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Variant Satisfiability of Parameterized Strings

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Rewriting Logic and Its Applications (WRLA 2020)

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Abstract

Two “knowingly incomplete,” yet useful, variant-based satisfiability procedures for QF formulas in the instantiations of two, increasingly more expressive, parameterized data types of strings are proposed. The first has four selector functions decomposing a list concatenation into its parts. The second adds a list membership predicate. The meaning of “parametric” here is much more general than is the case for decision procedures for strings in current SMT solvers, which are parametric on a finite alphabet. The parameterized data types presented here are parametric on a (typically infinite) algebraic data type of string elements. The main result is that if an algebraic data type has a variant satisfiability algorithm, then the data type of strings over such elements has a “knowingly incomplete,” yet practical, variant satisfiability algorithm, with no need for a Nelson-Oppen combination algorithm relating satisfiability in strings and in the given data type.

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Notes

  1. 1.

    If , with \(B_{0}\) associativity and/or commutativity axioms, and U identity axioms, the B-preregularity notion can be broadened by requiring only that: (i) \(\varSigma \) is \(B_{0}\)-preregular in the standard sense that \( ls (u\rho )= ls (v\rho )\) for all and substitutions \(\rho \); and (ii) the axioms U oriented as rules \(\vec {U}\) are sort-decreasing in the sense explained in Sect. 2.2.

  2. 2.

    In Maude, X would be the sort Elt of the TRIV parameter theory.

  3. 3.

    For more details about sufficient completeness of parameterized OS theories and methods for checking it see [16].

  4. 4.

    I am purposefully avoiding identity axioms because, thanks to the theory transformation in [8] mapping a convergent \(\vec {\mathcal {E}}\) with identity axioms U into a semantically equivalent convergent \(\vec {\mathcal {E}}_{U}\) where such axioms have been transformed into rewrite rules, this involves no real loss of generality.

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Acknowledgements

I cordially thank the referees for their very helpful suggestions to improve the paper. This work has been partially supported by NRL under contract N00173-17-1-G002.

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

    José Meseguer

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  1. José Meseguer
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Correspondence to José Meseguer .

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  1. Valencian Research Institute for Artificial Intelligence, Universitat Politècnica de València, Valencia, Spain

    Santiago Escobar

  2. Facultad de Informática, Universidad Complutense de Madrid, Madrid, Spain

    Narciso Martí-Oliet

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Meseguer, J. (2020). Variant Satisfiability of Parameterized Strings. In: Escobar, S., Martí-Oliet, N. (eds) Rewriting Logic and Its Applications. WRLA 2020. Lecture Notes in Computer Science(), vol 12328. Springer, Cham. https://doi.org/10.1007/978-3-030-63595-4_6

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  • DOI: https://doi.org/10.1007/978-3-030-63595-4_6

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