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Automatic Decidability: A Schematic Calculus for Theories with Counting Operators

Authors Elena Tushkanova, Christophe Ringeissen, Alain Giorgetti, Olga Kouchnarenko



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Elena Tushkanova
Christophe Ringeissen
Alain Giorgetti
Olga Kouchnarenko

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Elena Tushkanova, Christophe Ringeissen, Alain Giorgetti, and Olga Kouchnarenko. Automatic Decidability: A Schematic Calculus for Theories with Counting Operators. In 24th International Conference on Rewriting Techniques and Applications (RTA 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 21, pp. 303-318, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)
https://doi.org/10.4230/LIPIcs.RTA.2013.303

Abstract

Many verification problems can be reduced to a satisfiability problem modulo theories. For building satisfiability procedures the rewriting-based approach uses a general calculus for equational reasoning named paramodulation. Schematic paramodulation, in turn, provides means to reason on the derivations computed by paramodulation. Until now, schematic paramodulation was only studied for standard paramodulation. We present a schematic paramodulation calculus modulo a fragment of arithmetics, namely the theory of Integer Offsets. This new schematic calculus is used to prove the decidability of the satisfiability problem for some theories equipped with counting operators. We illustrate our theoretical contribution on theories representing extensions of classical data structures, e.g., lists and records. An implementation within the rewriting-based Maude system constitutes a practical contribution. It enables automatic decidability proofs for theories of practical use.
Keywords
  • decision procedures
  • superposition
  • schematic saturation

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