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Metalevel Algorithms for Variant Satisfiability

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Rewriting Logic and Its Applications (WRLA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9942))

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Abstract

Variant satisfiability is a theory-generic algorithm to decide quantifier-free satisfiability in an initial algebra \(T_{\varSigma /E}\) when the theory \((\varSigma ,E)\) 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.

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Notes

  1. 1.

    When the axioms B consist of a combination of associativity, commutativity, and (left and/or right) identity axioms, we can decompose B into the disjoint union \(B = B_{0} \uplus U\), where \(B_{0}\) are associativity and/or commutativity axioms, and U are left and/or right identity axioms. The equations in U, of the general form \(f(e,x)=x\) and/or \(f(x,e)=x\), can be oriented as rewrite rules R(U) of the form \(f(e,x) \rightarrow x\) and/or \(f(x,e) \rightarrow x\) to be applied modulo \( B_{0}\). The B-preregularity notion can then be broadened by requiring only that: (i) \(\varSigma \) is preregular; (ii) \(\varSigma \) is \(B_{0}\)-preregular in the standard sense that \( ls (u\rho )= ls (v\rho )\) for all \(u=v \in B_{0}\) and sort specializations \(\rho \); and (iii) the rules R(U) are sort-decreasing in the sense of Definition 1. Maude automatically checks B-preregularity of an OS signature \(\varSigma \) in this broader sense [8].

  2. 2.

    For a discussion of similar but not exactly equivalent versions of the variant notion see [7]. Here we follow the formulation in [18].

  3. 3.

    [27, 28, 34] give examples of constructor variants and constructor unifiers.

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Acknowledgements

Partially supported by NSF Grant CNS 13-19109.

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  1. Department of Computer Science, University of Illinois at Urbana-Champaign, Champaign, USA

    Stephen Skeirik & José Meseguer

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  1. Stephen Skeirik
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Correspondence to Stephen Skeirik .

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  1. Alexandru Ioan Cuza University , Iasi, Romania

    Dorel Lucanu

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Skeirik, S., Meseguer, J. (2016). Metalevel Algorithms for Variant Satisfiability. In: Lucanu, D. (eds) Rewriting Logic and Its Applications. WRLA 2016. Lecture Notes in Computer Science(), vol 9942. Springer, Cham. https://doi.org/10.1007/978-3-319-44802-2_10

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  • DOI: https://doi.org/10.1007/978-3-319-44802-2_10

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