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Variant-Based Decidable Satisfiability in Initial Algebras with Predicates

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Logic-Based Program Synthesis and Transformation (LOPSTR 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10855))

Abstract

Decision procedures can be either theory-specific, e.g., Presburger arithmetic, or theory-generic, applying to an infinite number of user-definable theories. Variant satisfiability is a theory-generic procedure for quantifier-free satisfiability in the initial algebra of an order-sorted equational theory \((\varSigma ,E \cup B)\) under two conditions: (i) \(E \cup B\) has the finite variant property and B has a finitary unification algorithm; and (ii) \((\varSigma ,E \cup B)\) protects a constructor subtheory \((\varOmega ,E_{\varOmega } \cup B_{\varOmega })\) that is OS-compact. These conditions apply to many user-definable theories, but have a main limitation: they apply well to data structures, but often do not hold for user-definable predicates on such data structures. We present a theory-generic satisfiability decision procedure, and a prototype implementation, extending variant-based satisfiability to initial algebras with user-definable predicates under fairly general conditions.

Partially supported by NSF Grant CNS 14-09416, NRL under contract number N00173-17-1-G002, the EU (FEDER), Spanish MINECO project TIN2015-69175-C4-1-R and GV project PROMETEOII/2015/013. Raúl Gutiérrez was also supported by INCIBE program “Ayudas para la excelencia de los equipos de investigación avanzada en ciberseguridad”.

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Notes

  1. 1.

    For a discussion of similar but not exactly equivalent versions of the variant notion see [6]. Here we follow the shaper formulation in [13], rather than the one in [9], because it is technically essential for some results to hold [6].

  2. 2.

    As pointed out at the beginning of Sect. 2, [16] treats the more general order-sorted case, where sorts form a poset \((S,\le )\) with \(s \le s'\) interpreted as set containment \(A_{s} \subseteq A_{s'}\) in a \(\varSigma \)-algebra A. All results in this paper hold in the order-sorted case.

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Authors and Affiliations

  1. Universitat Politècnica de València, Valencia, Spain

    Raúl Gutiérrez

  2. University of Illinois at Urbana-Champaign, Champaign, USA

    José Meseguer

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  1. Raúl Gutiérrez
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  2. José Meseguer
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Correspondence to Raúl Gutiérrez .

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  1. University of Chieti-Pescara, Pescara, Italy

    Fabio Fioravanti

  2. Roskilde University, Roskilde, Denmark

    John P. Gallagher

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Gutiérrez, R., Meseguer, J. (2018). Variant-Based Decidable Satisfiability in Initial Algebras with Predicates. In: Fioravanti, F., Gallagher, J. (eds) Logic-Based Program Synthesis and Transformation. LOPSTR 2017. Lecture Notes in Computer Science(), vol 10855. Springer, Cham. https://doi.org/10.1007/978-3-319-94460-9_18

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  • DOI: https://doi.org/10.1007/978-3-319-94460-9_18

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