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Superposition with Lambdas

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Automated Deduction – CADE 27 (CADE 2019)

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Abstract

We designed a superposition calculus for a clausal fragment of extensional polymorphic higher-order logic that includes anonymous functions but excludes Booleans. The inference rules work on \(\beta \eta \)-equivalence classes of \(\lambda \)-terms and rely on higher-order unification to achieve refutational completeness. We implemented the calculus in the Zipperposition prover and evaluated it on TPTP and Isabelle benchmarks. The results suggest that superposition is a suitable basis for higher-order reasoning.

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Notes

  1. 1.

    https://github.com/c-cube/zipperposition.

  2. 2.

    http://matryoshka.gforge.inria.fr/pubs/lamsup_results.tgz.

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Acknowledgment

Simon Cruanes patiently explained Zipperposition’s internals and allowed us to continue the development of his prover. Christoph Benzmüller and Alexander Steen shared insights and examples with us, guiding us through the literature and clarifying how the Leos work. Maria Paola Bonacina and Nicolas Peltier gave us some ideas on how to treat the extensionality axiom as a theory axiom, ideas we have yet to explore. Mathias Fleury helped us set up regression tests for Zipperposition. Ahmed Bhayat, Tomer Libal, and Enrico Tassi shared their insights on higher-order unification. Andrei Popescu and Dmitriy Traytel explained the terminology surrounding the \(\lambda \)-calculus. Haniel Barbosa, Daniel El Ouraoui, Pascal Fontaine, and Hans-Jörg Schurr were involved in many stimulating discussions. Christoph Weidenbach made this collaboration possible. Ahmed Bhayat, Mark Summerfield, and the anonymous reviewers suggested several textual improvements. We thank them all.

Bentkamp, Blanchette, and Vukmirović’s research has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 713999, Matryoshka). Bentkamp and Blanchette also benefited from the Netherlands Organization for Scientific Research (NWO) Incidental Financial Support scheme. Blanchette has received funding from the NWO under the Vidi program (project No. 016.Vidi.189.037, Lean Forward).

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Authors and Affiliations

  1. Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    Alexander Bentkamp, Jasmin Blanchette & Petar Vukmirović

  2. Max-Planck-Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany

    Jasmin Blanchette, Sophie Tourret & Uwe Waldmann

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  1. Alexander Bentkamp
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    Pascal Fontaine

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Bentkamp, A., Blanchette, J., Tourret, S., Vukmirović, P., Waldmann, U. (2019). Superposition with Lambdas. 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_4

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