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Constructive Completeness for Modal Logic with Transitive Closure

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Certified Programs and Proofs (CPP 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7679))

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Abstract

Classical modal logic with transitive closure appears as a subsystem of logics used for program verification. The logic can be axiomatized with a Hilbert system. In this paper we develop a constructive completeness proof for the axiomatization using Coq with Ssreflect. The proof is based on a novel analytic Gentzen system, which yields a certifying decision procedure that for a formula constructs either a derivation or a finite countermodel. Completeness of the axiomatization then follows by translating Gentzen derivations to Hilbert derivations. The main difficulty throughout the development is the treatment of transitive closure.

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Author information

Authors and Affiliations

  1. Saarland University, Saarbrücken, Germany

    Christian Doczkal & Gert Smolka

Authors
  1. Christian Doczkal
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  2. Gert Smolka
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Editor information

Editors and Affiliations

  1. Microsoft Research, Redmond, WA, USA

    Chris Hawblitzel

  2. INRIA Saclay and LIX, Ecole Polytechnique, Palaiseaux Cedex, France

    Dale Miller

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Doczkal, C., Smolka, G. (2012). Constructive Completeness for Modal Logic with Transitive Closure. In: Hawblitzel, C., Miller, D. (eds) Certified Programs and Proofs. CPP 2012. Lecture Notes in Computer Science, vol 7679. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35308-6_18

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  • DOI: https://doi.org/10.1007/978-3-642-35308-6_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35307-9

  • Online ISBN: 978-3-642-35308-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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