never be the same if _they_ do. (274-75, emphasis mine)
The cognitive framework of stories the community constructs in order to disremember20 Beloved creates an intermental web that connects the community members into what Palmer refers to as a group mind. This mind is shown to have an uncanny experience of the ghostly presence that keeps haunting this group's imagination. The haunting past becomes an integral part of the communal knowledge or the communal mind. It becomes their common knowledge that memory, even though it is recreated in the group's mind, may reincarnate and change the present.21
After years of misunderstanding, mutual rejection, and loneliness, not only Denver, but the community, too, realizes their mistaken assumptions about Beloved's murder. When it turns out that Sethe does not only feel guilty for her violent act, but may die immersing herself in her destructive remorse, Denver appeals to the community and receives help. In order to understand Sethe empathetically, Denver has to reject the community's original conception of her mother. When this is achieved, the daughter can reenter the communal intermental unit and change it from within. The individual's impact on the community is thus revealed—Denver explains her mother's mental states to others, thus changing the group's opinion about the murder.
When Denver must ask for help, she realizes that "nobody was going to help her unless she told it—told all of it" (253). The daughter makes amends for her mother's silent pride through verbal communication. She tells part of the story about Beloved's haunting to some women in the community, explaining her mother's deteriorating mental and physical condition. Although Denver revises the story of the haunting, representing Beloved as a cousin who came to visit, got sick, and bothered her mother (254), the women of the community reinterpret the story in much more supernatural terms:
> the news that Janey got hold of she spread among the other coloredwomen [sic]. Sethe's dead daughter . . . had come back to fix her. She was worn down, speckled, dying, spinning, changing shapes and generally bedeviled. . . . It took them days to get the story properly blown up and themselves agitated and then to calm down and assess the situation. (255)
By revising Denver's story and creating their own interpretation the women cognitively reprocess Sethe's past and present and alter their communal mind.
Using her enhanced mind-reading skills Denver eventually understands her mother's victimization; she empathizes with Sethe's suffering as a slave-mother who is forced into violence against her own children by the system of slavery. While Beloved has been the victim of her mother's violence, Sethe has been the victim of slavery and of her own ambiguous choice. Reinterpreting Sethe's motivation through empathy, Denver finally effaces the demonic mother figure she has imaginatively constructed. Ironically, by using her imagination, the daughter finally succeeds in coming to terms with her real mother.
### NOTES
1 Developing Abraham and Torok's argument, in _Family Secrets_ , Esther Rushkin suggests that phantoms or ghosts are an unconscious, repressed formation, too horrible or shameful to be represented in words (28). The phantom image generated in the subconsciousness of the traumatized parent is indirectly transmitted to the child by means of cryptic discourses, thus becoming an integral part of the child's subconscious worldview. According to Rushkin, the child that inherits the gap in her parent's speech actually inherits parts of the parent's subconsciousness (29).
2 After some members of Sethe's community witness Beloved's murder, they stop visiting Sethe's house and treat the whole family as social outcasts. Stamp Paid seems to be the only character who succeeds in interpreting Sethe's intentions empathically. In his words: "She love those children. She was trying to outhurt the hurter" (234). Even though, just like the rest of the community, Stamp Paid has been avoiding Sethe, toward the end of the novel he admits that unlike others he does not perceive Sethe as crazy, but rather as an extremely devoted mother who tried to overpower the system of slavery through the murder of her own child.
3 After the brothers leave Denver continuously has nightmares and fantasies about her mother's violence. She dreams about her mother cutting her head off "her pretty eyes looking at me like I was a stranger. Not mean or anything, but like I was somebody she found and felt sorry for. Like she didn't want to do it but she had to and it was not going to hurt. That it was just a thing grown-up people do" (206). Denver makes an attempt to understand her mother's violence by attributing it to the inexplicable realm of the grown-up's world. This, however, does not justify the mother's deed, and therefore, just like the readers, Denver experiences continuous difficulties in understanding her mother's motives and intentions.
4 While Sethe and her children mange to flee from the slaveholders, the father of the family, Halle, fails to escape. His fate remains a mystery for years until Paul D, who was enslaved on the same farm as Sethe and her family, arrives at her house. He tells Sethe that the last time he saw Halle on the farm, he was smearing butter over his face. Paul D assumes that "something broke" Halle, but it is only after Sethe tells him that prior to her escape she was violated and beaten by the slaveholders that he understands what happened to Sethe's husband. Paul D concludes that Halle must have witnessed the horrible scene of his wife's violation, and unable to protect her, he lost his mind.
5 It is possible that Sethe's deed remains unnamed because the grandmother, as well as other people in the community, lack suitable words to express it. Stamp Paid, who witnessed Sethe's offense, also experiences difficulty talking about the murder directly; he refers to it as "Sethe's Misery," thus expressing both the child's and the mother's victimization.
6 In _Is There a Text in This Class?_ Stanley Fish argues that "meanings are the property neither of fixed and stable texts not of free and independent readers but of interpretive communities that are responsible for the shape of a reader's activities and for the texts those activities produce" (322). According to Fish, our "consciousnesses are constituted by a set of conventional notions" that we assimilate from our socio-cultural environment. Using these notions to interpret our reality, we produce the culturally accepted meanings that allow us to participate in the interpretive community set by our social environment. The concept of "interpreting communities" in relation to _Beloved_ was initially brought to my attention by Rita Horvath, whose manuscript "Murdered Joy: A Reading of Henry James's 'The Figure in the Carpet'" discusses what it means to enter a reading community and what factors enable or prevent one to enter a community of interpreters / readers.
7 Powers claims that Morrison's narrative manipulates time by "fastforwarding and rewinding of the temporal setting through flashbacks and stories of the past related by characters or through the meandering, history-purveying eye of an omniscient narrator. Rather than a chronological, diachronic storyline that weaves the evidence of the past into an ordered, _de facto_ , meaningful narrative, time in Morrison synchronizes various pats into one level narrative space, which in turn is subjected more to the demands of characterization than to those of narration" (33).
8 The slaveholders perceive Sethe and her children as their commodity and therefore do not believe in Sethe's right to mother her children. In these circumstances, the only motherly act Sethe can impose on the white society is the murder of her own child.
9 For Sethe the truth was simple: she had to protect her children from the School-teacher even if she had to kill them. As the narrator puts it: "she just flew. Collected every bit of life she had made, all the parts of her that were precious and fine and beautiful, and carried, pushed, dragged them through the veil, out, away, over there where no one could hurt them" (163).
10 This, however, proves to be incorrect, since Beloved forces Sethe to explain her intentions again and again, draining Sethe's emotional and physical strength.
11 The community's stony silence after the murder contrasts with the silent mutuality between Baby Suggs and her congregation. Although both reactions involve paralinguistic communication and signify intermental thinking, the former silence testifies to utter misunderstanding between Sethe and her community, while the latter implies cooperation and understanding between the leader and the group.
12 Schwarz, Joel. "65% of school-age children have an imaginary companion by age 7," _Medical News Today_ 7 Dec. 2004, 21 Oct. 2007, http://www.medicalnewstoday. com/articles/17426.php. According to developmental psychologists, children who report having imaginary companions also have more developed ToM abilities. Research with 152 pre-schoolers found that having an imaginary companion or impersonating an imaginary character was positively correlated with ToM performance. School-age children who did not impersonate scored lower on emotion understanding (Taylor et al.)
13 The term "relational" or "collective identity" was developed in feminist criticism to describe the characteristics of female identity as different from a male individualistic, autonomous self. For a useful review of models of identity, see Paul Eakin's _How Our Lives Become Stories_ (46-53) .
14 Although Benjamin discusses the relationships between mothers and infants in general, her theory is particularly relevant to mother-daughter bonds. Since the daughters' identification processes are more problematic than the sons', they find alternative ways to relate to their mothers. Benjamin's intersubjective view illuminates the processes of identification and construction of relational identity through mutual recognition.
15 As Kay Young and Jeffrey Saver argue in "Neurology of Narrative," the ability to construct narrative is inseparable from personhood. Brain-injured individuals "who have lost the ability to construct narrative . . . have lost their selves" (780).
16 In "Narrative Empathy," Suzanne Keen suggests that empathy is not only an emotional, but also a cognitive process: "empathy itself clearly involves both feeling and thinking. Memory, experience, and the capacity to take another person's perspective (all matters traditionally considered cognitive) have roles in empathy" (213). Beloved's intervention in Denver's cognitive processes not only develops her ToM, but also her emotional capacities. By changing the narrative perspective of her birth story Denver empathizes with her mother. She begins to understand Sethe's emotional and physical difficulties as a slave mother, struggling to protect her children and keep her family together. This emotional and cognitive process finally allows Denver to arrive at an understanding of why Sethe killed Beloved.
17 I can become "myself only while revealing myself for another, through another, and with help of another" . . . "I must find myself in another by finding another in myself (in mutual reflection and mutual acceptance)" (Bakhtin 287).
18 "Expressive writing has been shown as a powerful therapeutic tool for resolving the intense effects engendered by traumatic experiences" (Robinson 207). By adopting an observer's perspective, at least while narrating the story, victims of trauma learn to reconstruct meaning and attain some sense of control over their experience. By splitting the narrating self from the experiencing self the traumatized person distances himself or herself from the past, transforming his or her participant perspective into a spectator's perspective. Denver, however, has to go through an almost reversed process: to add a participant's point of view, thus making her mother's story more emotional and relevant to herself.
19 It is unclear where the women gain the experience or knowledge of the sound. It is possible that the community has been using this sound at the spiritual meetings they held when Baby Suggs, or Baby Suggs Holy as they referred to her, was alive. The memory of Sethe's mother-in-law, to whom the community is spiritually indebted, also propels mutual forgiveness and reconciliation.
20 As opposed to forgetting, disremembering is a deliberate process of healing by putting the past behind. The horrors of slavery cannot be forgotten, but rather put aside in order to continue living.
21 According to psychological research, remembering is a creative, rather than a precise and unchanged retrieval of past events. As studies on flashbulb memories, Deese-Roediger-McDermott illusions, and eyewitness testimonies demonstrate, present recall may distort the original memories of past events. For further details see Roediger and McDermott as well as Schacter. However, as Daniel Schacter also demonstrates in _Searching for Memory_ , emotional and especially traumatic memories influence the rememberer's present and may even change this person's character. The great majority of individuals who experienced traumatic events repeatedly recollect these events to the point where they cannot fully enjoy their present lives. The traumatic memories "impose" themselves on the individual "cast[ing] a shadow" over the victims' lives (196, 203). Thus, past and present inform and shape each other. On the one hand, present events evoke the memories of the past, which may distort or change them, yet on the other hand, when the recall of the past occurs it may influence the present life of the remembering person.
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## Changing Minds: Theory of Mind and Propaganda in Egon Erwin Kisch's _Asien gründlich verändert_
SETH KNOX
The foremost goal of every propagandist is to affect audience cognitions. In order to accomplish this, the propagandist must construct a plausible and probable "audience mind" that has the capacity to be affected. The construction of an audience mind clearly falls within Flavell's definition of metacognition as "any knowledge or cognitive activity that takes as its object, or regulates, any aspect of any cognitive enterprise" (104). As propagandists take as their object of study the cognitions of others, the Theory of Mind (ToM) of cognitive psychology holds particular promise for the analysis of their techniques of mind reading and manipulation. This paper will examine Egon Erwin Kisch's propagandistic travel book _Asien gründlich verändert_ ( _Changing Asia_ ; 1932, translated 1935) through an approach informed by cognitive studies and ToM. Two mind reading and manipulation techniques that are deployed throughout _Asien gründlich verändert_ will be analyzed here: resonance and high source-tracking load. A cognitive approach to these techniques illuminates how they may create the illusion of independent reasoning and self-generated belief in the mind of the propagandee.
When discussing a topic as politically and emotionally sensitive as propaganda, it is helpful to have a working definition of the term. Unfortunately propaganda seems to resist a precise and non-controversial definition, despite the fact that propaganda studies have attracted scholarly work since the end of World War I from several disciplines, including political science, history, sociology, psychology, and communications. Defining propaganda is complicated by its associations in modern English with brainwashing and immoral attempts to manipulate mass audiences with clever falsehoods. Further, it may be tempting to label as propaganda any persuasive communication that one disagrees with (and deny as propaganda any persuasive communication that one finds appealing).
This paper seeks a definition of propaganda that focuses on communicative function. Taylor, in his historical study of propaganda, defines it as "the conscious, methodical and planned decisions to employ techniques of persuasion designed to achieve specific goals that are _intended to benefit those organizing the process_ " (6; emphasis in the original). Jowett and O'Donnel similarly define propaganda as "the deliberate and systematic attempt to shape perceptions, manipulate cognitions, and direct behavior to achieve a response that furthers the desired intent of the propagandist" (4). It is clear from these definitions that propaganda is a subcategory of persuasion; persuasion is, however, distinct from propaganda in that persuasion is an "interactive process in which a sender and a receiver are linked by symbols, verbal and nonverbal, through which the persuader attempts to influence the persuadee to adopt a change in a given attitude or behavior because the persuadee has had perceptions enlarged or changed" (O'Donnel and Kable 9). Whereas persuasion is interactive and ostensibly more voluntary, and it is even possible that the persuader could become the persuadee over the course of the exchange, propaganda does not attempt to engage the audience in a truly participatory dialogue. Instead, propaganda is one-sided communication that anticipates and attempts to manipulate the cognitions of the propagandee in a way that is beneficial to the propagandist. Whether this is ultimately beneficial to the propagandee is either secondary or irrelevant. The conclusions to be drawn from persuasive communication are somewhat flexible due to its interactional nature; the conclusions and attitudes of propagandistic communication are fixed and absolute.
Propaganda is typically classified into three shades: white (overt), gray, and black (covert). This classification is based on the degree to which the propaganda source is obscured and sometimes the accuracy of the information presented. _Asien gründlich verändert_ is an example of overt, or white, propaganda. This means that the source of the propaganda (Kisch) is clearly identified, the information presented is accurate in so far as the book describes an actual journey undertaken by Kisch, and the book's self-selecting audience would have been aware of Kisch's membership in the Communist Party and his pro-Soviet bias. _Asien gründlich verändert_ is a travel account of Kisch's 1931 journey with an international authors' brigade to the youngest Soviet Republics of Uzbekistan and Tajikistan. At the time, Kisch was working as a professor at the University of Kharkov in the Ukraine and as a speaker on Radio Moscow, whose propagandistic broadcasts could be heard in Germany (Patka 122). The deepening global economic crisis and the growing disillusionment of the German reading public with the United States as a viable economic model for Germany increased interest in and sales of his earlier travel books on the Soviet Union ( _Zaren, Popen, Bolschewiken_ 1927) and the United States ( _Paradies Amerika_ 1930) (Djukic-Cocks 26).
These three travel works formed a propagandistic trilogy that resonated with the vague, conspiratorial "anti-System" thinking that dominated political discourse in the Weimar Republic. Anti-System theories were embraced by both the nationalist right and the Communist left; they differed most noticeably in their choice of villains behind the international political system supposedly responsible for Germany's ruin (Lieberman 358-63). Kisch's travels to America and the Soviet Union provided readers with eyewitness accounts of the inevitable misery wrought by capitalism and the inspiring collective achievements of Communism. Kisch's earlier travel works, however, were not as overtly propagandistic as _Asien gründlich verändert_. _Zaren, Popen, Bolschewiken_ is an often dry, if highly selective, presentation of facts and statistics of the early Soviet Union. _Paradies Amerika_ is an entertaining mosaic of Kisch's frequently amusing and sometimes shocking experiences from his journey through "Gottes eigenes Land" ("God's own country"; _Gesammelte Werke_ 5: 9).
_Zaren, Popen, Bolschewiken_ and _Paradies Amerika_ give the reader the impression that they are travel works about their destinations (the propagandistic tendencies of these works are more subtly embedded in the travel narrative). _Asien gründlich verändert_ , however, bridges the two earlier travel narratives and makes clear that this travel trilogy is about the near future of the world (and, more importantly, Germany). Kisch's work on Central Asia also clearly promotes Communism as the most preferable system for the coming new world order. Whereas _Zaren, Popen, Bolschewiken_ and _Paradies Amerika_ were highly successful in financial terms (and, important for their propagandistic value, in reaching a large audience), _Asien gründlich verändert_ was available in only its first edition in Germany before being banned by the Nazi government (before the ban, Kisch was also able to publish a hastily compiled collection of travel writings on China). The second German-language edition was published in exile in Moscow. It is thus impossible to know whether _Asien gründlich verändert_ could have had the same popular appeal and market success as its two predecessors.
A mind reading technique that Kisch employs throughout _Asien gründlich verändert_ is the deployment of anchors to create the effect of resonance. Resonance refers to a complex of values expressed in propaganda that is shared by the propagandist's intended audience, and an anchor is a belief, value, or attitude that is already present in the cognitive system of the propagandee (Jowett and O'Donnell 25-26). For example, if one were creating propaganda targeting self-described environmentalists, anchors might include assertions of the efficacy of government regulation in reducing pollution, the scientific consensus on man-made causes of global warming, and so on. Anchors should repeat beliefs already held by the target audience. The net effect of anchors is resonance—the activation of beliefs, values, and attitudes that ideally creates the illusion that the propagandized message is coming from within the subject rather than as a message imposed from without.
Anchors may make the propagandee more susceptible to propagandized content by activating the believability effect observed by Evans, Barston, and Pollard, who found that people are likely to accept conclusions that appear to be based on their preexisting assumptions. If propagandized content is presented as conclusions derived logically from well-placed anchors (even if the logic is weak or faulty), the propagandee may be more likely to accept it. Of course, in order to determine the content that is suitable for use as anchors, the propagandist must first imagine the mind of the propagandee. This construction of the audience is necessarily an inexact abstraction, and fits well into the idea of problematization formulated by Michel Foucault (Bratich 243-44). In _Asien gründlich verändert_ , Kisch must problematize his imagined audience and anticipate probable group-identities in order to identify potentially effective anchors. Several anchors used by Kisch serve to encompass Kisch (and by extension his propagandized messages) and his audience in a common group-identity.
Throughout _Asien gründlich verändert_ , Kisch refers to himself as a "Western European," a "German," and a "Berliner" (this is despite the fact that he is actually from Prague). Such descriptors clearly link Kisch with the group-identity of his German-speaking audience. However, in _Zaren, Popen, Bolschewiken_ , Kisch went to great lengths to bring Soviet Russians into the sphere of the German group-self. In addition to emphasizing the ways in which Russians resembled Germans, both linguistically and in similar customs, Kisch attributed to Russians those qualities that most Germans (and most people) aspire to: dedication, nobility, honesty, and a strong work ethic (Knox 80-81). This attempt to include Soviet citizens in the group-identity of his German-speaking audience continues in _Changing Asia_ as Kisch depicts Soviet Communists as liberating Uzbeks and Tajiks from the tyrannies of czarist colonialism, capitalist exploitation, and Islamist sexism, terror, and oppression.
The representation of Soviets as existing within the German group-self is simultaneously reinforced through a starkly polarized depiction of others existing outside the group-self. To achieve this, Kisch calls upon an effective technique used by the British during World War I: atrocity propaganda. Two examples of Kisch's use of atrocity propaganda are particularly striking. After a visit to the Sindan, a former dungeon prison for enemies of the Emir of Bokhara, Kisch recounts the story of the imprisoned uncle of Olim Kahn. Conditions in the prison were so severe that this man attempted to gnaw off his shackled arm at the shoulder ( _Gesammelte Werke_ 4: 247, _Changing Asia_ 54-55). Later in the book, Kisch records the story of a Ukrainian woman's ordeal in 1918 during the civil war. When the Whites entered her village and broke into her home, they raped her as they murdered her three year-old daughter and five year-old son before shooting her husband, who was a local revolutionary leader ( _Gesammelte Werke_ 4: 283-85, _Changing Asia_ 111-14). Both stories, along with other atrocities, are presented as barbarous practices that are being stamped out by the Soviets.
The rigid moral dichotomy established in _Asien gründlich verändert_ serves to achieve resonance between Kisch's readership and the declared objectives of Soviet activities in Central Asia. Also, in anticipation of his readers developing a sense of hopelessness after reading his atrocity propaganda, Kisch highlights the accomplishments of the Soviets and the hopeful words of Uzbeks and Tajiks who are delighted with the Soviet presence and look forward to an increasingly bright future. Kisch assures his readers that, despite what they read in the press, the Central Asian Republics are achieving independence in cotton production. Increasing numbers of girls are attending Soviet-built schools, literacy rates are advancing at an impressive clip, more and more women are giving up their veils and working outside the home, and Islamist terrorists, secretly funded by imperialist England, are being decisively defeated. All of this strongly encourages resonance between the perceived motivations and attitudes of the Soviets with those of the propagandee.
In one chapter, "Was hat sich geändert in Chodschent?" ("What Are the Changes in Khojent?"), Kisch anticipates doubt in the reader caused by newspaper articles he is likely to have read. After referring to specific articles and answering the charges raised by each, Kisch shares his frustration with his reader (who, if Kisch's propaganda has been successful, also shares his frustration):
> Wenn sich nichts geändert hätte auf diesem Sechstel der Erdoberfläche, als daß die Ausbeutung weggefegt worden ist—auch ohne die Abschaffung der Arbeitslosigkeit, des Kinderelends, des Analphabetismus, der Religionsverdummung, der Korruption, auch ohne die Erfüllung von hundert Millionen Menschen mit werktätiger Begeisterung, mit Wissen und Kultur, auch ohne all dies, nur um dieser Beseitigung der Ausbeutung willen verlohnte es sich, Zeitgenosse unserer Zeit zu sein.
>
> Aber wer begreift das? Der in kapitalistische Verhältnisse hineingeborene und von ihrer Ideologie benebelte Mensch findet das furchtbarste Unheil nicht furchtbar; er hat sich an die soziale Ungleichheit gewöhnt, die lange vor der Geburt des Menschen beginnt und mit seiner Grablegung noch lange nicht endet. Er empfindet es nicht als grausam, er empfindet es bestenfalls als unabwendbar, daß Hunderttausende Menschen für einen roboten müssen; wer durchschaut die Mittel der Massenverdummung, die diese Verhältnisse künstlich und künstlerisch aufrechterhalten?
>
> Der Leser hat obige Abweichung ins Leitartikelhafte mit Unwillen gelesen. Diese ausnahmsweise Überschreitung ist jedoch notwendig, weil hier unerfreuliche Details nicht nur nicht verschwiegen, sondern oft kraß skizziert wurden und daher auch angegeben werden muß, daß die verschiedenen Skizzen verschiedene Maßstäbe haben: die Originalgröße der unerfreulichen Details ist verschwindlich klein im Verhältnis zum erfreulichen Gesamtbild. ( _Asien gründlich verändert_ 350-51)
>
> [If there had been no other changes on this sixth part of the surface of the globe but that exploitation has been done away with—leaving aside the eradication of unemployment, of the misery of children, of illiteracy, of religious superstition, of corruption, leaving aside the fact that a hundred million human beings have been filled with constructive enthusiasm, with knowledge and culture, leaving aside all this—if only for the sake of abolishing exploitation, it would be worth while to live in our day and age.
>
> But how many people understand this? A man born into capitalistic conditions, his mind befogged with capitalistic ideology, sees nothing horrible in the most horrible squalor. He is accustomed to a social inequality which begins before birth and continues after burial. He does not deem it cruel—at best he feels it to be regrettable, but inevitable—that hundreds of thousands must act as robots for a single man who is clever enough to exploit the stupidity of the masses.
>
> The reader has read this editorial digression with displeasure. But one such digression becomes obligatory to balance unfavourable evidence which cannot be glossed over—which indeed should be repeatedly and emphatically pointed out. In other words, the reader must be reminded that these pictures are drawn to scale, and details which are derogatory if studied as isolated phenomena take on minor proportions when regarded in their relation to the picture as a whole, which is decidedly favourable.] ( _Changing Asia_ 203-04)
In this remarkable chapter, Kisch draws the reader's attention to potential reasons to doubt the assertions of his travel narrative. This passage, however, simultaneously instructs the reader on how to process Kisch's claims in the context of competing sources of propaganda. By insisting that evidence that does not speak well of Soviet efforts will not be "glossed over," but rather emphasized, Kisch is explicitly manipulating the cognitions of the reader so that his text will be perceived as balanced and objective. (The verb "manipulate" is used here in a negative sense, since, despite Kisch's claim, evidence suggesting that Soviet efforts are anything but advantageous to Central Asians is quite scarce in the book.) The text has a good chance of appealing to the propagandee, since it is inherently flattering to the self-selecting readership of _Asien gründlich verändert_ : they, Kisch implies, can see clearly (without minds "befogged with capitalist ideology") and will not be manipulated like "robots" by a malicious individual who knows how to take advantage of the average person's stupidity. Kisch's readers, unlike the common crowd, will not be taken in by propaganda.
The lengthy passage above achieves resonance by appealing to the propagandee's vanity. Kisch suggests what most readers probably already believe: that their intelligence is above average, they are perceptive and able to grasp the big picture, and they are not gullible. If the propagandee already believes these things (and such self-flattering notions are probably reliable anchors), then it may seem as if Kisch is giving voice to the propagandee's own thoughts. This could further place the propagandee in a more suggestive state and more receptive to Kisch's representations of the out-group (men whose minds are befogged with capitalism).
Considering that Kisch has provided repeated stimulation of negative representations of capitalists in _Paradies Amerika_ and _Asien gründlich verändert_ , and considering that Kisch's audience is self-selecting, these propagandized representations could have an effect on the base-rate bias of the propagandee's allocation system. In other words, as a result of repeated exposure to the propagandized representations of capitalists, these representations could be evoked more and more frequently when encountering (or reading about) really existing capitalists. In his discussion of the cognitive allocation system, Brase refers to this phenomenon as "initial base-rate bias," also referred to as "chronically accessible schemas" in social psychology (19-20). Brase describes a chronically accessible schema as "a particular way of representing events that is used by a person in a habitual manner, such as when a person represents all social events as being related to race relations (or politics, or sex, or ethnicity, or any other specific frame of reference)" (19). If propagandized representations of capitalists are repeated often enough, then such representations will be most easily available in memory and habitually activated when capitalists (or discussions of capitalists) are encountered. Kisch's instinct that effective propaganda relies at least in part on repetition is well founded, as a study by Skurnik, Yoon, Park, and Schwarz found that previous exposure to a written statement increased the probability that a person would find the statement credible. In this manner, it might be possible for a book-length work of propaganda such as _Asien gründlich verändert_ to create new anchors in the propagandee that deviate subtly from already existing anchors.
In addition to achieving resonance (and possibly a chronically accessible schema), Kisch employs several techniques to tax the propagandee's source-tracking ability. This claim may seem strange, given that _Asien gründlich verändert_ is an example of overt propaganda. Kisch is clearly identified as the author (source) of the text. Kisch is also known to his readers as a Communist revolutionary and one-time leader of the Austrian Red Guard. In fact, Kisch could only function effectively as an overt propagandist precisely because of his carefully cultivated public image. His image as a journalist and celebrity was carefully controlled. His public persona was that of the _rasende Reporter_ , the tireless journalist who would do anything for his story, scribbling notes hastily to capture as many facts as possible. This, coupled with his association with the _Neue Sachlichkeit_ , or "New Objectivity," movement, gained him a reputation as a reliable and trustworthy source.
Also strongly linking Kisch to his work is the fact that _Asien gründlich verändert_ belongs to the non-fiction genre of travel literature. Readers of travel literature expect an account of an actual journey made by a really existing author. While the author's personal opinions are typically tolerated, there is an expectation that the author will describe the voyage essentially as it was experienced. Thus travel literature affords the reader a form of vicarious travel. It also affords the propagandist the assumption of a certain amount of objectivity on the part of the propagandee. Yet despite the seeming advantages of genre and reputation, Kisch employs techniques that weaken source tags linking him to the representations in his book. Applying the concept of mind reading to travel literature provides a plausible explanation for this behavior and why it is advantageous to the propagandist.
The model below is based on Baron-Cohen's description of the Shared Attention Mechanism in the human mind reading system (44-45). For a very basic understanding of the Shared Attention Mechanism, one can imagine a child and a parent sitting on a park bench and watching a bird. The Shared Attention Mechanism enables the child to realize that his or her parent is attentive to the same bird that he or she is watching. In regard to any representation in the text _Asien gründlich verändert_ , both Kisch and the reader are aware that the other is attending to the same representation. When, for example, Kisch describes the Sindan prison, the reader is aware that Kisch is the source of the representation (and hence aware of it), and the reader may also be aware that Kisch knows the reader will be attending to the representation. Conversely, Kisch is obviously attending the representation, and he is also aware that his reader will as well. Kisch is also aware that the reader will know he is attending the representation. This model, however, implies naïveté on the part of the reader. To borrow and adapt Baron-Cohen's explanation of the subject's search for "relevance," the reader "assumes that the meaning of [a representation] will be relevant to the [author's] current intentions" (27). In other words, the reader will likely ask the question, "Why is Kisch showing me this?"
_Figure 1. Naïve metarepresentation._
This probable search for relevance, especially with a text that attempts political persuasion, could easily lead to the realization illustrated in the second model: the reader realizes that Kisch determines what is seen and not seen, and who is heard and not heard on this journey. This realization, coupled with Kisch's representations of the minds of Uzbeks, Tajiks, and foreign Communists, could result in a suspicion of metarepresentation. When this situation occurs during the reading of a detective novel, it is part of the fun. The reader can play the game of asking, "Why is the author showing me _this_ and not _something else_? Is it a clue? Is it a trick?" This sort of intellectual game is stimulating and engaging in a detective novel. Such suspicion of metarepresentation in propaganda, however, can greatly weaken, if not negate, its effectiveness.
_Figure 2. Suspicion of metarepresentation._
It is thus in the best interest of the propagandist to discourage the creation of source tags that link him or her to propagandized representations. One technique used by Kisch to accomplish this is pronoun confusion. Except for a chapter detailing an illegal tiger hunt in Afghanistan, Kisch eschews the first-person singular pronoun and repeatedly uses the plural form "we." While this could often be interpreted in the text as his membership in the traveling authors' brigade, the pronoun is frequently ambiguous and seems to encompass author and reader. There is no ambiguity, however, at the conclusion of the final chapter, where Kisch compares the achievements and enthusiasm of Central Asian cotton farmers with the conditions of African-American cotton farmers:
> . . . und _wir_ denken an Dixie, den Baumwollgürtel Amerikas, das Land, wo die Sklaverei herrscht wie vor Lincolns Zeit. [. . .] _[W]ir_ denken an dieses Land, wo _wir_ zerlumpte, schlotternde, ausgehungerte, ausgebeutete Gestalten sahen, wo _wir_ von torkelnden Säufern angerempelt, auf Schritt und Tritt von Kindern angebettelt wurden, wo weißgelockte Neger die weißgelockte Ware schleppten, die hier wie dort gedeiht. (386-87; emphasis mine)
>
> . . . and _we_ think of Dixie, the cotton belt of America, where there are slaves, actually, just as before the time of Lincoln. [. . .] _We_ think of that country, where _we_ saw figures in rags, hungry, plundered figures, where staggering drunkards jostled into _us_ , children begged from _us_ at every step, and white-haired Negroes "toted" the white burden, which thrives both here and there. (258)
This is a reference to Kisch's earlier travel book _Paradies Amerika_. The "we" in this paragraph cannot possibly refer to the authors' brigade, as they did not accompany Kisch to America. This "we" includes Kisch and his readers, who have traveled with him vicariously from America to Russia to Central Asia. His experiences are his readers' experiences, thus weakening him as the ultimate source of representations in his travel works.
Kisch also introduces multiple sources of representations. This creates a high source-tracking load, which can further weaken the ability of the propagandee to assign Kisch as the source of all the representations in _Asien gründlich verändert_. The book is populated with characters who gesticulate and narrate their experiences of living in a Central Asian Soviet Republic. When they appear and speak, they are the sources of the information they provide (it would take an especially astute reader to continuously tag Kisch, the narrative editor, as the ultimate source of these representations). Kisch also quotes passages from _The New Republic_ that are quite critical of social and economic prejudice in the United States, thereby pitting an American media voice against the excesses of capitalism. In addition to contributing a sense of credibility to the work (i.e., multiple voices confirming Kisch's observations), these additional voices also stress the reader's source-tracking ability. With so many new minds on which to exercise mind reading skills, Kisch can fade as the ultimate source of the representations before the reader. Simultaneously, Kisch's selective presentation of Central Asian speakers has a high probability of distorting the propagandee's estimation of the popularity of Soviet activities in the Central Asian Republics. Weaver, Garcia, Schwarz, and Miller found in a recent study that the assertions of just one member of a group, if repeated often enough, were likely to be considered representative of the shared opinion of the entire group.
Another means by which Kisch achieves a high source-tracking load is to require the reader to monitor multiple levels of intentionality. In the chapter "Ein Bezirk am Pamir" ("A District on the Pamir"), Kisch recounts the story of visiting a few Tajik comrades in Garm. Kisch admits to the reader that his weight and the excessive heat of the region conspire to make the Tajik custom of squatting on the ground and drinking tea particularly uncomfortable (although Kisch specifically describes how uncomfortable the experience is for "us"). One long-winded Tajik comrade regales Kisch with the endless improvements the Soviets have made in the province of Pamir. Occasionally Kisch attempts to stand, which his comrade interprets as impatience to see these improvements. His comrade quickly insists that Kisch sit and enjoy his tea—as there will be plenty of time for exploring. One can easily imagine Kisch's urgency as this routine repeats itself several times.
The joke is dependent on the reader's understanding that Kisch knows that his comrade thinks that he is eager to venture off and verify what he is being told, and this motivates the comrade to insist that Kisch should enjoy his hospitality a while longer. Of course, the comrade's belief is false and Kisch is only behaving in this manner because he does not want his comrade to think him rude. This routine is sustained over several pages ( _Gesammelte Werke_ 4: 323-28; _Changing Asia_ 169-78). Humor of this type can stress the ToM mechanism with multiple levels of intentionality, thus distracting the propagandee from the ultimate source of the metarepresentations—the propagandist.
Most of the techniques that have been discussed here are not new to the study of propaganda from communications and social science perspectives. The concept of mind reading, however, offers an additional perspective on these techniques. Paul Kecskemeti has observed that in order "to be persuasive, the propaganda theme has to be perceived as coming from within" (849), and others have also noted that successful propaganda must seem as if it were merely reflecting beliefs and attitudes already present in the propagandee. ToM may explain how this is possible. The anchors used by propagandists to achieve resonance are, ideally, semantic memories (representations that lack a source tag and form part of a person's general, accepted knowledge about the world). Zunshine's literary application of Cosmides and Tooby's work on representational quarantine is especially relevant here (50-53; 54-58). She explains Katerina Ivanova Marmeladova's (from Fedor Dostoyevski's _Crime and Punishment_ ) striking confusion of fantasy and reality as a "failure to monitor properly the source of her representations," that is, herself (57). It is the task of the propagandist to induce a very similar cognitive failure in the propagandee. By using techniques that limit the propagandee's ability to track the propagandist as the ultimate source of propagandized representations, these representations have a greater chance of being stored without source tags. Lacking source tags and resembling semantic memories, these representations could then evade systems of representational quarantine and migrate more freely through the propagandee's cognitive architecture. Reinforced and made more accessible through repetition, such representations, lacking external source tags, would seem to come from within. Propaganda that succeeds in this manner would not appear as propaganda at all; rather it would appear merely as a mirror held up to reflect and affirm the preexisting beliefs, attitudes, and affects of the propagandee.
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## Functional Brain Imaging and the Problem of Other Minds
DAN LLOYD, VINCE CALHOUN, GODFREY PEARLSON, AND ROBERT ASTUR
Once upon a time a rainbow appeared to two friends, Bert and Ernie. Knowing all about rainbows, Bert and Ernie decided to set out for the pot of gold they knew to be at the rainbow's end. Bert clearly saw the bow fall between two hills, and proposed a path between them. Ernie just as clearly saw the rainbow intersect the hilltop to the right, and proposed to climb straight to the summit. An argument and wrestling match ensued, during which the rainbow disappeared, leaving the friends to ponder a deep mystery of the human condition. Or perhaps a not-so-deep mystery about optics, since their disagreement about the location of the bow was a consequence of their different viewpoints. Rainbows form as rays of light strike water droplets, each a tiny prism, refracting and reflecting the shades of the spectrum at fixed angles from about 40 degrees (violet) to 42 degrees (red). Every droplet reflects and refracts at exactly these angles and wavelengths; to catch a colored ray from any drop you need to be in its path. Accordingly, any one rainbow is really many—there is a rainbow seen from your standpoint, but myriad different rainbows to be seen from other spatial viewpoints. Bert and Ernie were in the unusual position of perceiving two contradictory visual worlds, but nonetheless perceiving these worlds accurately, as they were in fact. There really are rainbows. Indeed, the sky is filled with rainbows just waiting to be seen. We resolve the seeming inconsistency among rainbows by recognizing that their location is essentially perspectival.
Without an understanding of optics, Bert and Ernie might have drawn a different moral from their two rainbows. An object cannot be in two places at the same time. Their observations of the rainbow contradict this principle, so the rainbow must not be a real object. Its appearance, accordingly, must be entirely subjective, rooted in something about Bert and Ernie that is not a part of physical reality.1 Bert (or Ernie) might call this "the problem of other rainbows." "I can see a rainbow right there," declares Bert, "but I can't see the rainbow you claim to see. Of course I believe you are sincere in your rainbow claims—it's just that I cannot confirm them in anything like the way I confirm my own—directly, incorrigibly, and privately. I must infer your rainbow percept indirectly, via a complicated and dubious analogy. In fact you might not be experiencing anything at all like the rainbow I see." Bert might therefore call rainbows a type of mental event. Moreover, if either Bert or Ernie happened to be a student of philosophy, he might appeal to several current arguments that claim to show that the phenomenal, subjective appearances of things are beyond the reach of materialist, objective science. For example, Thomas Nagel writes:
> It is impossible to exclude the phenomenological features of experience from a reduction in the same way that one excludes the phenomenal features of an ordinary substance from a physical or chemical reduction of it—namely, by explaining them as effects on the minds of human observers. If physicalism is to be defended, the phenomenological features must themselves be given a physical account. But when we examine their subjective character it seems that such a result is impossible. The reason is that every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view. (437)
But with respect to rainbows these metaphysical gyrations turn out to be completely unnecessary. Rainbows are perfectly well-behaved natural phenomena. After a little optics, any special difficulty with rainbows simply evaporates, so completely do they reduce to their sole physical ingredients, light and water. As the subjectivity of rainbows collapses into the objectivity of optics, neither Bert nor Ernie need abandon his point of view, however. Their perceptual claims remain true and are in fact vindicated. In this case objectivity and point of view coexist.
Ever since Descartes, modern philosophy has grappled with the problem of other minds, much like the problem of other rainbows as confronted in Bert and Ernie's optical innocence. The experienced world is thoroughly perspectival, but the machinery that leads to alternative appearances has been mostly unknown. The classical problem of other minds confronted a double barrier: the metaphysical distinction between mind and brain mirrored a physiological accident, namely, that brains hid in skulls. The obvious correlation between mental and neurological conditions could be glimpsed only in snatches and indirectly. Lately, that has changed. Neuroscience has married the clever experimental strategies of cognitive psychology to some amazing observational technology, with a resulting explosion in research correlating brain and behavior. The brain, with its hemispheres, lobes, ridges, and folds is perhaps _the_ scientific icon of our generation, easily overshadowing the mighty atom of an earlier era and in a tight race with the double helix. The gray matter is rarely merely gray; upon it, like a screen, are projected the transient fires of cognition, typically as hot orange blobs, the simmering symptoms of human perception and action, in both normal and abnormal brains. The neuroimages thus reflect both the anatomy and function of the brain, in studies of gray and orange.
The impression that there is a great deal of research in this area is entirely true. Although functional magnetic resonance imaging is just one of many tools for brain exploration, a search of the PubMed database on the joint topic of "fMRI and brain" yields more than 68,000 articles, of which over 6,000 were published in the last year. The heralded "golden age of brain science" would seem to have arrived. Bear in mind that this is indeed science. The majority of these papers report on controlled experiments with multiple subjects. Their conclusions, accordingly, are generalizations that have a better than 95% chance of being true of the general population. For the most part, all these studies are true of you and me. This one consideration alone almost entirely defangs the traditional problem of other minds. In the parable of the rainbow, during their initial state of ignorance Bert and Ernie had no basis for comparing their experiences. Bert's was simply given, and from his point of view Ernie's was simply wrong, and insofar as it was wrong had no point of comparison with Bert's own. Without a circumspect view of their situation, their perceptions were simply contradictory. But, in the case of perception overall and functional neuroimaging, we do learn something important from the science, namely that there are identifiable commonalities between you and me in hundreds or even thousands of situations. That something is the "area of activity" represented in the characteristic brain images. (Ask not for whom the blob shines. . . .) It is easy to forget that this common ground among us all is a contingent fact, as is the fortunate interplay of neural function and blood flow, since it is the latter that functional neuroimaging usually detects.
However, in pursuit of this scientific common ground we can observe the very process that Nagel described. The "single point of view" of the individual has been "abandoned" for the sake of hypotheses about brains in general. Again the parable of the rainbow is instructive. Bert observed an unproblematic visual fact—there is a rain-bow—until his observation collided with Ernie's. The parable suggests that the problem of subjectivity is actually a problem of intersubjectivity. Or in other words, subjectivity is not encountered in the commonalities among subjects, but in their differences. In fact the friends need not enroll in a course in optics to determine that rainbows are real. All they need to do is trade places. Once they learn how the apparent location of the rainbow changes with one's standpoint, they can coordinate their differing perceptions. Prior to the reductive explanations of light and water, there appears to be a prereductive explication. We can refer to this prereductive inquiry as "standpoint mapping."2 In this example, standpoint mapping links spatial viewpoints with apparent positions of rainbows. Importantly, standpoint mapping describes how changes in standpoint lead to changes in what is perceived. Standpoints are related by describing how to exchange one point of view for another. That is, standpoint mapping describes the dynamics of points of view. Bert and Ernie can determine that rainbows are real (though insubstantial) by confirming that the rules of dynamic standpoint mapping are constant for each of them. Moreover, note that Bert and Ernie can establish that rainbows are both real and subjective without appeal to reductive explanation at all. The optics of water droplets need not be mentioned.
"Dynamic standpoint mapping" is a deliberately abstract characterization of an activity that pervades our lives. How will Bert and Ernie communicate the interrelated changes of standpoint and perception? Their most natural impulse will be to tell each other stories. In their simplest forms, their stories will narrate how moving oneself also moves the rainbow. Implicit in their first-person narratives will be a generalization to the second and third persons. That is, the audience for such a story is invited to see if the same story holds from her or his viewpoint as well. In short, one function of narrative is to exchange dynamic standpoint maps, and this in turn enables communities to resolve fundamental distinctions between the objective and subjective. Standpoint mapping—storytelling—is a general topic beyond this chapter, of course. Here two points are relevant. First, the myriad forms of narrative in the service of standpoint mapping remind us that the subjective is neither monolithic nor sharply demarcated from the "objective." Through stories individuals coordinate their worlds; some areas of agreement are solid, while others are fluid, subject to debate. Second, this intricate human activity precedes science. It sorts the universe into the personal and the interpersonal, the singular and the shared, and in so doing distinguishes what is to be explained from what is available to do the explaining. Over time, the landscape of knowledge and speculation shifts. As the mapping progresses, the personal and idiosyncratic elements are subsumed into a logical edifice of causal generalization, that is, science. But science arrives late to a world already sorted into objective facts and subjective experiences.
With standpoint mapping in mind we can identify a gap in contemporary neuroimaging. In the quest for common ground cognitive neuroscience reduces the subject to a single standpoint; the dynamic and transitive mapping from one viewpoint to another disappears in this process. There are many reasons for this: First, science itself requires uniformity. A relatively new science will search first for the common ground and later for the effects of individual change and differences. Second, the technology of neuroimaging is not exact. From image to image changes are usually too small to be reliably detected, and often effects only emerge when multiple subjects are averaged. Experiments depend on repeating conditions both by single subjects and across subjects. Accordingly, only large, stable, relatively static effects appear. Imaging is also relatively slow. Each 3-dimensional image takes a second or more to assemble, and because the signal is small, often many images are averaged together to cancel out the randomly varying noise and preserve the more consistent signal. This does not enable the detection of faster dynamic processes. Finally, to achieve uniformity in the face of noise, experiments in imaging generally describe brain activity in correlation with conditions in the task environment. (The "task environment" is the combination of stimuli and the task posed by the experimenters. In other words, it is the set of experimental conditions defined for the particular study.) For example, if experimenters are interested in face perception, they will present subjects with images of faces in alternation with some other stimulus or task (for example, a blank screen or a scrambled image of light and dark patches). For analysis, the face scans are grouped and averaged. Methodologically, this assumes that brains are clamped to the face perception task in the same way throughout the face presentations, and equally disengaged from face perception in the nonface blocks of the experiment.
In short, the path from data to analysis is one of merging and averaging; along this path individual standpoints and their dynamic variation disappear. These methodological blinders lead to skepticism about the prospects for imaging in application to the problem of subjectivity. But they are largely arbitrary conventions of the discipline, rather than a priori limits. These conventions are inevitable only if no other interpretive method yields significant results. Accordingly, one way forward is to try something different, and see what's there—alert to the possibility that the answer may be nothing.
If the standard approach is objective, then we might characterize the alternative as the subjective strategy for brain research. Exactly this will occupy the remainder of this chapter, a descent into the data of one neuroimaging study, which is ongoing at the Olin Neuropsychiatry Research Center in Hartford, Connecticut.3 We will examine this study in a nonstandard way, driven by a need to preserve, not blur, subjectivity. In this experiment, participants drive a simulated automobile through a virtual landscape, controlling their route with a steering wheel and brake and gas pedals. The experiment is divided into three runs, and each run has the same structure of three parts: First, subjects see a black screen with a plus sign, a "fixation cross," in the middle, for thirty seconds. Then the virtual reality simulation begins, and subjects drive through the simulated landscape, for ninety seconds. Then subjects move through the simulated landscape again, but it is as if they are passengers in the automobile, merely observing but not driving. This lasts for sixty seconds. This same sequence of Fixate, Drive, and Observe repeats three times without intermission, and is followed with a final thirty seconds of fixation. The whole session, then, comprises this sequence:
> Fixate—Drive—Observe—Fixate—Drive—Observe—Fixate—Drive— Observe—Fixate
With three runs, the experiment lasts about nine minutes. While the subjects are driving, a number of behavioral variables are recorded. This includes the current position of the car in the virtual landscape, steering, accelerating, braking, and others. Each session yields 375 whole-brain images, collected every 1.5 seconds.
Although no one would confuse the experience in the scanner with actual driving, it is nonetheless a compelling and immersing perceptual and behavioral task. Complex and open-ended, the task comes closer to simulating a natural interaction with the world than the host of button-pushing stimulus-response experiments designed to probe specific narrowly defined facets of perception or cognition. In addition, each session in the scanner comprises three very different subjective states, Fixation, Driving, and Observing, which interweave and repeat. In their phenomenology the three conditions are certainly diverse: one of them requires action, while two are passive. One presents a nearly blank screen, while two are visually rich and dynamic.
Facing this experiment, the standard neuroimaging approach would begin by assuming that the three main conditions will correlate with three distinct patterns of activity in the brain. Standardly, we would seek similarity among these patterns in all subjects, and from there localize brain activity, ultimately to explain the behaviors elicited by the experiment as the joint production of functionally specific brain regions. Here, however, we are interested in a less objective analysis, one revealing of subjective differences. We seek the interplay of standpoints. This suggests questions outside of (and prior to) the standard analysis. For example, in what respects do individuals differ in their brain responses during this experiment? In each subject, do the three conditions coordinate with three distinct patterns of activity in the brain? If so, then these correlations will link the brain activity to the objective conditions of the experiment, rather than to subjective variation.
The standard approach seeks to confirm the uniformity of brain activity in conformity to environmental conditions. In pursuit of this goal, standard interpretive methods are blind to deviations from the working assumptions.4 A different approach would begin with individual subjects and apply interpretations subject-by-subject. Moreover, a nonstandard approach would not assume that the three experimental conditions map perfectly onto three distinct brain patterns. Just as your rainbow and mine are different, the sufficient mark of subjectivity is difference in perception despite similarity of objective (environmental) conditions.
Considering subjects one by one is an easy emendation. But how can we probe for different subjective responses to the same stimuli? Some experiments in cognitive neuroscience use subject reports to flag perceptual shifts, but in this analysis we use a more direct approach of letting the brain "speak for itself." That is, we analyze each image independently of any assumption about its objective correlations with the task environment. To make sense of the series of images as a whole, they are grouped by similarity. Similar images reflect similar brain states realized at different time points in the experiment. On this basis, then, we can carve the experiment at its joints, as signaled by the brain, independently of the switches among driving, observing, and staring at a fixation cross.
In more detail, the alternative method works as follows. Like other digital images, functional brain images are ultimately number strings (vectors) and as such one can measure similarity and dissimilarity between any pair of images. Think of dissimilarity as an abstract "distance" between images. So, with any set of images, some will be relatively similar/near each other, and some will be dissimilar/far apart. The image series accordingly can be retooled as an abstract space of brain activity, with unknown constellations of brain patterns scattered about. The similarities that structure brain space take into account entire images, that is, global patterns of activity that involve the whole brain. Perhaps the distances derive from localized differences in the images, but this analysis does not assume localization. Instead, the measure of distance encompasses whole-brain patterns that may be partly focalized, or not. Importantly, relations of near and far in "brain space" are not necessarily related to the temporal order of images, nor are they necessarily aligned with objective environmental conditions in the experiment.
Brain space is not conveniently depicted in two or three dimensions. (Mathematically, an image series traverses as many dimensions as there are pixels in the images.) Instead, we can subdivide brain space into neighborhoods, and assign each image to its region by virtue of similarity with its neighbors. There are several methods for "cluster analysis," each seeking to maximize the similarity within neighborhoods while also maximizing the dissimilarities of separate neighborhoods from one another.5 We used one common method to perform this analysis with several subjects in the driving study described above. To get oriented to the method, we begin with a division of the 375 images for one subject (who happens to be the first author of this chapter) into two broad clusters (Fig. 1).
**_Fig. 1._** _Two-cluster division of 375 brain images for one subject in the driving study. Each plus sign is one image (i.e., a representation of brain activity at one point in time), and separate lines denote separate clusters in "brain activation space." (When a series of consecutive images fall in the same cluster, the plusses overlap in cross-hatching.) The Fixate, Drive, and Observe conditions are abbreviated, and elapsed time (in 1.5 second intervals, the time taken to collect each image) indicated. The second cluster appears to partially match the Drive condition._
The two clusters appear on separate lines, with each image (at each time point) symbolized with a small plus. (Plusses overlap when consecutive images fall in the same cluster.) Here we have divided the entire set of images (for one subject) into two groups or "clusters." Within each cluster, images are more similar to each other than to the images collected in the other cluster. In other words, the analysis assigns each image to a type, which we can think of as a global brain state or pattern of activity. In this case, then, there will be two distinct patterns or global brain states, and our subject oscillated between them over the course of the experiment. The analysis shows when the subject brain is in each of the two states. The first cluster comprises images collected at the time points marked on the first line, and the second image comprises the remaining time points. (The numbering of the clusters is arbitrary. The same analysis might be represented with lines 1 and 2 interchanged.) Reading the figure from left to right, we observe that this subject was in one broadly defined brain state for approximately the first forty scans (about sixty seconds), as shown in the dense cross-hatching of plusses on the first line. Then the brain activity changed to better resemble a second, different overall pattern, as shown by plusses along the second line. These two coarsely distinguished brain states alternate through the experiment. When the brain images are collected in two clusters, one cluster matches the third Drive epoch well, along with segments of the two preceding Drive periods.
A critical question, then, is whether a three-cluster interpretation maps well onto the three experimental conditions. If so, then the standard assumption that the brain is clamped to the objective task environment would be confirmed. A three-cluster analysis is depicted in Fig. 2.
**_Fig. 2._** _Division of images for one subject into three clusters. Although the image neighborhood boundaries often correspond to changes in task condition, only the Fixation condition is consistently represented by a single brain image type._
For this subject, only the Fixation condition has a single neighborhood to call home. The Drive and Observe conditions do not sort images to conform to experimental expectations. In other words, if we divide all the patterns of global brain activity in this subject's session into three types, we find that the types cut across the conditions of the experiment. The Drive condition is represented by two different patterns, as are the Observe conditions. Moreover, single patterns often encompass differing experimental conditions; there is no overall scheme of "meaning" to assign to the three global brain conditions. Nonetheless, the clustering is not entirely random. In general, the global brain activity remains similar for periods of time, and is most likely to switch to a new state at a transition in the experiment. The Fixation condition seems generally represented by a single type of brain image. But even these generalizations have exceptions. In this case, then, the brain has a mind of its own—experimental conditions in the task environment do not entirely determine global patterns of brain activity. This motivates some further subdivision to see in greater detail how the experimental sequence is represented in the brain. Fig. 3 divides one subject's brain space into eight compartments.
**_Fig. 3._** _Brain space (for one subject) divided into eight regions._
Five clusters—five global brain activity patterns—are devoted to the Drive conditions. One cluster spans the third Drive condition; the other Drive epochs are spread among clusters. At this level of analysis, several sporadic clusters appear (6 and 8, for example). These represent distinct patterns of brain activity that are realized briefly and intermittently. In short, for this subject major transitions in brain state only partially reflect differences in the task environment. Something "nonobjective" characterizes this brain.
Before continuing this interpretation, we examine the second issue of subjectivity, namely, individual differences. Figure 4 compares ten subjects, each image series divided into three clusters. To enable comparisons among the subjects, we symbolize the three image clusters with black, gray, and white bands. Accordingly, each numbered line of Fig. 4 represents a single subject, rather than a single cluster (as in figures 1 through 3). Subject 1, for example, moves through brain states that nicely correspond to the task conditions of the experiment, Fixation (in black), Drive (gray), and Observe (white). However, the other subjects march to different drummers. Most of them divide the Drive condition into two clusters, and conflate the other two conditions in a single cluster. However, three subjects display a distinctly different categorization of brain scans, dividing the experiment temporally into beginning, middle, and end.
**_Fig. 4._** _Image series for ten subjects, sorted in three clusters each. Each subject is represented by a gray-scale horizontal band spanning the experimental conditions (375 scans). The three clusters are arbitrarily encoded as black, gray, or white. Subject 1 comes closest to matching the experimental conditions of Fixation (black), Driving (gray), and Observing (white). Subject 2, who also appears in figures 1 through 3, displays a single recurring image type during the Fixation condition, but the other clusters do not correspond consistently to the experimental conditions. For subjects 3 through 7, the Drive condition is divided into two clusters and the other two conditions conflated. Finally, subjects 8 through 10 tend to divide the session by time._
So far, this interpretation has emphasized several departures from the generalized and generalizing assumptions of the standard approach. The standard approach would assume that the brain displays distinct patterns of activity specific to the three main conditions of the experiment. The subjects in Fig. 4 support this assumption at moments of transition from one condition to the next. Most of the time, the change from Fixation to Driving (for example) is reflected in a global shift of brain state. Similarly, in several subjects brain activity remains in the same neighborhood of global similarity during each condition. Each Fixation condition, for example, is characterized by a fairly continuous series of states within the same cluster. However, this interpretive strategy has also disclosed some interesting departures from the expectations built into the standard interpretation. During some experimental conditions, brains change from one global state to another. For some subjects, brain states endure across significant changes in the experimental conditions. Finally, in different ways in these image series we observe representations of the passage of time. Three subject brains almost entirely sort the time series into successive sections.
In this experiment, we might have expected that to drive is to drive is to drive— that is, as conditions repeat the brain should as well. Objectively, in the task environment, this is exactly what happens. But in the brains of volunteers for this experiment, something else is happening as well: brain states oscillate and interweave in patterns that are distinctive and ultimately unique to each subject. In short, when this complex experience is parsed according to each individual's trajectory in brain space, we observe an interplay of distinction, progression, and repetition. Following this interpretation, we return to Nagel's claim that "every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view." Our approach here was entirely physiological. The observations collected in the figures are derived from physical processes. Commentary notwithstanding, the figures above project from brain to page via an "objective" process from cell metabolism to magnetic resonance to signal reconstruction to cluster analysis to ink on these pages. But the subjective has not disappeared in this progression. The analysis displayed features of brain activity distinct to different individuals, and all subjects displayed activity independent of the conspicuous markers of task and condition. One could develop increasingly fine distinctions among individuals, enlarging both the space of brain activity but also demarcating finer distinctions in behavior. As this research program proceeds, one behavioral domain will be of special interest, namely, language. Our subjects can answer questions about their experiences. Indeed, they can tell their stories. These too can be unwoven in brain space.
Will the subjective disappear in a research program such as this? Yes, if the subjective is understood as essentially private, incommunicable, and ineffable. Nor does this research strategy achieve objectivity, if the objective identifies strict causal relationships between conditions in task environments and brain activity, where these one-to-one relations are conceived as common to all. But, as the parable of the rainbow suggests, the subjective/objective distinction is a negotiation between subjects, a continual pursuit of commonality and difference. It overcomes solipsism but precedes science. In the middle ground is standpoint mapping, employing new interpretive methods, of which this paper is just one illustration.6 The figures in this chapter are a new kind of text, which seem to share with sentences some regularities of structure and semantics, and with stories a temporal progress and beat. The patterns of activity in the brain are not a simple mirror of the world, but rather a subjective interpretation that, like the rainbow, refracts objective conditions into a spread of differing subjective responses. The temporal sequence of images encodes temporally ordered change and regularity in the environment.
In this middle ground of science and subjectivity, we cannot say exactly "what it is like" to be any of these experimental subjects in isolation. But we can say that what it is like changes at specific moments in the experiment, and we can further compare and contrast subjects as they move from one subjective condition to another. Their stand-points—their subjective take on the world—varies, and a subjective research strategy begins to map this dynamism. In this way, the negotiation between commonality and difference is displayed through the objective measures of functional neuroimaging, achieving an objective view of subjectivity. Other minds are ultimately other standpoints, and the many mappings among them are open even to brain scanning as a means of exploration. These neuroimages therefore invite us to apply the literary tools of poetics and especially narratology. In this form of brain reading, the task is not so much to translate the "language" of the brain as a fixed system of symbols and meanings. Even this preliminary investigation reveals broad variation in individuals' parsing of environmental conditions rather than a fixed semantics of brain activation patterns. Rather, we observe patterns of change, some driven by environmental changes but some driven by unknown factors in the brain itself. Without a lexicon for this obscure text, we can first look for clues in more formal and structural techniques of literary analysis, considering a brain image series as an extended text comprising various and shifting representations of the world and its changes. To declare this text a narrative is premature but plausible, considering the pervasive role of stories in every culture and medium. Brains produce narratives with great fluency; possibly narrative describes the dynamic changes in the brain itself.
Classically, philosophers conceived the problem of other minds as a metaphysical gulf between oneself and every other person, represented in papers like Nagel's as an inscrutable and insurmountable difference in viewpoint. But there is no single problem of other minds, but rather as many "problems" as there are minds. The problem of other minds does not dissolve by mining some hidden interiority, but rather by displaying the differences in the experienced worlds of different individuals. Each of these problems is open to partial solution. In practice we have continual access to partial pictures of the world from other points of view, and much of the activity of human life is dedicated to coordinating points of view. Literature and art are particularly devoted to presenting and communicating individual perspectives, to rendering the subjective in various physical, objective media. By pursuing a more subjective research strategy, humans can also display their subjectivity in the new medium of neuroimaging. The figures in this chapter are example texts in this new medium. They remain highly abstract and largely untranslated, but we hope they begin to break loose from the picture of the brain as clamped to its environment in a fixed relationship that does not vary from person to person. There are many rainbows; each is an objective fact that is nonetheless different for every person. Likewise there are many worlds, each objectively indexed to each individual. Science, with a little retooling, can join with literary studies in showing and exploring the richness and diversity of the human mind.
### NOTES
1 Bert and Ernie are not alone in reasoning from optical ignorance toward mystical hypotheses. See Epstein and Greenberg, "Decomposing Newton's Rainbow." A related spectral display also provoked transcendental speculations: one's shadow, cast on a cloud, will appear to be ringed with a rainbow with its exact center at one's point of view, a phenomenon known as a "glory," "Bhuddha's light," or "Brocken spectre." Wanderers at dawn on high ridges have wondered at this visual treat, and attributed it to their state of mind in concert with some other dimension of the natural world. See C. J. Wright, "The 'Spectre' of Science."
2 "Standpoint mapping" is a new coinage, but reflects some of the ideas of standpoint epistemology developed in recent feminist philosophy of science. Standpoint epistemology pays particular attention to the processes of scientific research in their institutional, social, and political contexts, noting that one's societal position (broadly, one's position in hierarchies of power and privilege) affects the questions one deems worthy of exploration, and the hypotheses and even data that one will find legitimate. "Standpoint" in standpoint epistemologies refers to social position. In this chapter, standpoint is used in an individual and phenomenological way to refer to the many ways in which experiences can differ, depending on concrete circumstances (like Bert and Ernie's physical location), differences in learned and innate cognitive capacities and "styles," and many other contingencies. Social position is just one among many aspects of one's standpoint in this sense. On standpoint epistemology, see the papers collected in Sandra Harding (ed.) _The Feminist Standpoint Theory Reader_.
3 Publications from this study include: K. N. Carvalho, G. D. Pearlson, R. S. Astur, and V. D. Calhoun, "Simulated Driving and Brain Imaging"; V. D. Calhoun, K. Carvalho, R. Astur, and G. D. Pearlson, "Using Virtual Reality to Study Alcohol Intoxication Effects on the Neural Correlates of Simulated Driving"; V. D. Calhoun, J. J. Pekar, and G. D. Pearlson, "Alcohol Intoxication Effects on Simulated Driving: Exploring Alcohol-Dose Effects on Brain Activation Using Functional MRI."
4 For discussion, see Dan Lloyd, _Radiant Cool: A Novel Theory of Consciousness_.
5 For an introduction, see Mark S. Aldenderfer and Roger K. Blashfield's _Cluster Analysis_.
6 Functional magnetic resonance imaging (fMRI) is just one of many technologies that could be harnessed in the exploration of subjectivity. Individual brains differ in structure, in connectivity among brain regions, in electromagnetic properties measured in milliseconds, and in many other ways; an expanding toolbox of technologies is available for cognitive neuroscience. All of these technologies invite diverse interpretive methods. Moreover, recent research has employed multiple methods simultaneously. For an example and discussion, see V. D. Calhoun, T. Adalid, G. Pearlson, and K. A. Kiehl, "Neuronal chronometry of target detection."
### WORKS CITED
Aldenderfer, Mark S., and Roger K. Blashfield. _Cluster Analysis_. Quantitative Applications in the Social Sciences 44. Beverly Hills, CA: Sage, 1985. Print.
Calhoun, Vince D., T. Adalid, G. Pearlson, and K.A. Kiehl. "Neuronal Chronometry of Target Detection: Fusion of Hemodynamic and Event-Related Potential Data." _NeuroImage_ 30.2 (2006): 544-53. Print.
Calhoun, Vince D., K. Carvalho, R. Astur, and G. D. Pearlson. "Using Virtual Reality to Study Alcohol Intoxication Effects on the Neural Correlates of Simulated Driving." _Applied Psychophysiol Biofeedback_ 30.3 (2005): 285-306. Print.
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Carvalho, K. N., G. D. Pearlson, R. S. Astur, and V. D. Calhoun. "Simulated Driving and Brain Imaging: Combining Behavior, Brain Activity, and Virtual Reality," _CNS Spectrum_ 11.1 (2006): 52-62. Print.
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## How Is It Possible To Have Empathy? Four Models
FRITZ BREITHAUPT
This essay asks what empathy is and what kind of assumptions we have to make to explain how empathy is possible. Four basic models of empathy will discussed. Three of these derive from recent work in cognitive science, psychology, evolutionary biology, and philosophy. The fourth model, based on narratology, will be first proposed in this paper. As will become clear, the different models do not fully contradict each other. Some authors (Thompson 2001, de Waal 2006) have suggested viewing different models of empathy as progressive steps in individual development or in evolutionary processes. Nevertheless, the difference between the models shows a core disagreement of what empathy is and which (mental, evolutionary, social, cognitive) requirements need to be met for empathy to be possible. Hence, this essay seeks to clarify the difference between these models. The models are not arranged according to disciplines or schools or thoughts, but rather according to more basic operative assumptions or prerequisites, such as similarity as the basis of empathy. All models fall under the most general definition of empathy as the capacity of an observer to receive access to the emotional state or intellectual awareness of another being or fictional construct. This definition covers phenomena that range from emotional contagion and passive associations of one's body with the body of someone else to mind reading and the imaginary transposition of oneself into the "shoes" of another individual or fictional construct.
A key question of this paper is whether empathy can be sufficiently described as a process between two people alone or whether empathy, at least complex human-primate empathy, requires a social setting with at least three agents. Most (but not all) work in cognitive science takes as its starting point for investigations of empathy settings of two people or animals. This is especially obvious in work that focuses on the so-called mirror neurons, where only two people or animals are needed, but also in work on mind reading and many forms of intersubjectivity. The fourth model proposes that there need to be at least three agents for empathy to take place. Empathy shifts from the interaction between two individuals (observer and observed) to an entire narrative scene.
Hence, the three basic models that will first be discussed start with a setting of two individuals (including the possibility of a fictional construct). Each model provides a different answer to the basic requirement for empathy, and hence each model changes what exactly we mean by empathy. These models derive from similarity as the basis of empathy (model 1), the expectation of some exchange or reciprocity of sympathy (model 2), or the intellectual construction of the other in my mind (model 3). The fourth model requires a temporal-narrative setting. The condition for empathy, the model suggests, is that an observer finds himself or herself reflected in a (complex) scene of at least two (other) agents.
Whereas the three models that will be discussed first are bottom-up models that start with the simplest elements of empathy, the fourth model holds that human empathy has to be seen as a complex phenomenon to be understood in its totality. Certainly, all-or-nothing theories are suspicious. Nevertheless, there are cases where the parts do not add up to the sum. This is the case with empathy, the fourth model suggests, as it is with language. Language is either there or not there and cannot be derived in a step-by-step approach (Chomsky). This does not mean that one could not break down language or the complex form of empathy into its components. It is also still likely that there was no evolutionary leap, but that there was a transition in which the components of empathy and language came about consecutively or by chance. However, once language or empathy are in existence, they have a transformatory function with effects that cannot be explained by their parts or the steps of their development. Hence, to understand them it would not be sufficent to simplify its basic model below the interaction of at least three people (or primates). At the same time, certain aspects of the other three models play a role in model 4.
This essay operates speculatively. Further research will have to be conducted to verify its claims, especially the ones concerning model 4 (for further evidence, see my forthcoming study, Breithaupt 2009). In regard to models 1-3, the essay will synthesize existing theories and models of empathy, mind reading, and related phenomena, such as pity and sympathy. Obviously, these sketches will have to be very brief and at times overly polemical to fit this short format.
This essay is also a bit unusual in the way it suggests treating literary fiction. There would probably not be narrative fiction without empathy, that is, a "slipping into the shoes" of the characters of the narration. Narrative fiction can only exist because it invites, triggers, channels, controls, and manages empathy. Fiction probably exists in response to the capacity for narrative or situational empathy. Hence, one can assume that narrative fiction contains a lot of knowledge about empathy. However, there is a complication. Works of fiction do not only provide access to fictitious situations for the reader / viewer, but also themselves "theorize" how such access could happen. Literature is a highly self-reflexive medium that produces "allegories of empathy" by duplicating the external reader-text or spectator-theater relation as an internal arrangement within its plot. Put differently, works of fiction produce an imaginary or implied reader. To say the same thing more technically: a work of narrative fiction is a second-order observation of its implied reader, who is posited as its empathizing first-order observer.
Hence, to access the knowledge fiction has about empathy, one needs to both understand how the narrative plot triggers empathy and how the work speculates about empathy. Both are deeply intertwined—which also suggests that the narrative empathy triggers can change historically and culturally. Narrative empathy is a developing capacity with its own history. It is therefore not evident how one can access the wealth of knowledge about empathy that fiction promises to contain. The contributions to this volume may pioneer some new approaches (see also Breithaupt 2008b).
### 1. THE MODEL OF SIMILARITY
In this model, the assumption is that we can have empathy because we are, or imagine we are, similar to other beings. We arrive at an understanding of the other individual because we derive it from ourselves, put in different circumstances. More recently, Vittorio Gallese, co-author with Giacomo Rizzolatti of many studies on mirror neurons, calls this assumption of similarity the "shared manifold hypothesis": human beings are, ultimately, deeply similar and thus can understand each other without reflection and rational thought (Gallese 2001). Enlightenment thinkers favored terms such as "universal humanity" for this deep similarity or commonality (for instance, Gottfried Lessing, Adam Smith, and, critically, Jean-Jacques Rousseau).
Assumptions of similarity have a large appeal among cognitive scientists; they allow us, among other things, to involve the idea of embodied cognition to explain how one individual seems to share, co-experience (Husserl), or understand the emotional state of another. The (imaginary) body is what we all share and hence this "body" can serve as a medium between us (Merleau-Ponty 1964). Despite its appeal, however, this model has two weaknesses. First, by postulating a deep similarity, a similarity that itself cannot be directly experienced, the model needs to slip into the register of the imaginary or projection. Second, the model is weakened by the fact that it has difficulties explaining the attraction of empathy: if we are all the same, there is no need to feel like the other. Put differently, if similarity is the core of empathy, empathy may be stuck on the level of mere emotional contagion, as Frans de Waal puts it, and would not lead to differentiations between self and other (de Waal 2006).
There is a lot of support for the assumption that similarity is the basis for empathy (and Theory of Mind, pity, and even identification). If one accepts the mirror neurons or a similar operator as the key basis for empathy that operate prior to reflection and rational motives (Gallese 2003), then one has to conclude that the readability of the emotional states of other individuals requires the immediate compatibility from one to another. It has to be noted that it is disputed which role if any the mirror neurons play in true empathy that involves emotions; thus far, we know that the mirror neurons lead observers of goal-directed movement to display similar brain patterns as if they would execute the same movement. Emotional empathy, however, is a contested area and the involvement of mirror neurons is not yet clear. Even if one employs a different notion of empathy that goes beyond the pre-reflexive mirror neurons as theorists of Theory of Mind (ToM) do, it seems impossible to object to the idea of similarity. Our imaginarily shared feeling, experiencing or understanding the emotions of another, seems to suggest that these feelings indeed can be shared and thus are the same for all of us, which means also that we are, mostly, the same. Similarity does not necessarily have to be on the level of bodily cognition. For example, the theory-theory of ToM holds that we engage in some shared treasure of cultural knowledge or folk mythology when we empathize.
Since the case for this model seems overwhelmingly strong, at least at first glance, I will skip over any further supporting evidence in this short sketch and instead turn quickly to what I perceive to be a central weakness within this model. We will come to this weakness by means of what may seem like a detour: through a reading of one of the most prominent aesthetic theories of the eighteenth century, developed by the key German Enlightenment thinker, Gottfried Lessing. The implications of our reading of Lessing, I believe, go far beyond the Enlightenment and aesthetics and are applicable for debates today.
In his _Hamburg Dramaturgy_ (1769), Lessing writes:
> He [Artistotle] speaks of pity and fear . . . and his fear is by no means the fear excited in us by misfortune threatening another person. It is the fear which arises for ourselves from the similarity of our position with that of the sufferer; it is the fear that the calamities impending over the sufferers might also befall ourselves; it is the fear that we ourselves might thus become objects of pity. In a word this fear is compassion referred back to ourselves. (Lessing 179)
For pity ( _Mitleid_ ) to occur there needs to be enough similarity between _alter_ and _ego_ , so that it is possible for _ego_ to put himself / herself in the position of _alter_. Only because _ego_ can do this, is it possible for _ego_ to feel the fear of _alter_ in a certain situation, since _ego_ sees the danger as if it were for himself / herself. Similarity ( _Ähnlichkeit_ ) is the condition of possibility for pity and empathy.
There is, however, another notion implicit in Lessing's primal scene of empathy. It is the notion of selfhood that emerges around 1770, that is, the idea that we are all different and unique (Wahrman 2004, Breithaupt 2008a). Lessing assumes that pity is the exception. We can access our human similarity only if we are put under a lot of stress, that is, when we feel pain or terror. We can feel like others and "with" others as in _Mit-leid_ only once our individual differences and our concepts of self are bypassed or short-circuited. There can be pity ( _Mitleid_ , literally with-suffering) as a ribbon of community, but no _Mitfreude_ (literally with-joy) since when one gains, only the jealousy of others will be aroused. The self prevents us from sharing the joy of others. Schadenfreude would be another of these disturbances.
There can be no union of selves for Lessing. Identification, empathy, pity are possible only in the complicated apparatus that we call fiction, an apparatus that operates like a nutcracker to crack the hard shell of selfhood in order to reveal the soft flesh that all beings share. There are two ways of reading this emergence of the self, that is, the emergence of a concept of selfhood as a defined entity marked by singularity, autonomy, and difference. One could think that selfhood is a _fundamental threat_ for empathy; successful fiction must break it down to accomplish the effect of similarity, thereby making empathy and reader engagement possible . However, I suggest viewing the emergence of "the self" in the mid-eighteenth century as a channeling or blocking device of empathy. The self of others and of oneself allows people _not_ to have empathy, not to succumb to the "compassion compulsion" (Edelman 2004), and thus allow the channeling of empathy. Narrative fiction, in this sense, would have the opportunity to mostly block empathy and thus heighten it for those special cases in which the common humanity would spark.
Hence, even if Lessing views similarity as a given underpinning of man, the perception of this similarity is not a given. The problem is that everyone thinks himself or herself to be different and as having an (individual) self. Lessing theorizes which conditions fiction needs to fulfill in order to bring about the _sensus communis_. Fiction has to bring about the similarity-effect. Fiction thus would have the task of creating those situations in which all feel the same: typically, scenarios of horror and of the overwhelming display of power. Tragedy, as it appears in Lessing's reading of Aristotle, is a site of _producing similarity,_ that is, the perception of similarity, between man by stripping man of that which holds man apart, namely selfhood. Selfhood operates as the regulatory device that limits the excess of pity.
What this means for the model of similarity is that similarity, as much as it may be posed as an existing ground of unity, is always (or can also be suspected to be) a merely produced or imagined similarity, a constructed communality that has heuristic value only because it cannot be disproved. Certainly, just as Lessing indirectly admits that we do not experience ourselves as similar but rather need to employ certain situations in drama that may lead us to forget our differences, so do current cognitive researchers readily accept and employ the register of simulation. As Gallese puts it: "Embodied simulation enables the constitution of a shared and common background of implicit certitudes about ourselves and, simultaneously, about others" (Gallese 2003, 521).
However, imagined similarity is a double-edged sword. It is very close to mere projection, in which a being assumes that all others are simply like-minded even when they are not. Instead of laying bare a level of similarity, projection erases the difference of the other. Even the apparent pain of another may not really be "pain," since each individual has a different history of pain and experiences pain differently. If we replace the word "similarity" by "projection," this amounts to more than a war of words. It changes what empathy is. If empathy is a matter of projection instead of similarity, then its key resource is some form of personal memory (since we derive our knowledge, such as the knowledge of pain, from our own experiences).
In short, if one understands a deep similarity to be the core of empathy, empathy consists of the inducing of an emotional state from one to another. Empathy in this sense has to be defined as the "imaginary sharing" of an emotional state when the cause of the emotion is only directly experienced by one or some of the group, but nevertheless as if experienced by the others. Grieving with someone who has lost a relative could fall under this definition. This "sharing" of an emotional state could be explained as the result of the mirror neurons. Or it could be explained by some other trigger that might be active in emotional contagion in group behavior, when one individual induces a matching or similar emotional state in others (de Waal 2006).
However, once a form of consciousness has been reached that distinguishes conceptually between self and other, similarity is not a given platform between beings any longer. Before one could reach a level of similarity again, the difference between self and other has to be erased (Lessing [1769] 1962). Similarity thus would not be the core basis of empathy, but rather the effect of complex arrangements that sidestep other mental states.
### 2. MODELS OF RECIPROCITY
Another school of thought argues that ego is most likely to have empathy with someone who could also empathize with ego and whom ego wants to have empathy with ego. This implies that empathy is intended to be a two-way process in which the empathizer reaches out with the intention of triggering a return. Consequently, the catalyst for empathy lies in the action or status of the empathizee that makes him or her of particular interest to the empathizer's hopes for a return of empathy. Empathy thus would be more a social than a heuristic process, as evolutionary biologists argue (Humphrey 1976, Dunbar 1996). This also means that empathy is directly tied to community building since it involves some simple form of exchange communication (I understand you, therefore you ought to understand me). This model is a central assumption underlying the belief that there is a close link between empathy, sympathy, and love. It also suggests some form of an economy of social emotions (Frijda 2007). One of the consequences of this model is that there would not be much room for neutral empathy. The following sketch will put the power relations into the center of the argument.
In August 1973, a man, on leave from prison, walked into a bank where he took four hostages. He demanded that a friend of his should be brought in too, and the two of them barricaded themselves with the hostages for more than five days before the police overcame them. What has made this incident famous is what happened afterward. The first man, incarcerated, was contacted by many female admirers who had seen him in the television coverage of the holdup. He later married one of them. The second man, who was cleared from charges for this bank robbery, formed a lasting friendship with one of the female hostages. The place was Stockholm and the resulting bonds gave rise to the name Stockholm syndrome.
What is interesting here for us is how a hostage situation apparently induces an empathetic or sympathetic response by the hostage toward the hostage-taker that can outlast the hostage situation. In many cases, the hostages assume the perspective of the hostage taker to such a degree that they come to fear the police more than the hostage takers (as in the 1973 Stockholm case). But how does this work?
A key feature of Stockholm syndrome is that the hostage is more likely to relate to the hostage-taker if some sign of hope is given, often as little as passing on some food, which can be read as a clue that the hostage-taker "is not that bad" after all. Psychologists call this element the "small kindness perception" (Carver 2007). Apparently, it is not the radical power asymmetry and the confinement alone that induce Stockholm syndrome, but the hope for a return of sympathy, a strategy to form a bond with the empathizee. The finding of this hope-for-a-return is also supported by the fact that positive bonding (Stockholm syndrome) is extremely rare in cases of active torture, where the hope of "positivizing" the torturer are minimal (Regner 2000).
Stockholm syndrome leads us to consider a specific element in the model of reciprocity that may otherwise not be visible: The empathizer reacts to (what he or she perceives to be) a prior act of the powerful offender / empathizee (Regner 2000). The reaction of the empathizer is fueled by the expectation or hope that the empathizee may react to the empathy by having empathy with the hostage / empathizer. In other words, empathy could be both described as an economic endeavor (the hope for a return) or as an act of manipulation in which one hopes to modulate the initial aggressive action of a person into a sympathetic action.
In contrast to the similarity model of empathy, the reciprocity model involves a strategic element of calculation even if in a crude non-conscious form. The question is whether this investment and hope for a return could indeed serve as a non-accidental basis for empathy. Such a starting point would allow one to see empathy as a means of survival that has significant evolutionary consequences.
Indeed, evolutionary biologists often use models of reciprocity and power asymmetry to describe the development of human empathy. Robin Dunbar in particular has presented the theory that a key human advance, namely the use of language, derived from the need for social networking and that empathy (in the sense of ToM) goes hand in hand with enabling human beings to operate in larger groups than apes or monkeys (Dunbar 1992, 1996, 1998; Dunbar, Barrett, and Lycett 2007). To abbreviate Dunbar's arguments sharply: empathy becomes the basis for human communication and thus replaces the much more time-intensive grooming of the monkeys. Whoever invests in empathy does so in the hope of his or her integration into the group. The key means for this new form of empathy, which bypasses the time intensity of grooming, is language. The evolutionary purpose of language, Dunbar argues, is thus social bonding, and not the exchange of information (Dunbar 1996).
Dunbar considers a variety of forms of gossip, ranging from the social control of "free riders" to mere chitchat. I would like to elaborate on one point by Dunbar, namely on his suggestion that the participants in social conversation and gossip are very careful not to hurt the feelings of the person with whom they are talking. We know not to "rub in" our successes, especially if we know that the other lost. In conversations, the conversant tends to take the perspective and also viewpoint of the other, especially if the other has higher social status. The conversant invisibilizes his or her position and adapts the perspective and viewpoint of the other, at least in the things that are said to please the other, or at least to be considerate of his or her emotional state.
Here we find a scenario that displays a similarity to Stockholm syndrome, even if in a very mild form. In fact, it seems not far fetched to argue that there is the hope for reciprocity for one's careful considerations of the other in conversation; the other seems more likely to help us, to accept us, and to help our social status if we show signs of respect and understanding of him or her or, at least, if we do not hurt his or her feelings. After all, paying back or reciprocity, positively or negatively, is a deep-rooted behavior (Trivers 1971, de Waal and Brosnan 2006).
Empathy according to the reciprocity model requires as its core an element of self-subjugation to assimilate the viewpoint of the other. The empathizer may hope that this (temporary) self-suppression leads to some form of payback. There is a slight paradox at the core of this form of empathy: We suppress or "invisibilize" our self to adapt the viewpoint of the other in order to strengthen our self. Certainly, this paradox can be resolved if we add the notion of a temporal delay of the return, if there will be any. However, the paradox may explain why this form of empathy is such an evolutionary exception that only a few species have adapted this form of behavior, since there is no instant gratification to be expected.
If we accept this model of reciprocity as the master model of empathy, then the tricky issue becomes to explain cases of empathy in which the hope or even desire for return seems far fetched. The monkeys of the original mirror-neuron experiment by Rizzolatti and others are a case in point (in a break of an experiment, one macaque who had an adapter helmet put on his head was given a peanut by the experimenters. His brain reaction, the monitor showed, was as if he had moved his hand). Here the semi-automatic, pre-reflexive sympathetic brain reaction of the observer seems to exclude any calculative act of empathy / sympathy. Still, there would be ways to defend the reciprocity model even for these mirror-reaction cases. One can always hope for a future return if one reveals empathy. There may be much more to empathy than mirror neurons. (Incidentally or ironically, it should be noted that the monkeys in Rizzolatti's experience are also "hostages" to the experimenters.)
Another interesting challenge to the reciprocity model derives from fiction. If empathy is indeed a form of cognitive appreciation of the other that includes a hope or expectation of a return, this seems absurd for fiction. What chance do we stand to get empathy from, say, a superhero or a complex protagonist in a novel? Where can there be reciprocity in fiction?
For one, there is a lot of empathy and mind reading going on between the characters of the novel. Hence, if we identify with one of the characters, we may be in a position to experience a lot of exchanges of empathy between various characters. More important still is the interaction between a reader / spectator (a human being) and a work of fiction (an artifact). Readers indeed do get a return. By adopting the perspective of a fictional character and looking back at our own life, it looks different, usually simpler. The character (or narrator) provides us with answers to our situations. If one chooses to empathize and identify with, say, a superhero, this superhero certainly would laugh about the business that hampers us. The situation is not that different with more sophisticated characters. They offer us a language that we can adapt and use to frame our own lives. And this is the gift of the hostage-taker character: relief, distance, freedom, even answers. One should add that the odd relation between fiction and reader shares more than one key element with Stockholm syndrome. Fiction captures us, we are absorbed. Even the least active or apathetic character in a work of fiction holds us hostage to his or her actions, thoughts, emotions. We may simply have to empathize to endure the ordeal, so to speak.
One of the weaknesses of the reciprocity model of empathy is that it can do without empathy. If the desire for the empathy / sympathy of the other drives our action, we may, after all, not need to empathize ourselves. Our wish is merely to be in the position of the other. This, as we know from Freud's drama of the Oedipus complex, is identification, the wish to re-place ( _beseitigen_ ) the other. Another problem is that if empathy is made visible to the other, and thereby becomes something like a token of exchange, we need to ask how exactly empathy is communicated. The empathizer's reaction needs to be fed back to the empathizer in such a way to prove correct emotional understanding. However, this could be faked by some form of compassion that does not derive from an emotional understanding.
Finally, it should be mentioned that a relationship between hostage and hostage-taker may involve a third faculty: the police. It has often been observed, as in the original Stockholm holdup, that hostages adapt the perspectives of the hostage-takers to such a degree as to share the fear of the police. This implies that the hostages also see the hostage-taker as a victim or at least as the weaker side in relation to the third faculty of government officials. Hence, it is not completely implausible to consider that the hostage can view a certain likeness of his or her existence as a hostage compared to the relationship of the hostage-taker toward the police. The hostage might opt to take the side of the hostage-taker, since they both share a structural position of being the weaker side of a relation of power. The small kindness perception makes it possible to view the hostage-taker as a good guy who would act differently if there were no police. As we will see later, such an explanation would move Stockholm syndrome closer to the narrative empathy of model 4.
### 3. MODELS OF APPROXIMATION
Many approaches produced within the ToM school use some model of approximation: When we deal with other people, we construct or simulate their actions and emotions and make predictions based on some decision-making operators that we see at work in them ("My friend tends to make rush decisions"). The original ToM constructed others as representation in our mind (Premack and Woodruff 1978). Empathy is here seen as a medium that does not assimilate the other, but rather negotiates his or her being different by making a mental model of the other in our mind. In literary aesthetics, this form of approximation corresponds to the demand of the consistency of characters: only consistent characters allow for this predictability.
There are at least three forms that approximation can take in regard to empathy. We could call these forms the: a) _constructionist_ , b) _speculative_ , or c) _strategic_ approach. In the following I will focus on the constructionist approach and only briefly refer to the two other variations.
The constructionist approach to empathy holds that we somehow piece together the key mental features of another being using various sources, ranging from precise observation of body language to memories of past encounters with the same person. The goal of constructionist activities is to produce a master model of the other being in our mind that lets us understand the emotional states of another or predict how the person will react in a variety of situations. These predictions can be based on simple yes / no knowledge that can be tested in false-belief scenarios. The well-known Smarties task illustrates such a form of basic positive knowledge (Gopnik and Slaughter 1991). When a child is shown a Smarties box and asked what is in the box, she will most likely say: Smarties. If it is then revealed to the child that this particular box contains pencils instead of Smarties, she has superior knowledge in regard to the box. Now comes the test: She is asked what another person who enters the room will think is in the box. If the child understands that the other person does not know what she knows, she has constructed a simple ToM of the other. Otherwise she will project her knowledge on all others and say: "Pencils." Typically, children acquire the ability to have a ToM in between 4 and 4 ½ years, whereas monkeys and apes do not.
However, this construction of the other's knowledge can also go beyond simple forms of positive knowledge, as in the Smarties task, and include complex intuitions of the psychic makeup of someone else. Such complex knowledge typically leads us to expect certain forms or styles of behavior from people if we cannot explicate our expectation in the simple form of positive knowledge. In regard to the reading of fiction, Keith Oatley has suggested that we "simulate" characters in our mind, run them in our mind like a computer program (Oatley 1994).
In fact, a trend in narrative fictions of the past two centuries can support this claim. Within the German literary canon it has become commonplace since the classics (Goethe, Schiller, Moritz) that characters can best be understood in light of their traumatic past that has shaped them and continues to haunt them (for the English context, see Pfau 2005). There are many famous works of fiction that culminate in the one moment in which readers / spectators finally get to learn the one past primal scene that has shaped a character and explains why he / she has behaved in a certain way. The first detective story of modern times, E. T. A. Hoffmann's _Mademoiselle Scudery_ (1819), shows this: everything falls into place once we know that the serial killer is endlessly repeating a (prenatal) childhood trauma. Not only does this revelation immediately solve a mysterious crime, it also allows the reader to create a ToM of the serial killer by taking his prenatal experiences into account (regardless of how believable we find it).
Whereas the constructionist approach attempts to build up the other from within, the speculative approach comes from the outside and aims to build a bridge to a being assumed to be more or less incomprehensible. Instead of slipping into the shoes of the other, we are supposed to sketch the contour of the footprint in order to realize that there must be someone there. The approximation of the other can be seen as an infinite process. In fact, contemporary Continental philosophy utilizes Kant's analysis of the sublime as a paradoxical approximation of empathy for the other. Whereas it appears to be impossible to ever comprehend the other, thinkers such as Emanuel Levinas have postulated an ethical demand to engage nevertheless in the impossible hermeneutic process that aims for the other.
The strategic approach to approximation does not necessarily involve a full-fetched representation of the other in my mind, but rather identifies the areas where I and the other will collide or cooperate. It is obvious that there can be an evolutionary advantage if I understand that the other desires the same object that I desire. Once I have this insight about the other, I can adapt my behavior and develop strategies that may lead me to the object faster.1
It should be added that the model of approximation and the theory of similarity described in section one do not necessarily exclude each other. One could always argue that every element of approximation (reading facial expressions, making sense of past trauma, adopting a perspective of the other, etc.) is deep down based on similarity. We can read each of these elements because we implicitly assume that the other person experiences these events and feelings just as we would. Nevertheless, the basic assumption of the approximation model is that the other person is different from us, perhaps even unknowable to us, and can only be comprehended based on operations of intelligent guesswork.
Another potentially weakening aspect of this model is that it downplays the special status of empathy. If the constructionist model of approximation holds true, then empathy is little more than one of the many arenas of abstract thought. Man can construct, piece together everything in his mind, houses, machines, people. The emergence of abstract thought and the ability to detach ourselves from ourselves then would be the key evolutionary device we need to look for, not social development.
### 4. NARRATIVE EMPATHY
There is at least some evidence for each of the three basic models of empathy discussed above. As we have noted, the different models do not necessarily contradict each other. Empathy may just be an imprecise concept capturing a variety of phenomena.2 Still, there is also some conflict between the different notions. For example, the calculation of the reciprocity model of empathy apparently is not limited to a core similarity. And the approximation model of empathy seems to come into play only because a core similarity and immediate understanding seems not to be given.
In the following, I want to suggest a fourth model that combines aspects of the three models described above. This fourth model cannot claim the beauty of radical simplicity. Instead, its beauty, so to speak, lies in providing a complex matrix from which all actual cases of empathy would be simplifications. Its starting assumption is that empathy occurs in a process that is deeply embedded in a temporal-narrative order. Another feature of this model is that it takes the basic frame of all models of empathy, namely someone observing someone (be it a person, a fellow species member, or a fictional construct), as a form that is reduplicated within the process of empathy as an internal distinction that appears in the observed action.3
The latter point will become clear below. For now it is enough to suggest that empathy may not just require a person A whom we observe, but an entire theatrical scene that binds an observer (empathizer) to an action executed by a character B and received by character A.
The starting point of empathy, I suggest, is an interaction that involves an active agent (B) and a passive agent (A), that is, subject and object, or perpetrator and victim. The empathizer observes this scene with two characters and their opposite tendencies (the one acting on the other). That is, already in its starting point, it is a social moment. And it is the core situation of fiction reading: the reader reads what B is doing to A. The observer (or reader) can relate to character / person A because A is in a specific setting (dominated by B) that is known to the observer. The bond between the observer and A is not necessarily based on some bodily or psychic deep similarity. Rather, there is a spontaneous commonality between the act of observing A and the suffering inflicted on A. The empathizer observes how A observes what B is doing with her or him.
This spontaneous commonality becomes more obvious when we look at what we mean when we say that we "observe" something. Observation is only partially an active process. In fact, it is possible only because of controlled inaction. We need to be quiet to observe, we do not usually intervene, and are waiting for the actions of others. Hence it would be more precise to say: "I am (passively) impressed by my (active) observations." That is, the observer shares, in a crude way, the setting with A: both are at the receiving end of the effects of actions initiated by B. Narrative empathy erects a bridge from the observer to the observed to the degree that the act of observation finds itself reflected, reduplicated in a social scene between two or more agents. Within the language of ToM we could say: empathy is possible when the observer registers what the other can see, feel, observe; the observer understands that the other is also an observer. From this it follows that one does not simply have empathy with A. It is more precise to say that one empathizes with the effect of B's action on A to the degree that it corresponds to the observer's position of observation. In other words, the (external) observer stops being a neutral bystander, but is drawn into or finds himself / herself involved in what he / she observes.
The question remains how this crude similarity or analogy between A's observation and the empathizer's observation of an observation triggers an emotional reaction by the empathizer. What allows the mere analogy of two observations to "skip" between people? Again, I can only offer a hypothesis. We have to assume that the observation of an analogy by the empathizer (which could be called a third-order observation, since it compares A's first-order observation of B to his second-order observation of A) creates the impression of identity. The observer projects identity of emotional reaction based on observed similar situations of passivity / victimization. Projection, in this sense, becomes a vehicle of cognition. If we follow this logic to its end, empathetic emotion would, thus, turn out to be the effect of affect or payback for our self-inflicted passivity of observation. Since we force ourselves to be quiet and observe, we experience the reward for our silence in the form of shared affections between our passivity and the observed objects (that we see as similar to us).
So far, I have suggested that the passive agent is the more likely recipient of empathy by an observer due to a basic passivity in observation. Our inability as observers to intervene, which is especially pronounced in media settings, may thus directly fuel our capacity to empathize. Perhaps this helps to explain why empathy favors victims and is not far from compassion or pity.4 Nevertheless, this structure also allows for empathy with agents (B) since, in the first place, (almost) every agent can be constructed as reacting to others' actions. Even the classical hero reacts to the prior actions of others and thus can be seen as the (passive) recipient of an action. Secondly, there is also an active element involved in the observation of the empathizer, and this active element can be reduplicated in the observed. The act of choosing what to observe is the active moment in my observing that I can experience as corresponding to the activity of B.
There are two prerequisites for this narrative empathy. Each could have been decisive in its evolutionary development. A word of caution has to be added first. Evolutionary narratives have a huge appeal, not only among scientists. Perhaps the success of Darwinism can partly be explained by the lure to invent stories that favor the fittest in various settings, such as social Darwinism. Certainly, we can expect that there is a true evolutionary story of empathy. However, it is easily possible to invent ten others that also seem to make sense. I mention this as a cautionary device, since I do not think I can justify offering such an evolutionary narrative for this fourth model of narrative empathy at this point. Nevertheless, if pressed, such a narrative of the origin of narrative empathy could involve the moment of taking sides.
Social beings often find themselves in the position of observing two (or more) other members of their clan engaged in a fight. This is obviously a case of the core situation of model 4. There are situations in which it is not obvious which side the bystander / observer should take. One strategy could be to side with the dominant individual who is likely to be the victor. The benefits for the observer are obvious. If the dominant individual wins the fight as expected, and the observer showed sign of support, this may help the observer. However, there is also a benefit in taking the side of the weaker individual involved in the fight. The more such seemingly unwise support is offered, the less attractive the victory is for the dominant individual. The payback for such sympathy with the loser could be a long-term weakening of social stratification. The less glorious the victory is, that is, the less sympathy it buys the winner from the group, the weaker the hierarchies would get. And weakened social hierarchy or, put positively, a more flexible social hierarchy adds a series of evolutionary advantages. What is important here for our context is that choosing sides aligns the bystander / observer with one of the fighting parties. Even though the observer is not himself / herself active, he or she experiences the fight from the perspective of one of the parties. Hence, it seems at least plausible that the ability to take sides by social beings has a key function in developing empathy.
There is at least one popular form of behavior today that cherishes the taking of sides: watching sports. Almost all sports that enjoy large spectatorships are competitive. And apparently, the interest of the spectators is significantly heightened if they choose a side (or have already chosen such a side long before). Sport spectators commonly report that "it is more fun" to follow the events of a match by aligning oneself with one or the other team. Hence, it seems reasonable that social beings such as humans have learned to choose sides even in arbitrary situations (such as sports). Hence, the fight that emerges within a clan will certainly affect all and they react to the commotion by choosing sides, thereby getting involved. Empathy could have been a further development of this capacity of choosing a side.
Another prerequisite for narrative empathy is the awareness of one's weakness. Model 4 assumes that one is more likely to empathize with the passive agent or victim since the observer shares the structural position of being more passive with the victim. Hence, one needs to assume that there is a certain core sensibility to one's weakness, an _Ur_ -experience or _Ur_ -feeling of being vulnerable.
To conclude: model 4 shifts empathy from a particular person (or fictional construct or species member) to an entire situation or scene. Such a scene or situation is composed of several individuals and perhaps could be described as the simultaneity of several perspectives, as in ToM of higher intensionality (as in: I suspect that you think that she wants to steal his cookie). Model 4 assumes that the core activity of empathy, namely observation, is not neutral to empathy, but rather provides the key structure that the empathizee finds reflected within the scene he or she observes. The external observation by the observer finds itself reduplicated within the observed scene as a basic distinction, such as activeness and passivity, expression and impression, or domination and subordination. Hence, the observed scene and the observer are linked by means of some spontaneous commonality that arises from the act of observation. Empathy is, first of all, a second-order observation in which the second-order observer is drawn back into the picture of first-order observation. Perhaps this, then, is my contribution to current discussions of empathy: to complicate the act of observation by the empathizer.
### NOTES
1 This view is held by many evolutionary biologists, as well as literary anthropologist René Girard, who argues that love derives from a competitive or mimetic desire, namely a situation of competition that will make objects desirable to a being only because he or she observes that his competitor wants the object (Girard 1966).
2 Evan Thompson, for example, distinguishes four levels of empathy, ranging from the passive association of bodies to ethical responsibility (Thompson 2001).
3 In system theories such a reduplication of an external distinction within the system is called "re-entry" (see Spencer-Brown 1969 and the works of Niklas Luhmann).
4 Martha Nussbaum disagrees with this assessment and argues that empathy is rather neutral whereas compassion is an active, ethical process (Nussbaum 2001).
### WORKS CITED
Bateson, Paul P., and Robert A. Hinde, eds. _Growing Points in Ethology._ Cambridge: Cambridge University Press, 1976. Print.
Berlant, Lauren, ed. _Compassion: The Culture and Politics of an Emotion_. New York and London: Routledge, 2004. Print.
Breithaupt, Fritz. _Kulturen der Empathie_. Frankfurt: Suhrkamp, 2009. Print.
\---. _Der Ich-Effekt des Geldes. Zur Geschichte einer Legitimationsfigur._ Frankfurt: Fischer Verlag, 2008. Print. (=2008a)
\---. "How I feel your Pain: Lessing's Mitleid, Goethe's Anagnorisis, Fontane's Quiet Perversion." _Deutsche Vierteljahrschrift_ (DVjs) (Sept. 2008): 400-23. Print. (=2008b)
Carver, Joseph M. "Love and Stockholm Syndrome: The Mystery of Loving an Abuser." 1 Oct. 2007. Web. http://counsellingresource.com/quizzes/stockholm/part-2.html.
Dunbar, Robin. "The Social Brain Hypothesis." _Evolutionary Anthropology_ 6 (1998): 178-90. Print.
\---. _Grooming, Gossip, and the Evolution of Language._ Cambridge: Harvard University Press, 1996. Print.
\---. "Neocortex size as a constraint on group size in primates." _Journal of Human Evolution_ 22 (1992): 469-93. Print.
Dunbar, Robert, Louise Barrett, and John Lycett. _Evolutionary Psychology_. Oxford: One-world, 2007. Print.
Edelman, Lee. "Compassion Compulsion." _Compassion: The Culture and Politics of an Emotion_. Ed. Lauren Berlant. New York and London: Routledge, 2004. 159-86. Print.
Frijda, Nico H. _The Laws of Emotion_. Mahwah and London: Lawrence Erlbaum, 2007. Print.
Gallese, Vittorio. "The Manifold Nature of Interpersonal Relations: The Quest for a Common Mechanism." _Philosophical Transactions of the Royal Society of London_ 358 (2003): 517-28. Print.
\---. "The Shared Manifold Hypothesis: From Mirror Neurons to Empathy." _Journal of Consciousness Studies_ 8. 5-7 (2001): 33-50. Print.
Girard, René. _Deceit, Desire and the Novel: Self and Other in Literary Structure_. Baltimore: Johns Hopkins University Press, 1966. Print.
Gopnik, Alison, and Virginia Slaughter. "Young Children's Understanding of Changes in their Mental States." _Child Development_ 62 (1991): 98-110. Print.
Humphrey, N. K. "The Social Function of Intellect." _Growing Points in Ethology_. Eds. Paul P. Bateson and Robert A. Hinde. Cambridge: Cambridge University Press, 1976. 303-17. Print.
Kappeler, Peter M., and Carel P. Schaik, eds. _Cooperation in Primates and Humans. Mechanisms and Evolution._ Berlin, Heidelberg: Springer, 2006. Print.
Lessing, Gottfried Ephraim. _Hamburg Dramaturgy_. Trans. Helen Zimmer. New York: Dover, 1962 [1769]. Print.
Merleau-Ponty, Maurice. _The Primacy of Perception_. Evanston: Northwestern University Press, 1964. Print.
Nussbaum, Martha. _Upheavals of Thought. The Intelligence of Emotions_. Cambridge: Cambridge University Press, 2001. Print.
Oatley, Keith. "A Taxonomy of Emotions of Literary Response and a Theory of Identification in Fictional Narrative." _Poetics_ 23 (1994): 53-74. Print.
Pfau, Thomas. _Romantic Moods: Paranoia, Trauma, and Melancholy, 1790-1840_. Baltimore: Johns Hopkins University Press, 2005. Print.
Premack, David, and Guy Woodruff. "Does the Chimpanzee have a Theory of Mind?" _Behavioral and Brain Science_ 1 (1978): 515-26. Print.
Regner, Freihart. "Unbewußte Liebesbeziehung zum Folterer? Kritik und Alternativen zu einer 'Psychodynamik der traumatischen Reaktion.'" _Zeitschrift für Politische Psychologie_ 8 (2000): 429-52. Print.
Spencer-Brown, George. _Laws of Form_. London: Allen and Unwin, 1969. Print.
Thompson, Evan. "Empathy and Consciousness." _Journal of Consciousness Studies_ 8. 5-7 (2001): 1-32. Print.
Trivers, Robert L. "The Evolution of Reciprocal Altruism." _Quarterly Review of Biology_ 46 (1971): 25-57. Print.
de Waal, Frans. _Primates and Philosophers: How Morality Evolved_. Princeton: Princeton University Press, 2006. Print.
\---, and Sarah F. Brosnan. "Simple and Complex Reciprocity in Primates." _Cooperation in Primates and Humans. Mechanisms and Evolution._ Eds. Peter M. Kappeler and Carel P. Schaik. Berlin, Heidelberg: Springer, 2006. 85-106. Print.
Wahrman, Dror. _The Making of the Modern Self: Identity and Culture in Eighteenth-Century England_. New Haven and London: Yale University Press, 2004. Print.
## Theory of Mind and the Conscience in _El casamiento engañoso_
JOSÉ BARROSO CASTRO
> "I have been taught candor from my birth," said Edith; "and, if I am to speak at all, I must utter my real sentiments. God only can judge the heart—men must estimate intentions by actions."
>
> —Walter Scott, _Old Mortality_
### _A BILLIARD TABLE IN THE REPUBLIC OF LITERATURE_
_The Deceitful Marriage_ and _Colloquy of the Dogs_ are two interrelated novels that belong to a collection that Cervantes calls _Novelas ejemplares_ ( _Exemplary Stories_ ). In the prologue of his work, Cervantes provides a fictional portrait of himself with clear physical traits and personality. He argues that the reader would be interested in knowing "el rostro y talle [de] quien se atreve a salir con tantas invenciones en la plaza del mundo" (16); "what kind of face and figure belong to him who has had the boldness to come out in the market place of the world and exhibit so many inventions" (3). This prologue brings the fictional realm closer to the frontier of the real world, an area that classic narratology has labeled as extradiegetic. In this frontier land, Cervantes seeks recognition as a distinguished member of the realm of fiction since he presents himself as the creator of inventions in what he calls "la plaza de nuestra república" ("the Republic of Literature"); inventions, not only in the ancient rhetorical sense of _inventiones,_ but above all, in the new literary model sense, exemplified by _Don Quijote_. With this identity card—or passport to the fictional world—he is ready to present the novelty of these stories using an interesting metaphor, that of a billiard table:
> Mi intento ha sido poner en la plaza de nuestra república una mesa de trucos, donde cada uno pueda llegar a entretenerse, sin daño de barras; digo, sin daño del alma ni del cuerpo, porque los ejercicios honestos y agradables, antes aprovechan que dañan. [. . .] Horas hay de recreación, donde el afligido espíritu descanse. (18)
>
> [My intention has been to set up in the public square of our country a billiard table where everyone may come to amuse himself without harm to body or soul; for decent and pleasing pastimes are profitable rather than harmful. [. . .] There is a time for recreation, when the tired mind seeks repose.] (5)
A prologue should be the first transition from real to possible worlds. Cervantes takes advantage of it, and sets the most fundamental basis of fiction by "crossing somehow the world boundary between the realms of the actual Težel
20). Consequently, he makes a "declaration of intentions" while proposing the metaphor of fiction as billiard table ("mesa de trucos"). This metaphor engenders an experience of intelligibility that provides the experience of positioning oneself in an arena of fiction. Therefore, Cervantes's main intention consists of putting the reader on the right track of that experience in such a way that his or her fictional mind ("body and soul") is ready to play inside a possible world ("a billiard table"). Now everyone has their identity card in hand and ready for a fictional world: the personal author (his figure and his soul) portrayed as an inventor and the reader engaged in an enterprise of entertainment "without harm for the soul and the body." In other words, the author has created a fictional mind for a fictional world. I insist on the term "fictional mind," for it is the author's same insistence on a body and a soul as protagonists of a possible world. In fact, Cervantes's creation of a fictional mind suggests the very definition of fiction: a "construct of the mind" (Ryan 19). However, we need to be more precise, since this fictional world belongs to the field of storyworlds, "possible worlds that are constructed by language through a performative force that is granted by cultural convention" (Palmer 33). That is why Cervantes prefers to place his billiard table in the Republic of Literature, that is, in a Republic of conventions where the user of this play has at his disposal a user's manual.
In addition, a billiard table is play, and as far as crossing boundaries of fiction is concerned, "play is a space between reality and imagination" (Bolton 220). And there is a final observation that the author emphasizes: this "performative force" granted by "cultural conventions" and available for possible worlds must keep the fictional mind immune to consequences; no moral damage should constitute the side effects of this play.
### A FICTIONAL MIND FOR A FICTIONAL WORLD
However, in his prologue it seems that the author insists more on the role of the "body and the soul" as a fictional mind interacting with this fictional billiard table, than on the issue of a real world of prototypes inserted in a fictional world of characters. On the one hand, Cervantes could pursue the Renaissance formula of _imitatio_ in the sense of "actual prototype" of universals being represented by "a fictional particular" (Doležel 9). On the other, since the "body and soul" is interacting with fiction, Cervantes could be pursuing a Neo-Aristotelian broader meaning of imitation, like the one stated by the Canon in _Don Quijote_ who prefers that "Hanse de casar las fábulas mentirosas con el entendimiento de los que las leyeren" ( _Quijote,_ 548); "stories must be suitable for the intellect of the reader."1 This comment reminds us of Doležel's warning that "persons with actual world 'prototypes' constitute a distinct semantic class within the set of fictional persons" (24-25).
In this prologue, Cervantes concerns himself with the mind, but not only the mind as a fictional agent, but also as a moral one. At first, it appears that Cervantes tries to escape a common transgression once denounced by Plato: the sin of evil _écriture_. In fact, most scholars agree that this "moral" urge to write is due to a desire to rid fiction of a bad Boccaccian "ambiguous moral reputation" of plots "full of sensuality and impudence" (García López xcv-xcvi). Indeed, Cervantes in his prologue poses a series of argumentative points that succeed, one after another, faulting all of them against this accusation of immoral transgression and defending his new literary model.
I propose that Cervantes's declaration of intentions is not designed to rid himself of the evil _écriture_ reputation, but instead, to explain how the fictional mind can interact in a new fictional medium of a possible world. In addition, Cervantes claims at the end of his argumentation that "my mind [my wit] conceived" my novels (5-6). Therefore, what Cervantes wants to claim is his own fictional mind, plus that of the reader, their own semantics of fictionality, their own tricks in this new billiard table. There is no risk of doing the wrong thing: "there is not justification for two semantics of fictionality, one designed for 'realistic' fiction, the other for fantasy" (Doležel 19). We must study Cervantes's own semantics for possible world fictionality. He does not employ a real world basis with a more "essentialist" semantic, but a "transworld identity" with a more "nonessentialist" semantics (Doležel 16-17).
Therefore, what are Cervantes's semantics of fictionality? It is worth examining the mind ("wit" in Cervantes's terms) that has created it. And this mind, that of the implied author and those of his characters, can be visible by their actions and, to be precise, by their mental functioning: "their motives, intentions, and the resulting behavior and action" (Palmer 16). In a more extended definition: it is the "emotions, sensations, dispositions, beliefs, attitudes, intentions, motives, and reasons for action" (Palmer 13) of these characters that are more relevant to us. This look at the mind in action constitutes the basis of Cervantes's transworld identity that ultimately claims sovereignty over a fictional world.
### AUTHOR'S INTENTIONS
The implied author says: "my intention has been to set up in the public square of our country a billiard table" for entertainment without damage to the body and soul. His mind chooses a track where the exercise of reading and writing might have moral consequences and ultimately suggests a final judgment about the morality of the author's action of writing, of the reader's action of reading, and of the character's actions of representing. In other words, this mind posits a moral conscience in both parties: from the implied reader and his risk of possible moral damage and from the implied author who admits that he, too, has to account to God for his literary activity at the day of his reckoning. The implied author constructs within his fictional mind his own basis for a moral and religious conscience. He wants to make sure that the glass of his conscience of _écriture_ (the glass of his fictional mind) is clean and transparent, and this is the primary condition under which he will claim his own authorship (in ethimological and moral terms "authority"). In this way, the reader imitates this same fictional semantics, and assigns a mental functioning for himself, the same as that of the implied author, implied reader, and the characters to follow.
### ASPECTUAL SHAPE OF THE MIND
Therefore, in his prologue Cervantes is constructing a fictional mind ready for a fictional world. We need to study the mental functioning of this mind on its own terms. Every fictional mind has its own state of mind. In the real world, if we want to go to metaphysical levels of existence, there has to be a basic state of mind that constitutes the human consciousness also and consequently, a human being. According to Heidegger the _Befindlichkeit_ is the basic mood by which human beings encounter (find) things and even their own existence (Heidegger, _Sein und Zeit_ §29). There is not a neutral mood; the _Befindlichkeit_ is the basic mood that constitutes our consciousness, and in turn, our own human being; we project our existence in the world ruled by this basic mood. This fundamental state of mind can be affirmative, negative, ironic, mystical, comic, parodist, tragic, and so on. I have argued in my book _Sobre la comprensión poética_ that:
> El modo de percibir las cosas y de percibirse uno a sí mismo no se hace desde un estado aséptico sino desde una actitud previa o Befindlichkeit, temple o disposición envolvente ( _SuZ,_ §29) que perfila desde un primer momento las significaciones pues crea un ámbito de encuentro en que son bien o mal recibidas. (Barroso 39)
>
> [ _Befindlichkeit_ consists of the mood or the disposition by which our encounter with meaning is sketched, in such a way that from the very point of entrance to fiction we construct an atmosphere where sense is welcomed or discouraged.]
The basic mood of a fictional mind is significant for the construction of a semantics of fiction. This mood or _Befindlichkeit_ has its own narrative counterpart. For instance, Alan Palmer reports Searle's theory that "every intentional state has what he calls an _aspectual shape._ This means that my conscious experiences are always from a point of view, are always perspectival . . . all seeing is seeing as" (Searle 131). Seen though Searle's perspective, Cervantes is setting a basic fictional perspective or aspectual shape (a basic state of mind) ruled by two interrelated axis: 1) an inner aspectual attitude of "play" and "recreation" of a "billiard table" ready for a mind—and in this case a soul—who is affected ("afligido" in the sense of "destressed") by the stresses of life ("donde el afligido espíritu descanse"); 2) a social aspectual disposition toward respecting a moral entertainment into which a reader's "body and soul" are welcomed into the public square of Literature and protected from any moral damage.
### WRITING AND READING CONSCIENCE
In the end, in his fiction Cervantes is claiming responsibility: he claims "authority" for the author and responsibility for the reader; that is, conscience in invention and in reading. In the earlier times during which Cervantes wrote, comedy—one of the basis of the origin of the novel—was less demanding, both for the author and for the reader: there was a rhetorical manual ready for everybody, and less freedom in creating fiction. Instead, Cervantes writes with more freedom, and that is why he is claiming sovereignty over his fiction; for this reason, both an authentic attitude in shape for entertainment and a disposition of active compromise in fiction are necessary. In this sense, Cervantes constructs a fictional mind for a fictional world. This attitude of compromise extends into a transcendental mentality that is concerned with preparing himself to account to God for his actions. In the end this responsibility turns out to be a writing and a reading conscience.
### READY FOR TRICKS: THREE LEVELS OF INVENTION
Now that the setting of fiction and fictional mind in the prologue has been analyzed, I would like to discuss _The_ _Deceitful Marriage_ and _Colloquy of the Dogs_ by placing the "mesa de trucos" on a simple level of narrative strategies, that is, those old and new tricks, accepted or not in the world of pure narrative conventions. Cervantes is very concerned with the art of narration, with speculation and experimentation in fictional invention, but he is also committed to the search of truth and sense in his fiction. Indeed, in these two interrelated novels that I analyze here, Cervantes can be very speculative with narrative, given that he is continuously playing with different levels of fiction. In fact, the _Colloquy of the Dogs_ is embedded in _The Deceitful Marriage._ One of the main characters of _The Deceitful Marriage_ transcribes the _Colloquy_ by memory and the other main character proceeds to read it. The reader must circulate through the different levels: the intradiegetic level of _The Deceitful Marriage_ , the metadiegetic level of _The Colloquy of the Dogs_ embedded in _The Deceitful Marriage_ and the extradiegetic level of the Prologue where the author portraits himself and surrounds fiction with his personal portrait and his fictional metaphor of the billiard table.
### A MORAL ISSUE: A CAPTIVE MIND
A moral issue is under discussion in _The Deceitful Marriage_ , since the Ensign Campuzano is forced to explain to his friend Peralta the reason why in the Hospital of Resurrection he received a sweat treatment for syphilis. Campuzano explains to his friend how his marriage was arranged between Doña Estefanía and him, a woman whom he met at an inn, and who suspiciously practiced a kind of "tapada" ("prostitution"). Campuzano is reporting to Peralta why this marriage failed. He has been deceived by Doña Estefanía but also she has been deceived by Campuzano; both of them committed to their marriage arrangement with bad intentions and with economic interests. No love and service are the basis of this marriage, if we consider these two last terms from a Christian point of view.
The levels of intelligibility of this marriage arrangement are set by Campuzano himself in terms of a mental functioning close to the Spanish Golden Age theory of the mind:
> Yo tenía entonces el juicio no en la cabeza, sino en los carcañares, haciéndoseme el deleite en aquel punto mayor de lo que en la imaginación le pintaba, y ofreciéndoseme tan a la vista la cantidad de hacienda, que ya la contemplaba en dineros convertida, sin hacer otros discursos de aquellos que daba lugar el gusto, que me tenía echados grillos al entendimiento, le dije que yo era el venturoso y bien afortunado en haberme dado el cielo, casi por milagro, tal compañera. (526)
>
> [At that time I had my judgment in my heels rather than in my head, and as the pleasure seemed just then greater even than I had imagined it, and all of that property in front of me as good as cash already, without stopping to look for any arguments beyond my own wishes, which had quite overcome my intellect, I told her that I was the most fortunate of persons to have been presented with this heaven-sent miracle, a companion like this to be the mistress of my will and possessions.] (184)
According to the Scholastic Theory of Mind (ToM), the imagination—a secondary faculty of the soul—is the intellectual process operation that results from the perception of something, a faculty that puts in mind the _phantasmata_ or mental data received from the senses; in other words, the physical data are replaced by an abstraction in the intellect in terms of the concept and the image of the thing itself.2 This mental functioning consists in an intellectual-centered practical reasoning. The thing perceived is Doña Estefanía's property, and the image and the concept in Campuzano's corrupt intellect is money ("hacienda"). On the other hand, the judgment plays its role in his mental process since it is the faculty of the soul that enables the intellect to accommodate universal intuitions of the wit to particular circumstances.3 So ultimately, Campuzano is unable to select and translate properly and morally the data (these particular circumstances) that have been offered by Estefanía, given that wit's only universal intuition is pleasure and own wishes. Consequently, the intellect is captive to a corrupt moral context of intelligibility. The universal intuition should be a love blessed by the Holy Spirit within the context of a matrimony declared a Holy Sacrament by the Council of Trent. But neither Estefanía nor the Ensign Campuzano had respect for this sacramental component of marriage. Estefanía left him, escaped with a lover, and took his jewels, although she did not know that these jewels (especially a chain) were artificial.4 Ultimately Campuzano faces two problems: forty days in hospital for syphilis treatment and the remorse of his conscience since he has to deal with "the evil intentions with which I had embarked on that affair" (185).5
Until this point, the reader has experienced the topical plot of a deceitful marriage that could be found in any Italian _novella._ However, ultimately Campuzano's imagination—as his secondary faculty of soul—is captive to his remorse in his conscience: that woman "even without looking for her, she's is always in my imagination and wherever I am my disgrace goes with me" (189).6 He has become the victim of a bad and unblessed use of intellect and this leaves his conscience to plague him: "I was caught in my own trap; but I can't help feeling sorry for myself."7 Later on we will see that Cervantes wants to complicate things, and open a new game in this billiard table: this attitude of remorse will become the basis for a new game: _The Colloquy of the Dogs._
### CAMPUZANO'S HEART
Now that Campuzano's mental functioning has been explained by its own Golden Age semantics, let us return to one of our earlier arguments, that of fictional minds involved in fictional worlds. The reader is also engaged in the task of ascribing a fictional mind to Campuzano, in order to make sense of his mental network of intentions and motives. In this way, the readers and Peralta witness a case of embedded narrative as defined by Alan Palmer as "total perceptual and cognitive view point; ideological worldview; memories of the past; and the set of intentions, desires, beliefs, motives, and plans for the future of each character in the story presented in the discourse" (Palmer 183-84). Campuzano's mental functioning turns out to involve both practical and moral reasoning. His set of intentions could be considered in terms of beliefs and faith. In an eighteenth-century novel, such as Jane Austen's _Emma,_ access to the fictional mind of a character is mostly regulated by emotions and feelings (Palmer 231). Cervantes's characters possess a fictional mind less psychologically sophisticated than _Emma,_ though in its fictional presentation, it is more complex ontologically. He is concerned with a consciousness in action more than a mind in action; in other words, he prefers the mind's reflexivity in itself, but with the ingredient of practical reasoning. And whenever something subjective is presented, the issue under discussion is examined in terms of moral conscience, rather than psychology.
Let me address this mental functioning in the context of both rhetoric and the ToM present in the Spanish Golden Age. In this way, I will take as an assignment Palmer's task of answering this question: "What are the features of the fictional-mind constructions of a particular historical period that are characteristic of that period and different from other periods?" (Palmer 241). One first question to address is to ask whether it is reasonable to apply cognitive studies' current connections between cognition and emotion to a Spanish Golden Age worldview. For instance, Antonio Damasio says that "no sharp line divides thinking from feeling, nor does thinking inevitably precede feeling or vice versa" (Damasio, 313). Alan Palmer reports that "Jon Elster in _Alchemies of the Mind_ (1999) continuously stresses interconnections between cognition and emotion, and argues that in practice they are difficult if not impossible to disentangle" (Palmer 19).
On the other hand, we have the medieval and Renaissance ToM of St. Augustine in which creatures are rational and intelligent in their function of being imperfect mirrors of the Trinity's mind. This divine mind is composed of memory, intellect, and will: "memoria," "intelligentia," "voluntas" ( _De Trinitate_ XV, 22, 42). If there is an emotion in the soul, it is the love for God, not in a psychological sense, but in an ontological sense. Therefore, intellect, memory, and love constitute human mind: "ego enim memini per memoriam, intellego per intelligentiam, amo per amorem" ( _De Trinitate_ XV, 22, 42: "For it is I who remember by memory, and understand by understanding, and love by love"). In St. Augustine, love and cognition are inextricably linked, since both of them are at the same time ontological mirrors of the Trinity of God's mind. In _The Deceitful_ _Marriage_ love is a religious and moral issue because human love became an economical transaction, and for this reason is not blessed by the Holy Spirit; that is, there is not the possibility to ascribe divine love to this human love.
At some point Campuzano's mental functioning is not pure practical reasoning since there is an "emotional" counterpart that it is worth analyzing: "I was beginning to despair, and doubtless would have done so if my guardian angel hadn't come to the rescue at that moment by reminding me that I was a Christian and that the worst sin a human being could fall into was that of despair, for it was a devilish sin" (188). The words "desesperarse" and "desesperación" can be interpreted in Old Spanish terms as "to commit suicide," but it is relevant to examine the word in the context in which it appears. Campuzano looks into his heart, which is the place where a Christian—following the Bible, St. Augustine, and other Church Fathers (De Magistro XII, 40, 82)— finds God's love, and he discovers that his heart turns out to be his own conscience speaking with great "candor." With this image, I remind the reader of Walter Scott's quotation that heads this essay, which I found in Alan Palmer's _Fictional Minds_. Campuzano addresses his own conscience, which responds in the form of his "guardian angel," and discovers that he—with his attitude—is moving away from the divine love to which his moral conscience is engaged. If we follow _Diccionario de Autoridades's_ definition of "deses-peración" we can understand the authentic semantics of Campuzano's attitude and mental functioning. This dictionary defines "desesperación" as "complete loss of hope, and par excellence can be meant to describe complete loss of Grace."8 In this sense, Kierkegaard illuminates very well the meaning and the mental functioning of "desesperarse": "the desperate renounces the possibility (for 'to despair' exactly means 'to renounce the possibility'); to be more precise: he insolently dares to suppose the impossibility of good" (Kierkegaard 305).9 In other words, this is the most specific definition of "sin;" in fact, Campuzano declares "la desesperación [es] pecado de demonios" ("dispair is a devilish sin"). In this sense, it is not by chance that the _Diccionario de Autoridades_ gives Alejo de Venegas's example of this particular aspect of sin present in the "despair": "since Evil displays them, his real intention is not love, but to send him into despair."10
In this precise moment of despair, Campuzano seeks desperately to take revenge on Doña Estefanía, but "luck, for better or worse, determined that I couldn't find Doña Estefanía, hard though I looked. I went to [the church of] San Llorente, commended myself to Our Lady, sat down on a bench, and was so upset that I fell into a deep sleep" (188). Here we find a very strong example of how Golden Age mental functioning does not relate cognition with emotion as a psychological ongoing of actions. Campuzano's cognition operates in terms of conscience, and conscience operates in terms of transcendental mediation; in other words, revelation. And the most powerful mediator is the Virgin Mary. Again, not by chance in _Diccionario de Autoridades'_ definition of "desesperación _,"_ we find a very apt example of the _Flos Sanctorum_ : "if you feel suffocated by your own offenses, and you fear falling into the profound abyss of despair, think of the Virgin Mary."11 Campuzano's cognitive mental functioning works within a supernatural belief structure. In this sense, mental functioning of revelation works within the confines of dream; there is no psychological process present. In fact, in the Bible there are many cases of dreams functioning as the repository of revelation. In conclusion, Campuzano's ToM is a combination of cognition, interaction between human and divine love, and divine revelation.
### CAMPUZANO'S MIND
However, his supernatural mental functioning inside a religious heart works in the context of practical reasoning, intellect and faculties of the soul. Considering the times in which Cervantes lived, the ToM applied to these times owes its theoretical standpoint to the Scholastics who ascribed to the Aristotelian and the Thomist anthropology of mind. Cervantes has much in common with this anthropology, but that said, he somehow breaks the limits of the ToM present in his times by extending the mental functioning of some of his characters to both mind and consciousness. According to Anthony Kenny in Achinas's ToM, intellect and will ("intelligence and rationality") are the two essential components of mind (Kenny 16). In contrast, in Cartesian ToM, the mind can be defined as consciousness in the sense of "the realm of whatever is accessible to introspection" (Kenny 17-18). Cervantes operates in a way in between: while using rationality in mental functioning, he plays with different levels of consciousness. However, consciousness in Cervantes is not psychological introspection, but ontological consciousness. Here Morón Arroyo clarifies that "between the theological worldview of Boccaccio and the psychological of Mme Lafayette, Cervantes should be placed in a ontological level" ( _Para entender_ 227). For instance, love in _Don Quijote_ is presented with its essential properties (ontology) and not as a psychological series of personal feelings.12 As far as Campuzano is concerned, he is a Christian; he follows a supernatural law that creates his own existence onthologically-speaking. For instance, when Campuzano is in despair, in the greatest depths of his Christian conscience he discovers a law inscribed in his heart, which is continuously listening to God ( _Romans_ 2, 14-16).13 In conclusion, the other side of Campuzano's practical reasoning is this law that creates the nature of his own existence; that is, when he examines his mind, he is exploring his own ontology.
### A NEW SET IN THE BILLIARD TABLE
Let us examine the consequences of this ontological law in Campuzano's own embedded narrative and fictional capacity. Campuzano tells Peralta that one night (the one before he had the sweat treatment for the last time) behind his bed "and in the middle of the night, when it was dark and I was lying awake, thinking of my past adventures and misfortunes, I heard talking close by [. . .] and soon after I realized by what they said that it was the two dogs 'Scipio' and 'Berganza' who were talking" (191).14 At this point, Campuzano decides to take a dare and to participate in the billiard table of possible worlds set by the implied author in the prologue. Campuzano defends himself from the impossibility of two talking dogs by explaining in detail what happened and how he has managed to overhear the conversation, and then transcribed it as the "Tale and Colloquy":
> "Well, there's another point", said the ensign. "Since I was listening so attentively, my mind [judgment] was in a delicate state, and my memory was also delicate, sharp and free of other concerns [. . .], I learnt it off by heart and wrote it down the next day in almost the same words in which I had heard it."15
In his report, Campuzano explains the situation in practical reasoning terms related to the levels of senses and faculties of the soul. He also displays his moral attitude in his explanation. His senses are attentive to voices since he declares that he was using "all of my five senses as God was pleased to give them to me" (192); but they are limited, given that his eyes cannot see the dogs. Two of his own faculties of the mind, judgment and memory, are unreliable engaged in "a delicate state" (192). If we take "delicado / delicate" to mean the secondary meaning of definitions "sickly or damaged," we realize these two faculties are limited. Furthermore his moral attitude could be on exercise of penance, since he admits that his thoughts were engaged "in my past adventures and misfortunes." In conclusion, this delicate situation deprives him again of a good use of his intellect.
However, in this bad state of both imagination and intellect, Campuzano dares to make a new strike for the reader and for his friend Peralta, with consequences in the teleology of the plot, which in the end, seems to be in continuous ups and downs as if one were playing at "a billiard table." In this new fictional situation or in this fictional trick of the billiard table, the reader has to fill gaps in Campuzano's own embedded narrative. The reader could be asking questions like: what are Campuzano's intentions in conveying these thoughts? How does he present his storyworld? What is his mental functioning? In my view, the penance for his sin could be a cue of his mental functioning and his fictional mind. Campuzano's report shows that both his mind and his fictional mind function intellectually, using practical reasoning and judgment. But whenever Campuzano wants to reach into the depths of his own mind, he does not embark on a psychological path. In contrast, during this self discovery, he confronts his own mind retreated into his own conscience, that is, looking for a final judgment in the morality of his actions. There is not a psychological process in this mental functioning. Instead, he experiences essential properties that he discovers in the ontology of both his soul and conscience. I argue that Campuzano's aspectual attitude toward his own storyworld is penance in the sense of his examination of those sins that made him turn aside from the natural and ontological law of divine love. In this sense, I agree with Ruth El Saffar in this statement: "the idea that evil exists so that man may choose his salvation is behind the Catholic belief in redemption through good works, and it is reflected in the _Casamiento-Coloquio_ " (20). Therefore, there is a religious attitude devoted to the penance for his sins and consequently to the return to the law that rules his own human being; this attitude is the best way to salvation. In the period of the novel, the Sacrament of Penance, in its three typical steps ( _contritio, confessio,_ and _satisfactio)_ was a common religious practice. In fact, this Sacrament was institutionalized by the Council of Trent. In this sense, I do not find in Cervantes's religious thought a secular and natural law as stated by Américo Castro and his followers. On the contrary, I agree with Muñoz Iglesias, who believes in "the increasing moral religious rigor of [Cervantes's] works" ("el creciente rigor moral de sus escritos" 326), but I cannot agree with Descouzis16 who claims that Cervantes was committed to the Catholic Church to the extent that he was spreading in his works the canons and decrees of the Council of Trent.17
### CAMPUZANO'S CONSCIOUSNESS
There is a final point on Campuzano's embedded narrative that should be discussed. I believe and agree with other scholars that Cervantes's ToM and precisely that Campuzano's ToM possesses two sides of the same coin: on one side is the practical reasoning, and the other, the "consciousness." This "consciousness" is not a Cartesian introspection, but a Baroque consciousness of self-reflection. In this sense, I propose that Cervantes's trick of placing this storyworld ( _The Colloquy of the Dogs)_ inside the fictional mind of a character (Campuzano) and then reading this storyworld through the fictional mind of another character (Peralta) as if it were two embedded narratives in plain interaction is the best example of autoreflexive fictional and ontological consciousness. This consciousness is auto-reflexive because it entails a play of mirrors of different fictional levels; it is ontological because of Campuzano's return to the law that governs his heart, a return that implies penance. In addition, these auto-reflexive effects are present in the conscience of both reader and author, since both of them are concerned with the possibility of an evil _écriture_.
### INTELLECT IN GOOD SHAPE
Now it is Peralta's turn to play at the billiard table. He opens the notebook that contains the _Colloquy,_ and starts to read while Campuzano naps. Peralta uses a mental functioning that is similar in its logic to that of Campuzano's: senses, intellect, and moral attitude. He employs his own senses by "hearing" ("oiré") the written _Colloquy,_ employs the faculties of his own soul. Peralta uses his intellect as he appreciates Campuzano's wit. And finally, has a moral attitude toward entertainment and toward the fictional world.
Peralta is skeptical about two talking dogs, but he wants to enjoy the fiction: "I shall be very happy to listen to this colloquy, which as it was written down from the notes of your ingenious self [wit], Ensign, I already adjudge to be good" (192).18 This declaration suggests that Peralta's mental functioning will circumscribe the operation of reading in the limits of pure rhetoric and fiction because he wants to limit his faculties of the soul to a rhetorical operation of fictional invention. In other words, Peralta appreciates that Campuzano's wit has made possible worlds where the _res_ selected does not come from real world (talking dogs), but from fiction with its own semantics; that is, his fictional mind is ready for a fictional world. In this operation, he uses wit and judgment in its rhetoric functions. Peralta's ToM operates in the rhetorical terms used in that period. For instance, according to Morón Arroyo's analysis on Juan de Valdes's _Diálogo de la lengua, intellect_ has two faculties: " _wit_ as the native capacity of invention, and the _judgment_ as the capacity of selection and organization of discourse."19 Finally, since Peralta does not accept real talking dogs in the realm of a fictional world, he takes the dialogue in an aspectual shape of "laughing and apparently making a joke of all that he had heard and of what he was expected to read" (193).20 Peralta's reaction shapes his embedded narrative and fictional mind. When he has finished reading Campuzano's _Colloquy,_ he does not want to discuss again the real possibility of two dogs talking. He remarks that his reading has entertained his intellect, enabling him to experience a fictional invention of a possible world: "Ensign, don't let's return to this argument. I appreciate the art of the colloquy and the invention you've shown, and that's enough. Let's go off on the Espolón and refresh our eyes, for my mind's [intellect] has been well refreshed" (252).21 Peralta's primary conclusion is that his _intellect_ is in good shape given that he has considered the process as one of "recreation." Peralta's point to Campuzano addresses the veracity of the material or _res_ selected in the invention. Peralta understands that "this colloquy may be a fake and may never have happened" (252).22 However, Peralta does not want to enter into the true material of the invention. He distinguishes what are the semantics of a real world and the semantics of a possible world very well. His intellect is in good shape.
### BAD OR GOOD USE OF INTELLECT, AND A TRANSWORLD IDENTITY
Interestingly Peralta proves to have followed the implied author's guidelines in the Prologue for access to a fictional world: that of an attitude of "recreation" and "repose." In the reading of the _Colloquy,_ there is ultimate reward and repose for Peralta and for the reader because both have used recreation to practice a "good" use of their intellect, a good use of their fictional mind. Campuzano employs his intellect with remorse, and within this remorse he invents the _Colloquy_ as a result of being exercised by his act of penance for his sins. In this sense, either Williamson's assumption that Campuzano is not a sincere Christian because "he couldn't forgive" ("no ha sabido perdonar" 190) or El Saffar's opinion that Campuzano is a "redeemed sinner" (20) must be examined taking into account the symbolic basis of the _Colloquy._ At the end of the dialogue between the two dogs, Cipión, who is most of the time the voice of the philosopher and moralist, emphasizes the good or bad use of intellect by saying, "Virtue and good intellect are one and indivisible, whether naked or clothed, alone or in company. Of course it is true that these things can suffer as far as people's estimation of them is concerned, but not in their real merit and worth" (252).23 Cipión's words about intellect must be taken symbolically and ethically. This symbolic semantics could be the cue for understanding Campuzano's intentions of his storyworld. In this sense, Cervantes creates his own semantics of fiction, not as a mimetic _imitatio,_ but as more autonomously _invention._ Cervantes does not use the semantics of the real world, but instead, invents a new semantics by overlaying storyworld participants who employ fictional tricks of a possible world. In terms of ToM, this embedding situation is a Cervantes's illustration of how a fictional mind should behave in a possible and fictional world. For those purposes, Cervantes has chosen the billiard table of fiction encouraging different fictional tricks. These tricks can be played by any fictional mind and yet require a high level of auto-reflexive consciousness. In this play where the fictional mind is the protagonist's, there are effects in the fictional and moral conscience of every other agent (author, reader, implied author, implied reader, characters). The "exemplary" lesson could be that reading and writing are both exercises of mental functioning and a moral functioning of conscience where every agent takes responsibility and during which there can be no "harm for the body and soul."
### NOTES
1 Translations are my own, unless indicated.
2 "imaginatio: alicuius rei conceptio et fictio, quae mente fit; vel imaginatio est quaedam motio per quam nobis simulacra quaepiam in anima gignuntur" ( _Diccionario Covarrubias_ ).
3 "el hábito de acomodar las intuiciones universales del ingenio a las circunstancias particulares" (Morón Arroyo "Prólogo" 133).
4 "como no es todo oro lo que reluce, las cadenas, cintillos, joyas, brincos con solo ser de alquimia se contentaron, pero estaban tan bien hechas, que solo el toque o el fuego podia descubrir su malicia" (532).
5 "iba mudando en buena la mala intención con que aquel negocio había comenzado" (528).
6 "la hallo en mi imaginación y adonde quiere que estoy tengo mi afrenta presente" (533).
7 "Bien veo que quise engañar y fui engañado, porque me hirieron por mis propios filos; pero no puedo tener tan a raya el sentimiento, que no me queje de mí mismo" (533).
8 "pérdida total de la esperanza, y por antonomasia se entiende de los bienes eternos, Lat. _Desperatio"_ ( _Diccionario de Autoridades_ ).
9 "El desesperado _sabe_ también lo que la posibilidad encierra, y sin embargo renuncia a la posibilidad (ya que desesperar no es otra cosa que renunciar a la posibilidad), o dicho con mayor exactitud, él tiene la osadía insolente de _suponer_ la imposibilidad del bien" ( _Las obras del amor_ 305).
10 "y pues el Diablo a tal tiempo se los ofrece, no lo hace de charidad, sino por trahelle a desesperación" (ibid.).
11 "si te comienzas a ahogar por la graveza de tus delitos, y temes caer en el profundo abysmo de la desesperación, piensa en María" (ibid.).
12 "el amor aparece en una serie de variables que son propiedades esenciales del amor (ontología), pero no como un sentimiento personal con altibajos, dudas, celos y reacciones provocadas por la persona amada (psicología)" (Morón Arroyo, _Para entender_ 228).
13 "For whenever the Gentiles, who do not have the law, do by nature the things required by the law, these who do not have the law are a law to themselves. They show that the work of the law is written in their hearts, as their conscience bears witness and their conflicting thoughts accuse or else defend them, on the day when God will judge the secrets of human hearts, according to my gospel through Christ Jesus" (Romans 2, 14-16).
14 "pensando en mis pasados sucesos y presentes desgracias, oí hablar allí junto . . . y a poco rato vine a conocer, por lo que hablaban, lo que hablaban, y eran los dos perros Cipión y Berganza" (535).
15 Pues en esto hay otra cosa—dijo el alférez—: que como yo estaba atento y tenía delicado el juicio, delicada, sotil y desocupada la memoria [. . .] todo lo tomé de coro, y casi por la mismas palabras que había oído lo escribí otro día (537).
16 Concerning the Sacrament of Penance, see Descouzis, "Meditaciones del Cautiverio. Decreto de la Penitencia," In _Cervantes, a nueva luz,_ 158-69.
17 I agree with Enrique Moreno Báez that Cervantes "los decretos del Concilio prueba que los conocía...pero no que la obra fuera determinada en su concepción o en su desarrollo por ninguno de ellos" (Moreno Báez 247).
18 "de muy buena gana oiré ese coloquio, que por ser escrito y notado del buen ingenio del señor alférez ya le juzgo por bueno" (537).
19 "el entendimiento se concreta en dos funciones: el ingenio, que es la capacidad nativa e inmediata de inventar; y el juicio, la capacidad de selección y disposición" (Morón Arroyo, "Prólogo" 133).
20 "sacó del pecho un cartapacio y le puso en manos del licenciado, el cual le tomó riyéndose y como haciendo burla de todo lo que había oído y de lo que pensaba leer" (537).
21 "Señor alférez, no volvamos más a esta disputa. Yo alcanzo el artificio del coloquio y la invención, y basta. Vámonos al Espolón a recrear los ojos del cuerpo, pues ya he recreado los del entendimiento" (623).
22 "Aunque este coloquio sea fingido y nunca haya pasado" (623).
23 "hay algunos hombrecillos que a la sombra de sus amos se atreven a ser insolentes; y si acaso la muerte o otro accidente de fortuna derriba el árbol donde se arriman, luego se descubre y manifiesta su poco valor; porque, en efeto, no son de más quilates sus prendas que los que les dan sus dueños y valedores. La virtud y el buen entendimiento siempre es una y siempre es uno, desnudo o vestido, solo o acompañado. Bien es verdad que puede padecer acerca de la estimación de las gentes, mas no en la realidad de lo que merece y vale" (622-23).
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## Contributors
**Robert S. Astur** , PhD, is the director of the Virtual Reality Laboratory, Olin Neuropsychiatry Research Center in Hartford, Connecticut, and assistant professor adjunct in the Department of Psychiatry of Yale University School of Medicine. His current projects include: the application of virtual spatial memory tests to psychiatric groups with specific brain abnormalities; examining the effects of abused drugs on memory processes and brain activity; and deciphering the effects of gender and aging on memory processes.
**José Barroso Castro** received his PhD from Cornell University. He is currently Director of Editorial Mendaur, a Spanish press that publishes books on literature, humanism, religion, and the history of Spain and Europe. He is the author of two books on the theory of reading and textual criticism: _Sobre la comprensión poética_ (Madrid: Visor-Antonio Machado Libros, 2001); and _Memorias da morte: A cultura clerical do pasamento_ (Pobra do Caramiñal: Editorial Mendaur, 2009). Among his recent articles are: "Las Meninas y Don Quijote: dos modos para la identificación en mundos ética y poéticamente posibles," _Critica del testo_ 9 (2006), 2-19; and "Conciencia en acto: autoextrañamiento y reminiscencias clásicas en "Adolescencia" de Vicente Aleixandre," _Iberoromania_ 63 (2006), 27-51.
**Ineke Bockting** holds doctoral degrees from the Universities of Amsterdam, The Netherlands, and Montpellier, France. She has taught at universities in The Netherlands, Norway, and France, and she is now a full professor at the University of Paris, France. Her publications include articles on various aspects of the American South, ethnic literatures, travel-narrative, autobiography, and literary stylistics and pragmatics, as well as a book-length study of the novels of William Faulkner, entitled _Character and Personality in the Novels of William Faulkner: A Study in Psychostylistics_ (Lanham: University Press of America, 1995).
**Fritz Breithaupt** received his PhD from Johns Hopkins University in 1997. He was the Distinguished Remak Scholar (2009-10) at Indiana University, where he teaches as associate professor of Germanic Studies and adjunct professor of Comparative Literature and Cognitive Science. He is the author of _Kulturen der Empathie_ (Frankfurt: Suhrkamp, 2009); _Der Ich-Effekt des Geldes: Zur Geschichte einer Legitimationsfigur_ (Frankfurt: Fischer, 2008); and a book on Johann Wolfgang Goethe entitled _Jenseits der Bilder: Goethes Politik der Wahrnehmung_ (Rombach: Freiburg, 2000). At Indiana University he has served as director of West European Studies and the official EU Center of Excellence. He also writes a column on academia for the magazine _Zeit Campus_.
**Diana Calderazzo** is a PhD candidate in Theatre Arts at the University of Pittsburgh, currently living in Manhattan and writing her dissertation on cognitive-based approaches to experiencing and understanding the work of Stephen Sondheim. Her writing on related topics has appeared in _Theatre Journal_ , _Theatre Symposium_ , and _The Sondheim Review_. She teaches for Fordham University College of Liberal Studies in Westchester, New York.
**Vince D. Calhoun** , PhD, is the Director of Image Analysis of MR Research at The Mind Research Network in Albuquerque, New Mexico, where he develops techniques for making sense of complex brain imaging data. Because each imaging modality has limitations, the integration of these data is needed to understand the healthy and especially the disordered human brain. Calhoun has created algorithms that map dynamic networks of brain function, structure, and genetics, and how these are impacted while being stimulated by various tasks or in individuals with mental illness such as schizophrenia. He is associate professor within the Department of Electrical and Computer Engineering, Neurosciences, and Computer Science, at the University of New Mexico, and associate professor, adjunct, in the Department of Psychiatry at Yale University
**Mikko Keskinen** is Professor of Comparative Literature at the University of Jyväskylä, Finland. He is the author of _Response, Resistance, Deconstruction: Reading and Writing in / of Three Novels by John Updike_ (Jyväskylä: Jyväskylä University Press, 1998) and _Audio Book: Essays on Sound Technologies in Narrative Fiction_ (Lanham: Lexington Books, 2008). He has published numerous articles on literary theory and contemporary American, British, and French literature in journals such as _Critique_ , _Journal of International Women's Studies_ , _PsyArt_ , _The Romanic Review_ , and _Imaginaires_. His recent chapter-length contributions to edited volumes on criticism have appeared in _Novels of the Contemporary Extreme_ (London: Continuum, 2006) and _Terrorism, Media, and the Ethics of Fiction: Transatlantic Perspectives on_ _Don DeLillo_ (London: Continuum, 2010).
**Seth Knox** received his PhD from the Department of Germanic and Slavic Languages at Wayne State University. He is currently Assistant Professor of German at Adrian College in Adrian, Michigan. Knox is the author of _Weimar Germany between Two Worlds:_ _The American and Russian Travels of Kisch, Toller, Holitscher, Goldschmidt, and Rundt_ (New York: Lang, 2006). He is also the author of several entries in _The Greenwood Encyclopedia of Folktales and Fairy Tales_ (Westport, CT: Greenwood, 2008) and "A Political Tourist Visits the Future: Ernst Toller's American and Russian Travels near the End of the Weimar Republic" in _Cross Cultural Travel: Papers from the Royal Irish Academy Modern Languages Symposium on Literature and Travel, National University of Ireland, Galway, November 2002_ (New York: Lang, 2003).
**Paula Leverage** is Associate Professor of French and Medieval Studies, and Director of the Center for Cognitive Literary Studies, at Purdue University. She is a graduate of Cambridge University and Toronto University, where she first discovered cognitive theory as a doctoral student and Commonwealth Scholar. She is the author of _Memory and Reception: A Cognitive Approach to the Chansons de Geste_ (Amsterdam: Rodopi, 2010) and has also published extensively on memory and literature in journals such as _Romania_ , _Romance Notes_ , _Dalhousie French Studies_ , _Olifant_ , and book collections.
**Dan Lloyd** is the Thomas C. Brownell Professor of Philosophy at Trinity College, Connecticut. He is the author of _Radiant Cool: A Novel Theory of Consciousness_ (Cambridge: MIT Press, 2004), a book joining noir fiction with a theory of consciousness, and _Simple Minds_ (Cambridge: MIT Press, 1989). His current projects include _Ghosts in the Machine_ (Rowan and Littlefield, forthcoming), a philosophical dialogue about minds, brains, and computers, and _Subjective Time_ (MIT Press, forthcoming), an anthology on the philosophy, psychology, and neuroscience of experienced temporality. He is the editor of the journal _Frontiers in Theoretical and Philosophical Psychology_ , and in his spare time he likes to convert brain scan data into music.
**Orley K. Marron** teaches at Bar Ilan University, Israel, from which she received her PhD in English Literature. She has a MSc degree in Computer Science from Marist College as well as a BSc in biology from the University of Houston, where she worked for several years in neurobiology research. Her dissertation explored the function of animated man-made objects in the literature of Nathaniel Hawthorne and Lewis Carroll, and how such objects are used to convey algorithmic problem solving or moral dilemma resolution. Her current work explores Theories of Mind in science fiction, visual art in fantastic literature, and fantastic models for dealing with trauma.
**Howard Mancing** is Professor of Spanish at Purdue University. He is the author of _The Chivalric World of Don Quijote: Style, Structure, and Narrative Technique_ (Columbia: University of Missouri Press, 1982); _Text, Theory, and Performance: Golden Age Come-dia Studies_ , co-edited with Charles Ganelin (West Lafayette: Purdue University Press, 1994); _The Cervantes Encyclopedia_ , 2 vols. (Westport, CT: Greenwood, 2004); and _Miguel de Cervantes' "Don Quixote": A Reference Guide_ (Westport, CT: Greenwood, 2006). In addition, he has published over forty articles and essays on Cervantes, Unamuno, _Lazarillo de Tormes_ , and the picaresque novel, narrative theory, comparative literature, the canon, the teaching of literature, academic administration, cognitive approaches to the study of literature, and other subjects, in a variety of books and in journals such as _Anales Cervantinos_ , _Cervantes_ , _Estudios Públicos_ , _Forum for Modern Language Studies_ , _Hispania_ , _Hispanic Review_ , _Modern Language Notes_ , _Modern Fiction Studies_ , _Publications of the Modern Language Association of America_ , and _Semiotica_. He is currently the President of the Cervantes Society of America. At the present time he is working on a book on cognitive science and literary theory tentatively titled _Voices in Everything: Restoring the Human Context to Literary Theory_.
**Keith Oatley** is Professor Emeritus of Cognitive Psychology at the University of Toronto, a Fellow of the Royal Society of Canada, and a Fellow of the British Psychological Society. His research focuses on the psychology of emotions and the psychology of fiction. He is the author or co-author of some 150 journal articles and chapters, and six books on psychology, including: _Best Laid Schemes: The Psychology of Emotions_ (Cambridge, New York: Cambridge University Press, 1992) and _Selves in Relation: An Introduction to Psychotherapy and Groups_ (London: Methuen, 1984). He has also written two novels, one of which, _The Case of Emily V._ (London: Secker and Warburg, 1993), won the 1994 Commonwealth Writers Prize for Best First Novel. He also edits and writes for _OnFiction: An Online Magazine on the Psychology of Fiction_.
**Alan Palmer** is an independent scholar living in London. He is the author of _Social Minds in the Novel_ (Columbus: Ohio State University Press, 2010), and _Fictional Minds_ (Lincoln: University of Nebraska Press, 2004). _Fictional Minds_ was a co-winner of the Modern Language Association Prize for Independent Scholars and also a co-winner of the Perkins Prize (awarded by the International Society for the Study of Narrative). His articles have been published in the journals _Narrative_ , _Semiotica_ , and _Style_ , and he has contributed chapters to several edited volumes. He is an honorary research fellow at the Department of Linguistics and English Language, Lancaster University.
**Godfrey Pearlson** , MD, is the director of the Olin Neuropsychiatry Research Center, a state-of-the-art fMRI facility in Hartford, Connecticut, and Professor of Psychiatry at the Yale University School of Medicine. His research projects include: identifying the structural and functional cerebral correlates of normal aging and of major neurodevelopmental and neurodegenerative psychiatric disorders; determining the functional cerebral actions of abused drugs; and exploring the interactions between genetics, neuroimaging, and pre-symptomatic expression of brain-based disorders in at-risk persons.
**Natalie Phillips** is a PhD candidate in English at Stanford University, where she is currently an American Council of Learned Societies / Mellon Fellow. Her current project, _Distraction: Problems of Attention in Eighteenth-Century Literature_ , argues that the emergence of competing theories of attention in the Enlightenment transformed both how writers portrayed the fictional mind and how they sought to capture readers' focus. She has received fellowships from the Stanford Humanities Center, the Mabelle McLeod Lewis Fund, and the Society for the History of Authorship, Reading, and Publishing. Her most recent article, "'Tis Tris': _Tristram Shandy_ and the Literary History of a Cognitive Slip," has been solicited by _Modern Philology_ , and her manuscript, "Economies of Attention: Readers and the Eighteenth-Century Essay," is forthcoming as a chapter in _The History of Reading_ (New York: Palgrave Macmillan, 2011).
**Klarina Priborkin** received her PhD from Bar Ilan University, Israel. Her thesis, _Ghostly Bridges: Cross-Cultural Mother / Daughter Storytelling in Postmodern Texts by Ethnic American Women Writers_ , focuses on problems of cross-cultural communication between ethnic mothers and their American-born daughters. Priborkin currently teaches English and American literature at Givat Washington College of Education. Her article "Mother's Dreams, Father's Stories: Family and Identity Construction in Chitra Banerjee Divakaruni's 'Queen of Dreams'" has been published in _South Asian Review_ 29 (2009), 199-219. She has also published "Cross-Cultural Mind-Reading or Coming to Terms with the Ethnic Mother in Maxine Hong Kingston's _The Woman Warrior_ " in _Toward a Cognitive Theory of Narrative Acts_ , ed. Frederick Luis Aldama (Austin: University of Texas Press, 2010).
**Richard Schweickert** is Professor of Psychological Sciences at Purdue University. He developed an interest in dreams while working on his bachelor's degree in mathematics at Santa Clara University. After working as a statistician at Bellevue Psychiatric Hospital, he received his master's degree in mathematics from Indiana University and his PhD in mathematical psychology from the University of Michigan. His research is on network models of human information processing, short-term memory, and dreams. He has served as Associate Editor of _Psychonomic Bulletin & Review_ and Editor of the _Journal of Mathematical Psychology_. He is a fellow of the American Association for the Advancement of Science and of the Association for Scientific Psychology.
**Mark Turner** is Institute Professor and Professor of Cognitive Science at Case Western Reserve University. He is the founding director of the Cognitive Science Network. His most recent book publication is an edited volume, _The Artful Mind: Cognitive Science and the Riddle of Human Creativity_ (New York: Oxford University Press, 2006). His other books and articles include: _Cognitive Dimensions of Social Science: The Way We Think about Politics, Economics, Law, and Society_ (New York: Oxford University Press, 2001); _The Literary Mind: The Origins of Thought and Language_ (New York: Oxford University Press, 1996); _Reading Minds: The Study of English in the Age of Cognitive Science_ (Princeton: Princeton University Press, 1991); and _Death is the Mother of Beauty: Mind, Metaphor, Criticism_ (Chicago: University of Chicago Press, 1987). He has been a fellow of the Institute for Advanced Study, the John Simon Guggenheim Memorial Foundation, the Center for Advanced Study in the Behavioral Sciences, the National Humanities Center, the National Endowment for the Humanities, and the Institute of Advanced Study of Durham University. He is a fellow of the Institute for the Science of Origins, external research professor at the Krasnow Institute for Advanced Study in Cognitive Neuroscience, and distinguished fellow at the New England Institute for Cognitive Science and Evolutionary Psychology. In 1996, the Académie française awarded him the Prix du Rayonnement de la langue et de la littérature françaises.
**Jennifer Marston William** is Associate Professor and Chair of German at Purdue University. She has published a book on German literature, _Killing Time: Waiting Hierarchies in the Twentieth-Century German Novel_ (Lewisburg: Bucknell University Press, 2010), in which she examines how cognitive metaphors have been used by German and Austrian novelists to reveal and criticize social hierarchical patterns. She has co-edited a critical edition of Anna Seghers's novella _Aufstand der Fischer von St. Barbara_ (Berlin: Aufbau Verlag, 2002) and has published articles on German literature and film in such journals as _German Studies Review_ , _The Germanic Review_ , and the _Journal of the Kafka Society of America_. Her current projects include serving as co-editor for entries on German literature in _The Literary Encyclopedia_ and a book on cognitive approaches to historical film.
**Allen Wood** , Professor of French, has been at Purdue University since 1984. He received his PhD from the University of Michigan in Comparative Literature. He has published _Literary Satire and Theory: A Study of Horace, Boileau, and Pope_ (New York: Garland, 1985) and _Le Mythe de Phèdre_ : _Les Hippolyte français du dix-septième siècle_ (Paris: Honoré Champion, 1996), an edition of three seventeenth-century French tragedies treating the Phaedra plot. He has written some thirty articles on various aspects of seventeenth-century French literature, including the more recent "Racine's _Esther_ and the Biblical / Modern Jew" in _Papers in Seventeenth Century French Literature_ 36 (2009), 209-18, and a forthcoming study, "Molière's Miser, Old Age, and Potency" for _Age on Stage_ , ed. Leni Marshall and Valerie Lipscomb (New York: Palgrave Macmillan, forthcoming). Professor Wood is also co-editor of the journal _Global_ _Business Languages_ and Editor in French for the Purdue Studies in Romance Literatures series.
**Zhuangzhuang Xi** is a graduate student in the Department of Psychological Sciences at Purdue University. She has a BS in Psychology and a BA in Economics, both from Peking University. She is a member of The Society for Mathematical Psychology and of The International Association for the Study of Dreams.
**Lisa Zunshine** is the Bush-Holbrook Professor of English at the University of Kentucky, Lexington. Her interests include eighteenth-century British literature, narrative theory, and cognitive cultural studies. She is the author of the first monograph on Theory of Mind and literature: _Why We Read Fiction: Theory of Mind and the Novel_ (Columbus: Ohio State University Press, 2006), and the following books: _Bastards and Foundlings: Illegitimacy in Eighteenth-Century England_ (Columbus: Ohio State University Press, 2005); _Strange Concepts and the Stories They Make Possible: Cognition, Culture, Narrative_ (Baltimore: The Johns Hopkins University Press, 2008). She is co-editor (with Jocelyn Harris) of _Approaches to Teaching the Novels of Samuel Richardson_ (New York: Modern Language Association, 2006), and editor of _Introduction to Cognitive Cultural Studies_ (Baltimore: The Johns Hopkins University Press, 2010).
## Index
Ableson, Robert,
Abraham, Nicolas, , 241n1,
absent character,
_Absorption and Theatricality_ (Fried), , 87n27, 87n32,
aesthetics, , ,
African-Americans, 199n13, , . _See also_ _Beloved_ (Morrison)
agents and agency, , , , , , , 86n9, , , 131n10, , , 273-74, 284-85. _See also_ Wood, Allen G.
agony, 63-64, ,
AIDS, , 88n33
_À la recherche du temps perdu_ (Proust), 13-14
_Alchemies of the Mind_ (Elster),
_Alice in Wonderland_ (Carroll), , , 193-96, 198n5, 198n7-9, 198n11, 207-8, 216n9, 216n11
allegory, ,
Allman, William F., ,
"Alternative Theory of Mind for Artificial Brains" (Marron), , 187-99
_The Ambassadors_ (James),
analogy, , , __ , , _57-58,_ , , , . _See also_ Turner
anchors and anchoring, , 178-79, , 249-56
anthropology, , 16-20, , , 286n1, . _See also_ Bateson, Gregory; Dunbar, Robin; Geertz, Clifford; Sperber, Dan
anti-System (theories),
Aristotle, , , , , ,
art (definition of), 19-20
_The Arthurian Handbook_ (Lacy, Ashe, and Mancoff), , 145n1
Arthurian literature, , 133-48
_Asien gründlich verändert_ (Kisch), , 247-56. _See alsoChanging Asia_
Asimov, Isaac, , 188-97
Asperger, Hans,
Asperger's syndrome, , ,
Astur, Robert, , 259-71
_As You Like It_ (Shakespeare),
attention: anthropomorphism and, ; dreaming and, ; as lexical item, , ; literary representations of, and as a literary device, , 106-14, 117-18, 119n5; theories of, 109-14, 119n6, 120n11, 167-68, 170-71, . _See also_ Shared Attention Mechanism; memory
_At the Crafter's Wheel_ (Midwood), 78-79
attractors, 7-8, 167-70,
attribution theory, 29-39, , , , , , 134-35, , 175-84, , 214n1, , , , . _See also_ Theory of Mind; mind reading
audiences: of children, ; emotions of, 93-97, , , ; imagination of, 99-100; individuals as, , , , 283-86; mass audience, 247-56; models of, 112-13, ; radio and, ; reading, , ; television and, ; theatre and, , , , , , 93-102,
Auerbach, Erich, , 129n3
"Aunt Charlotte and the NGA Portraits" (Turner), 47-54. _See also_ children's literature
Austen, Jane, 6-7, 70-72, , 105-20, 127-29, ,
Author Recognition Tests (Stanovich and West),
authority (definition of), 292-93
autism, , 7-8, , 86n6, 133-48, , 183n5, , 198n12, , 216n9
autofiction, 214n2
Baillie, Johanna, 87n17
Bakhtin, Mikhail, , 130n7, , 243n17
Ball, Lorraine V.,
Banfield, Ann, 153-55, , , 160n2, 161n8
Baron-Cohen, Simon, , , , 86n9, , , 147n12, , ,
Barroso Castro, José, , 289-302
Barston, Julie L.,
Bateson, Gregory,
Batson, C. Daniel,
"The Beast in the Jungle" (James), 183n9
Beck, Aaron T., 102n7
_Befindlichkeit_ (Heidegger),
behavior: autism and, 133-47; brain-imaging and, , , ; evolution and, ; of a group, ; interpretations of, , , , , , 63-70, , 84-85, 86n10, , , , 130n4, 187-99, 201-16, 232-33
behaviorism, ,
belief-desire psychology, , , , 107-8, 166-67,
_Beloved_ (Morrison), 8-9, 229-44
Benjamin, Jessica, , , 243n14
_Beowulf_ , 130n6
Bering, Jesse M.,
billiard table ( _mesa de trucos_ ), , 289-95, 297-300
blending (conceptual), 4-5, 41-61, _43-44_ , __ , _56-60_ , , 131n10
blindness: , ; to change, ; of mind,
_Blossom Time_ (Horsley), 79-80
Blotner, Joseph, 183n6
Boccaccio, Giovanni,
Bockting, Ineke, , 175-85
the body: absence of, , , , , 216n10; double position of, with regard to the mind, , , , , 201-16, ; metamorphosis of, 219-26; as object, , ; practice of, as equivalent to Theory of Mind, 201-16; prosthetic, 190-91, ; relation of, to the mind, , 63-88, ; relative transparency of, 63-88; as text, 67-68, ; simulation (mirror neurons) and, , 275-76; as "theatre of the emotions," , 96-97, ; torture and, 86n15. _See also_ facial expressions; the look; sign language body language, , 63-88, , 201-16, 233-34, ,
Bond, Christopher, 94-95
_The Bonds of Love_ (Benjamin),
Botton, Alain de,
the brain: artificial brains, , 187-99; changes in, with regard to ToM, , 259-70; chemistry of, ; dreaming brain, 225-26; executive functions of, ; experiments on, 260-70, ; implied equivalency of, with the mind, , 208-10; of infants, , , 102n4, ; injury and, ; insufficiency of, to account for "mind," ; language of, ; mind-brain, 19-20; nervous system and, ; the Problem of Other Minds and, , , , 259-72; representations of, in literature, 208-10; size of, ; social brain hypothesis, ; unchanged nature of, . _See also_ evolution; mirroring; mirror neurons
brain imaging, , , , 259-70, _265-67_
brainwashing,
Brantley, Ben,
Brase, Gary L.,
Breithaupt, Fritz, , 146n7, 273-86
_Bridget Jones_ (Fielding), 71-72,
_The Broadcast Tapes of Dr. Peter_ (film), , 88n33
Brown, Thomas E., 110-11,
Bruner, Jerome,
Brustein, Robert,
Bunyan, John,
Busby, Keith,
Butler, Judith,
Butler, Philip,
Butte, George, , 107-9, 118n2, 126-29, 130n6, 131n9, 131n10
Byrne, Richard,
Calderazzo, Diana, , 93-102
Calhoun, Vince, , 259-70
Canaletto (Canal, Giovanni Antonio), 52-54
capitalism, 84-85, 88n38, , 250-52,
Carroll, Lewis, , , 192-98, , 215-16n9, 216n11. _See also_ Dodgson, Charles Lutwidge
Carruthers, P., 147n17
_El casamiento engañoso_ (Cervantes), , 289-302
Cassidy, Kimberly Wright,
Castro, Américo,
catharsis, , , . _See also_ Aristotle
causal generalization, , 266-67
_La Celestina_ (Rojas), , 130n7
Center for Cognitive Literary Studies (at Purdue),
Certeau, Michel de,
Cervantes, Miguel de, , , 123-31, 289-302
Cerveris, Michael,
_The Chair_ (film),
Chalmers, David J.,
change blindness,
_Changing Asia_ (Kisch), , 247-56. _See alsoAsien gründlich verändert_
"Changing Minds" (Knox), , 247-56
characters and characterization (in literature): attribution theory and, 29-40, 72-73, 123-31; cognitive-theoretical approaches to, , 105-20; distraction as positive indicator in, 105-20; Free Indirect Discourse and, , , 175-84; mediating role of, with respect to the reader, 188-99, 201-16; mental states of, 105-20, ; paranormal natures of, 201-16; readers' interpretations of, , 175-84; reception of, from historical perspective, 133-34; relative depth of, 108-9, 112-13; representations of minds of, 133-47, 187-99, 289-302; Theory of Mind of, 125-32, 133-47, , 225-27, 229-44, ; understandings of,
Chekhov, Anton, 22-23
chiasmus,
children: , , , _60_ , , , 178-9, , , , ; autism and, , , 147n4; blending and, 42-47, 53-54; death and, , 214-15n4, 241n2, 242n5, 242n8, 242n9, 243n16, , ; education and, 192-94; as fictional characters, , , , , , , , 229-44; intersubjectivity of, ; language development and, 177-78; memory and, ; social deixis and, ; storytelling and, ; Theory of Mind of, , , 17-18, , 102n4, , , 196-97, , , , 243n12, ,
children's literature, __ , , 216n11, . _See also_ "Aunt Charlotte and the NGA Portraits" _and_ _Harold and the Purple Crayon_
_Chimpanzee Politics_ (de Waal),
chimpanzees, 2-4, 16-18, , , 146n11, . _See also_ the brain; cognitive evolutionary psychology; evolution
Chodorow, Nancy,
Chrétien de Troyes, , 133-47
Christianity (Christ), , , , , 296-300, 301n13
Chu, June Y.,
cinéma vérité, 81-84, 87n29
_Clarissa_ (Richardson), ,
Clark, Andy,
cluster analysis (brain imaging technique), 264-68, 270n5
coding, 222-23
cognitive evolutionary psychology, , 5-6, 16-18, 45-46, 63-70, , , , 126-27, , 130n5, 130n6, 131n9, 131n10, 195-97, , , , 273-74, 278-80, 283-86
cognitive linguistics, , 165-73, 175-84
_The Cognitive Neuroscience of Attention_ (Posner),
cognitive poetics, , ,
cognitive science and scientists: attention and, ; creativity and, 120n11; dream studies and, 220-24; empathy and, , ; hyperfocus and, 110-11; imagination and, , 118n4; laughter and, 100-101; limits of, as far as research questions, , ; literary, cultural studies and, 3-4, 6-7, , , , 84-85, ; music as subject of, , 120n12; neuroscience and, 64-65, 260-62, 270n6; propaganda and, 247-56. _See also_ brain; evolution; memory; mind
Cohen, Morton N., 198n5
Cohn, Dorrit, 87n18, , 183n10, 216n10
Cold War,
Coleridge, Samuel Taylor,
Collier, John,
_Colloquy of the Dogs_ (Cervantes), 289-302
colonialism,
_Commonweal_ ,
communism, 247-56
Communist Party,
compassion, , , , , , , , 286n4. _See also_ empathy
compression (of outer-space analogies in blending), , 54-60, _57-60_
compulsion, 98-99
_A Confession_ (Collier), , _77_
conscience, , 289-302
consciousness, 27-28, , , , 102n6, , , 120n14, , 130n6, , , , 161n9, , 198n10, , 216n10, , , 241n1, 242n6, , 292-300
_Conte del Graal_ (Chrétien de Troyes), , 133-47
contextual thought reports, 30-31
_Continuations_ (of Chrétien de Troyes's unfinished _Conte del Graal_ ), , 145n3
conversation (as form of language), 16-18, . _See also_ language
_Il Convivio_ (Dante),
Cosmides, Leda, 131n9,
Council of Trent, , 298-99
_Count Basil_ (Baillie), 87n17
Craik, Kenneth,
_Crime and Punishment_ (Dostoyevski),
cryptonymy, ,
cultural intelligence hypothesis,
Damasio, Antonio, , , 96-97, 99-100, 102n6,
_A Dance in the Country_ (Renoir), 79-80
Dante, Alighieri,
Darrieussecq, Marie, , 201-16, 216n9
daughters, , , , , , , , 229-44,
Davis, Lucy, 63-64
Davis, Lydia,
_The Deceitful Marriage_ (Cervantes), , 289-302
deep intersubjectivity, 106-9, 118n2, 126-29. _See also_ Butte, George
deictic projection, , 176-80, , 183n11
deixis, , , 175-84,
Dennett, Daniel, , , 86n9, 130n5
Derrida, Jacques, , , 215n6
Descartes, René, , ,
desires, , , , 88n38, , 107-8, , , 165-67, , , , 280-81, , 286n1, . _See also_ emotion; mental states; the mind
detective fiction, 15-16, 189-91, ,
de Waal, Frans, ,
_The Dialogic Imagination_ (Bakhtin), 130n7
dialogism, 30-31,
_Diálogo de la lengua_ (Valdés),
_Diaries_ (Kafka),
_Diccionario de Autoridades,_
Dickens, Charles, , 27-39
Diderot, Denis, , , 119n8
difference (as constitutive of subjectivity), , 259-70
direct cinema, , 87n29
disanalogy, 55-60, _57-58_
disinterest,
_La Disparition_ (Perec), 214n3
disremembering, , 244n20
"Distraction as Liveliness of Mind" (Phillips), , 105-20
distraction (character trait), 105-20
Djikic, Maja,
Dodgson, Charles Lutwidge, 188-99, 198n5. _See also_ Carroll, Lewis
Doležel, Lubomir,
Domhoff, G. William, 220-22
Donald, Merlin, 130n6
Donley, Corrine R.,
Donne, John,
Donnellan, Declan,
Donnelly, Liza,
_Don Quixote_ (Cervantes), , 123-31, 289-90,
Dostoyevski, Fedor,
double-scope integration, , 41-61, 131n10
Doyle, John, 93-103
_Dr. Jekyll and Mr. Hyde_ (Stevenson), , 219-27
drama (genre), 93-103. _See also_ _specific texts, productions, and playwrights_ dreams, 19-20, , 219-26
Drew, Robert,
Drew Associates,
Duckworth, Robinson, 198n5
Dunbar, Robin, , 16-19, 130n5, 130n8,
dyad, 28-29,
dynamic standpoint mapping. _See_ standpoint mapping
Eakin, Paul, , 243n13
education, 135-37, , 147n13,
electronic voice phenomena (EVP),
Elfenbein, Andrew, 120n13
Eliot, George, , ,
Ellis, Jack C., , 120n11
El Saffer, Ruth, ,
Elster, Jon,
embedded narratives, 297-300. _See also_ Palmer, Alan
embodied transparency, , 63-85, 87n18, 216n10, . _See also_ body; brain; mind
_Emma_ (Austen),
Emmott, Catherine, 162n14
emotion: attributions of, ; biological basis of, , 96-97, ; changes to, as a result of literary encounter, ; chimpanzees, of, ; contagiousness of, , ; conveyed by FID, , ; deictics and, ; dreams and, ; emotional resonance, ; empathy and, 243n16, 273-84; images' impacts on, ; imputed through ToM, , , 243n12, 243n16; literary characters, of, 113-16, , 137-38, , , , , , , , , , 243n16, 244n18; memory and, , 244n21; narrative simulation and, ; non-human characters, of, , ; physical indicators of, , 65-66, , 71-82, , 186n13, 187n17, ; propaganda and, ; readers' attribution of, ; relationship of, to mind, 295-96; response in spectator or reader, , 93-102, 165-66, 238-39, 242n10; transparency of, as sadistic, 73-74; understanding and predicting behavior, , . _See also_ brain; Damasio, Antonio
empathy: and ToM, , ; as a cognitive process, 243n16; as social ability, , 18-19, 22-23, ; by literary characters, , , 230-31, , 240-41, 243n16; by spectators , ; cognitive foundations of, ; definitions of, , ; mirroring and, , , , 102n4; models of, , 273-87; readers' for characters, , , ,
_Encyclopédie_ (Diderot), , 119n8
Enders, Jody, 146n8
_Engaging Audiences_ (McConachie),
England, , ,
the Enlightenment, , 275-76
Erasmus, Desiderius,
_erlebte Rede_. _See_ Free Indirect Discourse (FID)
_Essay on Human Understanding_ (Locke),
Evans, J. St. B. T.,
evolution: aesthetics and, ; brain development and, , 45-46; cognitive processes and, , , 63-70, , , 130n6, 131n9, 131n10, ; empathy and, 273-74, 278-81, , ; lag in, ; mental representation and, ; narrative theory and, . _See also_ the brain, chimpanzees, Theory of Mind evolutionary psychology: 64-66, , 130n5, 131n9
EVP (electronic voice phenomena),
"Explaining the Emergence of Autobiographical Memory in Early Childhood" (Nelson),
externalist perspective (on the mind), , 27-38
_Eyes of Love_ (Kern),
facial expressions, , , 33-34, 65-66, 102n4, ,
_Fallen Idol_ (Collier),
_Family Secrets_ (Rushkin), 241n1
fantasy, , , , , ,
Fauconnier, Gilles, , 41-42, 131n10
Faulkner, William, , 175-84
Feeny, Nohr,
feminism and feminist criticism, , 243n13
_Feminism and Psychoanalytic Theory_ (Chodorow),
fiction: as make-believe, ; as model of social world, 19-23; as simulation, , ; attribution theory and, ; diary, ; empathy's necessity to, , 276-77, ; epistolary, ; in the Renaissance, 130n7; internalist perspective on, , ; paranormal nature of, 201-16; representations of minds and, , , 229-44, 283-86; sentimental, ; social role of and responses to, , 16-19, 27-40, . _See also_ detective fiction; embodied transparency; empathy; literature; narrative; _specific works, authors, and theorists_
fiction, science. _See_ science fiction
fictional minds, , , , 108-9, , , , 127-28, , , 158-59, 198n10, 290-93, , 298-300
_Fictional Minds_ (Palmer), , , 86n2, , 160n2,
fictionality, 161n7,
_The Fictions of Language and the Language of Fictions_ (Fludernik),
FID. _See_ Free Indirect Discourse Fidler, Dorothy,
Fielding, Helen, 71-72,
Fielding, Henry,
Fielding, Sarah,
_Fight Club_ (Palahniuk), 73-74
Fillmore, Charles,
Fish, Stanley, 242n6
Fisher King, 142-46
Flaubert, Gustave, 153-54
Flavell, J. H.,
Flerx, Vicki C.,
Fletcher, Pamela M., 76-78
Fludernik, Monika, , , , 216n15
folk pyschology,
Forster, E. M., 108-9
Foucault, Michel,
Fowler, Roger, 162n15
France, ,
Frankel, Richard,
Frappier, Jean,
Free Indirect Discourse (FID; _erlebte Rede_ ), , 153-62
Freud, Sigmund, , 215n4, 215n7,
Freund, Charles Paul,
Fried, Michael, ,
Frith, Christopher D.,
Frith, Uta,
"Functional Brain Imaging and the Problem of Other Minds" (Lloyd, Calhoun, Pearlson, and Astur), , 259-71
functional magnetic resonance imaging (fMRI), , , , , 270n6. _See also_ brain imaging
Gallagher, Helen L.,
Gallese, Vittorio, ,
_The Game of Logic_ (Dodgson), 197n2
games (as analogy to representational strategy of literature),
games, 146n10, , 192-93
Garcia, S. M.,
Gardner, Elysa,
Garfield, Jay L.,
_Gattaca_ (film),
the gaze, , 77-78, 119n5, , , . _See also_ facial expressions; look
Geertz, Clifford,
gender(ing), , , , , 243n18
general crystallized intelligence (gC), 120n12
general fluid intelligence (gF), 120n12
Germany, 248-49
Gerould, Daniel,
Gervais, Ricky, , 80-81,
Gestalt theory, , ,
ghosts, 8-9, 203-7, 213-16, 229-31, 234-35, , 241n1
ghost stories, , ,
Gillespie, Nick,
_Gimme Shelter_ (film),
Girard, René, 286n1
_Gödel, Escher, Bach_ (Hofstadter),
_The Golden Ass_ (Apuleius),
Goldman, Alvin I., 86n3, 86n5, 146n10
Gombrich, Ernst,
Goodall, Jane,
Goodwin, Karen,
gossip,
Grand Guignol, 94-95
Grandin, Temple, , 198n12
Haining, Peter,
Hakemulder, Frank,
Hall, Calvin S.,
_Hamburg Dramaturgy_ (Lessing),
Hamilton, Craig,
_Hamlet_ (Shakespeare), ,
_Harold and the Purple Crayon_ (Johnson), 42-43, __ , . _See also_ children's literature
Harris, Jocelyn, 119n9
Hartman, Ernest,
hauntology, 215n6
Hawke, Ethan,
Headrick, Charlotte,
hearing and overhearing,
_Hearts and Minds_ (film),
Heidegger, Martin,
Hirsh, Jacob,
Hitchcock, Alfred,
Hobson, J. Allan, ,
Hoffmann, E.T.A., 87n18,
Hofstadter, Douglas, ,
holy or "Blessed" fools,
Horsley, John,
Horvath, Rita, 242n6
"How is it Possible to have Empathy? Four Models" (Breithaupt), , 273-88
_How Our Lives Become Stories_ (Eakin), , 243n13
_How Proust Can Change Your Life_ (de Botton),
Hume, David, , , 119n6
Humphrey, Hubert,
Husserl, Edmund,
Hutchins, Edwin,
hyperfocus, 110-11
hypothesis (linguistic function),
_I Know That You Know That I Know_ (Butte), , ,
_I, Robot_ (Asimov), , 188-90
Ickes, William, 15-16
ideational fluency, 118n4, 120n13
identity: group, ; personal, 88n38, 234-36, 243n13
imagination: and the mind's eye, ; and ToM, ; blending and, 41-42, ; conscience and, ; creative, ; empathy and, ; memory and, ; of an audience, , ; power of the, ; role of, with respect to the soul,
_imitatio_ , . _See also_ mimesis
"The Importance of Deixis and Attributive Style for the study of Theory of Mind" (Bockting), , 175-85
initial base-rate bias,
intentionality (intentional states, Intentional Stance), , , 17-19, , , 46-47, 64-70, , 86n9-10, , , 130n5, 131n8, 131n10, , 139-41, , , , 190-91, , , 214n1, 255-56, . _See also_ Dennett, Daniel
intermental thought, 4-5, 28-31, , . _See also_ intersubjectivity
internalist perspective, , 27-28, , , 38-39
the Internet, , , , 157n19
interpretive communities, 242n6
intersubjectivity, , 106-07, 118n2, 126-29, , 234-39, 243n14, , . _See also_ deep intersubjectivity; intermental thought
intertextuality, , 216n9
_In the Lake of the Woods_ (O'Brien), 214n3
intramental thought (private thought),
intuition,
irony, , ,
Irwin, John, 183n6
Isen, Alice M., 120n11
Isherwood, Charles,
Islam,
_Is There a Text in This Class?_ (Fish), 242n6
Jack the Ripper,
Jakobson, Roman,
James, Henry, , , 183n9
Johnson, Crockett,
Johnson, Mark, , 97-98
Johnson, Robert A., 146n8
Jowett, Garth,
judgment, , , 298-99
Kafka, Franz, , 219-20, 224-26
Kahn, David, ,
Kandel, Eric,
Kanner, Leo,
Kant, Immanuel,
Kauffmann, Walter,
Kawin, Bruce, 183n6
Kecskemeti, Paul,
Keen, Suzanne, 243n16
Keller, Jennie W.,
Kennedy, John F.,
Kenny, Anthony,
Kern, Stephen, 77-80
Keskinen, Mikko, , 201-17
Kierkegaard, Søren,
Kisch, Egon Erwin, , 247-56
knighthood, (knight errant), , , , , 136-43, 147n18
Knox, Seth, , 247-56
Kurosawa, Akira, 215n5
"The Lady with the Little Dog" (Chekov), 22-23
Lahr, John,
Lakoff, George,
language: anti-patriarchal versions of, ; blending and, , 54-55, _56-60_ ; brain imagining's hopes for, 268-69; conversation's requirements and, 16-18; empathy and, 174, ; in FID, , 158-59, 160n2-3; mistrust of, by the traumatized, ; necessity of, for ToM activities, 146n10, ; origins of, 16-20; schizophrenic, 179-80; social function of, 16-17, , , , ; un-naturalness of, . _See also_ analogy; body language; deictic position; deixis; metaphor; Saussure; sign language
_The Language of Psychosis_ (Rosenbaum and Sonne),
Larocque, Laurette,
Latta, Robert,
laughter, 100-101
_Lazarillo de Tormes_ (anonymous), , 130n7
Leech, Geoffrey, , 162n15
Leek, Frederike van der,
Lessing, Gottfried, 275-78
Leverage, Paula, , , 133-47
Levin, Daniel T.,
Levinas, Emanuel,
Levitin, Daniel,
"Liar!" (Asimov), 191-92
Liddell, Alice, , 198n5
_Light in August_ (Faulkner),
linguistics, , 216n13. _See also_ cognitive linguistics; _specific linguists_
_Linguistics and the Novel_ (Fowler), 162n15
literalness, . _See also_ autism; Chrétien de Troyes
_The Literary Mind_ (Turner),
literary studies: "cognitive turn" in, ;
ToM's functionality in,
literature: aesthetics and, 281-83; attribution theory and, , , 175-85; children's, 42-43, 47-61, ; as cultural representation, 70-85; effect of, on ToM, ; fictional minds in, 289-302; ghost literature, 201-16; mimetic focus on, ; place of, as human enterprise, ; positive (mental) effects of, 22-23; as rule-based system, 188-99; social (conversational) aspects of, 17-19; ToM's necessity to the functioning of, 1-3. _See also_ characters and characterization; fiction; language; narrative; _specific authors, works, and theorists_
Literature and Cognitive Science Conference,
_Little Dorrit_ (Dickens), , 29-40
Litz, A. Walton, 120n14
Lloyd, Dan, , 259-72
Locke, John, , , 119n6, 139n9
Lodge, David,
logic, , , , , , 187-99, , , , , , , . _See also_ symbolic logic
the look, , 36-38
love, 295-97
Lutz, Donna J.,
Lyons, John,
Machiavellian intelligence, 124-26
_Madame Bovary_ (Flaubert), 153-54
_Mademoiselle Scudery_ (Hoffmann),
magnetic resonance, . _See also_ functional magnetic resonance imaging (fMRI)
Mahler, Anna,
Mahler, Gustav,
Mahler-Werfel, Alma,
Mallory, Thomas, , 146n6
Mancing, Howard, , 123-31
Mar, Raymond, ,
Marlowe, Christopher,
Marron, Orley K., , 187-99
Martin, Jay, 183n6
_Master Flea_ (Hoffmann), 87n18
Mauron, Charles,
McConachie, Bruce, , 97-98, , 102n5
McCracken, Peggy, 147n21
McCullough, Ann, , 147n13
McEwan, Ian,
McLane, Betsy A.,
Medina, John,
melodrama, 93-95,
memory: , 102n6, 146n8, 198n9, 198-99n12, , , 243n16, 244n21, , , ; autobiographical memory, ; collective memory, 239-40; as externalized, ; working memory, 120n13. _See also_ trauma
mental model, 15-18, , ,
mental states: as impetus for bodily action, 68-70, ; intentional state, 17-18, , ; models' or representations' relation to, 16-17, 31-32, 41-61, , 146n10; novels and, ; physicality of, ; relative visibility of, 63-85. _See also_ brain; Dennett, Daniel; embodied transparency; intermental thought; mentalizing; mind
mentalizing, , , 146n10
Merchant, Stephen, 63-64, 80-81
Merleau-Ponty, Maurice, ,
metacognition,
_The Metamorphosis_ (Kafka), , 219-26
metaphor, , 19-20, , 208-11, 216n13, 289-90
metaphysics, , ,
metarepresentations, 187-99, , , __ , 255-56
metonyms, ,
Mical, Thomas, 215n6
_Middlemarch_ (Eliot), 36-38
_A Midsummer Night's Dream_ (Shakespeare), ,
Midwood, William,
Miller, D. T.,
mimesis, , . _See also_ representation _Mimesis_ (Auerbach),
the mind: alien and artificial minds, 187-99; analogous models of, 45-46; animals and, , , , ; autism's effect on the opacity of, 133-47; judgment, intuition, and, , , , , , ; mental spaces and, 41-60; "mind's eye" and, 153-63, ; as ontology, 297-99; projection and, , 43-46, , 275-78, 283-84; relationship of, to the body, , ; relative visibility of, through the body, 63-85, 201-16, , ; representations of, in characterizations, 107-20, 281-83, 290-92, , , , . _See also_ brain; cognitive evolutionary psychology; language; Theory of Mind (ToM)
mindblindness, , 138-39, . _See also_ autism
mind reading. _See_ Theory of Mind _Mindreading_ (O'Connell),
mirroring (cognitive), , 93-102, , , 275-80,
mirror neurons, 65-66, 86n5, 86n6, 215n9, , , ,
Mithen, Steven, 19-20
_Monty Python and the Holy Grail_ (Gilliam and Jones), , 145n1
morality (mind's relation to), 146n10, 291-303
Moreira-Slepoy, Garciela,
Moreno Báez, Enrique, 302n17
Morón Arroyo, Ciriaco, , , 301n3, 301n12, 302n19
Morrison, Toni, 8-9, 229-45
_Morte d'Arthur_ (Mallory),
"Mother-Child Reminiscing and Children's Understanding of the Mind" (Reese and Sutcliffe),
"Mother / Daughter Mind Reading and Ghostly Intervention in Toni Morrison's _Beloved_ " (Priborkin), , 229-44
Muñoz Iglesias, Salvador,
music, , , 98-99
_Nachwelt_ (Streeruwitz), , 153-62
Nagel, Thomas, 260-61, 268-69
_Naissance des fantômes_ (Darrieussecq), , 201-16
naïveté, 133-47
_Nameless and Friendless_ (Osborn),
_Narrating Modernity_ (Fletcher), 76-78
narrative and narrators: assimilation of, as schemas, , ; attribution theory and, 29-40; consistency of, as necessity, ; covert, 153-62; embedded narratives, , , , 198n10, , , 297-300; empathy and, , , , 243n16, 273-86; evolutionary theory and, ; first-person, , , , , , , , , ; Fludernik and Richardson on, , , 216n15; internalist and externalist perspectives on, ; internalist bias of, , ; master narratives and, ; narrative theory, , , , , , 162n14, , ; performativity and, 167-72; relative reliability of, 74-75, 87n16, , 203-4, , , 215n5; self-narration, 236-38; strategies of, , , ; third-person, , narrative and narrators _(continued)_ 153-60, 161n6, 162n14, , 238-39; threatened subsuming of, from without, ; travel narratives, 253-55. _See also_ fiction; Free Indirect Discourse (FID); literature
"Narrative Empathy" (Keen), 86n13, 243n16
National Public Radio, ,
Nelson, Katherine,
_Neue Sachlichkeit_ (movement),
"Neurology of Narrative" (Young and Saver), 243n15
neuroscience, , 260-69; relation of, to reading, . _See also_ cognitive science and scientists
"New Objectivity" (movement),
_New Republic,_ ,
_New York Times,_ 94-95,
Nichols, Nichelle, 199n13
Niven, Larry, 188-89, 197n3
_No Lies_ (film),
_Novelas ejemplares_ (Cervantes), , 289-302
novels. _See_ _specific authors and works_
Nussbaum, Martha, , , 286n4
Oates, Joyce Carol,
Oatley, Keith, , , 13-26,
objectivity, , , , , , , 268-69
O'Brien, Tim, 214n3
obsession, , , , 98-99,
O'Connell, Sanjida, ,
O'Connor, Flannery, 183n11
O'Donnel, Victoria,
Oedipus: , ; Oedipal complex, _The Office_ (Gervais and Merchant), 63-64, _64,_ 80-81, 83-84
"Of Heartache and Head Injury" (Richardson), 120n14
"Of Tragedy" (Hume),
Olim Kahn,
Olin Neuropsychiatry Research Center,
optics, 259-61
Orkney Islands, 46-60
Osborn, Emily Mary,
_Othello_ (Shakespeare), , ,
other minds, problem of, , 13-15, 29-31, 55-56, , 70-85, , 259-70
Pace-Schott, Edward,
paintings, 76-80
Palahniuk, Chuck, 73-75
Palmer, Alan: , , 86n8, 86n13, 146n9; embedded narratives and, ; on fiction, , , ; _Fictional Minds_ , , , 86n2, , 160n2, ; intentional states and, ; intermental thought and, , 27-40, 198n10, 231-32, ; "Intermental Thought in the Novel," ; mind-ruts and, ; narrative theory and, , ; "Social Minds in _Little Dorrit,_ " , 27-39; subjectivity and, 160n3; "theory theory" and, 214n1; on thought's active features, ; on thought's relative verbal nature, 161n8
_Pamela_ (Richardson),
"Pantaloon in Black" (Faulkner), 181-82
_Paradies Amerika_ (Kisch), , ,
paranormal, , 201-16. _See also_ ghosts
Parasuraman, Raja,
Park, D. C.,
"Parker's Back" (O'Connor), 183n11
_Parzival_ (play),
Pascal, Roy, , 160n3
patients (as opposed to agents),
Paz, Jennifer dela,
Pearlson, Godfrey, , 259-71
penance, 298-300
perceptions. _See_ senses and the sensory
Perceval, 133-47. _See alsoParzival_
Perec, Georges, 214n3
performative rhetoric,
performativity, 66-68, 80-84, 92-102, 167-72, 201-14, . _See also_ body; body language; language
Perkins, Alexis,
Perner, Josef,
Perry, Tricia,
perspective, , , ,
_Persuasion_ (Austen), , 118n2, 120n14,
persuasion (definition),
Peterson, Jordan B., ,
Petterson, Candida C.,
Pfau, Thomas,
_Phèdre_ (Racine), 7-8, 165-73
Phelan, James, 87n16, , 183n9
Phelan, Peggy,
phenomenology, , , , , 270n2
Phillips, Natalie, , 105-20
philosophy. _See_ other minds, _specific philosophers_
physicalism,
_The Pilgrim's Progress_ (Bunyan),
pity, , , 274-77,
Plato,
plays (genre), , , , , , 93-102, 165-73
_Poetics_ (Aristotle), ,
Pollard, Paul,
Posner, Michael,
possible worlds, 290-91, , 299-300
post-structuralism,
Powers, Christopher, , 242n7
practical reasoning, 294-99
practice of body theory, , , 201-16
_Praise of Folly_ (Erasmus),
Pratchett, Terry,
Premack, David, ,
Priborkin, Klarina, 8-9, 229-44
_Pride and Prejudice_ (Austen), , 70-72, , 105-20
_Primary_ (Drew),
Prince, Hal,
private thought, 28-31,
problematization (term),
propaganda, , 247-56. _See also_ metarepresentations
prototypes, 290-91
Proust, Marcel, 13-15, 22-23
Provine, Robert R.,
_Psychiatric Times_ ,
psychoanalytic theory, 135-36, ,
psychology: , , , , , , , 120n11, , , , , , , , , 216n14, , 243n12, 244n21, , , , , , , ; of characters' minds, , , , , , 117-18, , . _See also_ cognitive evolutionary psychology; _specific psychologists_
public mind. _See_ social minds
PubMed (database),
punctuation (role of, with respect to deixis), 178-79. _See also_ signs (graphic)
Purdue University, , , 146n7
Rabelais, François, 130n7
race: in explaining Perceval's behavior, 135-37; race relations and propagandized representation, ; racial discrimination (in _Beloved_ ), ; representation of interracial relationships, 199n13
Racine, Jean, 7-8, 165-73
Radio Moscow,
rationality theory,
reading and readers: attentions of, ; attribution theory and, 29-39, , ; Baron-Cohen's models and, , ; cognition's visibility and, 5-6, 105-20; as driving literary changes, ; moral danger to, 289-302; neuroscience and, ; reception and reception theories and, 6-8, 105-20, 133-48, 158-60, 165-73, 187-99; representations of, in reading and readers _(continued)_ fiction, , 111-16; ToM's necessity in, 1-2, 154-55, , 187-99; transparency's illusion and,
"Reading Phantom Minds" (Keskinen), , 201-16
reciprocity (of empathy), , 278-81,
"Récit de Théramène" (part of _Phèdre_ ), 7-8, 165-73
Red Guard, Austrian,
Reese, Elaine, ,
remorse, , , 294-95,
Renaissance, , , 130n7, 215n8, ,
Renoir, Pierre-Auguste,
representation: , , , 64-66, , 106-8, 120n14, , , , , 146n10, , , , , , , , , , , 252-56, , , , , , ; and blending, 48-54; cultural, 66-85. _See also_ literature; mimesis; mind; Theory of Mind (ToM)
_Reproduction of Mothering_ (Chodorow),
Republic of Literature, 289-90
the requirement hypothesis, 14-16,
resonance (manipulation technique), , , , 252-53,
responsibility: 286n2, ; of the reader, 292-93
Richardson, Alan, 87n17, 120n14, 146n9
Richardson, Brian, , , 215n5
Richardson, Samuel, , , 87n21
_Ring World_ (Niven), , 197n3
Rizzolatti, Giacomo, , ,
Roach, Joseph,
Robinson, John A., , 243n18
robot, , , 187-99, ,
Rockwell, John,
Roddenberry, Gene, , , 199n13
Rogers, Ronald W.,
Rohmer, Eric, , 145n1, 146n7
Rojas, Fernando de, , 130n7
_Romeo and Juliet_ (Shakespeare),
Rosenbaum, Bent, 178-80, 183n7
Rouch, Jean,
Rourke, Mary T.,
Rousseau, Jean-Jacques,
Royle, Nicholas, 216n14
rule, 87n18, , , 188-197, , ,
"Runaround" (Asimov), 189-90
_Rushes_ (film),
Rushkin, Esther, 241n1
Ryden, Hope, , 87n30
"Sancho Panza's Theory of Mind" (Mancing), , 123-31
_Santa Cruz Lectures on Deixis_ (Fill-more),
sarcasm,
Sargent-Baur, Barbara Nelson, 146n8
Saussure, Ferdinand de, , 216n13
Saver, Jeffrey L., 243n15
Schacter, Daniel, 244n21
schema, , , , , , , ,
schizophrenia, , , 179-80, 183n6
Schneider, Adam, 220-2
Scholasticism, , ,
Schwarz, Joel, 243n12
Schwarz, N., ,
Schweickert, Richard, , , 219-26
science, , , 160n1, 260-62, 268-69, 270n1, 270n2
science, cognitive. _See_ cognitive science and scientists
science fiction, , , 187-99
Scott, Sir Walter, , ,
script (cognitive science term), , 124-25
_Searching for Memory_ (Schacter), 244n21
Searle, John,
secret discourse,
the self-improvement hypothesis, 22-23
selkie (mythical being), 46-60, 60n1
senses and the sensory, , , , 86n15, , , , , , 161n4, , . _See also_ body; facial expressions; Merleau-Ponty, Maurice
Serres, Michel,
Shakespeare, William, , 18-20, 102n1, , 131n8, , ,
Shank, Roger,
Shared Attention Mechanism, ,
Short, Michael H., , 162n15
Skurnik, I.,
sign language, 33-36,
signs (graphic), , ,
Simenon, Georges, 14-15
simile,
Simons, Daniel J.,
Simons, Penny,
simulation, , 19-23, , , ,
simulation theory, 86n3, , 147n17, 214n1, 216n9. _See also_ Theory of Mind
Sindan (prison), ,
slavery, , 229-44,
small-clause, 180-4
small kindness perception, ,
Smarties task,
Smith, Adam, ,
Smith, P., 147n17
social brain hypothesis,
social constructionism,
the social improvement hypothesis, 18-19,
social minds, , 27-39. _See also_ intermental thought; intersubjectivity
solipsism, , 198n10, . _See also_ other minds
Sondheim, Stephen, , 93-102
Sonne, Harly, 178-80, 183n7
_The Sound and the Fury_ (Faulkner), , , 177-80
source-monitoring, . _See also_ source-tracking
source-tracking, , , 253-56. _See also_ source-monitoring
Soviet Union, , , , ,
Spain,
Spanish Golden Age, , 289-301
spectator(s): , 67-68, , , 93-102, 165-67, 244n18, , , , . _See also_ reading and readers
_Specters of Marx_ (Derrida),
Sperber, Dan, ,
Spitzer, Leo,
Spolsky, Ellen, , 86n11, 86n12, 87n19
Squire, Larry,
St. Augustine, , 295-96
St. Thomas, ,
standpoint mapping, 261-63, 268-69, 270n2
Stanovich, Keith,
_Star Trek_ (Roddenberry), , , 199n13
Steiner, Peter,
Stevenson, Helen, 215n9, 216n10
Stevenson, Robert Louis, , , 219-26
Stewart, Avril,
Stinson, Linda, 15-16
Stockholm syndrome, 278-81
Stockwell, Peter, ,
Streeruwitz, Marlene, , 153-62
_style indirect libre._ _See_ Free Indirect Discourse (FID)
subjectivity, , , , , , , 118n1, , , , , , , , 160n3, , , , , , , , 259-70, , . _See also_ deep subjectivity; intersubjectivity
Sutcliffe, Emily, ,
_Sweeney Todd_ (Sondheim), , 93-102
symbolic logic, 191-93
_Symbolic Logic_ (Dodgson), , 197n4
synecdoche,
Tajik, Tajikistan, , , 254-55
Tanner, Tony,
Taylor, B.A.,
Taylor, Marjorie, 243n12
Taylor, Philip M.,
telepathy, , , , , 216n14. _See also_ Grandin, Temple
_The Tempest_ (Shakespeare),
theater, , , 93-102, , , , . _See also_ _specific works, productions, and playwrights_
"Theory of a Murderous Mind" (Calderazzo), , 93-102
Theory of Artificial Mind (ToAM), , 187-99
Theory of Mind (ToM), 1-3: and brain size, ; child development and, , , , , 243n12; comparative, ; connections with free indirect discourse, ; conscience and, 289-301; deficiencies of, in literary characters, 133-48, 178-80, ; definitions of, , 13-15, , , 64-65, 134-35, , ; dispositions and, , , , ; dreams and, , 219-26; empathy and, , , , , , , 243n16, 273-86; evolution and, , , 16-19, , 63-65, , 130n5, 131n10, ; experiments on, , , , ; as "hungry" adaptation, 64-67, ; intermental, 28-39; levels of, in Dunbar, ; literary theory and, , 153-54, , ; literature's role in stimulating, , ; mental models or representations and, 16-17, 41-60, 133-48, ; as misnomer, ; models of approximation and, 281-83; narrator's, , 181-82, 206-07, ; as necessary to understand intentionality of others, 17-18, 63-85, 131n10; in non-humans, 2-3, ; as Practice of Body, 201-16; propaganda and, , 247-56; rationality theory, ; of readers, , , , , , , 154-55, , , , ; reasoning and, , , , 295-99; requirement hypothesis and, 14-16, ; simulation theory, 86n3, , 147n17, ; social improvement hypothesis and, 18-19; Spanish Golden Age and, 293-95; temporal organization, ; "theory theory," , 214n1, ; visual processing capabilities and, 140-42. _See also_ body; brain; language; literature; mind; narrative; reading; _specific theorists and researchers_
"Theory of Mind and Literature" (conference), 4-5,
"Theory of Mind and Metamorphoses in Dreams, _Jekyll & Hyde,_ and _The Metamorphosis_ " (Schweickert and Xi), , 219-26
"Theory of Mind and the Conscience in _El casamiento engañoso_ " (Barroso Castro), , 289-302
Theory of Minds (plural), 19-23
Thompson, Evan, , 286n2
thought (as preceding language), ,
_Through the Looking Glass_ (Carroll), 193-99
_Tom Jones_ (Fielding),
Tomasello, Michael, , , 131n10, 146n11
Tomlin, Russell,
Tooby, John, 131n9,
Torok, Maria, , 241n1
Torrance, Ellis Paul, 120n11
_Totem and Taboo_ (Freud), 215n7
tragedy, , , , , , , , . _See also_ _specific tragedies_
_Tragedy and Philosophy_ (Kauffmann),
trajectors, 7-8, 170-72
trauma, , , , 230-35, , 241n1, 243n18, 244n21, 282-83
travel literature,
travel narrative, , 247-56
truth-judgment, 181-82
Turner, Mark, 4-6, , 41-60, 131n10, 160n1, 216n13
Turner, Michelle A., 119n4
Ukraine, Ukrainian, ,
University of Kharkov,
UpDater,
_USA Today,_
Uzbek, Uzbekistan, , , 254-55
Les Vacances de Maigret (Simenon), 14-15
Valdés, Juan de,
Van de Castle, Robert L., ,
Venegas, Alejo de,
_Vile Bodies_ (Waugh), , 198n10
visual processes, 140-42, 161n4, , , , , , , , , 259-63
Watt, Ian, , 130n7
Waugh, Evelyn, , 198n10
_The Way by Swann's_ (Proust), 13-14
_The Way We Think_ (Turner and Fauconnier), , , , , 60n2, 131n10
Weaver, K.,
Weimar Republic,
Werner, Rebecca Stetson,
Wertsch, James,
West, Richard,
Whiten, Andrew,
"Whose Mind's Eye" (William), , 153-62
"Why Jane Austen Was Different, and Why We May Need Cognitive Science to _See_ It" (Zunshine), , ,
_Why We Read Fiction_ (Zunshine), , , 86n2, , , 161n5, , 214n1, 216n14
Wild Boy of Aveyron,
Wilde, Oscar,
William, Jennifer Marston, , , 153-62
Williams, Gladys,
Williams, Harry F., 147n22
Williamson, Edwin,
Wilson, Robert, , , 146n7
Wimmer, Heinz,
Woloch, Alex, , 119n10
Wood, Allen G., 7-8, 165-73
Woodruff, Guy, , ,
Woolf, Virginia,
World War I, ,
Wren, Celia, ,
Xi, Zhuangzhuang, , 219-26
Yoon, C.,
Young, Kay, 243n15
Zadan, Craig, ,
_Zaren, Popen, Bolschewiken_ (Kisch), 248-50
Zoeterman, Sara,
Zunshine, Lisa: , , , , 130n6, 131n10, 146n9, 162n17, , , , 214n1, 216n10, , ; background of, 131n9; on detective fiction, , ; embodied transparency, ; enjoyment of mind reading, ; representation's role in literature, 107-8, , ; "Theory of Mind and Fictions of Embodied Transparency," 5-6, 63-88; ToM's definition, . _See also_ "Why Jane Austen Was Different, and Why We May Need Cognitive Science to _See_ It,"; _Why We Read Fiction_
| {
"redpajama_set_name": "RedPajamaBook"
} | 5,384 |
\section{Ответы к некоторым упражнениям}
\begin{enumerate}
\addtocounter{enumi}{3}
\item
\begin{enumerate}
\item
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=1em]{
\inetcell(F){$F$}[R] &
\inetcell(And){$\text{And}$}[L] \\ };
\inetwirefree(And.left pax)
\inetwirefree(And.right pax)
\inetwire(F.pal)(And.pal)
\node (p) [right=of And.pal] {$\phantom{p}$};
\node (m) [right=of And.above right pax] {$m$};
\node (x) [right=of And.above left pax] {$x$};
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(F){$F$}[R] \\
\node (p) {$\phantom{p}$}; \\
\inetcell(e){$\epsilon$}[R] \\ };
\inetwirefree(F.pal)
\inetwirefree(e.pal)
\node (m) [right=of F.above pal] {$m$};
\node (x) [right=of e.above pal] {$x$};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=1em]{
\inetcell(T){$T$}[R] &
\inetcell(And){$\text{And}$}[L] \\ };
\inetwirefree(And.left pax)
\inetwirefree(And.right pax)
\inetwire(T.pal)(And.pal)
\node (p) [right=of And.pal] {$\phantom{p}$};
\node (m) [right=of And.above right pax] {$m$};
\node (x) [right=of And.above left pax] {$x$};
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\node (m) {$m$}; \\
\node (p) {$\phantom{p}$}; \\
\node (x) {$x$}; \\ };
\inetwirecoords(m)(x)
\end{tikzpicture}
$$
\item Обойдемся без иллюстрации.
Cеть содержит активную пару $T \bowtie \text{And}$, которая порождает $F \bowtie \text{And}$.
Это взаимодействие приводит к созданию нового агента $F$ и активной пары $T \bowtie \epsilon$.
Через три шага получаем $\text{False}$.
\end{enumerate}
\addtocounter{enumi}{1}
\item
\begin{align*}
\text{Mult}[\epsilon, 0] &\bowtie 0; \\
\text{Mult}[\delta(x, y), z] &\bowtie S[\text{Mult}(y, \text{Add}(x, z))].
\end{align*}
\item Сети взаимодействия детерминированны по своей сути, обладая свойством так называемой сильной конфлюэнтности.
Из этого свойства следует, что любые последовательности редукций приводят сеть к одному и тому же состоянию.
\end{enumerate}
\section{Комбинаторы взаимодействия}
Подобно тому, как $K$ и $S$ в комбинаторной логике способны представить любые вычислимые функции, агенты $\delta$, $\gamma$ и $\epsilon$ порождают универсальную систему взаимодействия.
Эти три агента показаны на рисунке~\ref{inetcomb}.
Первые два из них осуществляют мультиплексирование (т.~е. объединение двух связей в одну), а третий~---~удаление.
Оказывается, соединяя определенным образом исключительно эти три комбинатора взаимодействия, можно заменить вообще любую сеть взаимодействия.
\begin{figure}[h]
$$
\begin{tikzpicture}[baseline=(c)]
\inetcell(c){$\gamma$}
\inetwirefree(c.pal)
\inetwirefree(c.left pax)
\inetwirefree(c.right pax)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(c)]
\inetcell(c){$\delta$}
\inetwirefree(c.pal)
\inetwirefree(c.left pax)
\inetwirefree(c.right pax)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(c)]
\inetcell(c){$\epsilon$}
\inetwirefree(c.pal)
\end{tikzpicture}
$$
\caption{комбинаторы взаимодействия ($\gamma$, $\delta$ и $\epsilon$).}
\label{inetcomb}
\end{figure}
Система комбинаторов имеет шесть правил взаимодействия, которые показаны на рисунке~\ref{combrule}.
Очевидно, что агент $\epsilon$ ведет себя как операция удаления, <<съедая>> все, с чем он взаимодействует.
В свою очередь, мультиплексирующие агенты либо аннигилируют (если агенты совпадают), либо дублируют друг друга (если они различны).
Заметим, что в последнем правиле взаимодействия правая часть является пустой сетью.
Система комбинаторов взаимодействия \textit{универсальна} в том смысле, что внутри нее можно представить любую другую систему взаимодействия.
Впрочем, известны также и другие универсальные системы взаимодействия, хотя они так или иначе сложнее системы комбинаторов.
\begin{figure}[h]
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(t){$\delta$}[D] \\
\node (p) {$\phantom I$}; \\
\inetcell(b){$\delta$}[U] \\ };
\inetwirefree(t.left pax)
\inetwirefree(t.right pax)
\inetwirefree(b.left pax)
\inetwirefree(b.right pax)
\inetwire(t.pal)(b.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell[opacity=0](t){$\delta$}[D] \\
\node (p) {$\phantom I$}; \\
\inetcell[opacity=0](b){$\delta$}[U] \\ };
\inetwirecoords(t.above left pax)(b.above left pax)
\inetwirecoords(t.above right pax)(b.above right pax)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(t){$\gamma$}[D] \\
\node (p) {$\phantom I$}; \\
\inetcell(b){$\gamma$}[U] \\ };
\inetwirefree(t.left pax)
\inetwirefree(t.right pax)
\inetwirefree(b.left pax)
\inetwirefree(b.right pax)
\inetwire(t.pal)(b.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell[opacity=0](t){$\gamma$}[D] \\
\node (p) {$\phantom I$}; \\
\inetcell[opacity=0](b){$\gamma$}[U] \\ };
\inetwirecoords(t.above left pax)(b.above right pax)
\inetwirecoords(t.above right pax)(b.above left pax)
\end{tikzpicture}
$$
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(t){$\gamma$}[D] \\
\node (p) {$\phantom I$}; \\
\inetcell(b){$\delta$}[U] \\ };
\inetwirefree(t.left pax)
\inetwirefree(t.right pax)
\inetwirefree(b.left pax)
\inetwirefree(b.right pax)
\inetwire(t.pal)(b.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(tl){$\delta$}[U] & & \inetcell(tr){$\delta$}[U] \\
& \node (p) {$\phantom I$}; & \\
\inetcell(bl){$\gamma$}[D] & & \inetcell(br){$\gamma$}[D] \\ };
\inetwirefree(tl.pal)
\inetwirefree(tr.pal)
\inetwirefree(bl.pal)
\inetwirefree(br.pal)
\inetwire(tl.left pax)(bl.right pax)
\inetwire(tl.right pax)(br.right pax)
\inetwire(tr.left pax)(bl.left pax)
\inetwire(tr.right pax)(br.left pax)
\end{tikzpicture}
$$
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(t){$\delta$}[D] \\
\node (p) {$\phantom I$}; \\
\inetcell(b){$\epsilon$}[U] \\ };
\inetwirefree(t.left pax)
\inetwirefree(t.right pax)
\inetwire(t.pal)(b.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(t){$\epsilon$}[U] &
\node (p) {$\phantom I$}; &
\inetcell(b){$\epsilon$}[U] \\ };
\inetwirefree(t.pal)
\inetwirefree(b.pal)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(t){$\gamma$}[D] \\
\node (p) {$\phantom I$}; \\
\inetcell(b){$\epsilon$}[U] \\ };
\inetwirefree(t.left pax)
\inetwirefree(t.right pax)
\inetwire(t.pal)(b.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(t){$\epsilon$}[U] &
\node (p) {$\phantom I$}; &
\inetcell(b){$\epsilon$}[U] \\ };
\inetwirefree(t.pal)
\inetwirefree(b.pal)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(t){$\epsilon$}[D] \\
\node (p) {$\phantom I$}; \\
\inetcell(b){$\epsilon$}[U] \\ };
\inetwire(t.pal)(b.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell[opacity=0](t){$\epsilon$}[U] &
\node (p) {$\phantom I$}; &
\inetcell[opacity=0](b){$\epsilon$}[U] \\ };
\end{tikzpicture}
$$
\caption{правила взаимодействия для комбинаторов.}
\label{combrule}
\end{figure}
\section{Полнота по Тьюрингу}
\label{complete}
Модель вычислений полна по Тьюрингу, если любая вычислимая функция представима в ней.
В случае сетей взаимодействия, доказать полноту по Тьюрингу можно, например, представив комбинаторную логику~---~систему комбинаторов с константами $S$ и $K$ и двумя правилами редукции, имеющую ту же мощность с точки зрения вычислений, что и $\lambda$-исчисление:
\begin{align*}
K\ x\ y &\rightarrow x; \\
S\ x\ y\ z &\rightarrow x\ z\ (y\ z).
\end{align*}
Чтобы представить эту систему в сетях взаимодействия, нам потребуется агент $@$, соответствующий аппликации, а также несколько агентов, представляющих сами комбинаторы.
В частности, комбинатор $K$ будет представлен двумя агентами $K_0$ и $K_1$ со следующими правилами взаимодействия.
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=1em]{
\inetcell(k){$K_0$}[R] &
\inetcell(a){$@$}[L] \\ };
\node (p) [right=of a.pal] {$\phantom x$};
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwire(a.pal)(k.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(k)]
\inetcell(k){$K_1$}[U]
\inetwirefree(k.pal)
\inetwirefree(k.middle pax)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(a){$@$}[D] \\
\node (p) {$\phantom x$}; \\
\inetcell(k){$K_1$}[U] \\ };
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwirefree(k.middle pax)
\inetwire(a.pal)(k.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix[row sep=2em]{
\inetcell[opacity=0](k){$\phantom{K_1}$}[U] &
\inetcell(e){$\epsilon$}[U] \\ };
\node (p) [left=of e] {$\phantom x$};
\inetwirefree(e.pal)
\inetwirecoords(k.above middle pax)(k.above pal)
\end{tikzpicture}
$$
Комбинатор $S$ может быть представлен аналогичным образом тремя агентами и тремя правилами; мы оставим эту задачу читателю в качестве упражнения.
Сети взаимодействия используются и для реализации $\lambda$-исчисления.
Действительно, первая реализация оптимальной стратегии для $\lambda$-исчисления (т.~е. стратегии, которая выполняет наименьшее число $\beta$-редукций, необходимое для достижения нормальной формы терма) формально являлась системой взаимодействия.
Сети взаимодействия также используются и для других (неоптимальных, но в некоторых случаях более эффективных) реализациях $\lambda$-исчисления.
На самом деле, если мы ограничимся линейным $\lambda$-исчислением, то нам потребуются только два агента $@$ и $\lambda$, представляющих, аппликацию и абстракцию, соответственно; при этом роль переменных играют связи.
В этом случае $\beta$-редукция будет соответствовать следующему правилу взаимодействия.
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(a){$@$}[D] \\
\node (p) {$\phantom x$}; \\
\inetcell(l){$\lambda$}[U] \\ };
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwirefree(l.left pax)
\inetwirefree(l.right pax)
\inetwire(a.pal)(l.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell[opacity=0](a){$\phantom @$}[D] \\
\node (p) {$\phantom x$}; \\
\inetcell[opacity=0](l){$\phantom \lambda$}[U] \\ };
\inetwirecoords(a.above left pax)(l.above left pax)
\inetwirecoords(a.above right pax)(l.above right pax)
\end{tikzpicture}
$$
Чтобы представить произвольные $\lambda$-термы, нам пришлось бы ввести агенты для копирования и удаления.
Также нам понадобились бы дополнительные агенты, которые бы могли контролировать область действия абстракций.
\section{Упражнения}
\label{exercise}
\begin{enumerate}
\item В сетях взаимодействия, определить следующие арифметические операции для натуральных чисел, представленных нулем $0$ и функцией следования $S$:
\begin{itemize}
\item проверка на нуль $\text{Zero}$, которая приводит к результату <<истина>> $\text{True}$, если число равно нулю $0$, и к результату <<ложь>> $\text{False}$~---~в противном случае;
\item минимум $\text{Min}$, которая вычисляет наименьшее из двух чисел;
\item факториал $\text{Fact}$ произвольного натурального числа.
\end{itemize}
\item Построить сеть взаимодействия, приводящую к бесконечному циклу.
\item Дополнить определение системы взаимодействия для комбинаторной логики в параграфе~\ref{complete}.
Точнее, определите наборы агентов и правил, необходимые, чтобы представить комбинатор $S$ (достаточно три агента и три правила).
\item
\begin{enumerate}
\item Построить систему взаимодействия для вычисления логического <<и>> $\text{And}$.
\item Изобразить сеть взаимодействия, представляющую выражение
$$
(\text{True}\ \text{And}\ \text{False})\ \text{And}\ \text{True}.
$$
Сколько шагов потребуется, чтобы получить нормальную форму?
\item Модифицировать систему так, чтобы результат был $\text{True}$ тогда и только тогда, когда оба аргумента имеют одинаковое значение (т.~е. либо оба $\text{True}$, либо оба $\text{False}$).
\end{enumerate}
\item Представить функцию, которая, для двух данных списков $l_1$ и $l_2$, строит список, содержащий элементы $l_1$, чередующиеся элементами $l_2$.
Например, результатом чередования списков $[0, 2, 4]$ и $[1, 3]$ является список $[0, 1, 2, 3, 4]$.
\item В примере~\ref{calcnat} представлена текстовая запись правил для сложения чисел.
Как аналогичным образом записать правила для умножения чисел из параграфа~\ref{numbers}?
\item Почему сети взаимодействия в чистом виде не могут применяться в качестве модели нетерминированных вычислений?
\item Определить функцию <<параллельное и>> $\text{ParallelAnd}$ с помощью агента $\text{Amb}$.
Функция $\text{ParallelAnd}$ отличается от обычной тем, что немедленно возвращает <<ложь>> $\text{False}$, как только один из аргументов принимает значение $\text{False}$.
\end{enumerate}
\section{Представление списков}
Упорядоченные последовательности элементов, то есть списки, можно представить разными способами.
В частности, мы можем построить список с помощью бинарного агента $\text{Cons}$, который будет соединять первый элемент списка с остальной его частью.
Пустой же список может быть представлен с помощью агента $\text{Nil}$.
Данный способ соответствует традиционному определению этой структуры данных в функциональных языках программирования.
Соединение двух списков, то есть операцию конкатенации, определяют следующим образом:
\begin{align*}
\text{Append}(\text{Nil}, l) &= l; \\
\text{Append}(\text{Cons}(x, l), l') &= \text{Cons}(x, \text{Append}(l, l')).
\end{align*}
Для конкатенации двух списков, определенных таким образом, требуется время $O(n)$, пропорциональное длине $n$ первого из списков.
Читателю не составит труда представить определение выше в сетях взаимодействия с помощью трех агентов $\text{Nil}$, $\text{Cons}$ и $\text{Append}$.
Однако, если мы примем во внимание тот факт, что сети взаимодействия представляют собой графы и не обязаны быть деревьями, то получим более эффективную реализацию.
Идея заключается в том, чтобы иметь связи как с началом списка, так и с его концом, то есть в использовании так называемых разностных списков.
Это можно сделать с помощью бинарного агента $\text{Diff}$, который бы имел связи с двумя агентами $\text{Cons}$, соответствующими первому и последнему элементам списка.
При этом пустой список $\text{Nil}$ будет представляться следующей сетью.
$$
\begin{tikzpicture}
\inetcell(d){$\text{Diff}$}[U]
\inetwirefree(d.pal)
\inetwire(d.left pax)(d.right pax)
\end{tikzpicture}
$$
Итак, представим эффективную версию конкатенации списков с помощью двух правил взаимодействия.
Заметим, что конкатенация двух списков будет выполняться за фиксированное время $O(1)$ вне зависимости от их длины.
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(a){$\text{Append}$}[D] \\
\node (p) {$\phantom x$}; \\
\inetcell(l){$\text{Diff}$}[U] \\ };
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwirefree(l.left pax)
\inetwirefree(l.right pax)
\inetwire(a.pal)(l.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(a){$\text{Diff}$}[U] &
\node (p) {$\phantom x$}; &
\inetcell(l){$\text{Open}$}[U] \\ };
\inetwirefree(a.left pax)
\inetwirefree(l.right pax)
\inetwirefree(a.pal)
\inetwirefree(l.pal)
\inetwire(a.right pax)(l.left pax)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(a){$\text{Open}$}[D] \\
\node (p) {$\phantom x$}; \\
\inetcell(l){$\text{Diff}$}[U] \\ };
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwirefree(l.left pax)
\inetwirefree(l.right pax)
\inetwire(a.pal)(l.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell[opacity=0](a){$\text{Open}$}[D] \\
\node (p) {$\phantom x$}; \\
\inetcell[opacity=0](l){$\text{Diff}$}[U] \\ };
\inetwirecoords(a.above left pax)(l.above right pax)
\inetwirecoords(a.above right pax)(l.above left pax)
\end{tikzpicture}
$$
Проиллюстрируем полученную систему взаимодействия примером конкатенации двух списков: вычисление будет состоять всего из двух редукций.
$$
\begin{tikzpicture}[baseline=(d)]
\matrix[row sep=1em]{
& & & \inetcell[rotate=-45](a){$\text{Append}$}[D] & \\
& & \inetcell(d){$\text{Diff}$}[U] & &
\inetcell(l){$\text{Diff}$}[U] \\
\inetcell(1){$1$}[D] &
\inetcell(c1){$\text{Cons}$}[U] & & &
\inetcell(c3){$\text{Cons}$}[L] \\
\inetcell(2){$2$}[D] &
\inetcell(c2){$\text{Cons}$}[U] &
\inetcell[opacity=0](p){$\text{Diff}$}[U] & &
\inetcell(3){$3$}[U] \\ };
\inetwirefree(a.right pax)
\inetwire(l.pal)(a.left pax)
\inetwire(d.pal)(a.pal)
\inetwirecoords(d.right pax)(p.right pax)
\inetwire(c1.pal)(d.left pax)
\inetwire(c2.pal)(c1.right pax)
\inetwire(c3.pal)(l.left pax)
\inetwire(1.pal)(c1.left pax)
\inetwire(2.pal)(c2.left pax)
\inetwire(3.pal)(c3.left pax)
\inetwire(p.right pax)(c2.right pax)
\inetwire(l.right pax)(c3.right pax)
\end{tikzpicture}
\rightarrow
$$
$$
\begin{tikzpicture}[baseline=(d)]
\matrix[row sep=1em]{
& \inetcell(d){$\text{Diff}$}[U] &
\inetcell(o){$\text{Open}$}[U] &
\inetcell(l){$\text{Diff}$}[U] \\
\inetcell(1){$1$}[D] &
\inetcell(c1){$\text{Cons}$}[U] & &
\inetcell(c3){$\text{Cons}$}[L] \\
\inetcell(2){$2$}[D] &
\inetcell(c2){$\text{Cons}$}[U] &
\inetcell[opacity=0](p){$\text{Open}$}[U] &
\inetcell(3){$3$}[U] \\ };
\inetwirefree(d.pal)
\inetwire(l.pal)(o.pal)
\inetwire(d.right pax)(o.left pax)
\inetwirecoords(o.right pax)(p.right pax)
\inetwire(c1.pal)(d.left pax)
\inetwire(c2.pal)(c1.right pax)
\inetwire(c3.pal)(l.left pax)
\inetwire(1.pal)(c1.left pax)
\inetwire(2.pal)(c2.left pax)
\inetwire(3.pal)(c3.left pax)
\inetwire(p.right pax)(c2.right pax)
\inetwire(l.right pax)(c3.right pax)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(d)]
\matrix[row sep=1em]{
& & \inetcell(d){$\text{Diff}$}[U] \\
\inetcell(1){$1$}[D] & \inetcell(c1){$\text{Cons}$}[U] & \\
\inetcell(2){$2$}[D] & \inetcell(c2){$\text{Cons}$}[U] & \\
\inetcell(3){$3$}[D] & \inetcell(c3){$\text{Cons}$}[U] & \inetcell[opacity=0](p){$\text{Diff}$}[U] \\ };
\inetwirefree(d.pal)
\inetwirecoords(d.right pax)(p.right pax)
\inetwire(c1.pal)(d.left pax)
\inetwire(c2.pal)(c1.right pax)
\inetwire(c3.pal)(c2.right pax)
\inetwire(1.pal)(c1.left pax)
\inetwire(2.pal)(c2.left pax)
\inetwire(3.pal)(c3.left pax)
\inetwire(p.right pax)(c3.right pax)
\end{tikzpicture}
$$
\section{Числа и арифметические операции}
\label{numbers}
Натуральные числа могут быть представлены с помощью нуля $0$ и функции следования $S$.
Например, числу $3$ соответствует выражение $S(S(S(0)))$.
Теперь рассмотрим определение стандартной операции сложения:
\begin{align*}
\text{Add}(0, y) &= y; \\
\text{Add}(S(x), y) &= S(\text{Add}(x, y)).
\end{align*}
Действительно, сложение любого числа $y$ с нулем дает в результате $y$, а для того чтобы прибавить $x + 1$ к $y$, можно вычислить $x + y$ и увеличить результат на $1$.
Чтобы представить этот алгоритм на языке сетей взаимодействия, введем три агента, соответствующих $\text{Add}$, $S$ и $0$.
$$
\begin{tikzpicture}[baseline=(0)]
\inetcell(0){$0$}[R]
\inetwirefree(0.pal)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(S)]
\inetcell(S){$S$}[R]
\inetwirefree(S.pal)
\inetwirefree(S.middle pax)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(Add)]
\inetcell(Add){$\text{Add}$}[R]
\inetwirefree(Add.pal)
\inetwirefree(Add.left pax)
\inetwirefree(Add.right pax)
\end{tikzpicture}
$$
Теперь определим правила взаимодействия.
В нашем случае, мы можем тривиальным образом отобразить определение $\text{Add}$.
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=1em]{
\inetcell(0){$0$}[R] &
\inetcell(Add){$\text{Add}$}[L] \\ };
\node (p) [right=of 0.pal] {$\phantom x$};
\inetwirefree(Add.left pax)
\inetwirefree(Add.right pax)
\inetwire(0.pal)(Add.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix[row sep=1em]{
\node (t) {$\phantom x$}; \\
\node (p) {$\phantom x$}; \\
\node (b) {$\phantom x$}; \\ };
\inetwirecoords(t)(b)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=1em]{
\inetcell(S){$S$}[R] &
\inetcell(Add){$\text{Add}$}[L] \\ };
\node (p) [right=of S.pal] {$\phantom x$};
\inetwirefree(S.middle pax)
\inetwirefree(Add.left pax)
\inetwirefree(Add.right pax)
\inetwire(0.pal)(Add.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\inetcell(Add){$\text{Add}$}[L]
\inetcell[node distance=1em, right=of Add.right pax](S){$S$}[R]
\node (p) [right=of Add.middle pax] {$\phantom x$};
\inetwirefree(Add.pal)
\inetwirefree(Add.left pax)
\inetwirefree(S.pal)
\inetwire(Add.right pax)(S.middle pax)
\end{tikzpicture}
$$
Обращаем внимание на сохранение интерфейса при взаимодействии.
Теперь рассмотрим сеть, соответствующую выражению $\text{Add}(S(0), S(0))$.
$$
\begin{tikzpicture}
\matrix[row sep=1em]{
\inetcell[rotate=-45](Add){$\text{Add}$}[D] & \\
\inetcell(S1){$S$}[U] & \inetcell(S2){$S$}[U] \\
\inetcell(01){$0$}[U] & \inetcell(02){$0$}[U] \\ };
\inetwirefree(Add.right pax)
\inetwire(S1.pal)(Add.pal)
\inetwire(S2.pal)(Add.left pax)
\inetwire(01.pal)(S1.middle pax)
\inetwire(02.pal)(S2.middle pax)
\end{tikzpicture}
$$
В этом примере возможна лишь одна последовательность редукций, так как на каждом шаге имеется лишь одна активная пара.
Полная последовательность редукций показана ниже.
Результатом будет сеть, представляющая выражение $S(S(0))$.
$$
\begin{tikzpicture}[baseline=(S2)]
\matrix[row sep=1em]{
\inetcell[rotate=-45](Add){$\text{Add}$}[D] & \\
\inetcell(S1){$S$}[U] & \inetcell(S2){$S$}[U] \\
\inetcell(01){$0$}[U] & \inetcell(02){$0$}[U] \\ };
\inetwirefree(Add.right pax)
\inetwire(S1.pal)(Add.pal)
\inetwire(S2.pal)(Add.left pax)
\inetwire(01.pal)(S1.middle pax)
\inetwire(02.pal)(S2.middle pax)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(S2)]
\matrix[row sep=1em]{
\inetcell(S1){$S$}[U] & \\
\inetcell[rotate=-45](Add){$\text{Add}$}[D] & \\
\inetcell(01){$0$}[U] & \inetcell(S2){$S$}[U] \\
& \inetcell(02){$0$}[U] \\ };
\inetwirefree(S1.pal)
\inetwire(01.pal)(Add.pal)
\inetwire(S2.pal)(Add.left pax)
\inetwire(S1.middle pax)(Add.right pax)
\inetwire(02.pal)(S2.middle pax)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(S2)]
\matrix[row sep=1em]{
\inetcell(S1){$S$}[U] \\
\inetcell(S2){$S$}[U] \\
\inetcell(02){$0$}[U] \\ };
\inetwirefree(S1.pal)
\inetwire(S2.pal)(S1.middle pax)
\inetwire(02.pal)(S2.middle pax)
\end{tikzpicture}
$$
Предыдущий пример слишком прост, чтобы показать существенные особенности сетей взаимодействия.
Более интересной будет система взаимодействия, реализующая операцию умножения чисел:
\begin{align*}
\text{Mult}(0, y) &= 0; \\
\text{Mult}(S(x), y) &= \text{Add}(\text{Mult}(x, y), y).
\end{align*}
Чтобы представить этот алгоритм, нам потребуется новый агент $\text{Mult}$ перемножения чисел, правила взаимодействия для которого сложнее, чем у агента $\text{Add}$, так как умножение не является линейной операцией (как в случае со сложением).
Для того чтобы оставаться в соответствии с определением правил взаимодействия, мы должны сохранить интерфейс.
Следующая схема иллюстрирует правила для $\text{Mult}$.
$$
\begin{tikzpicture}[baseline=(Mult)]
\matrix[row sep=1em]{
\inetcell(Mult){$\text{Mult}$}[D] \\
\inetcell(0){$0$}[U] \\ };
\inetwirefree(Mult.left pax)
\inetwirefree(Mult.right pax)
\inetwire(0.pal)(Mult.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(0)]
\matrix[column sep=1em]{
\inetcell(0){$0$}[U] &
\inetcell(e){$\epsilon$}[U] \\ };
\inetwirefree(0.pal)
\inetwirefree(e.pal)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(Mult)]
\matrix[row sep=1em]{
\inetcell(Mult){$\text{Mult}$}[D] \\
\inetcell(S){$S$}[U] \\ };
\inetwirefree(Mult.left pax)
\inetwirefree(Mult.right pax)
\inetwirefree(S.middle pax)
\inetwire(S.pal)(Mult.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(Add)]
\inetcell(Mult){$\text{Mult}$}[D]
\inetcell[node distance=1em, above=of Mult.right pax](Add){$\text{Add}$}[D]
\inetcell[node distance=3em, right=of Add.above left pax](d){$\delta$}[R]
\inetwirefree(Mult.pal)
\inetwirefree(Add.right pax)
\inetwirefree(d.pal)
\inetwire(Mult.right pax)(Add.pal)
\inetwire(Add.left pax)(d.left pax)
\inetwire(Mult.left pax)(d.right pax)
\end{tikzpicture}
$$
Чтобы сохранить интерфейс, мы воспользовались удаляющим $\epsilon$ и дублирующим $\delta$ агентами, которые были введены в примере~\ref{erasedup}.
Последний пример демонстрирует один из самых важных аспектов сетей взаимодействия.
Заметим, что дублирование активных пар невозможно, поэтому взаимодействие активных пар происходит лишь однажды; иначе вычисление по сути производилось бы дважды и, следовательно, было бы неоптимальным.
Действительно, чтобы продублировать сеть, $\delta$ должен провзаимодействовать со всеми агентами в сети, но если $\alpha$ и $\beta$ соединены своими главными портами, то они не могут взаимодействовать с $\delta$ и, стало быть, не могут быть продублированы.
Приведем еще один пример арифметической операции.
\begin{example}
\label{maxexample}
Рассмотрим функцию, возвращающую наибольшее из двух чисел:
\begin{align*}
\text{Max}(0, y) &= y; \\
\text{Max}(x, 0) &= x; \\
\text{Max}(S(x), S(y)) &= S(\text{Max}(x, y)).
\end{align*}
С данным определением связана одна проблема: в нем используются одновременно оба аргумента функции $\text{Max}$.
Дело в том, что если мы попытаемся действовать аналогично двум предыдущим примерам операций на числах, нам потребуется два главных порта для агента $\text{Max}$, но это невозможно в сетях взаимодействия.
Тем не менее, мы можем изменить определение $\text{Max}$, введя новую функцию $\text{Max}'$, чтобы получить эквивалентную систему, где каждая операция определена с использованием только одного аргумента:
\begin{align*}
\text{Max}(0, y) &= y; \\
\text{Max}(S(x), y) &= \text{Max}'(x, y); \\
\text{Max}'(x, 0) &= S(x); \\
\text{Max}'(x, S(y)) &= S(\text{Max}(x, y)). \\
\end{align*}
Начнем с правил взаимодействия для агента $\text{Max}$.
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=1em]{
\inetcell(0){$0$}[R] &
\inetcell(Max){$\text{Max}$}[L] \\ };
\node (p) [right=of 0.pal] {$\phantom x$};
\inetwirefree(Max.left pax)
\inetwirefree(Max.right pax)
\inetwire(0.pal)(Max.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix[row sep=1em]{
\node (t) {$\phantom x$}; \\
\node (p) {$\phantom x$}; \\
\node (b) {$\phantom x$}; \\ };
\inetwirecoords(t)(b)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=1em]{
\inetcell(S){$S$}[R] &
\inetcell(Max){$\text{Max}$}[L] \\ };
\node (p) [right=of S.pal] {$\phantom x$};
\inetwirefree(Max.left pax)
\inetwirefree(Max.right pax)
\inetwirefree(S.middle pax)
\inetwire(S.pal)(Max.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(Max)]
\inetcell[rotate=-45](Max){$\text{Max}'$}[R]
\inetcell[draw=none, node distance=2em, above right=of Max.left pax](t){$\phantom S$}[R]
\inetwirefree(Max.pal)
\inetwirefree(Max.right pax)
\inetwire(Max.left pax)(t.middle pax)
\end{tikzpicture}
$$
Теперь нам остается определить правила для $\text{Max}'$.
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=1em]{
\inetcell(Max){$\text{Max}'$}[R] &
\inetcell(0){$0$}[L] \\ };
\node (p) [right=of 0.pal] {$\phantom x$};
\inetwirefree(Max.left pax)
\inetwirefree(Max.right pax)
\inetwire(0.pal)(Max.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(S)]
\inetcell(S){$S$}[U]
\inetwirefree(S.pal)
\inetwirefree(S.middle pax)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=1em]{
\inetcell(Max){$\text{Max}'$}[R] &
\inetcell(S){$S$}[L] \\ };
\node (p) [right=of S.middle pax] {$\phantom x$};
\inetwirefree(Max.left pax)
\inetwirefree(Max.right pax)
\inetwirefree(S.middle pax)
\inetwire(S.pal)(Max.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\inetcell[rotate=45](Max){$\text{Max}$}[L]
\inetcell[node distance=2em, above left=of Max.right pax](S){$S$}[L]
\node (p) [left=of S.pal] {$\phantom x$};
\inetwirefree(S.pal)
\inetwirefree(Max.pal)
\inetwirefree(Max.left pax)
\inetwire(S.middle pax)(Max.right pax)
\end{tikzpicture}
$$
Построение же системы взаимодействия, способной вычислять наименьшее из двух натуральных чисел, мы оставим читателю (см.~параграф~\ref{exercise}).
\end{example}
Пример~\ref{maxexample} дает нам способ представлять произвольные арифметические операции.
На самом деле, в следующем параграфе мы увидим, что любые <<чистые>> функции можно компилировать в сети взаимодействия, которые сами таким образом оказываются своеобразным универсальным языком программирования.
\section{Парадигма взаимодействия}
Системы сетей взаимодействия определяются множеством $\Sigma$ символов, из которых сеть может состоять, и множеством $\mathcal R$ так называемых \textit{правил взаимодействия}~---~правил перезаписи для сетей взаимодействия, удовлетворяющих определенным условиям, обсуждаемым ниже.
Каждый символ $\alpha \in \Sigma$ будет иметь некоторую (фиксированную) \textit{арность}~---~натуральное число (с нулем).
Условимся, что вне зависимости от рассматриваемого множества $\Sigma$ функция $\text{Ar}: \Sigma \rightarrow \mathbb N$ будет возвращать арность, связанную с данным символом из $\Sigma$.
Более точно сети определяются следующим образом.
\begin{definition}[cеть]
Для данного множества $\Sigma$ \textit{сеть} $N$ является ориентированным графом (граф не обязан быть связным), в котором вершины помечены символами из $\Sigma$.
Помеченная таким образом вершина называется \textit{агентом}; дуга между двумя агентами называется \textit{связью}; т.~е. сети состоят из агентов и связей.
Точки присоединения связей к агентам называются \textit{портами}.
Если агент $\alpha$ имеет арность $n$, то каждая вершина, помеченная $\alpha$ должна иметь $n + 1$ портов: один порт называется \textit{главным}, а остальные $n$~---~\textit{дополнительными} портами.
Количество дополнительных портов каждого агента равно арности соответствующего символа.
\end{definition}
Принимая во внимание ориентированность графа, пронумеруем порты против часовой стрелки, начиная с главного порта.
Если $\text{Ar}(\alpha) = n$, то агент можно изобразить следующим образом.
$$
\begin{tikzpicture}[baseline=(i.base)]
\inetcell(a){$\phantom X\alpha\phantom X$}[R]
\inetwirefree(a.pal)
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\node (0) [right=of a.above pal] {$x_0$};
\node (1) [left=of a.above left pax] {$x_1$};
\node (i) [left=of a.above middle pax] {$\vdots$};
\node (n) [left=of a.above right pax] {$x_n$};
\end{tikzpicture}
$$
Если $\text{Ar}(\alpha) = 0$, то агент $\alpha$ не будет иметь дополнительных портов, но каждый агент всегда имеет главный порт.
В сети взаимодействия дуги соединяют агентов вместе так, что на каждый порт приходится не более одной дуги, при этом дуги также могут соединять порты одного и того же агента.
Порты агента, которые не связаны с каким-либо другим портом в сети, называются \textit{свободными}.
Стоит обозначить два особых вида сетей.
Сеть может состоять исключительно из связей (без агентов); такие сети мы будем называть \textit{системами связей}, причем концы дуг в этом случае мы также будем называть портами.
Если в системе связей имеется $n$ дуг, то она содержит $2n$ свободных портов.
Если же сеть не содержит ни агентов, ни связей, то мы будем называть ее \textit{пустой сетью}.
Наконец, \textit{интерфейсом} сети назовем множество ее свободных портов.
\begin{definition}[правило взаимодействия]
Пара агентов $(\alpha, \beta) \in \Sigma \times \Sigma$, соединенных вместе своими главными портами называется \textit{активной парой}; мы будем обозначать ее как $\alpha \bowtie \beta$.
Активные пары являются аналогами редексов в $\lambda$-исчислении.
\textit{Правило взаимодействия} $\alpha \bowtie \beta \rightarrow N$ в $\mathcal R$ состоит из активной пары слева и сети $N$~---~справа;
при этом правила взаимодействия должны удовлетворять следующим двум сильным условиям.
\begin{enumerate}
\item В правиле взаимодействия левая и правая части должны иметь один и тот же интерфейс, то есть взаимодействие сохраняет все свободные порты.
Это свойство можно проиллюстрировать на схеме.
$$
\begin{tikzpicture}[baseline=(yi.base)]
\matrix[column sep=1em]{
\inetcell(a){$\phantom X\alpha\phantom X$}[R] &
\inetcell(b){$\phantom X\beta\phantom X$}[L] \\ };
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwire(a.pal)(b.pal)
\inetwirefree(b.left pax)
\inetwirefree(b.right pax)
\node (x1) [left=of a.above left pax] {$x_1$};
\node (xi) [left=of a.above middle pax] {$\vdots$};
\node (xn) [left=of a.above right pax] {$x_n$};
\node (y1) [right=of b.above left pax] {$y_1$};
\node (yi) [right=of b.above middle pax] {$\vdots$};
\node (yn) [right=of b.above right pax] {$y_n$};
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(xi.base)]
\matrix[column sep=2em]{
\node (x1) {$x_1$}; &
\node (t) {$\phantom x$}; &
\node (yn) {$y_n$}; \\
\node (xi) {$\vdots$}; &
\node (n) {$N$}; &
\node (yi) {$\vdots$}; \\
\node (xn) {$x_n$}; &
\node (b) {$\phantom x$}; &
\node (y1) {$y_1$}; \\ };
\inetbox{(b) (t)}(box)
\inetwirecoords(x1)(intersection cs:
first line={(box.north west)--(box.south west)},
second line={(x1)--(yn)})
\inetwirecoords(xn)(intersection cs:
first line={(box.north west)--(box.south west)},
second line={(xn)--(y1)})
\inetwirecoords(yn)(intersection cs:
first line={(box.north east)--(box.south east)},
second line={(x1)--(yn)})
\inetwirecoords(y1)(intersection cs:
first line={(box.north east)--(box.south east)},
second line={(xn)--(y1)})
\end{tikzpicture}
$$
Заметим, что сеть $N$ сама может содержать агенты $\alpha$ и $\beta$.
Также $N$ может быть системой связей, если только суммарное количество свободных портов в активной паре было четным.
В случае активной пары вовсе без свободных портов $N$ может (но не обязана) быть пустой сетью.
\item Множество $\mathcal R$ может содержать не более одного правила для каждой неупорядоченной пары агентов, то есть только одно правило для $\alpha \bowtie \beta$, которое совпадает с правилом для $\beta \bowtie \alpha$.
\end{enumerate}
\end{definition}
Правила взаимодействия порождают отношение редукции на сетях.
\begin{definition}[редукция]
Шаг редукции по правилу $\alpha \bowtie \beta \rightarrow N$ заменяет одну из активных пар $\alpha \bowtie \beta$ сетью $N$.
Более точно, отношение \textit{редукции} $W \rightarrow W'$ между сетями $W$ и $W'$ имеет место, если сеть $W$ содержит активную пару $\alpha \bowtie \beta$ и множество $\mathcal R$ содержит правило взаимодействия $\alpha \bowtie \beta \rightarrow N$, такую, что сеть $W'$ получается заменой активной пары $\alpha \bowtie \beta$ в сети $W$ сетью $N$ (после такой замены не появляется <<висящих>> дуг, так как $N$ имеет тот же интерфейс, что и $\alpha \bowtie \beta$).
Рефлексивное транзитивное замыкание отношения редукции обозначим <<$\rightarrow^*$>>.
Другими словами, мы пишем $N \rightarrow N'$, если $N'$ может быть получена из $N$ редукцией одной активной пары, а $N \rightarrow^* N'$ означает, что существует последовательность из нуля или более шагов взаимодействия, приводящих сеть $N$ к виду $N'$.
\end{definition}
Вообще говоря, не требуется, чтобы имелось правило взаимодействия для любой пары агентов, но если сеть содержит активную пару, для которой нет правила взаимодействия, то эта пара не будет редуцироваться (она будет \textit{заблокирована}).
Заметим, что интерфейс сети \textit{упорядочен}.
Используя это свойство, мы может избавиться от обозначений для свободных портов.
Например, правило взаимодействия
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=1em]{
\inetcell(a){$\alpha$}[R] &
\inetcell(b){$\beta$}[L] \\ };
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwire(a.pal)(b.pal)
\inetwirefree(b.left pax)
\inetwirefree(b.right pax)
\node (x1) [left=of a.above left pax] {$x_1$};
\node (xn) [left=of a.above right pax] {$x_2$};
\node (y1) [right=of b.above left pax] {$y_1$};
\node (yn) [right=of b.above right pax] {$y_2$};
\node (p) [right=of b.middle pax] {$\phantom x$};
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=2em]{
\node (x1) {$x_1$}; & & \node (y2) {$y_2$}; \\
& \node (p) {$\phantom x$}; & \\
\node (x2) {$x_2$}; & & \node (y1) {$y_1$}; \\ };
\inetwirecoords(x1)(y1)
\inetwirecoords(x2)(y2)
\end{tikzpicture}
$$
соединяет $x_1$ с $y_1$ и $x_2$ с $y_2$, что эквивалентно следующей схеме.
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=1em]{
\inetcell(a){$\alpha$}[R] &
\inetcell(b){$\beta$}[L] \\ };
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwire(a.pal)(b.pal)
\inetwirefree(b.left pax)
\inetwirefree(b.right pax)
\node (x1) [left=of a.above left pax] {$x_1$};
\node (xn) [left=of a.above right pax] {$x_2$};
\node (y1) [right=of b.above left pax] {$y_1$};
\node (yn) [right=of b.above right pax] {$y_2$};
\node (p) [right=of b.middle pax] {$\phantom x$};
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix[column sep=2em]{
\node (x1) {$x_1$}; & & \node (y1) {$y_1$}; \\
& \node (p) {$\phantom x$}; & \\
\node (x2) {$x_2$}; & & \node (y2) {$y_2$}; \\ };
\inetwirecoords(x1)(y1)
\inetwirecoords(x2)(y2)
\end{tikzpicture}
$$
Однако, в последней записи существены метки на концах свободных портов, так как мы изменили их порядок.
В дальнейшем мы всегда будем сохранять порядок свободных портов при записи правил взаимодействия, чтобы избежать необходимости в метках для интерфейсов.
Сеть взаимодействия находится в \textit{полной нормальной форме} (обычно мы будем называть ее просто \textit{нормальной формой}), если в ней нет активных пар.
Обозначим через $N \downarrow N'$ тот факт, что существует конечная последовательность взаимодействий $N \rightarrow^* N'$, такая, что $N'$ находится в нормальной форме.
Если $N \downarrow N'$, то сеть $N$ будем называть \textit{нормализуемой}; если же все последовательности взаимодействий, начинающиеся с сети $N$, конечны, то $N$~---~\textit{сильно нормализуема}.
Одним из прямых следствий определения систем взаимодействия, в частности ограничений, накладываемых на правила взаимодействия, является следующее полезное свойство редукции на сетях взаимодействия.
Если возможны две различные редукции сети $N$ (т.~е. $N \rightarrow N_1$ и $N \rightarrow N_2$), то существует сеть $M$, такая, что обе сети $N_1$ и $N_2$ могут быть редуцированы к $M$ за \textit{один шаг}: $N_1 \rightarrow M$ и $N_2 \rightarrow M$.
Это свойство, иногда называемое \textit{сильной конфлюэнтностью} или свойством ромба, сильнее конфлюэнтности; из него следует конфлюэнтность.
Таким образом, мы получаем следующий результат.
\begin{theorem}
Пусть $N$~---~сеть в системе взаимодействия $(\Sigma, \mathcal R)$.
\begin{enumerate}
\item Если $N \downarrow N'$, то сеть $N$ сильно нормализуема, т.~е. любые последовательности редукций, начинающиеся с $N$, завершаются.
\item Если $N \downarrow N'$ и $N \downarrow N''$, то $N' = N''$ (нормальные формы уникальны).
\end{enumerate}
\end{theorem}
Представим две простые, но важные операции на сетях взаимодействия.
\begin{example}
\label{erasedup}
Чаще других рассматриваются \textit{удаляющий} агент $\epsilon$, который приводит к удалению любых агентов, с которыми он взаимодействует, и \textit{дублирующий} агент $\delta$, который, наоборот, дублирует любого агента, с которым он взаимодействует.
Точнее, правила взаимодействия для этих двух агентов выглядят следующим образом.
$$
\begin{tikzpicture}[baseline=(a)]
\matrix[row sep=1em]{
\inetcell(e){$\epsilon$}[D] \\
\inetcell(a){$\phantom x\alpha\phantom x$}[U] \\ };
\node (i) [below=of a.middle pax] {$\dots$};
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwire(e.pal)(a.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(i.base)]
\matrix{
\inetcell(1){$\epsilon$} &
\node (i) {$\dots$}; &
\inetcell(n){$\epsilon$} \\ };
\inetwirefree(1.pal)
\inetwirefree(n.pal)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(a)]
\matrix[row sep=1em]{
\inetcell(d){$\delta$}[D] \\
\inetcell(a){$\phantom x\alpha\phantom x$}[U] \\ };
\node (i) [below=of a.middle pax] {$\dots$};
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwirefree(d.left pax)
\inetwirefree(d.right pax)
\inetwire(d.pal)(a.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(i.base)]
\matrix[row sep=2em]{
\inetcell(l){$\phantom x\alpha\phantom x$}[U] & &
\inetcell(r){$\phantom x\alpha\phantom x$}[U] \\
\inetcell(1){$\delta$} &
\node (i) {$\dots$}; &
\inetcell(n){$\delta$} \\ };
\node (li) [below=of l.middle pax] {$\dots$};
\node (ri) [below=of r.middle pax] {$\dots$};
\inetwire(l.left pax)(1.right pax)
\inetwire(l.right pax)(n.right pax)
\inetwire(r.left pax)(1.left pax)
\inetwire(r.right pax)(n.left pax)
\inetwirefree(1.pal)
\inetwirefree(n.pal)
\inetwirefree(l.pal)
\inetwirefree(r.pal)
\end{tikzpicture}
$$
На каждой из двух схем выше символом $\alpha$ обозначается произвольный агент из $\Sigma$, поэтому они представляют собой одновременно несколько правил взаимодействия.
Первая схема определяет взаимодействие активной пары $\epsilon \bowtie \alpha$, а вторая~---~$\delta \bowtie \alpha$.
Первое правило определяет взаимодействие между произвольным агентом $\alpha$ и удаляющим агентом $\epsilon$ как удаление $\alpha$ и создание удаляющих агентов на всех свободных портах $\alpha$.
Заметим, что если $\alpha$ имеет нулевую арность, то правая часть правила будет пустой сетью; в этом случае процесс удаления завершается.
Частным случаем является взаимодействие двух удаляющих агентов, когда $\alpha$ есть сам $\epsilon$.
Эти правила обеспечивают механизм так называемой <<сборки мусора>> для сетей взаимодействия.
Во втором правиле мы видим, что агент $\alpha$ дублируется, при этом на всех свободных портах $\alpha$ создаются дублирующие агенты $\delta$, которые таким образом могут продолжить дублировать остальную часть сети.
\end{example}
\section{Недетерминированные расширения}
Системы взаимодействия можно считать моделью распределенных вычислений в том смысле, что правила взаимодействия применимы в любом порядке или даже одновременно независимо друг от друга в любых частях сети (синхронизация при этом не требуется благодаря свойству сильной конфлюэнтности у редукции).
Но сети взаимодействия сами по себе не способны представлять недетерминированные вычисления, которые характерны для параллельных вычислительных систем.
Были предложены несколько расширений понятия о системах взаимодействия, чтобы стало возможным моделировать внутри них недетерминированный выбор.
В частности, мы могли бы допустить несколько правил взаимодействия для одной и той же активной пары, причем так, чтобы выбор одного из этих правил происходил произвольным образом.
Но сети взаимодействия не содержат выразительных средств для полноценной модели недетерминированных вычислений.
Если же обобщить понятие агента так, чтобы взаимодействие могло иметь место на двух и более портах (т.~е. допустить агенты с несколькими главными портами), то полученная система уже будет обладать необходимой выразительностью.
На самом деле, достаточно иметь всего один агент с двумя главными портами.
Он и будет представлять собой неоднозначный выбор.
Обычно такой агент обозначается $\text{Amb}$.
Правила взаимодействия для этого агента выглядят следующим образом.
$$
\begin{tikzpicture}[baseline=(b.base)]
\matrix[row sep=2em]{
\inetcell(a){$\text{Amb}$}[D] \\
\inetcell[opacity=0](p){$X$}[D] \\ };
\node (m) [above=of a.above right pax] {$m$};
\node (t) [above=of a.above left pax] {$a$};
\node (b) [below=of p.left pax] {$b$};
\inetcell[below=of p.right pax](x){$\phantom x\alpha\phantom x$}[U]
\node (d) [below=of x.middle pax] {$\dots$};
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwirefree(x.left pax)
\inetwirefree(x.right pax)
\inetwire(a.pal)(p.left pax)
\inetwire(a.pal)(x.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(x)]
\matrix[row sep=1em]{
\node (m) {$m$}; & \node (t) {$a$}; \\
\inetcell(x){$\phantom x\alpha\phantom x$}[U] & \\
& \node (b) {$b$}; \\ };
\node (d) [below=of x.middle pax] {$\dots$};
\inetwirefree(x.left pax)
\inetwirefree(x.right pax)
\inetwirecoords(x.pal)(m)
\inetwirecoords(t)(b)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(b.base)]
\matrix[row sep=2em]{
\inetcell(a){$\text{Amb}$}[D] \\
\inetcell[opacity=0](p){$X$}[D] \\ };
\node (m) [above=of a.above right pax] {$m$};
\node (t) [above=of a.above left pax] {$a$};
\node (b) [below=of p.right pax] {$b$};
\inetcell[below=of p.left pax](x){$\phantom x\alpha\phantom x$}[U]
\node (d) [below=of x.middle pax] {$\dots$};
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\inetwirefree(x.left pax)
\inetwirefree(x.right pax)
\inetwire(a.pal)(p.right pax)
\inetwire(a.pal)(x.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(x)]
\matrix[row sep=1em]{
\node (t) {$a$}; & \node (m) {$m$}; \\
& \inetcell(x){$\phantom x\alpha\phantom x$}[U] \\
\node (b) {$b$}; & \\ };
\node (d) [below=of x.middle pax] {$\dots$};
\inetwirefree(x.left pax)
\inetwirefree(x.right pax)
\inetwirecoords(x.pal)(m)
\inetwirecoords(t)(b)
\end{tikzpicture}
$$
Когда агент $\alpha$ связан своим главным портом с одним из главных портов $\text{Amb}$, может иметь место взаимодействие, в результате которого $\alpha$ оказывается связанным с \textit{основным} свободным портом $m$ в интерфейсе на стороне агента $\text{Amb}$.
Если оба главных порта $\text{Amb}$ связаны с главными портами других агентов, то выбор одного из правил взаимодействия выше недетерминирован.
Воспользуемся агентом $\text{Amb}$, чтобы представить функцию <<параллельное или>> $\text{ParallelOr}$~---~это необычная логическая операция, которая возвращает <<истину>> $\text{True}$, если один из ее двух аргументов равен $\text{True}$, причем вне зависимости от того, вычислен ли другой аргумент.
\begin{example}
Функция $\text{ParallelOr}$ должна немедленно возвращать результат $\text{True}$, как только один из ее аргументов принял значение $\text{True}$, даже когда другой из ее аргументов вовсе не определен.
С помощью агента $\text{Amb}$ мы можем представить эту функцию в виде сети, изображенной на следующей схеме.
$$
\begin{tikzpicture}
\matrix[column sep=2em]{
\inetcell(o){$\text{Or}$}[D] &
\inetcell(a){$\text{Amb}$}[R] &
\inetcell[opacity=0](p){$X$}[R] \\ };
\inetwirefree(o.right pax)
\inetwire(o.pal)(a.right pax)
\inetwire(o.left pax)(a.left pax)
\inetwire(a.pal)(p.left pax)
\inetwire(a.pal)(p.right pax)
\end{tikzpicture}
$$
Здесь $\text{Or}$ представляет собой обычное логическое <<или>>, определяемое следующими двумя правилами взаимодействия.
$$
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(t){$T$}[R] &
\node (p) {$\phantom x$}; &
\inetcell(o){$\text{Or}$}[L] \\ };
\inetwirefree(o.left pax)
\inetwirefree(o.right pax)
\inetwire(o.pal)(t.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(t){$T$}[U] &
\node (p) {$\phantom x$}; &
\inetcell(e){$\epsilon$}[D] \\ };
\inetwirefree(t.pal)
\inetwirefree(e.pal)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(p.base)]
\matrix{
\inetcell(f){$F$}[R] &
\node (p) {$\phantom x$}; &
\inetcell(o){$\text{Or}$}[L] \\ };
\inetwirefree(o.left pax)
\inetwirefree(o.right pax)
\inetwire(o.pal)(f.pal)
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}[baseline=(p.base)]
\matrix[row sep=1em]{
\node (t) {$\phantom x$}; \\
\node (p) {$\phantom x$}; \\
\node (b) {$\phantom x$}; \\ };
\inetwirecoords(t)(b)
\end{tikzpicture}
$$
\end{example}
Модель вычислений, соответствующая сетям взаимодействия с дополнительным агентом $\text{Amb}$ обладает в определенном смысле большей мощностью, чем обычные системы взаимодействия, позволяя представить недетерминированные вычисления и параллельные функции, такие как <<параллельное или>> $\text{ParallelOr}$.
\section{Дополнительная литература}
Оригинальная статья~\cite{inet} вводит понятие сетей взаимодействия и содержит много примеров их использования.
Больше информации о комбинаторах взаимодействия, а также доказательство их универсальности читатель может получить в~\cite{icomb}.
В~\cite{impl} обсуждаются реализации сетей взаимодействия.
Описанная выше текстовая запись для сетей взаимодействия впервые была введена в оригинальной статье~\cite{inet}, в то время как исчисление, основанное на этой нотации, определяется в~\cite{calc}.
Также~\cite{calc} содержит другие интересные результаты о нормальных формах и стратегиях.
\section{Текстовая запись и стратегии}
Для сетей взаимодействия вполне естественно графическое представление, и зачастую схемы гораздо легче понять, чем их текстовое определение.
Однако, формальная текстовая запись имеет значительные преимущества: она упрощает написание программ, так как графические редакторы не всегда применимы, а также дает возможность задавать свойства сетей более кратко и элегантно.
Были разработаны уже несколько форм текстовой записи для сетей взаимодействия.
Ниже мы пройдем путь от примитивных, но избыточных обозначений к более компактной, но нетривиальной записи, а затем введем на основе последней исчисление взаимодействия.
В качестве примера мы будем использовать одну и ту же сеть, показанную на рисунке~\ref{lterm}, применяя правило взаимодействия из параграфа~\ref{complete}, которое соответствует $\beta$-редукции в линейном $\lambda$-исчислении.
\begin{figure}[b]
$$
\begin{tikzpicture}
\matrix[column sep=1em]{
\inetcell(a){$@$}[D] &
\inetcell(p){$\lambda$}[U] &
\inetcell(f){$\lambda$}[D] \\ };
\inetwirefree(a.right pax)
\inetwire(a.pal)(f.pal)
\inetwire(a.left pax)(p.pal)
\inetwire(p.left pax)(p.right pax)
\inetwire(f.left pax)(f.right pax)
\end{tikzpicture}
$$
\caption{$\lambda$-терм, представленный в виде сети.}
\label{lterm}
\end{figure}
Естественные обозначения можно получить простым перечислением всех агентов, участвующих в сети, используя определенные соглашения.
Например, мы могли бы договориться всегда перечислять порты агентов против часовой стрелки, начиная с главного порта.
Связь можно задать, используя одно и то же имя для двух портов.
Тогда сеть, представленная на рисунке~\ref{lterm}, будет выглядеть следующим образом:
$$
@(a, b, c), \lambda(a, d, d), \lambda(b, e, e).
$$
Действительно, здесь два агента $\lambda$, представляющих абстракцию, и один агент $@$, представляющий аппликацию.
Заметим, что вхождение одного и того же имени для двух разных портов обозначает связь между ними; например, в записи $\lambda(a, e, e)$ вхождение $e$ дважды означает, что в сети имеется связь между двумя дополнительными портами данного агента $\lambda$.
Те же обозначения можно использовать и для записи правил взаимодействия.
К примеру, правило взаимодействия для $\beta$-редукции в линейном $\lambda$-исчислении мы могли бы записать как
$$
@(a, b, c), \lambda(a, d, e) \rightarrow I(b, d), I(c, e),
$$
где символ $I$ используется для представления связей с интерфейсом сети (а не агентами).
Следует обратить внимание, что в левой и правой частях правила используются одни и те же имена для портов и связей, так как правила взаимодействия по определению обязаны сохранять интерфейс сети.
Еще один способ записи заключается в том, чтобы использовать порядковые номера портов вместо имен, начиная с $0$ для главного порта.
Произвольную сеть можно преставить в виде упорядоченной пары $(A, W)$, где $A$~---~множество входящих в сеть агентов, а $W$~---~множество связей.
Например, агенты, из которых состоит сеть на рисунке~\ref{lterm}, составляют множество
$A = \{@_1, \lambda_1, \lambda_2\}$, поэтому сеть записывается следующим образом:
$$
(\{@_1, \lambda_1, \lambda_2\}, \{@_1.0 = \lambda_1.0, @_1.1 = \lambda_2.0, \lambda_1.1 = \lambda_1.2, \lambda_2.1 = \lambda_2.2\})
$$
В свою очередь, правило взаимодействия для $\beta$-редукции будет иметь вид
$$
(\{\lambda_i, @_j\}, \{\lambda_i.0 = @_j.0\}) \rightarrow (\varnothing, \{\lambda_i.1 = @_j.1, \lambda.2 = @_j.2\}).
$$
Наконец, третий способ записи, который оказывается более кратким, основан на представлении активных пар в виде уравнений.
В нашем случае пример имеет форму
$$
\lambda(a, a) = @(\lambda(b, b), c),
$$
где знак равенства означает связь между главным портом агента $\lambda$ в левой части уравнения и главным портом агента $@$~---~в правой его части (т.~е. уравнение представляет активную пару $\lambda \bowtie @$).
Левая часть уравнения $\lambda(a, a)$ означает, что оба дополнительных порта этого агента связаны, ровно как и в случае $\lambda(b, b)$.
Воспользуемся аналогичным обозначением для правил взаимодействия, заменив знак равенства на <<$\bowtie$>>, а круглые скобки~---~квадратными, чтобы отличать активную пару от результата взаимодействия.
Тогда правило $\beta$-редукции примет вид
$$
@[x, y] \bowtie \lambda[x, y].
$$
Будучи чрезвычайно сжатой формой записи, введенные обозначения активных пар и правил взаимодействия оказываются более удобными при реализации систем взаимодействия.
Они также послужат основой для исчисления взаимодействия, к построению которого мы сейчас перейдем.
\subsection{Исчисление взаимодействия}
Ранее уже говорилось, что системы взаимодействия обладают свойством сильной конфлюэнтности, но, как и в других системах, выполняющих редукцию, для систем взаимодействия существуют различные определения нормальных форм и стратегий (например, когда говорят о так называемых <<ленивых>> вычислениях, часто имеют в виду слабую нормальную форму).
Ниже мы увидим, что исчисление взаимодействия дает возможность точно определять такие понятия.
Сети взаимодействия оказали наибольшее влияние на $\lambda$-исчисление, в котором стратегии играют важную роль: только нормализующие стратегии там гарантируют получение нормальной формы произвольного $\lambda$-терма, если таковая существует.
Начнем с описания синтаксиса исчисления взаимодействия.
\begin{description}
\item[Агенты.] Пусть $\Sigma$~---~множество символов $\alpha, \beta, \dots$ и каждый символ имеет \textit{арность} (мы предполагаем наличие функции $\text{Ar}: \Sigma \rightarrow \mathbb N$, возвращающей арность).
Вхождение символа будем называть \textit{агентом}.
Арность символа соответствует числу дополнительных портов соответствующего агента.
\item[Имена.] Пусть $N$~---~множество имен $x, y, z, \dots$, которое не пересекается с $\Sigma$.
\item[Термы.] Термы строятся с помощью агентов из $\Sigma$ и имен из $N$ согласно грамматике
$$
t ::= x\ |\ \alpha(t_1, \dots, t_n),
$$
где $x \in N$, $\alpha \in \Sigma$ и $\text{Ar}(\alpha) = n$, причем каждое имя может иметь не более двух вхождений в терм.
Если $n = 0$, то скобки опускаются.
Если имя имеет два вхождения, говорят, что оно \textit{связано}; в противном случае имя \textit{свободно}.
Свободные имена могут иметь лишь одно вхождение, поэтому термы \textit{линейны} в том же смысле, что и термы в линейном $\lambda$-исчислении.
Последовательность термов $t_1, \dots, t_n$ мы будем сокращенно обозначать $\vec t$.
Терм вида $\alpha(\vec t)$ можно представить в форме дерева: главный порт $\alpha$ служит корнем, а термы $t_1, \dots, t_n$, в свою очередь, играют роль поддеревьев, которые связаны с дополнительными портами агента $\alpha$.
Если имена входят дважды, то соответствующие листья дерева будут связаны дополнительными дугами.
Заметим, что терм $\alpha(\vec t)$ не может содержать активных пар.
\item[Уравнения.] Пусть $t$ и $u$~---~некоторые термы.
Тогда неупорядоченную пару $t = u$ будем называть \textit{уравнением}.
С помощью $\Delta, \Theta, \dots$ обозначим мультимножества (т.~е. множества, допускающие многократное вхождение элементов) уравнений.
Примерами уравнений служат $x = \alpha(\vec t)$, $x = y$, $\alpha(\vec t) = \beta(\vec u)$.
Они позволят нам представлять сети с активными парами.
\item[Правила.] \textit{Правилами} будем называть неупорядоченные пары вида
$$
\alpha[t_1, \dots, t_m] \bowtie \beta[u_1, \dots, u_n],
$$
где $(\alpha, \beta) \in \Sigma \times \Sigma$, $m = \text{Ar}(\alpha)$, $n = \text{Ar}(\beta)$, а $t_i$ и $u_i$~---~некоторые термы, причем каждое имя в правиле может иметь только два вхождения (свободные имена в правилах запрещены).
Неупорядоченная пара $(\alpha, \beta)$ является \textit{активной парой} этого правила (она соответствует левой части правила взаимодействия).
\end{description}
\begin{definition}[имена, входящие в терм]
Обозначим через $\mathcal N(t)$ множество имен, входящих в терм $t$, которое определяется следующим образом:
\begin{align*}
\mathcal N(x) &= \{x\}; \\
\mathcal N(\alpha(t_1, \dots t_n)) &= \mathcal N(t_1) \cup \dots \cup \mathcal N(t_n).
\end{align*}
Данное определение тривиальным образом распространяется на правила и мультимножества уравнений.
\end{definition}
В любом терме можно заменить свободное вхождение одного имени другим, если только сохраняется линейность терма.
\begin{definition}[переименование]
\textit{Переименование} $t[x := y]$ есть результат замены в терме $t$ свободного вхождения имени $x$ новым именем $y$.
Данная операция тривиальным образом распространяется на уравнения и мультимножества уравнений.
\end{definition}
Теперь обобщим переименование до \textit{подстановки}, которая заменяла бы свободное вхождение имени в терме другим термом, как и раньше, сохраняя линейное свойство.
\begin{definition}[подстановка]
\textit{Подстановка} $t[x := u]$ есть результат замены в терме $t$ свободного вхождения имени $x$ термом $u$, если при такой замене сохраняется линейность термов.
\end{definition}
Заметим, что переименование является частным случаем подстановки.
При этом подстановка обладает следующими двумя полезными свойствами.
\begin{theorem}
Пусть $x \not\in \mathcal N(w)$.
\begin{enumerate}
\item Если $y \in \mathcal N(u)$, то $t[x := u][y := w] = t[x := u[y := w]]$.
\item Если $y \not\in \mathcal N(u)$, то $t[x := u][y := w] = t[y := w][x := u]$.
\end{enumerate}
\end{theorem}
Итак, мы ввели достаточно определений, обозначений и свойств, чтобы теперь рассматривать сети в исчислении взаимодействия.
\begin{definition}[конфигурации]
\textit{Конфигурация}~---~это пара $c = (\mathcal R, \langle\vec t\ |\ \Delta\rangle)$, где $\mathcal R$~---~множество правил, $\vec t$~---~последовательность термов $t_1, \dots, t_n$, а $\Delta$~---~мультимножество уравнений.
Каждое имя может иметь не более двух вхождений в $c$.
Если имя имеет одно вхождение, то оно называется \textit{свободным}; в противном случае оно считается \textit{связанным}.
Для простоты иногда будем опускать $\mathcal R$, если из контекста понятно, о каком именно множестве правил взаимодействия идет речь.
Ниже мы обычно будем обозначать конфигурации как $c$ или $c'$.
Последовательность термов $\vec t$ будем называть \textit{головой} конфигурации.
\end{definition}
По существу $\langle\vec t\ |\ \Delta\rangle$ представляет собой сеть взаимодействия, которую мы можем редуцировать, используя правила из $\mathcal R$, а $\Delta$ содержит переименования и активные пары в сети.
Свободные вхождения имен и голова конфигурации вместе формируют интерфейс сети.
Как и в $\lambda$-исчислении, мы будем считать равными конфигурации с точностью до $\alpha$-конверсии, т.~е. замены связанных имен при сохранении линейного свойства.
Действительно, конфигурации, отличающиеся лишь выбором связанных имен, соответствуют одним и тем же сетям взаимодействия.
Преобразовать произвольную сеть взаимодействия в конфигурацию можно, в частности, следующим способом.
Сначала выделим в сети все деревья и направим их главные порты в одном направлении.
Каждую пару деревьев, которые связаны главными портами, представим как уравнение, а деревья, чьи главные порты свободны, добавим в голову конфигурации.
Чтобы проиллюстрировать такое преобразование, ниже мы приводим простой пример.
\begin{example}
\label{encode}
Операция сложения для натуральных чисел была представлена в сетях взаимодействия (см. параграф~\ref{numbers}) с помощью множества символов $\Sigma = \{0, S, \text{Add}\}$, где $\text{Ar}(0) = 0$, $\text{Ar}(S) = 1$, $\text{Ar}(\text{Add}) = 2$.
На следующей схеме показана сеть, соответствующая выражению $1 + 0$, а затем мы выделяем в ней два дерева и направляем их главными портами вниз.
$$
\begin{tikzpicture}[baseline=(a)]
\matrix[column sep=1em]{
\inetcell(s){$S$}[R] &
\inetcell(a){$\text{Add}$}[L] \\ };
\inetcell[right=of a.above left pax](z){$0$}[L]
\inetcell[left=of s.above middle pax](0){$0$}[R]
\inetwirefree(a.right pax)
\inetwire(s.pal)(a.pal)
\inetwire(s.middle pax)(0.pal)
\inetwire(a.left pax)(z.pal)
\end{tikzpicture}
=
\begin{tikzpicture}[baseline=(a)]
\matrix[column sep=1em]{
\inetcell(a){$\text{Add}$}[D] &
\inetcell(s){$S$}[D] \\ };
\node (x) [above=of a.above right pax] {$x$};
\inetcell[above=of s.above middle pax](0){$0$}[D]
\inetcell[above=of a.above left pax](z){$0$}[D]
\inetwirefree(a.right pax)
\inetwire(s.pal)(a.pal)
\inetwire(s.middle pax)(0.pal)
\inetwire(a.left pax)(z.pal)
\end{tikzpicture}
$$
Теперь несложно получить конфигурацию $\langle x\ |\ \text{Add}(0, x) = S(0)\rangle$, где единственный порт интерфейса помечен $x$, который мы добавили в голову конфигурации.
\end{example}
Обратное преобразование проще: следует построить деревья, соответствующие всем термам в конфигурации, связать порты, соответствующие одинаковым именам, и связать главные порты деревьев, участвующих в уравнениях.
\begin{definition}[редукция]
\label{red}
Отношение \textit{редукции} <<$\rightarrow$>> на конфигурациях порождается следующими тремя правилами.
\begin{description}
\item[Взаимодействие.] Если $\alpha[u'_1, \dots, u'_m] \bowtie \beta[w'_1, \dots, w'_n] \in \mathcal R$ и имеется конфигурация $c = \langle\vec t\ |\ \alpha(u_1, \dots, u_m) = \beta(w_1, \dots, w_n), \Delta\rangle$, то
$$
c \rightarrow \langle\vec t\ |\ u_1 = u'_1, \dots, u_m = u'_m, w_1 = w'_1, \dots, w_n = w'_n, \Delta\rangle.
$$
\item[Разыменование.] Если $x \in \mathcal N(u)$, то
$$
\langle\vec t\ |\ x = t, u = w, \Delta\rangle \rightarrow \langle\vec t\ |\ u[x := t] = w, \Delta\rangle.
$$
\item[Стягивание.] Если $x \in \mathcal N(\vec t)$, то
$$
\langle\vec t\ |\ x = u, \Delta\rangle \rightarrow \langle\vec t[x := u]\ |\ \Delta\rangle.
$$
\end{description}
Рефлексивное транзитивное замыкание отношения редукции обозначим <<$\rightarrow^*$>>.
\end{definition}
Редукция в исчислении взаимодействия демонстрирует фактические затраты на реализацию одного шага взаимодействия, который, как мы видим, заключается не только в построении сети, соответствующей правой части правила взаимодействия, но также разыменовании и стягивании связей.
То же самое происходит и при работе с графическим представлением сетей взаимодействия, хотя обычно взаимодействие кажется атомарной операцией.
\begin{example}[натуральные числа]
\label{calcnat}
Натуральные числа и операция сложения ниже представлены двумя различными способами с использованием исчисления взаимодействия.
Первый из них~---~стандартное представление, тогда как второй вариант эффективнее, так как выполняет сложение двух чисел за фиксированное время $O(1)$.
\begin{enumerate}
\item Пусть $\Sigma = \{0, S, \text{Add}\}$ при $\text{Ar}(0) = 0$, $\text{Ar}(S) = 1$, $\text{Ar}(\text{Add}) = 2$ и множество $\mathcal R$ содержит следующие два правила взаимодействия:
\begin{align*}
\text{Add}[x, S(y)] &\bowtie S[\text{Add}(x, y)]; \\
\text{Add}[x, x] &\bowtie 0.
\end{align*}
Как было показано в примере~\ref{encode}, выражение $1 + 0$ соответствует конфигурации $(\mathcal R, \langle a\ |\ \text{Add}(0, a) = S(0)\rangle)$.
Одна из возможных последовательностей редукций для этой конфигурации выглядит следующим образом:
\begin{align*}
&\langle a\ |\ \text{Add}(0, a) = S(0)\rangle \\
&\rightarrow \langle a\ |\ 0 = x, a = S(y), 0 = \text{Add}(x, y)\rangle \\
&\rightarrow^* \langle S(y)\ |\ 0 = \text{Add}(0, y)\rangle \\
&\rightarrow \langle S(y)\ |\ 0 = x, y = x\rangle \\
&\rightarrow^* \langle S(0)\ |\ \varnothing\rangle.
\end{align*}
\item Пусть $\Sigma = \{S, C, C^*, \text{Add}\}$, причем $\text{Ar}(S) = 1$, $\text{Ar}(C) = \text{Ar}(C^*) = \text{Ar}(\text{Add}) = 2$.
Натуральное число представим в виде разностного списка агентов $S$, так, что агент $C$ будет иметь связи с началом и концом такого списка.
Тогда нуль $0$ будет соответствовать конфигурации $\langle C(x, x)\ |\ \varnothing\rangle$, а произвольное положительное число $n$ примет форму $\langle C(x, S^n(x))\ |\ \varnothing\rangle$.
Множество $\mathcal R$ будет содержать следующие два правила взаимодействия:
\begin{align*}
C[x, y] &\bowtie \text{Add}[C^*(z, y), C(x, z)];\\
C[x, y] &\bowtie C^*[y, x].
\end{align*}
Приведем в качестве примера выражение $m + n$, чтобы показать, что сложение выполняется за время $O(1)$ вне зависимости от самих чисел $m$ и $n$:
\begin{align*}
&\langle z\ |\ C(x, S^m(x)) = \text{Add}(C(y, S^n(y)), z)\rangle \\
&\rightarrow \langle z\ |\ x = x', S^m(x) = y', C(y, S^n(y)) = C^*(z', y'), z = C(x', z')\rangle \\
&\rightarrow^* \langle C(x, z')\ |\ C(y, S^n(y)) = C^*(z', S^m(x))\rangle \\
&\rightarrow \langle C(x, z')\ |\ y = x', S^n(y) = y', z' = y', S^m(x) = x'\rangle \\
&\rightarrow^* \langle C(x, z')\ |\ S^n(S^m(x)) = z'\rangle \\
&\rightarrow \langle C(x, S^{m + n}(x))\ |\ \varnothing\rangle.
\end{align*}
\end{enumerate}
\end{example}
Исчисление взаимодействия~---~Тьюринг-полная модель вычислений.
Стало быть, в нем имеет место и проблема останова (т.~е. в общем случае нельзя определить, будет ли конечной последовательность редукций).
В следующем примере рассматривается конфигурация, которая приводит к бесконечной последовательности редукций.
\begin{example}[бесконечный цикл]
\label{endless}
Рассмотрим конфигурацию $\langle x, y\ |\ \alpha(x) = \beta(\alpha(y))\rangle$ с правилом взаимодействия $\alpha[x] \bowtie \beta[\beta(\alpha(x))]$.
Тогда мы имеем последовательность редукций, которая оказывается бесконечной:
\begin{align*}
&\langle x, y\ |\ \alpha(x) = \beta(\alpha(y))\rangle \\
&\rightarrow \langle x, y\ |\ x = x', \alpha(y) = \beta(\alpha(x'))\rangle \\
&\rightarrow \langle x', y\ |\ \alpha(y) = \beta(\alpha(x'))\rangle \\
&\rightarrow \dots
\end{align*}
\end{example}
Обратим внимание, что исчисление взаимодействия может представлять системы взаимодействия с двумя ограничениями.
Во-первых, в исчислении взаимодействия нельзя записать правило с активной парой в правой части.
Впрочем, это ограничение не является проблемой, так как несложно показать, что класс систем взаимодействия без активных пар в правой части правил имеет ту же мощность с вычислительной точки зрения.
Во-вторых, любое правило без интерфейса неизбежно имеет пустую сеть в правой части.
Однако, в дальнейшем нас не будут интересовать несвязные части сети, так как результат вычисления определяется только относительно интерфейса.
\subsection{Свойства исчисления взаимодействия}
Данный параграф посвящен обсуждению свойств редукции, которая порождается правилами взаимодействия, разыменования и стягивания (см. определение~\ref{red}).
Мы уже рассматривали свойства систем взаимодействия в их графической форме.
Все те же свойства справедливы и для исчисления взаимодействия.
\begin{theorem}[конфлюэнтность]
Отношение редукции <<$\rightarrow$>> на конфигурациях обладает свойством сильной конфлюэнтности: если $c \rightarrow d$ и $c \rightarrow e$, где $d$ и $e$~---~различные конфигурации, то существует конфигурация $c'$, такая, что $d \rightarrow c'$ и $e \rightarrow c'$.
\end{theorem}
Пусть $c$~---~конфигурация.
Если не существует конфигурации $c'$, такой, что $c \rightarrow c'$, то $c$ называется \textit{нормальной формой}.
Пусть теперь $c$~---~некоторая конфигурация, а $c'$~---~нормальная форма.
Тогда $c \downarrow c'$, если $c \rightarrow^* c'$.
Из свойства сильной конфлюэнтности непосредственно следует единственность нормальной формы, если она существует: если $c \downarrow d$ и $c \downarrow e$, то $d = e$.
Заметим, что проблема останова (см. пример~\ref{endless}) возникает лишь из-за правила взаимодействия, но не правил разыменования и стягивания.
Разыменование и стягивание порождают только конечные последовательсти редукций, так как они всегда уменьшают число уравнений в конфигурации.
\subsection{Нормальные формы и стратегии}
Хотя мы неоднократно подчеркивали свойство сильной конфлюэнтности систем взаимодействия, в общем случае существуют различные последовательности редукций, приводящие к нормальной форме, если она существует.
Более того, существуют и другие понятия нормальных форм, в частности для так называемых <<ленивых>> вычислений.
Ниже мы определим более слабое понятие нормальной формы, которая является аналогом головной нормальной формы в $\lambda$-исчислении.
Последние часто используются в реализациях функциональных языков программирования.
\begin{definition}[головная нормальная форма]
Конфигурация $(\mathcal R, \langle\vec t\ |\ \Delta\rangle)$ есть \textit{головная нормальная форма}, если каждый $t_i$ в $\vec t$ имеет одну из следующих форм.
\begin{itemize}
\item $\alpha(\vec s)$.
Например, $\langle S(x)\ |\ x = 0\rangle$.
\item $x$, где $x \in \mathcal N(t_j)$ для $j \neq i$.
При этом связь, соответствующую имени $x$, будем называть \textit{открытой}.
Например, $\langle x, x\ |\ \Delta\rangle$.
\item $x$, если мультимножество $\Delta$ содержит уравнение вида $y = s$, где $s$~---~терм, причем $x \in \mathcal N(s)$ и $y \in \mathcal N(s)$.
В этом случае уравнение $y = s$ и соответствующую этому уравнению часть сети называют \textit{циклическим деревом}.
Например, конфигурация $\langle x\ |\ y = \alpha(\beta(y), x), \Delta\rangle$ содержит циклическое дерево $y = \alpha(\beta(y), x)$.
\end{itemize}
\end{definition}
Иными словами, конфигурация находится в головной нормальной форме, если ее голова представляет собой интерфейс, каждый свободный порт которого связан либо с главным портом, либо с другим портом этого интерфейса, либо с дополнительным портом, который никогда не станет главным после последовательности редукций.
Проиллюстрируем понятие головной нормальной формы на нескольких схемах.
На первой~---~часть сети, соответствующей конфигурации $\langle\alpha(t_1, \dots, t_n)\ |\ \Delta\rangle$, которая состоит из агента $\alpha$ со свободным главным портом и термов $t_i$, связанных с дополнительными портами $\alpha$.
На второй схеме показана открытая связь (через агент $\delta$).
$$
\begin{tikzpicture}[baseline=(0.base)]
\inetcell(a){$\phantom X\alpha\phantom X$}[U]
\inetwirefree(a.pal)
\inetwirefree(a.left pax)
\inetwirefree(a.right pax)
\node (0) [above=of a.above pal] {$\langle\alpha(t_1, \dots, t_n)\ |\ \Delta\rangle$};
\node (1) [below=of a.above left pax] {$t_1$};
\node (i) [below=of a.above middle pax] {$\dots$};
\node (n) [below=of a.above right pax] {$t_n$};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(0.base)]
\inetcell(a){$\phantom x\delta\phantom x$}[U]
\inetcell[below=of a.above right pax](z){$0$}[U]
\inetwirefree(a.pal)
\inetwirefree(a.left pax)
\inetwire(z.pal)(a.right pax)
\node (0) [above=of a.above pal] {$\langle x, \delta(x, 0)\ |\ \varnothing\rangle$};
\node (1) [below=of a.above left pax] {$x$};
\end{tikzpicture}
$$
Следующие две схемы являются примерами конфигураций, которые содержат циклические деревья: уравнения $\alpha(x, \beta(y)) = y$ и $\delta(S(x), y) = y$.
$$
\begin{tikzpicture}[baseline=(c.base)]
\matrix[column sep=1em]{
\inetcell(a){$\alpha$}[U] &
\inetcell(b){$\beta$}[D] \\ };
\node (x) [below=of a.above left pax] {$x$};
\node (c) [above=of a.above pal] {$\langle x\ |\ \alpha(x, \beta(y)) = y\rangle$};
\inetwirefree(a.left pax)
\inetwire(a.pal)(b.middle pax)
\inetwire(b.pal)(a.right pax)
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(c.base)]
\matrix[column sep=1em]{
\inetcell(a){$\delta$}[U] &
\inetcell[opacity=0](b){$\beta$}[D] \\ };
\inetcell[below=of a.above left pax](s){$S$}[U]
\node (x) [below=of s.above middle pax] {$x$};
\node (c) [above=of a.above pal] {$\langle x\ |\ \delta(S(x), y) = y\rangle$};
\inetwirecoords(b.pal)(b.middle pax)
\inetwirefree(s.middle pax)
\inetwire(a.left pax)(s.pal)
\inetwire(a.pal)(b.middle pax)
\inetwire(b.pal)(a.right pax)
\end{tikzpicture}
$$
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 795 |
Q: What's the difference between incrementing a counter once and twice I am knew to coding and practicing on problems so I can get better. I was doing a problem on CodingBat but I got stuck on problem that I dont understand why my code isn't working.
The problem is: Given a string, return a string where for every char in the original, there are two chars.
doubleChar("The") → "TThhee"
doubleChar("AAbb") → "AAAAbbbb"
doubleChar("Hi-There") → "HHii--TThheerree"
The code I wrote was
public String doubleChar(String str) {
char[] STR = str.toCharArray();
char[] two = new char[STR.length*2];
int counter=0;
for(int i=0;i<STR.length;i++){
two[counter]=STR[i];
two[counter+1]=STR[i];
counter++;
}
String b= new String(two);
return b;
}
output results // im guessing the counter cant increment counter+1 but only though counter++. Could i get a better explanation?
After messing around it for awhile, I got it to work but I still dont understand why the original does not. I am also new to coding so I would much appreciate the help!
working:
public String doubleChar(String str) {
char[] STR = str.toCharArray();
char[] two = new char[STR.length*2];
int counter=0;
for(int i=0;i<STR.length;i++){
two[counter]+=STR[i];
counter++;
two[counter]=STR[i];
counter++;
}
String b= new String(two);
return b;
}
A: In your original solution you increment your counter variable only once.
"counter + 1" doesn't increment your counter value, it's just an addition of a variable and a number (counter += 1 can increment the value of the counter variable).
So, when you write this:
two[counter]=STR[i];
two[counter+1]=STR[i];
counter++;
It means (when counter = 0)
two[0]=STR[i];
two[0+1]=STR[i];
0++; //So the value of the counter variable is still 0 here
And in your proper solution
two[counter]+=STR[i];
counter++;
two[counter]=STR[i];
counter++;
You increment your counter variable twice, so:
two[0]+=STR[i];
0++;
two[1]=STR[i];
1++;
A: functionally,
counter++
is the same as
counter = counter + 1;
In your original code, you use the values counter and counter + 1, but you do not change the value in counter. You only change it in the single counter++ operation, but there have to be 2 of them. Alternatively, you could have written the following:
two[counter]=STR[i]; //does NOT change the value of counter
two[counter+1]=STR[i]; //does NOT change the value of counter
counter += 2; //DOES change the value of counter
A: in your original:
counter++;
should be:
counter+=2;
A: You need to increment counter by 2 in your original code and it works fine. Draw the output on paper:
for(int i=0;i<STR.length;i++){
two[counter]=STR[i];
two[counter+1]=STR[i];
counter++;
}
For STR = "The":
i = 0:
counter = 0;
two[counter] = two[0] = "T";
two[counter+1] = two[1] = "T";
i = 1:
counter = 1;
two[counter] = two[1] = "h"; (You just overwrote two[1] here, it was "T" but is now "h");
two[counter+1] = two[2] = "h";
etc..
Now with corrected code:
for(int i=0;i<STR.length;i++){
two[counter]=STR[i];
two[counter+1]=STR[i];
counter+=2;
}
i = 0:
counter = 0;
two[counter] = two[0] = "T";
two[counter+1] = two[1] = "T";
i = 1:
counter = 2;
two[counter] = two[2] = "h";
two[counter+1] = two[3] = "h";
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,226 |
from pyticketswitch.cost_range import CostRange
from pyticketswitch.discount import Discount
from pyticketswitch.seat import Seat, SeatBlock
from pyticketswitch.commission import Commission
from pyticketswitch.mixins import JSONMixin, SeatPricingMixin
class PriceBand(SeatPricingMixin, JSONMixin, object):
"""Describes a set of tickets with the same ticket type and price.
Attributes:
code (str): the price band identifier.
default_discount (:class:`Discount <pyticketswitch.discount.Discount>`):
this is the discount that will be assumed if no other discount
is specified at reservation time. It holds the prices for the price
band.
allows_leaving_single_seats (str): indicates if this price band will
allow customers to leave a single seat with no neighbours. Values
are 'always', 'never', or 'if_necessary'.
description (str): human readable description of the price band if
available.
cost_range (:class:`CostRange <pyticketswitch.cost_range.CostRange>`):
summary data for the price band including offers.
no_singles_cost_range (:class:`CostRange <pyticketswitch.cost_range.CostRange>`):
summary data for the price band including offers.
example_seats (list): list of :class:`Seats <pyticketswitch.seat.Seat>`
that can be used as examples of what the user might get when they
reserved tickets in this price band.
example_seats_are_real (bool): when :obj:`True` this field indicates
that the example seats are in fact real seats and will be the ones
we attempt to reserve at the reservation stage. When :obj:`False`
these seats merely examples retrieved from cached data, and have
likely already been purchased.
seat_blocks (list): list of
:class:`SeatBlocks <pyticketswitch.seat.SeatBlock>`. When available
this are the contiguous seats that are available for purchase.
:class:`SeatBlocks <pyticketswitch.seat.SeatBlock>` contain
:class:`Seats <pyticketswitch.seat.Seat>`.
user_commission (:class:`Commission <pyticketswitch.commission.Commission>`):
the commission payable to the user on the sale of tickets in this
price band. Only available when requested.
gross_commission (:class:`Commission <pyticketswitch.commission.Commission>`):
the gross commission payable. This is not generally available.
availability (int): the number of tickets/seats available in this price band.
percentage_saving (int): the total percentage saving of the combined seatprice
and surcharge
is_offer (bool): whether or not the priceband has an offer
"""
def __init__(
self,
code,
default_discount,
description=None,
cost_range=None,
no_singles_cost_range=None,
example_seats=None,
example_seats_are_real=True,
seat_blocks=None,
user_commission=None,
discounts=None,
allows_leaving_single_seats=None,
availability=None,
seatprice=None,
surcharge=None,
non_offer_seatprice=None,
non_offer_surcharge=None,
percentage_saving=0,
absolute_saving=0,
is_offer=None,
tax_component=None,
):
self.code = code
self.description = description
self.cost_range = cost_range
self.allows_leaving_single_seats = allows_leaving_single_seats
self.no_singles_cost_range = no_singles_cost_range
self.default_discount = default_discount
self.example_seats = example_seats
self.example_seats_are_real = example_seats_are_real
self.seat_blocks = seat_blocks
self.user_commission = user_commission
self.discounts = discounts
self.availability = availability
self.seatprice = seatprice
self.surcharge = surcharge
self.non_offer_seatprice = non_offer_seatprice
self.non_offer_surcharge = non_offer_surcharge
self.percentage_saving = percentage_saving
self.absolute_saving = absolute_saving
self.is_offer = is_offer
self.tax_component = tax_component
@classmethod
def from_api_data(cls, data):
"""Creates a new **PriceBand** object from API data from ticketswitch.
Args:
data (dict): the part of the response from a ticketswitch API call
that concerns a price band.
Returns:
:class:`PriceBand <pyticketswitch.order.PriceBand>`: a new
:class:`PriceBand <pyticketswitch.order.PriceBand>` object
populated with the data from the api.
"""
api_cost_range = data.get('cost_range', {})
api_no_singles_cost_range = api_cost_range.get(
'no_singles_cost_range', {}
)
cost_range = None
no_singles_cost_range = None
if api_cost_range:
api_cost_range['singles'] = True
cost_range = CostRange.from_api_data(api_cost_range)
if api_no_singles_cost_range:
api_no_singles_cost_range['singles'] = False
no_singles_cost_range = CostRange.from_api_data(
api_no_singles_cost_range
)
discount = Discount.from_api_data(data)
kwargs = {
'code': data.get('price_band_code'),
'description': data.get('price_band_desc'),
'availability': data.get('number_available'),
'cost_range': cost_range,
'no_singles_cost_range': no_singles_cost_range,
'default_discount': discount,
'example_seats_are_real': data.get('example_seats_are_real', True),
'allows_leaving_single_seats': data.get(
'allows_leaving_single_seats'
),
'percentage_saving': data.get('percentage_saving'),
'is_offer': data.get('is_offer'),
}
example_seats_data = data.get('example_seats')
if example_seats_data:
example_seats = [
Seat.from_api_data(seat) for seat in example_seats_data
]
kwargs.update(example_seats=example_seats)
seat_block_data = data.get('free_seat_blocks')
if seat_block_data:
separators_by_row = seat_block_data.get('separators_by_row')
restricted_view_seats = seat_block_data.get('restricted_view_seats')
seats_by_text_message = seat_block_data.get('seats_by_text_message')
blocks_by_row = seat_block_data.get('blocks_by_row')
seat_blocks = []
if blocks_by_row:
for row_id, row in blocks_by_row.items():
for block in row:
separator = separators_by_row.get(row_id)
seat_block = SeatBlock.from_api_data(
block=block,
row_id=row_id,
separator=separator,
restricted_view_seats=restricted_view_seats,
seats_by_text_message=seats_by_text_message,
)
seat_blocks.append(seat_block)
kwargs.update(seat_blocks=seat_blocks)
user_commission_data = data.get('predicted_user_commission')
if user_commission_data:
user_commission = Commission.from_api_data(user_commission_data)
kwargs.update(user_commission=user_commission)
discounts_data = data.get('possible_discounts', {}).get('discount')
if discounts_data:
discounts = [
Discount.from_api_data(discount_data)
for discount_data in discounts_data
]
kwargs.update(discounts=discounts)
tax_component = data.get('sale_combined_tax_component')
if tax_component is not None:
kwargs.update(tax_component=tax_component)
kwargs.update(SeatPricingMixin.kwargs_from_api_data(data))
return cls(**kwargs)
def get_seats(self):
"""Get all seats in child seat blocks
Returns:
list: list of :class:`Seats <pyticketswitch.seat.Seat>`.
"""
if not self.seat_blocks:
return []
return [
seat
for seat_block in self.seat_blocks
for seat in seat_block.seats or []
]
def __repr__(self):
if self.description:
return u'<PriceBand {}:{}>'.format(
self.code, self.description.encode('ascii', 'ignore')
)
return u'<PriceBand {}>'.format(self.code)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,471 |
{"url":"http:\/\/ilovephilosophy.com\/viewtopic.php?f=2&t=193689&start=125","text":"## My Surrogate Activities.\n\nThis is the place to shave off that long white beard and stop being philosophical; a forum for members to just talk like normal human beings.\n\n\"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$$\n\nZero_Sum\nEvil Neo-Nazi Extraordinaire.\n\nPosts: 2560\nJoined: Thu Nov 30, 2017 7:05 pm\nLocation: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America.\n\n\"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. You see...a pimp's love is very different from that of a square. Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too. What exactly is logic? -Magnus Anderson Support the innocence project on AmazonSmile instead of Turd's African savior biker dude. http:\/\/www.innocenceproject.org\/ Mr Reasonable resident contrarian Posts: 25948 Joined: Sat Mar 17, 2007 8:54 am Location: pimping a hole straight through the stratosphere itself ### Re: My Surrogate Activities. Mr Reasonable wrote: What the fuck is that? \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$$\n\nZero_Sum\nEvil Neo-Nazi Extraordinaire.\n\nPosts: 2560\nJoined: Thu Nov 30, 2017 7:05 pm\nLocation: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America.\n\n### Re: My Surrogate Activities.\n\nYou see...a pimp's love is very different from that of a square.\nDating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too.\n\nWhat exactly is logic? -Magnus Anderson\n\nSupport the innocence project on AmazonSmile instead of Turd's African savior biker dude.\nhttp:\/\/www.innocenceproject.org\/\n\nMr Reasonable\nresident contrarian\n\nPosts: 25948\nJoined: Sat Mar 17, 2007 8:54 am\nLocation: pimping a hole straight through the stratosphere itself\n\n### Re: My Surrogate Activities.\n\nYou see...a pimp's love is very different from that of a square.\nDating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too.\n\nWhat exactly is logic? -Magnus Anderson\n\nSupport the innocence project on AmazonSmile instead of Turd's African savior biker dude.\nhttp:\/\/www.innocenceproject.org\/\n\nMr Reasonable\nresident contrarian\n\nPosts: 25948\nJoined: Sat Mar 17, 2007 8:54 am\nLocation: pimping a hole straight through the stratosphere itself\n\n### Re: My Surrogate Activities.\n\nWho said that? Do you secretly listen to that crap man? I mean, I knew you listened to music I couldn't stand before but that's a new low even for you.\n\"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. Hanson is a highly talented, highly underrated band. In 100 years they'll be teaching about them in music departments at fine universities. They're ahead of their time you just can't see it. You see...a pimp's love is very different from that of a square. Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too. What exactly is logic? -Magnus Anderson Support the innocence project on AmazonSmile instead of Turd's African savior biker dude. http:\/\/www.innocenceproject.org\/ Mr Reasonable resident contrarian Posts: 25948 Joined: Sat Mar 17, 2007 8:54 am Location: pimping a hole straight through the stratosphere itself ### Re: My Surrogate Activities. Yeah, I can totally see you listening to that 90's Hip Hop. That's more of your thing. \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. Mr Reasonable wrote:Hanson is a highly talented, highly underrated band. In 100 years they'll be teaching about them in music departments at fine universities. They're ahead of their time you just can't see it. Thanks man for that this evening, I almost pissed myself laughing just now. \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. You see...a pimp's love is very different from that of a square. Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too. What exactly is logic? -Magnus Anderson Support the innocence project on AmazonSmile instead of Turd's African savior biker dude. http:\/\/www.innocenceproject.org\/ Mr Reasonable resident contrarian Posts: 25948 Joined: Sat Mar 17, 2007 8:54 am Location: pimping a hole straight through the stratosphere itself ### Re: My Surrogate Activities. You see...a pimp's love is very different from that of a square. Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too. What exactly is logic? -Magnus Anderson Support the innocence project on AmazonSmile instead of Turd's African savior biker dude. http:\/\/www.innocenceproject.org\/ Mr Reasonable resident contrarian Posts: 25948 Joined: Sat Mar 17, 2007 8:54 am Location: pimping a hole straight through the stratosphere itself ### Re: My Surrogate Activities. Wow man, I didn't know transgendered metal was a thing. At least it isn't Hip Hop..... \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$$\n\nZero_Sum\nEvil Neo-Nazi Extraordinaire.\n\nPosts: 2560\nJoined: Thu Nov 30, 2017 7:05 pm\nLocation: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America.\n\n### Re: My Surrogate Activities.\n\nWe can agree on Cannibal Corpse. Knew one of their guitarists back in the day because their girlfriend was a friend of mine. True story.\n\"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$Zero_Sum Evil Neo-Nazi Extraordinaire. Posts: 2560 Joined: Thu Nov 30, 2017 7:05 pm Location: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America. ### Re: My Surrogate Activities. I've met all kinds of death metal band types over the years. I have a friend who started a chapter of a death metal club here in Alabama after going and meeting these dudes who started the club in Brooklyn. So they've got chapters in every state and he happened to be the guy who started the one here. I went to a death metal bar up there once and met one of the guys from type o negative, and a bunch of us rode around in this guy's hearse. My friend promoted a ton of shows all over the place from here to texas and god knows where else and over the years I've just known a bunch of these guys. He played with some band at warped tour once. A lot of times I meet these people and they're \"famous\" in death metal circles, but I have no idea who they are. I was backstage at a morbid angel show once in Atlanta and didn't realize that I was talking to two of the guys from the band. I saw ozzy with sepultura in the 90s, Danzig, a bunch of others. You see...a pimp's love is very different from that of a square. Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too. What exactly is logic? -Magnus Anderson Support the innocence project on AmazonSmile instead of Turd's African savior biker dude. http:\/\/www.innocenceproject.org\/ Mr Reasonable resident contrarian Posts: 25948 Joined: Sat Mar 17, 2007 8:54 am Location: pimping a hole straight through the stratosphere itself ### Re: My Surrogate Activities. \"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-$$$\n\nZero_Sum\nEvil Neo-Nazi Extraordinaire.\n\nPosts: 2560\nJoined: Thu Nov 30, 2017 7:05 pm\nLocation: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America.\n\n### Re: My Surrogate Activities.\n\nI saw Manson in 1995 or so when he was opening for danzig. Danzig, Manson and Korn for 15 bucks in a small venue. Crazy shit man.\nYou see...a pimp's love is very different from that of a square.\nDating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too.\n\nWhat exactly is logic? -Magnus Anderson\n\nSupport the innocence project on AmazonSmile instead of Turd's African savior biker dude.\nhttp:\/\/www.innocenceproject.org\/\n\nMr Reasonable\nresident contrarian\n\nPosts: 25948\nJoined: Sat Mar 17, 2007 8:54 am\nLocation: pimping a hole straight through the stratosphere itself\n\n### Re: My Surrogate Activities.\n\n\"I'm sorry, but the lifestyle you've ordered that you've grown accustomed to is completely out of stock. Have a nice day! \"-\\$\n\nZero_Sum\nEvil Neo-Nazi Extraordinaire.\n\nPosts: 2560\nJoined: Thu Nov 30, 2017 7:05 pm\nLocation: U.S.S.A- Newly lead Bolshevik Soviet block. Also known as Weimar America.\n\nPreviousNext","date":"2019-12-05 15:38:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.429988294839859, \"perplexity\": 11384.304798396313}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540481076.11\/warc\/CC-MAIN-20191205141605-20191205165605-00319.warc.gz\"}"} | null | null |
Sichuan Airlines () – chińskie linie lotnicze z siedzibą w Chengdu. Obsługują połączenia krajowe. Głównym węzłem jest Port lotniczy Chengdu-Shuangliu i Port lotniczy Chongqing-Jiangbei. Zostały założone 19 września 1986, a rozpoczęły działalność 14 lipca 1988 lotami do Chengdu i Wanzhou.
Agencja ratingowa Skytrax przyznała liniom trzy gwiazdki.
Flota
Flota Sichuan Airlines składa się ze 185 samolotów o średnim wieku 8 lat (stan na luty 2023 r.).
Przypisy
Chińskie linie lotnicze | {
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Q: How to use reshape2's dcast to select only one of several observations(values) I have the following dataset
> dataset2
ID ATCcode date
1 1 N06AA 2001-01-01
2 1 N06AB 2001-04-01
3 1 N06AB 2001-03-01
4 1 N06AB 2001-02-01
5 1 N06AC 2001-01-01
6 2 N06AA 2001-01-01
7 2 N06AA 2001-02-01
8 2 N06AA 2001-03-01
9 3 N06AB 2001-01-01
10 4 N06AA 2001-02-01
11 4 N06AB 2001-03-01
It's in long format and I'd like it to be in wide format. However, I only want the earliest date for each ATCcode - and not any of the later dates. Thus I'd like to end up here:
> datasetLong
ID N06AA N06AB N06AC
1 1 2001-01-01 2001-02-01 2001-01-01
2 2 2001-01-01 <NA> <NA>
3 3 <NA> 2001-01-01 <NA>
4 4 2001-02-01 2001-03-01 <NA>
(This is just a sample of the real dataset, it has more variation in the values and more observations than this).
I've managed to cast the dataset somewhat, but not in the manner which I want to:
dataset3 <- reshape2::dcast(dataset2, ID ~ ATCcode)
gives me the length of each vector/list:
> dataset3
ID N06AA N06AB N06AC
1 1 1 3 1
2 2 3 0 0
3 3 0 1 0
4 4 1 1 0
Instead of the length, I'd like just one value, and that value should be the smallest value (or, the earliest date).
I've found a similar question asked on StackOverflow, but I was unable to use that in any way without getting various errors. I have not used melt in the above attempt, is that maybe necessary?
Any help is appreciated.
A: This answer uses tidyverse methods.
One way would be would be to select minimum date from each ID and ATCcode and convert the data to wide format.
library(dplyr)
df %>%
mutate(date = as.Date(date)) %>%
group_by(ID, ATCcode) %>%
slice(which.min(date)) %>%
tidyr::pivot_wider(names_from = ATCcode, values_from = date)
# ID N06AA N06AB N06AC
# <int> <date> <date> <date>
#1 1 2001-01-01 2001-02-01 2001-01-01
#2 2 2001-01-01 NA NA
#3 3 NA 2001-01-01 NA
#4 4 2001-02-01 2001-03-01 NA
data
df <- structure(list(ID = c(1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 3L, 4L,
4L), ATCcode = structure(c(1L, 2L, 2L, 2L, 3L, 1L, 1L, 1L, 2L,
1L, 2L), .Label = c("N06AA", "N06AB", "N06AC"), class = "factor"),
date = structure(c(1L, 4L, 3L, 2L, 1L, 1L, 2L, 3L, 1L, 2L,
3L), .Label = c("2001-01-01", "2001-02-01", "2001-03-01",
"2001-04-01"), class = "factor")), class = "data.frame", row.names = c("1",
"2", "3", "4", "5", "6", "7", "8", "9", "10", "11"))
| {
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News World US and Greece reaffirm excellent bilateral relations
US and Greece reaffirm excellent bilateral relations
Greece and the United States reaffirmed their excellent bilateral relations and mutual desire to further deepen cooperation in a series of areas, from defence and security, to investment and energy, and to education and culture, according to a joint statement released on Monday after the meeting of Foreign Minister Nikos Dendias and his American counterpart Mike Pompeo in Thessaloniki.
The two sides conducted a high-level review of the US-Greece Strategic Dialogue, ahead of the third Strategic Dialogue to be held in Washington in 2021.
The joint statement follows:
"The United States and Greece shared views on the Eastern Mediterranean and reaffirmed their belief that maritime delimitation issues should be resolved peacefully in accordance with international law.
In this regard, the United States welcomed Greece's confirmed readiness to engage with other countries in the region to achieve maritime delimitation agreements through dialogue and in accordance with international law.
The United States and Greece also reiterated their dedication to enhancing their close cooperation as NATO Allies, using all appropriate means at their disposal, to safeguard stability and security in the wider region.
The two sides reiterated their desire to continue to strengthen their cooperation in various sectors in the framework of the 3+1 format (Cyprus, Greece, Israel, and the United States), which was launched in Jerusalem in March 2019, as this partnership can contribute to the promotion of peace, stability, security, and prosperity in the Eastern Mediterranean and the wider region.
The United States and Greece reiterate their support for the integration of all the countries of the Western Balkans into European and Transatlantic institutions according to the choice of their citizens.
They highlighted Greece's efforts to this end, noting the continued relevance of the historic Prespes Agreement and North Macedonia's subsequent accession to NATO, while underscoring the importance of its consistent implementation in good faith.
Greece and the United States expressed their intent to further enhance their strategic defence and security partnership by expanding and deepening the US-Greece Mutual Defence Cooperation Agreement, which was last updated in October 2019 and greatly contributes to the security of both nations.
In the coming Strategic Dialogue, both sides intend to discuss how to further assist each other in maintaining strong, capable, and interoperable militaries.
The United States and Greece took note of their security cooperation based upon a shared interest in protecting the safety and security of both nations from terrorists, transnational criminal organisations, and other threat actors.
Despite the global pandemic, outstanding US-Greece law enforcement cooperation on arrests, seizures, and extraditions has continued unabated.
Moreover, the two countries took stock of continued improvement in border security and counterterrorism information sharing, and collaboration according to international standards.
They committed to full implementation of the 2016 Joint Statement on Greece's participation in the Visa Waiver Program, including beginning issuance of an updated national identification card and fully collecting and analysing air passenger data.
The United States and Greece look forward to enhancing trade and investment between our two countries.
The two governments noted that Greece substantially improved its protection and enforcement of intellectual property in 2019, which resulted in Greece's removal from the Office of the United States Trade Representative's Special 301 Report this year.
The United States and Greece welcomed the inauguration of new investments and acquisitions by US companies in Greece and reaffirmed their commitment to encourage that trend.
The two governments look forward to increasing cooperation on trade.
The United States and Greece attach great importance on the cooperation on artificial intelligence, cyber security, 5G, and privatisation of strategic infrastructure. The two countries remain committed to enhancing the capacity of women to prosper in the workforce, succeed as entrepreneurs, and be better positioned to foster economic growth.
The United States and Greece welcomed the completion of the Greek section of the Trans Adriatic Pipeline and looked forward to discussing at the coming Strategic Dialogue their mutual support for the Interconnector Greece-Bulgaria, the floating storage regasification unit project at Alexandroupoli, the privatisation of the Kavala underground gas storage, the Interconnection Greece-North Macedonia project, and other commercially viable projects, which could include the EastMed Gas Pipeline.
Both sides welcomed the September 22 formal establishment of the EastMed Gas Forum as a regional organisation and the ongoing successful cooperation on energy through the 3+1 process.
Greece and the United States underlined the participation of ExxonMobil in partnership with Total and Hellenic Petroleum in offshore exploration blocks off the coast of Crete, as well the potential opportunities for US investment in the renewable energy sector, with the recently ratified Greek Law 'Modernisation of Environmental Legislation'.
The United States and Greece reaffirmed the centrality of people-to-people ties in their bilateral relations.
They look forward to marking Greece's independence bicentennial in 2021, as well as exchanging exhibitions and cooperation between museums.
The United States and Greece reaffirmed their intent to strengthen links between US and Greek universities and create new joint, dual-degree, and study abroad programs.
Both countries will continue to support ongoing educational and academic exchanges, including the Fulbright Program in Greece, which marked its 72nd anniversary in 2020.
The United States and Greece agreed to continue their longstanding cooperation to protect Greece's cultural property.
Both countries also agreed to cooperate on capacity building within the creative industries, including in the film sector.
The United States and Greece also agreed to explore public-private partnerships in culture and technology."
See also: Meeting between US Secretary of State Pompeo and FM Dendias concluded
(amna.gr)
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