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Extending SMT Solvers to Higher-Order Logic

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Book cover Automated Deduction – CADE 27 (CADE 2019)

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Abstract

SMT solvers have throughout the years been able to cope with increasingly expressive formulas, from ground logics to full first-order logic (FOL). In contrast, the extension of SMT solvers to higher-order logic (HOL) is mostly unexplored. We propose a pragmatic extension for SMT solvers to support HOL reasoning natively without compromising performance on FOL reasoning, thus leveraging the extensive research and implementation efforts dedicated to efficient SMT solving. We show how to generalize data structures and the ground decision procedure to support partial applications and extensionality, as well as how to reconcile quantifier instantiation techniques with higher-order variables. We also discuss a separate approach for redesigning an HOL SMT solver from the ground up via new data structures and algorithms. We apply our pragmatic extension to the CVC4 SMT solver and discuss a redesign of the veriT SMT solver. Our evaluation shows they are competitive with state-of-the-art HOL provers and often outperform the traditional encoding into FOL.

This work was partially supported by the National Science Foundation under award 1656926.

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Notes

  1. 1.

    http://matryoshka.gforge.inria.fr/pubs/hosmt/.

  2. 2.

    Since veriT does not parse TPTP, its reported results are on the equivalent benchmarks as translated by CVC4 into the HOSMT language [3].

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Acknowledgments

We are grateful to Jasmin Blanchette and Pascal Fontaine for numerous discussions throughout the development of this work, for providing funding for research visits and for suggesting many improvements. We also thank Jasmin for generating several of the benchmarks with which we evaluate our approach; Simon Cruanes and Martin Riener for many fruitful discussions on the intricacies of HOL; Andres Nötzli for help with the table and plot scripts; Mathias Fleury, Hans-Jörg Schurr and Sophie Tourret for suggesting many improvements. This work was partially supported by the National Science Foundation under Award 1656926 and the European Research Council (ERC) under starting grant Matryoshka (713999).

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Authors and Affiliations

  1. The University of Iowa, Iowa City, USA

    Haniel Barbosa, Andrew Reynolds & Cesare Tinelli

  2. University of Lorraine, CNRS, Inria, and LORIA, Nancy, France

    Daniel El Ouraoui

  3. Stanford University, Stanford, USA

    Clark Barrett

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  1. Haniel Barbosa
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Correspondence to Daniel El Ouraoui .

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  1. University of Lorraine, Villers-lès-Nancy, France

    Pascal Fontaine

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Barbosa, H., Reynolds, A., El Ouraoui, D., Tinelli, C., Barrett, C. (2019). Extending SMT Solvers to Higher-Order Logic. In: Fontaine, P. (eds) Automated Deduction – CADE 27. CADE 2019. Lecture Notes in Computer Science(), vol 11716. Springer, Cham. https://doi.org/10.1007/978-3-030-29436-6_3

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