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f5135cdabcce655edda88445d82dca5f7e27f758 | subsection | 38 | 69 | Soundness of Andromeda | Universes
The final example shows how to use coercions and operations to implement a universe à la
Tarski. We postulate a universe U, whose elements should be thought of as
names of types, with an operation El that converts the names to the
corresponding types:
constant U : Type
constant El : U → Type
Because El is a... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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3796de46271e53b23736668ac2df908c4bcf15a1 | subsection | 39 | 69 | Soundness of Andromeda | It is
difficult to do justice to all of them. Its overall design follows the tradition of
LCF and its
descendants , , . However, LCF and
many of its descendants use Church's simple type
theory , whereas Andromeda is based on the
dependent type theory of Martin-Löf . Consequently, Andromeda cannot
advantageously integr... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 120,
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"raw": "Michael J. Gordon, Arthur J. Milnter, and Christopher P. Wadsworth. Edinburgh LCF: A Mechanized Logic of Computation. Number 78 in Lecture Notes in Computer Science. Springer-Verlag, 1979.",
"... | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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21b3a8ea115320c7d7bdb5f62ffd0fc7e3923d12 | subsection | 40 | 69 | Soundness of Andromeda | This can be accomplished with a minor modification of AML: if we implement
all calls of AML to the nucleus as operations (whose default handler is the
nucleus), then the user can intercept them and do whatever they like: record them,
communicate them, or modify them to obtain a proof translation.
Alternatively, the |ju... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1414,
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"raw": "Paolo Capriotti. Notions of type formers. In 23rd International Conference on Types for Proofs and Programs (TYPES). Budapest, Hungary, May 29–June 1 2017.",
"source_ref_id": "7f1fd75a19aa109... | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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42e2743dd178ec85b29037340953c20c6336eb5b | subsection | 41 | 69 | Soundness of Andromeda | Syntax
{Contexts}
\Gamma , {}& \bullet & & \text{empty context}\\
\ \mid \ \ {}& \Gamma ,\, x\, {:}\, A & & \text{context $\Gamma $ extended with $x: A$}
{Terms and types}
s, t, A, B\ \mathrel {{:}{:}{=}}\ {}& \mathsf {Type}& & \text{universe}\\
\ \mid \ \ {}& \mathop {\textstyle \prod _{(x {:} A)}} B& & \text{product... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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c150fcf3acef45b3ab7596ccbc1d60a31e8d8c1f | subsection | 42 | 69 | Soundness of Andromeda | We first define the map function to show how AML syntax works, and the
auxiliary apply function that folds application of a function over a list of
arguments:
let rec map f xs =
match xs with
| [] ⇒ []
| ?x :: ?xs ⇒ (f x) :: (map f xs)
end
let rec apply f xs =
match xs with
| [] ⇒ f
| ?x :: ?xs ⇒ apply (f x) xs
end
N... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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2b1e37507d70e6f0409a860e3f77d0cdabff7473 | subsection | 43 | 69 | The standard library | To test the viability of our design we implemented a small standard library in AML. By
design, anything that is implemented in AML is safe: it may not work as expected, or
diverge, but it will never produce an invalid judgment, or derive an invalid equality. (Of
course, this does not say much until we have dealt with \... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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395d12b620d6c24a260c8b4501c100e17275b8a1 | subsection | 44 | 69 | Equality checking | The most substantial part of the library is a user-extensible equality checking algorithm
with rudimentary support for implicit arguments, based on similar
ones by Stone and Harper and Coquand . It
computes a witness of equality \Gamma \vdash s ~ \equiv ~ t : A in two phases:The type directed phase computes the weak h... | {
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{
"arxiv_id": "",
"doi": "10.1145/1183278.1183281",
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"raw": "Christopher A. Stone and Robert Harper. Extensional equivalence and singleton types. ACM Transactions on Computational Logic, 7(4):676–722, Oct... | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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3a4edf11bfe712ae070b65a131f4dec25c6b2190 | subsection | 45 | 69 | Equality hints | The equality checking algorithm can be extended by the user with new rules, which we call
equality hints. There are three kinds:an \eta -hint, or an extensionality hint, is a term whose type has the form
\mathop {\textstyle \prod _{(x_1 {:} A_1)}} \cdots \mathop {\textstyle \prod _{(x_n {:} A_m)}}
\mathop {\textstyle ... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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1ee3c98221fa634368273be5204b0e4625fef6f8 | subsection | 46 | 69 | Equality hints | For
instance, when we axiomatize simple products A\times B, the extensionality rule\mathop {\textstyle \prod _{(A {:} \mathsf {Type})}}
\mathop {\textstyle \prod _{(B {:} \mathsf {Type})}}
\mathop {\textstyle \prod _{(x, y {:} A\times B)}} \\
\mathsf {Eq}_{A}(\mathtt {fst}\,A\,B\,x,\mathtt {fst}\,A\,B\,y) \rightarrow \... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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f17793fe66655fe2617e7dc8f7a4be7249173e62 | subsection | 47 | 69 | Implicit arguments | The standard library provides basic support for implicit arguments. In other
systems these are usually implemented with meta-variables, which are not
available in AML. In their place, we use ordinary fresh variables generated
using the assume construct. We refer to these as implicit
variables. We collect constraints th... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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b043e6072de159a04cb96f667df39eca7860c298 | subsection | 48 | 69 | Examples | In this section we show Andromeda at work through several examples. | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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25ca745f11f11084e618a7a5dc3d81d7bcce01da | subsection | 49 | 69 | Proving equality with handlers | As explained in §REF , when AML is faced with proving a
non-trivial equality, it delegates it to user code by triggering the operation
|equal|. To see how this works, let us walk through a computation that
constructs a term witnessing symmetry of equality (without the standard library
installed):
λ (A : Type) (x y : A... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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d7a9f5dc24f7a201ac7d13669d82e758fd33a4b6 | subsection | 50 | 69 | Dependent sums | Our second example shows how to axiomatize dependent sums. This time we use the standard
library and rely on its equality checking. We start by postulating the type and term
constructors:
constant Σ : Π (A : Type) (B : A → Type), Type
constant existT : Π (A : Type) (B : A → Type) (a : A), B a → Σ A B
Next, we postula... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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467bff0e1d89f65bc03706eb9890c287e93409b4 | subsection | 51 | 69 | Natural numbers | The standard library provides functions for calculating weak head-normal forms that can be used
as a computation device at the level of type-theoretic terms. We show how this is done
by axiomatizing natural numbers and computing with them.We postulate the type of natural numbers and its constructors
constant nat : Typ... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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67c488d291ab744e5a12539a41fc1bc9fb05dd92 | subsection | 52 | 69 | Natural numbers | The overall usability of the system depends on
having a good standard library with sensible default settings.The definition of multiplication and its Peano axioms are derived similarly:
constant ( * ) : nat → nat → nat
constant multdef :
∏ (n m : nat), n * m ≡ natrect (λ , nat) O (λ x, x + n) mlet multO =
now betas = ... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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372413554038ce2512d973b4634291fb5916719a | subsection | 53 | 69 | Untyped | An example that cannot be done easily in proof assistants based on intensional type
theory is in order. Let us axiomatize the untyped \lambda -calculus as a type that is
judgmentally equal to its own function space, and show that it possesses a fixed-point
operator.We first postulate that there is a type equal to its f... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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fbed347323633c79ed4799fa114faa48bca75923 | subsection | 54 | 69 | Universes | The final example shows how to use coercions and operations to implement a universe à la
Tarski. We postulate a universe U, whose elements should be thought of as
names of types, with an operation El that converts the names to the
corresponding types:
constant U : Type
constant El : U → Type
Because El is an eliminat... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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08cc23f3af13d518ca99024efdce65f2be139875 | subsection | 55 | 69 | Related work | Andromeda draws heavily on the experience and ideas from other proof assistants. It is
difficult to do justice to all of them. Its overall design follows the tradition of
LCF and its
descendants , , . However, LCF and
many of its descendants use Church's simple type
theory , whereas Andromeda is based on the
dependent... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 201,
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"raw": "Michael J. Gordon, Arthur J. Milnter, and Christopher P. Wadsworth. Edinburgh LCF: A Mechanized Logic of Computation. Number 78 in Lecture Notes in Computer Science. Springer-Verlag, 1979.",
"... | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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cf83ba92b7b51e6f929bfabaad13da4cfdee6d1d | subsection | 56 | 69 | Future work | We feel that Andromeda shows a promising way to design a proof assistant based on
type theory with equality reflection, but much remains to be done. | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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5ed4a3dace2fead57dffd3e0090964301a8e1d7f | subsection | 57 | 69 | Syntactic sugar and end-user support | AML turned out to be a useful tool for the implementers of the standard library. If we
imagine that the end-user is a mathematician who just wants to do mathematics, without
learning the intricacies of operations and handlers, then we need further support for
creating a more user friendly environment. There ought to be... | {
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{
"arxiv_id": "",
"doi": "",
"end": 528,
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"raw": "William J. Bowman. Growing a proof assistant. In Higher-Order Programming with Effects, 2016.",
"source_ref_id": "5bdd94a3b7fe0379915c493551b22095b81a2a70",
"start": 480
}
]
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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203e1b0159e3892f1fa76c7f65bc406932978c8b | subsection | 58 | 69 | Formal verification of the meta-theoretic properties | While we carefully designed the underlying type theory and made sure it is precisely clear
what the type-theoretic rules are, we have not formally verified that the system has the
desired meta-theoretic properties, such as uniqueness of typing, validity of inversion
principles, and well-behaved context joins. We expect... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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2fc7f24ec986d3ff3dddfce80d334f35b2844a70 | subsection | 59 | 69 | Recording derivations | Like all LCF-style proof assistants, Andromeda does not record derivations, only their
conclusions. (In fact, all practical proof assistants do this, though some implement
type theories that allow derivations to be reconstructed.) There
might be situations in which we wish to record or communicate the derivation. For
i... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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64d645dcd98731e75b4cf38446f5c3fade178d01 | subsection | 60 | 69 | Removal of | A major forthcoming modification of the current system is elimination of \mathsf {Type}: \mathsf {Type}. The syntax of AML takes advantage of \mathsf {Type}: \mathsf {Type} to conflate term
and type judgments within a single abstract type \mathtt {judgment}.But the nucleus does not rely on \mathsf {Type}: \mathsf {Type... | {
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{
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"raw": "Paolo Capriotti. Notions of type formers. In 23rd International Conference on Types for Proofs and Programs (TYPES). Budapest, Hungary, May 29–June 1 2017.",
"source_ref_id": "7f1fd75a19aa1099... | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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5417e6a6402ad3d06485ffee8e42f5cc87e21c72 | subsection | 61 | 69 | The rules of type theory | In this appendix we give the formulation of type theory in a declarative way that
minimizes the number of judgments, and so is better suited for a semantic account. We omit formal
treatment of bound variables and substitution, which is standard. | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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] | 2,018 | en | Computer Science | [
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cda8e15eae9623c8167c702948c0cb8f36801e78 | subsection | 62 | 69 | Syntax | {Contexts}
\Gamma , {}& \bullet & & \text{empty context}\\
\ \mid \ \ {}& \Gamma ,\, x\, {:}\, A & & \text{context $\Gamma $ extended with $x: A$}
{Terms and types}
s, t, A, B\ \mathrel {{:}{:}{=}}\ {}& \mathsf {Type}& & \text{universe}\\
\ \mid \ \ {}& \mathop {\textstyle \prod _{(x {:} A)}} B& & \text{product}\\
\ \m... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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e50de7606fc0a5c16803846edffa35231d8f5119 | subsection | 63 | 69 | Judgments | & \Gamma \ \mathsf {ctx} & & \text{$\Gamma $ is a well formed context} \\
& \Gamma \vdash t : A & & \text{$t$ is a well formed term of type $A$ in context $\Gamma $} \\
& \Gamma \vdash s ~ \equiv ~ t : A & & \text{$s$ and $t$ are equal terms of type $A$ in context $\Gamma $}
{We use the following abbreviations:}
& \Gam... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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0f97ee6bc2404fcb827f0a5bdd7c191fec188896 | subsection | 64 | 69 | Contexts | [ctx-empty]
ctx[ctx-extend]
ctxA typexdom()
(, x : A) ctx | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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5d2b448105bec4eaa626fa0588203044f9beaf73 | subsection | 65 | 69 | Terms and types | Conversion
[term-ty-conv]
t : AA B
t : BVariable
[term-var]
(, x : A) ctx
, x : A x : A[term-var-skip]
(, y : B) ctxx : A
, y : B x : AUniverse
[ty-type]
ctx
Type typeProduct
[ty-prod]
A type, x : A B type
(x : A)B type[term-abs]
, x : A t : B
(x : A.B . t) : (x : A)B[term-app]
s : (x ... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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d901fa85e1f2a0cc1ff5e24bf6dbdde2a485a52e | subsection | 66 | 69 | Equality | General rules
[eq-refl]
t : A
t t : A[eq-sym]
t s : A
s t : A[eq-trans]
s t : At u : A
s u : AConversion
[eq-ty-conv]
s t : AA B
s t : BEquality reflection
[eq-reflection]
u : EqA(s,t)
s t : AComputation
[prod-beta]
, x : A s : Bt : A
(x : A.B . s)@x:A.B t s[t/... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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73aacd734a4076b3849f10e1ff3f5bd7f6964620 | subsection | 67 | 69 | Congruences | Type formers
[cong-prod]
A C, x : A B D[x/y]
(x : A)B (y : C)D[cong-eq]
A Bs u : At v : A
EqA(s,t) EqB(u,v)Products
[cong-abs]
A C, x : A B D[x/y], x : A s t[x/y] : B
(x : A.B . s) (y : C.D . t) : (x : A)B[cong-app]
A C, x : A B D[x/y]s u : (x : A)Bt v : A
(s@x:A.B t) ... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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f3cb9990066c9b301d33dbfd35b1c9f5ae2f1a2f | subsection | 68 | 69 | The auto tactic | We include here the complete code for implementing a simple auto tactic
from § .We first define the map function to show how AML syntax works, and the
auxiliary apply function that folds application of a function over a list of
arguments:
let rec map f xs =
match xs with
| [] ⇒ []
| ?x :: ?xs ⇒ (f x) :: (map f xs)
end... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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16c94c8b52255689a68376a589ed06ea253c67cf | abstract | 0 | 25 | Abstract | Accurately predicting the performance of coronagraphs and tolerancing optical
surfaces for high-contrast imaging requires a detailed accounting of
diffraction effects. Unlike simple Fraunhofer diffraction modeling, near and
far-field diffraction effects, such as the Talbot effect, are captured by
plane-to-plane propaga... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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187c72f66ab457e14cad31426c45faee79d67165 | subsection | 1 | 25 | INTRODUCTION | The Physical Optics Propagation in PYthon module (POPPY) , is an open-source library for modeling optical diffraction. POPPY was originally developed as the back-end diffraction code for the package WebbPSF, which simulates PSFs for the James Webb Space Telescope (JWST) and now also the Wide Field Infrared Survey Teles... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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8cc891f459b6eb8dfe22f235ff62d965115d97b2 | subsection | 2 | 25 | Implementing Fresnel Propagation Methods | POPPY supports optical simulations in two regimes, the standard Fraunhofer (far-field) and Fresnel (near-field) approximations . The methods used in Fraunhofer calculations have been previously presented. To quickly generate results in the Fraunhofer regime, for coronagraphs with a small focal plane mask size POPPY typ... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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77cbd89c2b24f86ed7141094041822d753b8d641 | subsection | 3 | 25 | Implementing Fresnel Propagation Methods | These effects must be taken into account when modeling high performance coronagraphs.
[Figure: Example of the Talbot Effect calculated using the POPPY Fresnel module.In addition to edge effects, phase errors in the plane at Z=0 (top row) are partially converted into a similar pattern of amplitude errors at an intermedi... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
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147752962f2616488c21978fbe569385cb73ff34 | subsection | 4 | 25 | Computation Needs for Modeling Optical Propagation | Such calculations are dominated primarily by a sequence of Fourier transforms of large 2D arrays of complex floating point numbers. Additional significant computation costs come from evaluations of complex exponential functions, and the general array operations for creating and manipulating the arrays representing coro... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
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3cd09452f6ac2ba19c2de5d2418aafffe23c62b8 | subsection | 5 | 25 | Achieving High Performance in Scientific Python | As the Python language has become widespread in scientific, engineering, and educational computing, , , there have been widespread efforts towards enabling high performance numerical work. In fact astronomers played a significant role, in the early development and establishment of the numpy library, which has now becom... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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7ad9df13f1035c6dde660928859ccb1204a0eddc | subsection | 6 | 25 | Achieving High Performance in Scientific Python | OpenCL works on a wider range of hardware, in particular the AMD GPUs available in typical Apple computers which are often used by many astronomers.Previous authors have implemented diffraction simulations directly in CUDA. We have combined the user friendly POPPY optical system and wavefront classes with optional high... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
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01b924635f874599119eef158d494db6bd73f133 | subsection | 7 | 25 | Profiling Enables Prioritization of Optimization Efforts | Efficiently targeting optimization work requires an accurate understanding of which parts of program dominate the costs.Hence the famous aphorism attributed to Donald Knuth: “Premature optimization is the root of all evil”. In order to decrease the overall run-time of POPPY calculations we therefore ran a set of test c... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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a7846c45a88bbc7832119ce2f6d1b8291ee90043 | subsection | 8 | 25 | Accelerating the Discrete Fast Fourier Transform: GPUs and/or Intel MKL | The first area to optimize is the calculation of 2D Fourier transforms of large complex arrays.
The default FFT option provided with NumPy is implemented via an interface to the FFTPACK C library.
A generally faster library, FFTW offers parallelized computation of the discrete Fourier transform in O(N\log N) for any N ... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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f879f9285942ae704b7c02232936fbe9ca7a6a76 | subsection | 9 | 25 | Accelerating the Discrete Fast Fourier Transform: GPUs and/or Intel MKL | That said, ensuring that you have installed and are using a version of numpy that uses MKL can still be somewhat opaque and confusing. Users should observe the number of cores in use during FFT operations and check that conda has installed both the mkl and mkl_fft packages, and that numpy.__config__.show() reports that... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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c5e569e68e25403240eb30ac2cf20d80e0160102 | subsection | 10 | 25 | Accelerating Evaluations of Exponential Expressions: NumExpr | After optimizing FFT runtime, the next most time-consuming operation is evaluations of exponential functions.
This occurs both as the exponential term in the quadratic phase factor,\exp [i(x^2 + y^2)/z],where x, y and are N\times N arrays and z is a real number, and also in evaluating the complex phasor of a given opti... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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1469ab16fac0da3355983dce74ac9c1190895ec8 | subsection | 11 | 25 | Accelerating Fourier Array Shifting on the GPU | The FFT algorithm imposes certain fixed relationships on the locations of spatial frequencies within the output array, in particular locating the origin in the extreme corners of the array rather than in the center.
Shifting the array contents to relocate the zero-frequency origin to the center of a Fourier transformed... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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1def2c9589242ff692566f32b4cebef11760f072 | subsection | 12 | 25 | Reproducibility | The benchmarking results and figures reported here were generated in a Jupyter notebook which the reader is invited to review and test on other system.
Benchmarking results were measured using the IPythonversion 6.1.0 in Python 3.6 %timeit magic command on default settings. For each configuration a mean and standard de... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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7cd1085b61d3c4a472197d9a713bf143ffe0208f | subsection | 13 | 25 | Fresnel Optical System Run Times | Measured mean unoptimized run times for the Fresnel coronagraphic test system, Appendix , are plotted and tabulated in Fig. REF .
The runtime increases approximately as the area of the array (\propto N^2).
Notably, the N=2800 runs faster than the next power of two, indicating significant speed gains maybe found by not ... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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ff2fc308baa082defab7863e0064d02ffe7b3fbe | subsection | 14 | 25 | Conclusions | In the initial, pre-optimization code state, calculation times were typically dominated by the FFTs (in plain NumPy or FFTW). After the optimizations described in this paper, the FFT has been accelerated enough that the calculations of complex exponentials are often a more substantial contributor.In the era of readily ... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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2a57c5e9b283ca10184c99b0956fe601ec28331e | subsection | 15 | 25 | Summary | For a realistic 10 plane optical system:An approximately 3\times speed-up was realized using optimized math libraries and a 5\times speed up was found using research-grade GPUs for some calculations.
NumExpr library’s chunked, multicore array math provides significant speed gains.
FFTW and Intel MKL-FFT provide large... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
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0792b0dca163983d0a5eefb003d82eb66ff2948a | subsection | 16 | 25 | Summary | \ref{fig:talbot}.\begin{lstlisting}[frame=single]import poppyimport astropy.units as uwf_f = poppy.FresnelWavefront(beam_radius=2*u.cm,wavelength=0.5*u.um,npix=256,oversample=8)sineWFE = poppy.wfe.SineWaveWFE(spatialfreq=500,amplitude=5e-9)wf_f* = sineWFEwf_f *= poppy.CircularAperture(radius=wf_f.diam/2)Z_t = 2*((1/sin... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
"Marshall D. Perrin"
] | [
"astro-ph.IM"
] | 2,018 | en | Physics | [
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8811f30fa0e99dd5e1c23ed309c680e1200ddafd | subsection | 17 | 25 | Summary | It makes use of data files on optical elements available in the WebbPSF reference data (\url{http://www.stsci.edu/~mperrin/software/webbpsf/webbpsf-data-0.7.0.tar.gz}).\begin{small}\begin{lstlisting}[frame=single]import os#export environment variable:os.environ['WEBBPSF_PATH'] = os.path.expanduser('~/STScI/WFIRST/webbp... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
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] | [
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2ff73f4f0d497d64c57e6ea8e6bef3c089da3225 | subsection | 18 | 25 | Summary | 2016).\newblock \url{http://adsabs.harvard.edu/abs/2016ascl.soft02018P}.\bibitem{the_astropy_collaboration_astropy_2013}{The Astropy Collaboration}, Robitaille, T.~P., Tollerud, E.~J., Greenfield,P., Droettboom, M., Bray, E., Aldcroft, T., Davis, M., Ginsburg, A.,{Price-Whelan}, A.~M., Kerzendorf, W.~E., Conley, A., Cr... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
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8d596d4ece898a5a4be56f6e4742315d9952d0e3 | subsection | 19 | 25 | Summary | 2007).\newblock \url{http://www.opticsexpress.org/abstract.cfm?URI=oe-15-24-15935}.\bibitem{lawrence_optical_1992}Lawrence, G.~N., ``Optical {{Modeling}},'' in [{\em Applied {{Optics}} and{{Optical Engineering}}.}{\nolinebreak\hspace{0.1em}]}, Shannon, R.~R. andWyant., J.~C., eds., {\bf XI}, {Academic Press}, New Yor... | {
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} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
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3d4e5a6e9767b56677d0d76079fccd08f88afe43 | subsection | 20 | 25 | Summary | 2017).\newblock\url{https://www.spiedigitallibrary.org/conference-proceedings-of-spie/10400/1040010/Optical-tolerances-for-the-PICTURE-C-mission--error-budget/10.1117/12.2274105.short}.\bibitem{lumbres_modeling_2018}Lumbres, J. and {et al.}, ``Modeling coronagraphic extreme wavefront controlsystems for high contrast im... | {
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} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
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2774b2280a69fd702751de8b6f505d551a4d0c81 | subsection | 21 | 25 | Summary | Rep. JPL Document D-74425, {JPL} (2013).\newblock \url{available\%0020from\%0020exep.jpl.nasa.gov/technology}.\bibitem{macintosh_gemini_2008}Macintosh, B.~A., Graham, J.~R., Palmer, D.~W., Doyon, R., Dunn, J., Gavel,D.~T., Larkin, J., Oppenheimer, B., Saddlemyer, L., and Sivaramakrishnan, A.,``The {{Gemini Planet Image... | {
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} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
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08a28ac195b724836a081c617558c420abce33f6 | subsection | 22 | 25 | Summary | 2017).\newblock\url{https://www.spiedigitallibrary.org/conference-proceedings-of-spie/10398/103980A/The-Large-UV-Optical-Infrared-LUVOIR-surveyor--Decadal-Mission/10.1117/12.2273853.short}.\bibitem{cooley_1965}Cooley, J.~W. and Tukey, J.~W., ``{An algorithm for the machine calculation ofcomplex Fourier series},'' {\em ... | {
"cite_spans": []
} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
"Ewan S. Douglas",
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7dd8fc20fade39460e36ab7bb89c2950f76e46d8 | subsection | 23 | 25 | Summary | Opt. A: Pure Appl. Opt.}~{\bf10}(7), 075308 (2008).\newblock \url{http://stacks.iop.org/1464-4258/10/i=7/a=075308}.\bibitem{pavlyk_accelerating_2017}Pavlyk, O., Nagorny, D., {Guzman-Ballen}, A., Malakhov, A., Liu, H., Totoni,E., Anderson, A., and Maidanov, S., ``Accelerating {{Scientific Python}} with{{Intel Optimizat... | {
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} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
Optical Systems | [
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1b694c5e7add74921ae4c618380dcdf4d331bab0 | subsection | 24 | 25 | Summary | 2014).\bibitem{douglas_douglase/poppy_benchmarking_2018}Douglas, E.~S., ``Douglase/poppy\_benchmarking: {{Initial}} release of notebookfor {{SPIE AST}} 2018.'' \url{https://zenodo.org/record/1286650} (June 2018).\bibitem{yamamoto_computational_2004}Yamamoto, A., Kitamura, Y., and Yamane, Y., ``Computational efficiencie... | {
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} | 10.1117/12.2313441 | 1806.06467 | Accelerated Modeling of Near and Far-Field Diffraction for Coronagraphic
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14f52a8371480147be5f9bd272275fabe11146c1 | abstract | 0 | 23 | Abstract | In this paper, we present an analysis of more than 500K comments from
open-source repositories of software systems. Our aim is to empirically
determine how developers interact with each other under certain psychological
conditions generated by politeness, sentiment and emotion expressed in
developers' comments. Develop... | {
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} | 1803.06502 | Analysing Developers Affectiveness through Markov chain Models | [
"Giuseppe Destefanis",
"Marco Ortu",
"Steve Counsell",
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"Michele Marchesi"
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407752cc719d1595a65a29633eba496718a07559 | subsection | 1 | 23 | Introduction | The study of emotions and psychological status of developers and people involved in the software-building system is gaining the attention of both practitioners
and researchers .
Feldt et al. focused on personality as one important psychometric factor and presented initial results from an empirical study investigating
t... | {
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"raw": "W. Ke and P. Zhang. The effects of extrinsic motivations and satisfaction in open source software development. Journal of the Association for Info... | 1803.06502 | Analysing Developers Affectiveness through Markov chain Models | [
"Giuseppe Destefanis",
"Marco Ortu",
"Steve Counsell",
"Stephen Swift",
"Roberto Tonelli",
"Michele Marchesi"
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3a4daa9f64583329a95ed84060a39ad3310ccde2 | subsection | 2 | 23 | Introduction | Section 3 describes the dataset used for this study and our approach/rationale to evaluate affectiveness of comments posted by developers. In section 4, we present the methodology used for building the Markov chains. In Section 5, we present the results and elaborate on the research questions we address in Section 6. S... | {
"cite_spans": []
} | 1803.06502 | Analysing Developers Affectiveness through Markov chain Models | [
"Giuseppe Destefanis",
"Marco Ortu",
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fdcea9f2d3751e67cff5c9834317c93c1effead7 | subsection | 3 | 23 | Related Work | Several recent studies have demonstrated the importance and relationship of productivity and quality to human aspects associated with the software development process.
Ortu et al. studied the effect of politeness , and emotions on the time required to fix any given issue. The authors
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5f6982023c40ccd49d864eac35d6df96311116a7 | subsection | 4 | 23 | Related Work | Jurado and Rodriguez gathered the issues of nine high profile software projects hosted on GitHub. Through an analysis of the occurrence of Ekman's basic emotions among the projects and issues, the authors discovered that in open source projects, sentiments expressed in the form of joy are almost one magnitude of orde... | {
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f932d7964c7a3d3b8402a752f1d745018bd49182 | subsection | 5 | 23 | Related Work | This would permit the team members to feel satisfied with the work carried out by the team without reducing the quality of the software products developed.Hidden Markov Models (HMMs) have been largely used in application for speech recognition, handwriting recognition, bioinformatic, textures and patterns detection. Ma... | {
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4534d059afb190054cc81c061234e1b94a91fd35 | subsection | 6 | 23 | Dataset | We built our dataset from fifteen open-source, publicly available projects from a dataset proposed by Ortu et al. , extracted from the Jira ITS of four popular open source ecosystems (as well as the tools and infrastructure used for extraction), i.e., the Apache Software Foundation, Spring, JBoss and CodeHaus communiti... | {
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f9fa01a513249c08db8dc2a4f5148a49b2eb15b1 | subsection | 7 | 23 | Affective Metrics | Henceforward, we consider the term “affective metric” as a definition indicating all
those measures linked to human aspects and obtained from text written by developers (i.e., comments posted on issue tracking systems).
This study is based on the affective metrics sentiment, politeness and emotions used by Ortu et al. ... | {
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0c5d4bb02b877611e3f3f7ff230af08ca3f9f316 | subsection | 8 | 23 | Sentiment. | We measured sentiment using the SentiStrengthhttp://sentistrength.wlv.ac.uk tool, which is able to estimate the degree of positive and negative sentiment
in short texts, even for informal language. SentiStrength, by default, detects two sentiment polarizations:Negative: -1 (slightly negative) to -5 (extremely negative)... | {
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8d791ebf00e541496a9bea094cc92f7e573638ad | subsection | 9 | 23 | Politeness. | To evaluate the level of politeness of comments related to a given issue, we used the tool developed by Danescu et al. ; the tool uses a machine learning approach and calculates the politeness of sentences providing, as a result, one of two possible labels: polite or impolite.
The tool provides a level of confidence re... | {
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b8560c9a71a313e36097277c0072f80ab3057ebb | subsection | 10 | 23 | Emotion. | The presence of emotion in software engineering artifacts have been analysed by Destefanis et al. and Ortu et al. .Ortu et al.
provided a machine learning based approach for emotion detection in developers'
comments based on Parrott’s emotional framework, which consists of six basic emotions: joy, sadness, love, anger... | {
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823e7ceb8368f7ae631a1f3752853386c4d936b8 | subsection | 11 | 23 | Affective Markov Chains | Markov Chains (MC) have been used to model behavioural aspects in social sciences , .
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93db0de6efe78e341c72519d369360c15a419f53 | subsection | 12 | 23 | Affective Markov Chains | Let us also consider other two transition sequences obtained from other two issues reports, [I,I] and [P,P].
Finally, we counted the frequency of each politeness-transition
obtaining the corresponding MC.
In our example, if we consider the
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6b0db3aeb59487db5a9cdcca662d71e7ddc0ab9a | subsection | 13 | 23 | Results | Existing research has already explored links between
productivity (as measured by issue fixing time) and discrete emotions, sentiment and
politeness , . The dynamic of an issue resolution
involves complex interactions between different stakeholders such as users,
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a554b0f7513716533e684c71cf849cadf5f98ad0 | subsection | 14 | 23 | Politeness | Figure REF shows the general Politeness' MC describing the probability of changing from a state to another.
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2ca51e1a29fec53414dfb7e7659bf0d01b2a4ffb | subsection | 15 | 23 | Sentiment | Figures REF , REF , REF , show the boxplots and beanplots obtained considering all the projects in the corpus. Each single boxplot and beanplot shows the statistics for an array of 15 probabilities (one probability for project).
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dc8de9209932e49a389f5a7766f5aa0a1967966a | subsection | 16 | 23 | Sentiment | It is interesting to analyze if the same behaviours occur for emotions.We built the MCs for emotions as presented in
Sec. to analyze the probabilities of
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12d81b7d9f98901e827f435712f5d240408d8102 | subsection | 17 | 23 | Discussion | In this paper, we presented MC models describing how developers interacted with each other, analyzing comments posted on an issue tracking system. We selected the 15 most commented projects from the Jira issue report dataset of Ortu et al. , , which contained 700,000 issues from 1000 open source projects and two millio... | {
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cb137e0d3ac189729100edb90e3927be03b9eae0 | subsection | 18 | 23 | Discussion | On the other hand, New Feature issue typologies were those with higher politeness; Trivial issue typologies were those characterised by lower politeness (these might be related to minor programming mistakes and/or poor knowledge of programming practices). A further factor could be related to the number of developers in... | {
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ca8138072ef02fcf20ac08be86e7f2c933c471d0 | subsection | 19 | 23 | Threats to validity | Several threats to validity need to be considered. Threats to external validity are related to generalisation of
our conclusions. With regard to the system studied in this work, we
considered only open-source systems and this could affect
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"Michele Marchesi"
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] | 2,018 | en | Computer Science | [
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d23e4dbd18153677b95a21c2ab45bba27f4da017 | subsection | 20 | 23 | Conclusions and future work | This paper presented an analysis of more than 500K comments
from open-source issue tracking system repositories. We empirically determined
how developers interacted with each other under certain psychological conditions generated
by politeness, sentiment and emotions of a comment posted on a issue tracking
system.
Resu... | {
"cite_spans": []
} | 1803.06502 | Analysing Developers Affectiveness through Markov chain Models | [
"Giuseppe Destefanis",
"Marco Ortu",
"Steve Counsell",
"Stephen Swift",
"Roberto Tonelli",
"Michele Marchesi"
] | [
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34d631749d4eacb6c942bacf52c9ca6597c0b4a7 | subsection | 21 | 23 | Appendix | In this section we present all the matrices for all the fifteen systems in our corpus.
P is the transition matrix for Politeness, S for Sentiment and E for Emotions. | {
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} | 1803.06502 | Analysing Developers Affectiveness through Markov chain Models | [
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"Marco Ortu",
"Steve Counsell",
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"Michele Marchesi"
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1f3d1d6c47426203ee0df7a2fb287c9db734ebb2 | subsection | 22 | 23 | Appendix | The states for P, S and E are the following:P=\begin{bmatrix}
polite&neutral&impolite\\
\end{bmatrix}S=\begin{bmatrix}
positive&neutral&negative\\
\end{bmatrix}E=\begin{bmatrix}
sadness&anger&joy&love&neutral\\
\end{bmatrix}The following matrices are the transition matrices for Politeness, Sentiment and Emotion for HBa... | {
"cite_spans": []
} | 1803.06502 | Analysing Developers Affectiveness through Markov chain Models | [
"Giuseppe Destefanis",
"Marco Ortu",
"Steve Counsell",
"Stephen Swift",
"Roberto Tonelli",
"Michele Marchesi"
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efb1dec6e0d934636cca0d93e775b483ad8593cd | abstract | 0 | 57 | Abstract | We define operators on semistandard shifted tableaux and use Stembridge's
local characterization for regular graphs to prove they define a crystal
structure. This gives a new proof that Schur $P$-polynomials are Schur
positive. We define queer crystal operators (also called odd Kashiwara
operators) to construct a conne... | {
"cite_spans": []
} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
"Ezgi Kantarci Oguz"
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09c9cc3bb547ab9b09a841ded09dac6fcf970ae0 | subsection | 1 | 57 | Introduction | Kashiwara introduced crystal bases to study representations of quantized universal enveloping Lie algebras. Canonical bases, developed independently by Lusztig , study the same problem from a geometric viewpoint. A crystal graph is a directed, colored graph with vertex set given by the crystal basis and directed edges ... | {
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... | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
"Ezgi Kantarci Oguz"
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ef5994498513aef0070897d74bd72140124bd43a | subsection | 2 | 57 | Introduction | Lie superalgebras are algebras with a \mathbb {Z}/2\mathbb {Z} grading, allowing for two families of variables (one commuting and one not) to interact. Originally arising from mathematical physics in connection with supersymmetry, Lie superalgebras were formalized mathematically and classified by Kac . One well-studied... | {
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3009cec01a416744268ace44c355079f2b316546 | subsection | 3 | 57 | Introduction | Our results and constructions were first announced in , where we conjectured that our axioms were sufficient. While that conjecture is false, we expect that a refinement might be true. | {
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a9217874bd0c2e04448e8a953fba6389afaa843c | subsection | 4 | 57 | Crystals for the quantum general linear Lie algebra | Kashiwara , introduced crystal bases in his study of the representation theory of quantized universal enveloping algebra U_q(\mathfrak {g}) for Lie algebra \mathfrak {g}. In this section, we review the theory of crystal bases and crystal graphs, focusing solely on the case of U_q(\mathfrak {gl}(r)) to simplify the expo... | {
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b288be820d2dec19ec15624876d96bcf5645a7e0 | subsection | 5 | 57 | Crystal bases and crystal graphs | We use the language of root systems to define crystal bases of type A_{r+1}, though the exposition is self-contained and no familiarity with Lie theory is assumed; see for further details. Let \mathbf {e}_1, \mathbf {e}_2, \ldots , \mathbf {e}_{r+1} be the standard basis for V = \mathbb {R}^{r+1} with the usual inner p... | {
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763e86b23d9d08f9c7541f435fae8cce7beabc50 | subsection | 6 | 57 | Crystal bases and crystal graphs | The weight lattice is \Lambda = \mathbb {Z}^{r+1}, and the dominant weights \Lambda ^{+} \subset \Lambda are those \lambda \in \Lambda such that \lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _{r+1} \ge 0.Definition 2.1 A crystal of dimension r+1 is a nonempty set \mathcal {B} not containing 0 together with crystal o... | {
"cite_spans": []
} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
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] | [
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ca35083c55db7b3a4769242c79dd403c4f092e66 | subsection | 7 | 57 | Crystal bases and crystal graphs | The weight map is \mathrm {wt}\left(\,\raisebox {-0.3}{\vtop {
{&\hbox{t}o 0pt{\usebox 2\hss }\vbox to 12{
\vss \hbox{t}o 12{\hss #\hss }
\vss }\cr i\crcr }}}\,\right) = \mathbf {e}_i.Definition 2.2 The character of a crystal \mathcal {B} is the polynomial\mathrm {ch}(\mathcal {B}) = \sum _{b \in B} x_1^{\mathrm {wt}... | {
"cite_spans": []
} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
"Ezgi Kantarci Oguz"
] | [
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e8df989a587a6a1b013e209cc0b7a8b83a97331a | subsection | 8 | 57 | Crystal bases and crystal graphs | The i-string through b is the maximal pathe_i^{\varepsilon _i} x {\stackrel{i}{\longrightarrow }} \cdots {\stackrel{i}{\longrightarrow }} e_i x {\stackrel{i}{\longrightarrow }} x {\stackrel{i}{\longrightarrow }} f_i x {\stackrel{i}{\longrightarrow }} \cdots {\stackrel{i}{\longrightarrow }} f_i^{\varphi _i} x .For examp... | {
"cite_spans": []
} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
"Ezgi Kantarci Oguz"
] | [
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21e4ec9a3e0e3d264ce887a24f71e97e8d351f75 | subsection | 9 | 57 | Crystal bases and crystal graphs | Then \mathrm {ch}(\mathcal {B}(\lambda )) is precisely the character of the irreducible representation indexed by \lambda , which corresponds to the Schur polynomial s_{\lambda }(x_1,\ldots ,x_{r+1}) defined in §REF . Even more compelling is the remarkable fact that the following combinatorial procedure on crystals cor... | {
"cite_spans": []
} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
"Ezgi Kantarci Oguz"
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c4c83a4ffb510018344ab3db8d7152c24095e9a0 | subsection | 10 | 57 | Crystals on Young tableaux | A partition \lambda is a weakly decreasing sequence of positive integers, \lambda = (\lambda _1,\lambda _2, \ldots , \lambda _{\ell }), where \lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _{\ell } > 0. By extending \lambda with trailing 0s until it has length r+1, we may identify partitions with dominant weights \La... | {
"cite_spans": []
} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
"Ezgi Kantarci Oguz"
] | [
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ccf6a8b4ddcdef47b7553761be47b4d48cf25f58 | subsection | 11 | 57 | Crystals on Young tableaux | This was done by Kashiwara and Nakashima and Littelman , though our presentation is again simplified to the case of the general linear group.For a word w of length k, a positive integer r \leqslant k, and a positive integer i, definem_i(w,r) = \mathrm {wt}(w_{1} w_{2} \cdots w_{r})_{i} - \mathrm {wt}(w_{1} w_{2} \cdots... | {
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0ae50238b65179c0f456bbb64b175d15c3239db9 | subsection | 12 | 57 | Crystals on Young tableaux | The string length \varepsilon _i(T) is given by\max \lbrace r \mid \mathrm {wt}(w_{r} w_{r+1} \cdots w_{n})_{i+1} - \mathrm {wt}(w_{r} w_{r+1} \cdots w_{n})_{i} \rbrace ,which also coincides with the number of i-free entries equal to i+1. Moreover, the largest index to attain the maximum in (REF ), if positive, is the ... | {
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1494d0bb54f9e6c603d7c34f6b86b83f59d65ee1 | subsection | 13 | 57 | Local characterization of crystals | Establishing that a given directed graph structure on a combinatorial set is a normal crystal is difficult. To simplify this greatly, Stembridge gave a local characterization of crystals that arise from representations for simply-laced types.In order to define Stembridge's axioms for type A_{r+1}, we first introduce no... | {
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deb194d3b83828c4962c835ee2cfa33efcd9cae1 | subsection | 14 | 57 | Local characterization of crystals | \end{array}Definition 2.12 ()
A directed, colored graph \mathcal {X} is regular if the following hold:(A1)
all monochromatic directed paths have finite length;
(A2)
for every vertex x, there is at most one edge x {\stackrel{i}{\longleftarrow }} y and at most one edge x {\stackrel{i}{\longrightarrow }} z;
(A3)
assuming... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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a517e02b72703cade050eb076e3ccbcb865bba66 | subsection | 15 | 57 | Local characterization of crystals | When |i-j| \ge 2, there is no change (left case), but when j = i\pm 1, the length changes by 1, either by decreasing the tail when \Delta _i \varepsilon (x,j) = -1 (middle case) or by increasing the head when \Delta _i \varphi (x,j) = -1 (right case).
[Figure: An illustration of axioms A3 and A4, where f_j {\swarrow },... | {
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"raw": ", A local characterization of simply-laced crystals, Trans. Amer. Math. Soc. 355 (2003), no. 12, 4807–4823 (electronic).",
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8bc96f093025019d31e086cf6fc42b6b42e9f9ca | subsection | 16 | 57 | A crystal for shifted tableaux | Schur P-polynomials arise as characters of tensor representations of the queer Lie superalgebra , characters of projective representations of the symmetric group , and representatives for cohomology classes dual to Schubert cycles in isotropic Grassmannians . Stanley conjectured that Schur P-polynomials are Schur posit... | {
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"raw": "A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras {\\rm Gl}(n,\\,m) and Q(n), Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422–430.",
"source_ref_i... | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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de0f324707edd041d31c9ba3c6e7b102241128d0 | subsection | 17 | 57 | Shifted tableaux | A partition \gamma is strict if \gamma _1 > \gamma _2 > \cdots > \gamma _{\ell } > 0. We identify a strict partition \gamma with its shifted Young diagram, the collection of cells with \gamma _i cells in row i shifted \ell (\gamma )-i cells to the left.A semistandard shifted tableau of shape \gamma is a filling of the ... | {
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"raw": "Bruce E. Sagan, Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley, J. Combin. Theory Ser. A 45 (1987), no. 1, 62–103. M... | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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ebffc14caa2580b0a63f753d6be644002a551605 | subsection | 18 | 57 | Shifted tableaux | That is, Schur P-polynomials are Schur positive.For example, our previous computation can we written asP_{(3,1)}(x_1,x_2,x_3) & = & s_{(3,1)}(x_1,x_2,x_3) + s_{(2,2)}(x_1,x_2,x_3) + s_{(2,1,1)}(x_1,x_2,x_3).Stembridge expanded on the idea of shifted insertion in his study of projective representations of the symmetric ... | {
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"raw": "John R. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), no. 1, 87–134. MR 99141... | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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e49efe4d844cc6fcad1d083dd38110064a7de2b8 | subsection | 19 | 57 | Crystals on shifted tableaux | We give a new proof of the Schur positivity of Schur P-polynomials by constructing a crystal graph on semistandard shifted tableaux. We note that Hawkes, Paramonov, and Schilling recently constructed a crystal on semistandard shifted tableaux in the context of type B/C Stanley symmetric functions. Their construction us... | {
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"raw": "Graham Hawkes, Kirill Paramonov, and Anne Schilling, Crystal analysis of type C Stanley symmetric functions, Electron. J. Combin. 24 (2017), no. 3, Paper... | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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e5da7e2f2ce134efc3522b7cc62ad78fc0cfbc9b | subsection | 20 | 57 | Crystals on shifted tableaux | This observation helps to make the following well-defined.Definition 3.5 The shifted lowering operators, denoted by \overline{f}_i, act on semistandard shifted tableaux by: \overline{f}_i(T)=0 if m_i(w(T)) \leqslant 0; otherwise, letting p be the smallest index such that m_i(w(T),p) = m_i(w(T)), letting x denote the en... | {
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