Generalized rewrite theories, coherence completion, and symbolic methods

Abstract

A new notion of generalized rewrite theory suitable for symbolic reasoning and generalizing the standard notion in [19] is motivated and defined. Also, new requirements for symbolic executability of generalized rewrite theories that extend those in [33] for standard rewrite theories, including a generalized notion of coherence, are given. Symbolic executability, including coherence, is both ensured and made available for a wide class of such theories by automatable theory transformations. Using these foundations, several symbolic reasoning methods using generalized rewrite theories are studied, including: (i) symbolic description of sets of terms by pattern predicates; (ii) reasoning about universal reachability properties by generalized rewriting; (iii) reasoning about existential reachability properties by constrained narrowing; and (iv) symbolic verification of safety properties such as invariants and stability properties.

Programming and symbolic computation in Maude

“…Among the many symbolic reasoning features that can be supported by Maude, in this paper we focus on order-sorted equational unification (this section) and order-sorted narrowing-based symbolic reachability analysis (Section 7). For a broader discussion of other symbolic reasoning methods and tools in rewriting logic and Maude, see Section 9 and [101,103].…”

“…But we can go further by considering more expressive constrained patterns of the form u | ϕ, where u is an Ω-term and ϕ is a QF Σ-formula, so that u | ϕ specifies the ground instances of u for which ϕ holds. State predicates having such constrained patterns u | ϕ as atomic predicates and closed under conjunction and disjunction in an effectively, symbolically computable manner have been studied in [125,103]. Such a language of pattern predicates is very useful to specify sets of states both in reachability logic (see below), and in the constrained style of narrowing-based reachability analysis defined in [103].…”

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

“…The set of constructor variants of the form [[ l, r ]] Ω E,B , where l and r are, respectively, the lefthand and righthand sides of a rewrite rule in a rewrite theory, play a crucial role in the following theory transformation R → R Ω l,r from [28].…”

“…Our protocol transformation relies on a program transformation from [28] for rewrite theories in Maude that we have improved by relaxing some of its applicability conditions. Such program transformation relies on constructor term variants [27], which is an extension of term variants [8,17].…”

“…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.…”