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Normalisation by Completeness with Heyting Algebras

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Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9450))

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Abstract

Usual normalization by evaluation techniques have a strong relationship with completeness with respect to Kripke structures. But Kripke structures is not the only semantics that fits intuitionistic logic: Heyting algebras are a more algebraic alternative.

In this paper, we focus on this less investigated area: how completeness with respect to Heyting algebras generate a normalization algorithm for a natural deduction calculus, in the propositional fragment. Our main contributions is that we prove in a direct way completeness of natural deduction with respect to Heyting algebras, that the underlying algorithm natively deals with disjunction, that we formalized those proofs in Coq, and give an extracted algorithm.

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Notes

  1. 1.

    namely, the programming language in which the evaluation function is written.

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Acknowledgments

The authors would like to thanks the reviewers for their insightful and constructive comments and pointers. Unfortunately we lacked time to include them all.

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Authors and Affiliations

  1. ENS Lyon, Lyon, France

    Gaëtan Gilbert

  2. Inria Paris, Paris, France

    Gaëtan Gilbert

  3. MINES ParisTech, PSL Research University, Paris, France

    Olivier Hermant

  4. Wesleyan University, Middletown, CT, USA

    Olivier Hermant

Authors
  1. Gaëtan Gilbert
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  2. Olivier Hermant
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Corresponding author

Correspondence to Gaëtan Gilbert .

Editor information

Editors and Affiliations

  1. New York University, New York, New York, USA

    Martin Davis

  2. University of the South Pacific, Suva, Fiji

    Ansgar Fehnker

  3. Macquarie University, Sydney, New South Wales, Australia

    Annabelle McIver

  4. The University of Manchester, Manchester, United Kingdom

    Andrei Voronkov

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Gilbert, G., Hermant, O. (2015). Normalisation by Completeness with Heyting Algebras. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2015. Lecture Notes in Computer Science(), vol 9450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48899-7_33

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  • DOI: https://doi.org/10.1007/978-3-662-48899-7_33

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-48898-0

  • Online ISBN: 978-3-662-48899-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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