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Definition 13 (Weak Pushout Step) Let [MATH] and [MATH] be graphs in [MATH] , and let [MATH] be a graph rule. We say that [MATH] is a weak pushout step for [MATH] if the following hold: |
[MATH] is a subgraph of [MATH] There are embeddings [MATH] and [MATH] such that [MATH] for all vertices in [MATH] If there are embeddings [MATH] and [MATH] such that [MATH] for all vertices in [MATH] , then there is an embedding [MATH] such that [MATH] for all vertices in [MATH] |
Like in our variation of graph rules, we use a weak pushout step as a variation of the categorical pushout that is typically used in graph rewriting, to ensure that chains are formed. In such a (weak) pushout, the requirement of subgraphs is missing, making the entire definition symmetrical ( [MATH] and [MATH] can be s... |
Definition 14 ((Simple) Weak Pushout Chain) Let [MATH] be a chain with [MATH] , and let [MATH] be a set of graph rules. If for each [MATH] , either [MATH] or there exists an [MATH] such that [MATH] is a weak pushout step for [MATH] , then [MATH] is a simple weak pushout chain under [MATH] |
Weak pushout chains are inductively defined: 1. every simple weak pushout chain under [MATH] is a weak pushout chain under [MATH] |
2. if for each [MATH] , there exists an [MATH] , which is a weak pushout chain under [MATH] with [MATH] and [MATH] , then [MATH] is a weak pushout chain under [MATH] |
3. nothing else is a weak pushout chain. For most of this paper, it suffices to consider simple weak pushout chains. There is a way to draw weak pushout steps that is convenient in practice, although it can leave parts implicit. On a weak pushout step [MATH] for [MATH] , as drawn in Figure 2(a) , large vertices indicat... |
A weak pushout chain does not necessarily have a consequence graph as its supremum: we can construct a weak pushout chain with [MATH] for any graph [MATH] However, the following holds: |
Lemma 4 If [MATH] is a weak pushout chain under [MATH] and [MATH] is a consequence graph maintaining [MATH] , then [MATH] is a least consequence graph of [MATH] maintaining [MATH] |
Proof Let [MATH] be a consequence graph. We first consider the case in which [MATH] is a simple weak pushout chain. By induction on [MATH] , we prove that [MATH] is maintained in [MATH] For [MATH] [MATH] is trivially maintained in any graph. For [MATH] , assume [MATH] is maintained in [MATH] by induction. If [MATH] , t... |
A chain that is both fair and a weak pushout chain is called a fair weak pushout chain. A fair weak pushout chain has a least consequence graph as its supremum. This gives a way to create least consequence graphs, which we’ll come back to in Section |
Translation between Sentences and Graph Rules This section shows how to turn sentences into graph rules. For every sentence, there is a corresponding graph rule that is maintained if and only if the sentence holds. This allows us to use graph rules in order to reason about sentences. We introduce a translate function [... |
Definition 15 (Translation) Given a term [MATH] , we say that [MATH] is the translation of [MATH] We define [MATH] as follows: [EQUATION] |
For notational convenience, [MATH] It follows that [MATH] is a graph rule. As an example of how the translation works, the graphs in Figure 1(b) are [MATH] and [MATH] respectively. As a whole, the graph rule in Figure 1(b) is [MATH] |
The vertices [MATH] and [MATH] of [MATH] can intuitively be understood as the variables [MATH] and [MATH] as in Definition Lemma makes this precise: |
Lemma 5 [MATH] if and only if there is an [MATH] such that [MATH] with [MATH] for [MATH] Proof The statement follows by induction on [MATH] , using that the vertices in [MATH] are [MATH] .∎ |
We can use Lemma to show a connection between graph rules and sentences: Lemma 6 A sentence [MATH] holds in [MATH] if and only if [MATH] is maintained in [MATH] |
Proof Suppose the sentence holds in [MATH] , and [MATH] It follows from Lemma that [MATH] As the sentence holds, [MATH] Using Lemma , take [MATH] with [MATH] and [MATH] for [MATH] Following Definition 15 , construct [MATH] such that [MATH] and [MATH] for [MATH] in the vertices of [MATH] |
For the other direction, suppose [MATH] is maintained in [MATH] , and let [MATH] By Lemma , there is an [MATH] such that [MATH] with [MATH] and [MATH] Since the graph rule is maintained, there is a [MATH] such that [MATH] with [MATH] and [MATH] Again using Lemma [MATH] , so the sentence holds in [MATH] . ∎ |
We use graph rules to deal with the requirements in Definition to make a graph standard, in a way similar to Lemma We give a set of graph rules that make checking if a standard graph exists easy: A standard graph exists provided that [MATH] , and that a set of additional graph rules, which we will call the standard-rul... |
Definition 16 (Conflict (Free)) Let [MATH] The relation symbol [MATH] stands for an empty relation. A graph for which [MATH] is conflict free If [MATH] is conflict free, we have [MATH] so we call any edge [MATH] conflict |
Definition 17 (Top-rule) Let [MATH] The relation symbol [MATH] stands for the full relation. We refer to the graph rule [MATH] as the top-rule , since any graph [MATH] satisfies [MATH] if and only if [MATH] maintains the top-rule. |
Definition 18 (Nonempty-rule) Let [MATH] The graph rule [MATH] is called the nonempty-rule A graph [MATH] maintains the nonempty-rule if and only if [MATH] |
A conflict-free graph [MATH] that maintains the top-rule, satisfies [MATH] if and only if it maintains the nonempty-rule. The relation symbol models the identity relation [MATH] However, we do not let [MATH] represent this relation directly. Instead, we let stand for an equivalence relation and ensure that we can make ... |
Definition 19 (Identity-rules) Given a set of relation symbols [MATH] , we say that the following set of graph rules are the identity-rules for [MATH] |
[EQUATION] Identity-rules ( ) to ( ) can be understood as ensuring is reflexive, symmetric, transitive, and a congruence respectively. The identity-rules hold under the standard semantics of , that is: if for some graph [MATH] , we have [MATH] then the identity-rules are maintained in [MATH] The following lemma speaks ... |
Lemma 7 Let [MATH] be a graph in which the identity-rules for [MATH] are maintained. There is an idempotent [MATH] such that [MATH] , and [MATH] |
Proof Since the first three identity-rules for [MATH] are maintained in [MATH] [MATH] is an equivalence relation on [MATH] Let [MATH] be some function that takes a canonical element from the equivalence class. It follows that [MATH] , and it remains to be shown that [MATH] For the vertices, this is immediate. For the e... |
Lemma gives us exactly the desired semantics for : for [MATH] , we have [MATH] Furthermore, it states that [MATH] and [MATH] are mutually embedded ( [MATH] holds for all [MATH] ). |
We now proceed to introduce constants, through a set of sentences. This characterisation is similar to how points are characterized in relation algebra, see for instance work by Schmidt and Ströhlein |
If [MATH] is a constant, then [MATH] is a point (sometimes called a right ideal). The corresponding constant can be retrieved from a point: [MATH] Our presentation here in terms of constants rather than points is a matter of personal preference. These rules state that the relation [MATH] should be nonempty, the cross-p... |
Definition 20 (Constant-rules, Standard-rules) Let [MATH] be a standard set of labels with constants [MATH] , we say that the following set of graph rules are the constant-rules for [MATH] |
[EQUATION] The top-, nonempty-, identity-, and constant-rules together are called the standard-rules for [MATH] and [MATH] , written [MATH] |
Similar to our treatment of , we would like to find an [MATH] such that [MATH] The [MATH] of Lemma gives us a graph that is isomorphic to one in which [MATH] holds, provided that [MATH] is conflict free and maintains the standard-rules. Lemma says that, for finding a model with ‘standard semantics’, it suffices to find... |
Lemma 8 Let [MATH] be a standard set of labels with constants [MATH] Let [MATH] be a set of sentences over [MATH] of the shape [MATH] We define [MATH] Let [MATH] be a conflict free consequence graph maintaining [MATH] , then there is a graph [MATH] , and functions [MATH] and [MATH] such that: |
1. [MATH] , and [MATH] 2. The graph [MATH] is standard. 3. Every sentence in [MATH] holds in [MATH] Proof We begin the proof by constructing [MATH] and [MATH] , based on [MATH] By Lemma , there is an idempotent function [MATH] with [MATH] Top- and nonempty-rules are maintained in [MATH] , so by constant-rule ( ), there... |
A Procedure to Find a Standard Graph A set of sentences is satisfiable if and only if there is no [MATH] such that [MATH] contains a conflict in a corresponding fair weak pushout chain. This follows from the previous sections as follows: Given a set of sentences [MATH] with relation symbols [MATH] , Lemma shows that we... |
5.1 An Algorithm for Fair Weak Pushout Chains Assume that the set of sentences [MATH] is finite. Consequently, only finitely many relation symbols [MATH] are used in those sentences. We restrict [MATH] to those relation symbols that are actually used in [MATH] This makes the corresponding set of graph rules [MATH] (inc... |
function ProduceChain [MATH] Input : A set of edges [MATH] such that [MATH] is a graph. A finite set of finite graph rules [MATH] with relation symbols [MATH] |
Effect : Produces an infinite list of graphs that are a fair weak pushout chain starting in [MATH] Let [MATH] , produce [MATH] Let [MATH] be our worklist; |
for [MATH] do Take [MATH] such that [MATH] for [MATH] such that [MATH] do if There is no [MATH] such that [MATH] with [MATH] then |
Let [MATH] be the maximum of [MATH] Add [MATH] to [MATH] 10 11 end if 12 13 end for 14 15 end for 16 if [MATH] is empty then 17 ProduceChain [MATH] |
18 19 else 20 Take [MATH] such that [MATH] is minimal; 21 Take [MATH] such that [MATH] and [MATH] 22 Let [MATH] 23 Let [MATH] 24 |
Take [MATH] such that [MATH] for [MATH] and [MATH] for [MATH] such that [MATH] if [MATH] for [MATH] 25 Take [MATH] such that [MATH] |
26 ProduceChain [MATH] 27 28 end if Algorithm 1 Construct a fair weak pushout chain starting in its input Lemma 9 Algorithm constructs a fair weak pushout chain starting in [MATH] under [MATH] , the limit of which is a least consequence graph of [MATH] under [MATH] |
Proof The algorithm constructs a weak pushout chain, because the graph constructed on Line is part of a weak pushout step for a graph rule in [MATH] Let [MATH] describe the weak pushout chain generated (with [MATH] ). Pick an arbitrary [MATH] Since the set of graph rules is finite, also the number of functions [MATH] w... |
The algorithm can be changed into a semi-decision procedure to decide whether the limit contains a conflict: If [MATH] contains a conflict, then any limit in which [MATH] occurs will contain the conflict. Therefore, if we are only interested in whether the limit has a conflict, we can abort the algorithm as soon as [MA... |
The same procedure can be used to prove entailment. Say we wish to determine if [MATH] entails [MATH] for a problem on a standard set of labels [MATH] , for [MATH] equal to [MATH] Assume without loss of generality that [MATH] We introduce a new label [MATH] [MATH] Let [MATH] Let [MATH] be the standard rules plus the de... |
There is another case in which we can abort: once the graph maintains all graph rules in [MATH] , we hit Line , and [MATH] is equal to the limit. In such a case, we have found the limit of the chain given by Algorithm , and can immediately decide whether or not it is conflict free. Unfortunately, even if conflict free ... |
5.2 Optimizations for Implementations We discuss some possible optimizations for the purpose of showing correctness of the algorithm described by the author in an earlier paper |
The earlier algorithm is not Algorithm , but an optimized version thereof. We only describe a few optimizations, that suffice to show that the algorithm presented earlier is correct as well. |
As optimizations, we allow changing the outcome of the algorithm, but require that the proof of Lemma remains valid. In particular, instead of the graph [MATH] constructed on Line , we can make a larger graph [MATH] if [MATH] and [MATH] is the limit of a (not necessarily fair) weak pushout chain. Through this change, t... |
As an instance of this, observe that we can combine graph rules, as this is a form of combining weak pushout steps: suppose [MATH] and [MATH] are graph rules in [MATH] , such that [MATH] Then we can find an [MATH] such that [MATH] is a weak pushout step of [MATH] We can then safely replace the graph rule [MATH] for [MA... |
Apart from changing the set of graph rules [MATH] , we can change the algorithm such that the standard-rules are always maintained after each step. Let [MATH] be a graph constructed in that way. According to Lemma , we represent the graph [MATH] by the graph [MATH] , making it such that we do not need to store the rela... |
, showing it is a semi-decision procedure for deciding whether a least consequence graph contains a conflict. 5.3 Example Run of the Optimized Algorithm |
We return to one of the examples given in Section : the entailment problem that asks whether [MATH] entails [MATH] We construct a [MATH] for the entailment problem as described in Section 5.1 |
[EQUATION] Using the translation of Section , Figure gives the graph rules we work with. We use the optimizations just described, and do not restate the standard-rules. |
We start the procedure with [MATH] and [MATH] Note that per our optimizations, the self loops [MATH] and [MATH] are implicitly there, as well as all [MATH] edges. Only one rule does not hold: [MATH] , and consequently only one graph rule is not maintained. A pushout step for it gives [MATH] and [MATH] as the next call ... |
[EQUATION] Since we did not use [MATH] yet, and all other rules are satisfied up to this point, we are exactly in the place we would have been if [MATH] wasn’t present. This is (minus the [MATH] ) the graph given in Section as a possible graph our algorithm could give. If we would have handled [MATH] before [MATH] inst... |
We now proceed by applying [MATH] The pushout step adds [MATH] We have not described precisely how our optimizations proceed at this point, but we need to renumber the nodes such that [MATH] and [MATH] are identified. For preserving fairness, we renumber high to low: the node [MATH] is relabeled to [MATH] This can caus... |
[EQUATION] At this point, [MATH] does not hold, and the resulting action is to insert [MATH] Subsequently, [MATH] does not hold and we insert a conflict. We abort concluding that the entailment holds. |
While we needed several iterations to conclude entailment, we saved many iterations by treating the standard rules separately. If we had applied [MATH] earlier, we would have derived the contradiction sooner. |
5.4 Presentation of the Algorithm We conclude this section with a note on the presentation in this paper. In the earlier paper, we presented the efficient implementation |
as discussed in the previous paragraph. This does not allow us to talk about the limit of the procedure. Using the same presentation would have alleviated the need for Lemma However, the simpeler presentation used in this paper allows us to argue that the limit of a chain always exists. This simplifies many of the othe... |
We give an example that shows why it is problematic to describe limits in the more involved presentation: Given the graph rules [MATH] [MATH] and the identity-rules, Figure shows a part of a weak pushout chain. Following the procedure for the given rules, we obtain the graphs in Figure 5(b) and 5(c) After every step, w... |
A Proof of Undecidability Lemma 10 The following decision problem is undecidable: given a set of sentences [MATH] , is there a standard graph [MATH] in which every sentence in [MATH] holds? |
Proof This proof closely follows a proof by Krisnadhi and Lutz on ‘conjunctive query answering’. We use a reduction from the undecidable problem whether two context free grammars have an empty intersection. This problem is given by two grammars with non-terminals [MATH] and [MATH] , a common set of terminals [MATH] , a... |
We make an encoding by choosing [MATH] and [MATH] such that there is a standard graph in which the sentences in [MATH] hold if and only if the context free grammars have an empty intersection. We encode every terminal and nonterminal with corresponding relation symbols, and use the constant symbol [MATH] for the empty ... |
[EQUATION] We show that there is a standard graph in which [MATH] holds if the grammars have an empty intersection, and there is no such graph if the grammars share a word. First suppose the grammars have an empty intersection. We construct a graph as follows: The vertices are words over [MATH] where [MATH] is the empt... |
[EQUATION] It can be checked that [MATH] is standard and all sentences in [MATH] hold. In particular, [MATH] as there is no word [MATH] such that [MATH] parses [MATH] and [MATH] parses [MATH] |
Now suppose for a proof by contradiction that [MATH] is a standard graph in which all sentences in [MATH] hold, and that [MATH] is a word that is parsed by [MATH] and [MATH] Since [MATH] holds, there is a path in [MATH] with vertices [MATH] and edges [MATH] By induction on the parse-tree of how [MATH] parses [MATH] , t... |
Undecidability of entailment follows as a corollary: there are no standard graphs in which every sentence in [MATH] holds iff the sentence [MATH] is entailed. |
We end with a final remark about the proof we presented. The relation symbol [MATH] and the operation [MATH] were not used in the proof. Thus, this proof of undecidability holds if [MATH] is restricted to sentences of a simpler shape. By application of Lemma , we conclude that deciding whether a conflict free consequen... |
Conclusion In this paper, we have given a translation of sentences into graph rules, and have proven that for a graph [MATH] , a sentence is maintained in [MATH] if and only if the translated graph rule holds in [MATH] Furthermore, when allowing the sentences to use extra relation-symbols with a dedicated meaning, we c... |
Our procedure can partially automate preserving invariants in information systems. Its implementation and evaluation is foreseen in Ampersand, but considered outside the scope of this paper. |
Acknowledgements We thank Wolfram Kahl and Stef Joosten for their thoughts and comments on this paper and earlier versions. Part of the research presented in this paper was performed at the University of Innsbruck, Austria, supported by the Austrian Science Fund (FWF) project Y757. Supported by The Netherlands Organisa... |
# Source: arxiv 1806.09503 # Title: Adjoint chaos via cumulant truncation # Sections: all # Downloaded: 2026-03-03T01:48:11.172050+00:00 |
Adjoint chaos via cumulant truncation John Craske Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK |
(Last updated: ) We describe a simple and systematic method for obtaining approximate sensitivity information from a chaotic dynamical system using a hierarchy of cumulant equations. The resulting forward and adjoint systems yield information about gradients of functionals of the system and do not suffer from the conve... |
1 Introduction Introduction Complete information about a particular solution of an engineering problem is often less useful than knowledge of the way in which a small number of functionals of the solution change with respect to input parameters. An example in fluid mechanics is the effect that a body’s shape has on the... |
(Foures et al. 2014 and flow optimisation more generally (Lions, 1971 . In these situations it is natural to focus on adjoint variables, which represent the derivative of a given functional with respect to the problem’s constraints or governing equations. With adjoint variables the derivative of the functional with res... |
Giles & Pierce ( 2000 Whilst adjoint analysis is well established and used successfully in many fields, the problem of obtaining functional gradients from chaotic dynamical systems, such as turbulence, is an open question |
(Vishnampet et al. 2015 . Whether such gradients are well-defined depends on the properties of the dynamical system. For example, if the system is uniformly hyperbolic (Smale, 1967 ; Eckmann & Ruelle, 2004 then linear response theory provides the required formula (Eyink et al. 2004 ; Ruelle, 2009 . In all chaotic syste... |
(Lea et al. 2000 ; Eyink et al. 2004 . In addition to the requirement of having to obtain a potentially large ensemble, the difficulty of the latter approach is in determining an appropriate time interval a priori . Consequently, probability density functions have also received attention as a reliable source of gradien... |
(e.g. Cooper & Haynes, 2011 , and typically relies on an assumption about the underlying probability density function. Recent efforts to reconcile adjoint techniques and chaotic systems have focused on deriving sensitivities from shadow trajectories, which are defined as remaining uniformly close to a given trajectory ... |
The issue regarding divergent trajectories in tangent and adjoint systems can be circumvented altogether by computing sensitivities of unstable periodic orbits |
(Lasagna, 2018 . Perturbations of unstable periodic orbits, which behave like a skeleton around chaotic orbits (see e.g. Auerbach et al. 1987 provide a proxy for the latter’s sensitivity. In general, each unstable period orbit returns a different sensitivity. In certain cases, however, the sensitivities are closely ali... |
The need to overcome the sensitive dependence on perturbations inherent in chaotic systems might be regarded as unnecessary, in view of the fact that one is often interested in gradients of |
ensemble-averaged quantities. Indeed, following Hopf ( 1952 and Lorenz ( 1967 , it is possible to directly simulate a system’s statistics or cumulant dynamics. With the use of the original governing equations, the cumulant equations can be derived from a single flow functional |
(Hopf, 1952 and provide a direct means of understanding the behaviour of a flow’s statistics. In addition to their evolution being slower and not necessarily chaotic, the cumulant equations can be used to investigate statistically unsteady problems, statistical stability and to provide an analytical means of determinin... |
(Farrell & Ioannou, 2014 . The evolution of a finite set of dependent variables corresponds to an infinite hierarchy of cumulant equations. The benefits of focusing on the evolution of statistics are therefore offset by the requirement of finding a suitable closure (Rothmayer & Black, 1993 . Fortunately, heterogeneous ... |
and wall-bounded shear-flow (Farrell et al. 2016 . Whilst strongly nonlinear systems, such as the model for Rayleigh-Bénard convection given by the Lorenz equations (Lorenz, 1963 require a more sophisticated treatment that accounts for the role of cumulants beyond second order, direct statistical simulation can neverth... |
(Allawala & Marston, 2016 The approach that we describe combines the desirable features of the statistical state equations with observations from direct simulation and classical adjoint techniques. In § |
we describe the problems associated with the adjoint analysis of chaotic systems, before deriving a well-conditioned adjoint operator from a system’s cumulant equations in . In § we apply the approach to the sensitivity analysis of thermal convection via the Lorenz equations, and consider their extension to a 9-dimensi... |
2 The problem The problem Consider a dynamical system whose state, [MATH] , evolves according to [EQUATION] where [MATH] is a vector of system parameters. If the dynamical system ( ) is chaotic then an understanding of the system’s statistics becomes crucial. Fortunately, engineers and scientists are typically interest... |
each element of the parameter vector [MATH] We focus our attention on the Lorenz equations as a specific example. Lorenz ( 1963 derived the following system of equations from a truncated description of Rayleigh-Bénard convection between hot and cold horizontal surfaces: |
[EQUATION] The state [MATH] describes the strength of the velocity field, the difference in temperature between ascending and descending fluid, and the strength of the horizontally averaged temperature with respect to a state of pure conduction, respectively. The parameters [MATH] are the Prandtl number, the aspect rat... |
We will focus on statistics [MATH] that correspond to a finite time average [MATH] of a function [MATH] [EQUATION] which depends implicitly on the parameters [MATH] |
via [MATH] . Under the assumption of ergodicity, the estimator [MATH] using the finite time average in ( corresponds to a phase-average of the function [MATH] when [MATH] Following previous work on the sensitivity analysis of chaotic systems (Lea et al. 2000 ; Wang, 2013 , we will focus on [MATH] , such that [MATH] est... |
and [MATH] . In spite of the chaotic dynamics described by equation ( ), [MATH] appears to vary linearly with respect to [MATH] , for the values of [MATH] considered, where [MATH] is a critical value of [MATH] . At |
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