Constructive Completeness for Modal Logic with Transitive Closure

  • Christian Doczkal
  • Gert Smolka
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7679)


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.


modal logic completeness decision procedures constructive proofs Hilbert Systems Gentzen systems Coq Ssreflect 


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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Christian Doczkal
    • 1
  • Gert Smolka
    • 1
  1. 1.Saarland UniversitySaarbrückenGermany

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