On Forward Closure and the Finite Variant Property

Bouchard, Christopher; Gero, Kimberly A.; Lynch, Christopher S.; Narendran, Paliath

doi:10.1007/978-3-642-40885-4_23

Cited by 24 publications

(19 citation statements)

References 16 publications

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“…This is a sizable class, including many, but not all, theories of interest to cryptographic protocol analysis (see [6]). In many cases known characterizations of theories with the finite variant property [12], [5] depend on conditions on E and R that can be checked without further reference to Σ, and so for these cases the finite variant property still holds after the addition of uninterpreted function symbols. Thus general asymmetric unification algorithms exist.…”

Section: Discussionmentioning

confidence: 99%

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On Asymmetric Unification and the Combination Problem in Disjoint Theories

Erbatur

Kapur

Marshall

et al. 2014

Lecture Notes in Computer Science

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Abstract. Asymmetric unification is a new paradigm for unification modulo theories that introduces irreducibility constraints on one side of a unification problem. It has important applications in symbolic cryptographic protocol analysis, for which it is often necessary to put irreducibility constraints on portions of a state. However many facets of asymmetric unification that are of particular interest, including its behavior under combinations of disjoint theories, remain poorly understood. In this paper we give a new formulation of the method for unification in the combination of disjoint equational theories developed by Baader and Schulz that both gives additional insights into the disjoint combination problem in general, and furthermore allows us to extend the method to asymmetric unification, giving the first unification method for asymmetric unification in the combination of disjoint theories.

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Section: Discussionmentioning

confidence: 99%

“…The next step considers the set of variable partitions, one of which is the following partition {{x 0 , x 3 }, {x 2 , x 4 }, {x 5 , z 1 }, {x 1 , z 0 , x 7 }, {x 6 }} Choosing a representative for each set and doing the replacement the Algorithm would produce the following Γ 3 from that partition:…”

Section: Lemma 3 (Baader-schulz [2])mentioning

confidence: 99%

On Asymmetric Unification and the Combination Problem in Disjoint Theories

Erbatur

Kapur

Marshall

et al. 2014

Lecture Notes in Computer Science

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“…The theory BOOL is FVP, whereas the theory NAT-VARIANT does not have the finite variant property since there is an infinite number of variants in NAT-VARIANT for the term X:Nat + 0. It is generally undecidable whether an equational theory has the FVP (Bouchard et al 2013); a semi-decision procedure is given by Meseguer (2015) that works well in practice and another technique based on the dependency pair framework is given by Escobar et al (2012). The procedure by Meseguer (2015) works by computing the variants of all flat terms f (X 1 , .…”

Section: Examplementioning

confidence: 99%

Inspecting Maude variants with`GLINTS`

Alpuente

Escobar

Sapiña

et al. 2017

Theory and Practice of Logic Programming

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This paper introducesGLINTS, a graphical tool for exploring variant narrowing computations in Maude. The most recent version of Maude, version 2.7.1, provides quite sophisticated unification features, including order-sorted equational unification for convergent theories modulo axioms such as associativity, commutativity, and identity. This novel equational unification relies on built-in generation of the set ofvariantsof a termt, i.e., the canonical form oftσ for a computed substitution σ. Variant generation relies on a novel narrowing strategy calledfolding variant narrowingthat opens up new applications in formal reasoning, theorem proving, testing, protocol analysis, and model checking, especially when the theory satisfies thefinite variant property, i.e., there is a finite number of most general variants for every term in the theory. However, variant narrowing computations can be extremely involved and are simply presented in text format by Maude, often being too heavy to be debugged or even understood. TheGLINTSsystem provides support for (i) determining whether a given theory satisfies the finite variant property, (ii) thoroughly exploring variant narrowing computations, (iii) automatic checking of nodeembeddingandclosednessmodulo axioms, and (iv) querying and inspecting selected parts of the variant trees.

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“…i.e., there is a substitution ρ such that t = E t ρ and σ | Var(t) = E (θ ρ)| Var(t) . A decomposition (Σ, B, E) has the finite variant property (FVP) [19] (also called a finite variant decomposition) iff for each Σ-term t, there exists a complete and finite set [[t]] E,B of variants of t. Note that whether a decomposition has the finite variant property is undecidable [5] but a technique based on the dependency pair framework has been developed in [19] and a semi-decision procedure that works well in practice is available in [6].…”

Section: Introductionmentioning

confidence: 99%

Most General Variant Unifiers

Escobar

Sapiña

2019

Electron. Proc. Theor. Comput. Sci.

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Equational unification of two terms consists of finding a substitution that, when applied to both terms, makes them equal modulo some equational properties. Equational unification is of special relevance to automated deduction, theorem proving, protocol analysis, partial evaluation, model checking, etc. Several algorithms have been developed in the literature for specific equational theories, such as associative-commutative symbols, exclusive-or, Diffie-Hellman, or Abelian Groups. Narrowing was proved to be complete for unification and several cases have been studied where narrowing provides a decidable unification algorithm. A new narrowing-based equational unification algorithm relying on the concept of the variants of a term has been developed and it is available in the most recent version of Maude, version 2.7.1, which provides quite sophisticated unification features. A variant of a term t is a pair consisting of a substitution σ and the canonical form of tσ . Variant-based unification is decidable when the equational theory satisfies the finite variant property. However, it may compute many more unifiers than the necessary and, in this paper, we explore how to strengthen the variantbased unification algorithm implemented in Maude to produce a minimal set of most general variant unifiers. Our experiments suggest that this new adaptation of the variant-based unification is more efficient both in execution time and in the number of computed variant unifiers than the original algorithm available in Maude. We follow the classical notation and terminology from [35] for term rewriting, from [3] for unification, and from [27] for rewriting logic and order-sorted notions.We assume an order-sorted signature Σ = (S, ≤, Σ) with a poset of sorts (S, ≤). The poset (S, ≤) of sorts for Σ is partitioned into equivalence classes, called connected components, by the equivalence relation (≤ ∪ ≥) + . We assume that each connected component [s] has a top element under ≤, denoted [s] and called the top sort of [s]. This involves no real loss of generality, since if [s] lacks a top sort, it can be easily added. We also assume an S-sorted family X = {X s } s∈S of disjoint variable sets with each X s countably infinite. T Σ (X ) s is the set of terms of sort s, and T Σ,s is the set of ground terms of sort s. We write T Σ (X ) and T Σ for the corresponding order-sorted term algebras. Given a term t, Var(t) denotes the set of variables in t.A substitution σ ∈ S ubst(Σ, X ) is a sorted mapping from a finite subset of X to T Σ (X ). Substitutions are written as σ = {X 1 → t 1 , . . . , X n → t n } where the domain of σ is Dom(σ ) = {X 1 , . . . , X n } and the set of variables introduced by terms t 1 , . . . ,t n is written Ran(σ ). The identity substitution is id. Substitutions are homomorphically extended to T Σ (X ). The application of a substitution σ to a term t is denoted by tσ or σ (t). For simplicity, we assume that every substitution is idempotent, i.e., σ satisfies Dom(σ ) ∩ Ran(σ ) = / 0. The restriction of σ to a set of var...

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On Forward Closure and the Finite Variant Property

Cited by 24 publications

References 16 publications

On Asymmetric Unification and the Combination Problem in Disjoint Theories

On Asymmetric Unification and the Combination Problem in Disjoint Theories

Inspecting Maude variants with`GLINTS`

Most General Variant Unifiers

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On Forward Closure and the Finite Variant Property

Cited by 24 publications

References 16 publications

On Asymmetric Unification and the Combination Problem in Disjoint Theories

On Asymmetric Unification and the Combination Problem in Disjoint Theories

Inspecting Maude variants withGLINTS

Most General Variant Unifiers

Contact Info

Product

Resources

About

Inspecting Maude variants with`GLINTS`