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First-Order Automated Reasoning with Theories: When Deduction Modulo Theory Meets Practice

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Abstract

We discuss the practical results obtained by the first generation of automated theorem provers based on Deduction modulo theory. In particular, we demonstrate the concrete improvements such a framework can bring to first-order theorem provers with the introduction of a rewrite feature. Deduction modulo theory is an extension of predicate calculus with rewriting both on terms and propositions. It is well suited for proof search in theories because it turns many axioms into rewrite rules. We introduce two automated reasoning systems that have been built to extend other provers with Deduction modulo theory. The first one is Zenon Modulo, a tableau-based tool able to deal with polymorphic first-order logic with equality, while the second one is iProverModulo, a resolution-based system dealing with first-order logic with equality. We also provide some experimental results run on benchmarks that show the beneficial impact of the extension on these two tools and their underlying proof search methods. Finally, we describe the two backends of these systems to the Dedukti universal proof checker, which also relies on Deduction modulo theory, and which allows us to verify the proofs produced by these tools.

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Notes

  1. This does not strictly conform to Definition 9, as two equivalents terms \(t \equiv u\) must receive the same interpretation. This can be technically fixed by letting D be composed of equivalence classes of terms, or by interpreting terms by their normal forms (it requires confluence and termination), or by dropping the \({\llbracket t \rrbracket }_\varphi = {\llbracket u \rrbracket }_\varphi \) constraint over terms in Definition 9.

  2. This benchmark is publicly available at: http://bware.lri.fr/.

  3. Available at: http://zenon.gforge.inria.fr/.

  4. Available at: http://www.ensiie.fr/~guillaume.burel/blackandwhite_autotheo.html.en.

  5. The corresponding generated Dedukti files are available at: https://cloud.lsv.ens-cachan.fr/public.php?service=files&t=59f1cdee894ea25967a51bcadc76052a.

  6. The rewrite systems that we designed to present these theories are given at: http://www.ensiie.fr/~guillaume.burel/empty_iProverModulo.html.en.

  7. Available at: https://github.com/c-cube/zipperposition/.

  8. Available at: https://github.com/Gbury/archsat/.

  9. Available at : https://alt-ergo.ocamlpro.com/.

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We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions, which helped us improve and clarify this manuscript.

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Authors and Affiliations

  1. ENSIIE and Samovar, Télécom SudParis and CNRS, Université Paris-Saclay, Évry, France

    Guillaume Burel

  2. Inria and LSV, CNRS and ENS Paris-Saclay, Université Paris-Saclay, Cachan, France

    Guillaume Burel

  3. Inria/LSV/ENS Paris-Saclay, Cachan, France

    Guillaume Bury

  4. Irif/Université Paris-Diderot, Paris, France

    Raphaël Cauderlier

  5. LIRMM, Université de Montpellier, CNRS, Montpellier, France

    David Delahaye

  6. EDF R&D (OSIRIS), Palaiseau, France

    Pierre Halmagrand

  7. CRI, MINES ParisTech, PSL University, Paris, France

    Olivier Hermant

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  1. Guillaume Burel
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A part of this work has been supported by the BWare Project [56, 107] (ANR-12-INSE-0010) funded by the INS programme of the French National Research Agency (ANR).

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Burel, G., Bury, G., Cauderlier, R. et al. First-Order Automated Reasoning with Theories: When Deduction Modulo Theory Meets Practice. J Autom Reasoning 64, 1001–1050 (2020). https://doi.org/10.1007/s10817-019-09533-z

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  • DOI: https://doi.org/10.1007/s10817-019-09533-z

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