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Sequent Calculus in the Topos of Trees

  • Ranald CloustonEmail author
  • Rajeev Goré
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9034)

Abstract

Nakano’s “later” modality, inspired by Gödel-Löb provability logic, has been applied in type systems and program logics to capture guarded recursion. Birkedal et al modelled this modality via the internal logic of the topos of trees. We show that the semantics of the propositional fragment of this logic can be given by linear converse-well-founded intuitionistic Kripke frames, so this logic is a marriage of the intuitionistic modal logic KM and the intermediate logic LC. We therefore call this logic KM lin . We give a sound and cut-free complete sequent calculus for KM lin via a strategy that decomposes implication into its static and irreflexive components. Our calculus provides deterministic and terminating backward proof-search, yields decidability of the logic and the coNP-completeness of its validity problem. Our calculus and decision procedure can be restricted to drop linearity and hence capture KM.

Keywords

Modal Logic Intuitionistic Logic Sequent Calculus Internal Logic Classical Propositional Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceAarhus UniversityAarhusDenmark
  2. 2.Research School of Computer ScienceAustralian National UniversityCanberraAustralia

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