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Superposition for Higher-Order Logic

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Abstract

We recently designed two calculi as stepping stones towards superposition for full higher-order logic: Boolean-free \(\lambda \)-superposition and superposition for first-order logic with interpreted Booleans. Stepping on these stones, we finally reach a sound and refutationally complete calculus for higher-order logic with polymorphism, extensionality, Hilbert choice, and Henkin semantics. In addition to the complexity of combining the calculus’s two predecessors, new challenges arise from the interplay between \(\lambda \)-terms and Booleans. Our implementation in Zipperposition outperforms all other higher-order theorem provers and is on a par with an earlier, pragmatic prototype of Booleans in Zipperposition.

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Notes

  1. https://github.com/sneeuwballen/zipperposition.

  2. https://doi.org/10.5281/zenodo.4534759.

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Acknowledgements

Uwe Waldmann provided advice and carefully checked the completeness proof. Visa Nummelin led the design of the \(\hbox {o}\)Sup calculus, which was a crucial stepping stone for this work. Simon Cruanes helped us with the implementation. Martin Desharnais generated (and regenerated) the Sledgehammer benchmarks. Ahmed Bhayat explained the higher-order Vampire schedule and suggested textual improvements. Christoph Benzmüller provided us insight into the pitfalls of higher-order reasoning. Alexander Steen explained the details of Leo-III scheduling. Geoff Sutcliffe let us use StarExec Miami for our experiments. Simon Cruanes, Herman Geuvers, Mathias Fleury, Nicolas Peltier, Giles Reger, Mark Summerfield, and the anonymous reviewers suggested 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’s research has received funding from a Chinese Academy of Sciences President’s International Fellowship for Postdoctoral Researchers (grant No. 2021PT0015). Blanchette’s research has received funding from the Netherlands Organization for Scientific Research (NWO) under the Vidi program (project No. 016.Vidi.189.037, Lean Forward).

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Authors and Affiliations

  1. State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, No. 4, South Fourth Street, Zhong Guan Cun, Beijing, 100190, China

    Alexander Bentkamp

  2. Department of Computer Science, Section of Theoretical Computer Science, Vrije Universiteit Amsterdam, De Boelelaan 1111, 1081 HV, Amsterdam, The Netherlands

    Alexander Bentkamp, Jasmin Blanchette & Petar Vukmirović

  3. Université de Lorraine, CNRS, Inria, LORIA, 615, rue du Jardin Botanique, 54600, Villers-lès-Nancy, France

    Sophie Tourret

  4. Max-Planck-Institut für Informatik, Saarland Informatics Campus E1 4, 66123, Saarbrücken, Germany

    Jasmin Blanchette & Sophie Tourret

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Bentkamp, A., Blanchette, J., Tourret, S. et al. Superposition for Higher-Order Logic. J Autom Reasoning 67, 10 (2023). https://doi.org/10.1007/s10817-022-09649-9

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