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International Conference on Automated Deduction

CADE 2017: Automated Deduction – CADE 26 pp 77–94Cite as

On the Combination of the Bernays–Schönfinkel–Ramsey Fragment with Simple Linear Integer Arithmetic

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10395))

Abstract

In general, first-order predicate logic extended with linear integer arithmetic is undecidable. We show that the Bernays-Schönfinkel-Ramsey fragment (\(\exists ^* \forall ^*\)-sentences) extended with a restricted form of linear integer arithmetic is decidable via finite ground instantiation. The identified ground instances can be employed to restrict the search space of existing automated reasoning procedures considerably, e.g., when reasoning about quantified properties of array data structures formalized in Bradley, Manna, and Sipma’s array property fragment. Typically, decision procedures for the array property fragment are based on an exhaustive instantiation of universally quantified array indices with all the ground index terms that occur in the formula at hand. Our results reveal that one can get along with significantly fewer instances.

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Notes

  1. 1.

    For any free-sort variable v that occurs in a clause \((\varLambda \,\Vert \, \varGamma \rightarrow \varDelta ) \in N\) exclusively in equations, we pretend that \(\varDelta \) contains an atom \(\text {False}_v(v)\), for a fresh predicate symbol \(\text {False}_v : \mathcal {S}\). This is merely a technical assumption. Without it, we would have to treat such variables v as a separate case in all definitions. The atom \(\text {False}_v(v)\) is not added “physically” to any clause.

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Authors and Affiliations

  1. Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

    Matthias Horbach, Marco Voigt & Christoph Weidenbach

  2. Saarbrücken Graduate School of Computer Science, Saarland Informatics Campus, Saarbrücken, Germany

    Marco Voigt

Authors
  1. Matthias Horbach
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  2. Marco Voigt
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  3. Christoph Weidenbach
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Corresponding authors

Correspondence to Matthias Horbach , Marco Voigt or Christoph Weidenbach .

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Editors and Affiliations

  1. Microsoft Research, Redmond, Washington, USA

    Leonardo de Moura

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Horbach, M., Voigt, M., Weidenbach, C. (2017). On the Combination of the Bernays–Schönfinkel–Ramsey Fragment with Simple Linear Integer Arithmetic. In: de Moura, L. (eds) Automated Deduction – CADE 26. CADE 2017. Lecture Notes in Computer Science(), vol 10395. Springer, Cham. https://doi.org/10.1007/978-3-319-63046-5_6

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  • DOI: https://doi.org/10.1007/978-3-319-63046-5_6

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