Metalevel algorithms for variant satisfiability

Abstract

Variant satisfiability is a theory-generic algorithm to decide quantifier-free satisfiability in an initial algebra T Σ{E when the theory pΣ, Eq has the finite variant property and its constructors satisfy a compactness condition. This paper: (i) gives a precise definition of several meta-level sub-algorithms needed for variant satisfiability; (ii) proves them correct; and (iii) presents a reflective implementation in Maude 2.7 of variant satisfiability using these sub-algorithms.The material is adapted from [25,18, 28]. Due to space limitations the following elementary notions, which can be found in [25], are assume known: (i) ordersorted (OS) signature Σ; (ii) set p S of connected components (each denoted rss p S) of a poset of sorts pS, ¤q; (iii) sensible OS signature; (iv) order-sorted Σalgebras and homomorphisms, and its associated category OSAlg Σ ; and (v) the construction of the term algebra T Σ and its initiality in OSAlg Σ when Σ is sensible. Furthermore, for connected components rs 1 s, . . . , rs n s, rss p S, f rs1s...rsns rss tf : s I 1 . . . s I n Ñ s I Σ | s I i rs i s, 1 ¤ i ¤ n, s I rssu denotes the family of "subsort polymorphic" operators f . T Σ will (ambiguously) denote: (i) the term algebra; (ii) its underlying Ssorted set; and (iii) the set T Σ sS T Σ,s . For rss p S, T Σ,rss s I rss T Σ,s I.An OS signature Σ is said to have non-empty sorts iff for each s S, T Σ,s r.We will assume throughout that Σ has non-empty sorts. An OS signature Σ is called preregular [19] iff for each t T Σ the set ts S | t T Σ,s u has a least element, denoted lsptq. We will assume throughout that Σ is preregular.An S-sorted set X tX s u sS of variables, satisfies s s I ñ X s X s I r, and the variables in X are always assumed disjoint from all constants in Σ. The Σ-term algebra on variables X, T Σ pXq, is the initial algebra for the signature ΣpXq obtained by adding to Σ the variables X as extra constants. Since a ΣpXqalgebra is just a pair pA, αq, with A a Σ-algebra, and α an interpretation of the constants in X, i.e., an S-sorted function α rXÑAs, the ΣpXq-initiality of T Σ pXq can be expressed as the following theorem: Theorem 1. (Freeness Theorem). If Σ is sensible, for each A OSAlg Σ and α rXÑAs, there exists a unique Σ-homomorphism, α : T Σ pXq Ñ A extending α, i.e., such that for each s S and x X s we have xα s α s pxq.In particular, when A T Σ pXq, an interpretation of the constants in X, i.e., an S-sorted function σ rXÑT Σ pXqs is called a substitution, and its unique homomorphic extension σ : T Σ pXq Ñ T Σ pXq is also called a substitution. Define dompσq tx X

Generalized Rewrite Theories and Coherence Completion

“…is finite and can be effectively computed according to the algorithm in [23], which has been implemented in Maude. Both the sets t E,B and t…”

“…Closely related work falls into three categories: (i) the already-mentioned symbolic reasoning techniques for rewrite theories, e.g., [10,2,22,1,12,16,25,17,24]; (ii) executability techniques for standard rewrite theories, including [27,8]; and (iii) variant-based symbolic computation, including [5,11,20,23], and also [7], where a limited form of "equational coherence completion" was introduced. In relation to all the work in (i)-(iii), the main contributions of this paper are: (1) a new notion of generalized rewrite theory, of rewriting in a generalized rewrite theory, and an initial model semantics for such theories; (2) new symbolic executability requirements, including a new notion of coherence that is a substantial generalization of the standard notions in [27,8]; and (3) new automatable theory transformations both to ensure ground coherence of generalized rewrite theories by coherence completion, and to make symbolic executability applicable to a widest possible class of such theories.…”

“…This can easily be done by computing variants in Maude, and constructor variants in the Maude implementation of [23]. This will enable new applications, both in symbolic reasoning and in equational abstraction.…”

“…Note that the answer to question (i) is new even for standard rewrite theories and can be quite useful to semi-automate equational abstractions [21]. This automation method is an interesting instance of what might be called theoretical dogfooding, where the new symbolic methods of variant computation [11,20,23] are applied to complete a rewrite theory into a ground coherent one. The answer to question (ii) is very general: under mild conditions symbolic executability can be ensured for a wide class of generalized theories by two theory transformations.…”

An Optimizing Protocol Transformation for Constructor Finite Variant Theories in Maude-NPA

“…The main contributions of this work are: (i) we provide a non-trivial protocol transformation based on [28]; (ii) since the protocols of Section 5 do not satisfy the conditions of [28], we provide a more powerful protocol transformation that we implemented, made available online, and pays off in practice; (iii) we provide an encoding of bilinear pairing that can handle all the protocols of Section 5 that Tamarin cannot handle; (iv) we implemented the algorithm of [32] for the computation of constructor variants [27] from scratch; and (v) there was no implementation of the program transformation of [28] and we implemented it.…”