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Implementation and Evaluation of Contextual Natural Deduction for Minimal Logic

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Perspectives of System Informatics (PSI 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9609))

Abstract

The contextual natural deduction calculus (\({\mathbf{ND }^\mathbf{c }}\)) extends the usual natural deduction calculus (\(\mathbf{ND }\)) by allowing the implication introduction and elimination rules to operate on formulas that occur inside contexts. It has been shown that, asymptotically in the best case, \({\mathbf{ND }^\mathbf{c }}\)-proofs can be quadratically smaller than the smallest \(\mathbf{ND }\)-proofs of the same theorems. In this paper we describe the first implementation of a theorem prover for minimal logic based on \({\mathbf{ND }^\mathbf{c }}\). Furthermore, we empirically compare it to an equally simple \(\mathbf{ND }\) theorem prover on thousands of randomly generated conjectures.

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Notes

  1. 1.

    e.g. Isabelle (www.cl.cam.ac.uk/research/hvg/Isabelle/) and Coq (http://coq.inria.fr).

  2. 2.

    \(\mathrm {depth}(A) = 1\), for an atomic A; \(\mathrm {depth}(B \rightarrow C) = \mathrm {max}(\mathrm {depth}(B), \mathrm {depth}(C)) + 1\).

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Acknowledgements

This work was supported by an Stipendium of the Österreichische Akademie der Wissenschaften (APART).

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Authors and Affiliations

  1. Vienna University of Technology, Vienna, Austria

    Bruno Woltzenlogel Paleo

  2. Australian National University, Canberra, Australia

    Bruno Woltzenlogel Paleo

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  1. Bruno Woltzenlogel Paleo
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Correspondence to Bruno Woltzenlogel Paleo .

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Editors and Affiliations

  1. Innopolis University, Innopolis, Russia

    Manuel Mazzara

  2. The University of Manchester, Manchester, United Kingdom

    Andrei Voronkov

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Woltzenlogel Paleo, B. (2016). Implementation and Evaluation of Contextual Natural Deduction for Minimal Logic. In: Mazzara, M., Voronkov, A. (eds) Perspectives of System Informatics. PSI 2015. Lecture Notes in Computer Science(), vol 9609. Springer, Cham. https://doi.org/10.1007/978-3-319-41579-6_24

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  • DOI: https://doi.org/10.1007/978-3-319-41579-6_24

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