Cut-elimination and proof search for bi-intuitionistic tense logic

Rajeev Goré*, Linda Postniece, Alwen Tiu

*Corresponding author for this work

    Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

    19 Citations (Scopus)

    Abstract

    We consider an extension of bi-intuitionistic logic with the traditional modalities ◇, □, ◆ and ■ from tense logic Kt. Proof theoretically, this extension is obtained simply by extending an existing sequent calculus for bi-intuitionistic logic with typical inference rules for the modalities used in display logics. As it turns out, the resulting calculus, LBiKt, seems to be more basic than most intuitionistic tense or modal logics considered in the literature, in particular, those studied by Ewald and Simpson, as it does not assume any a priori relationship between the modal operators ◇ and □. We recover Ewald's intuitionistic tense logic and Simpson's intuitionistic modal logic by modularly extending LBiKt with additional structural rules. The calculus LBiKt is formulated in a variant of display calculus, using a form of sequents called nested sequents. Cut elimination is proved for LBiKt, using a technique similar to that used in display calculi. As in display calculi, the inference rules of LBiKt are "shallow" rules, in the sense that they act on top-level formulae in a nested sequent. The calculus LBiKt is ill-suited for backward proof search due to the presence of certain structural rules called "display postulates" and the contraction rules on arbitrary structures. We show that these structural rules can be made redundant in another calculus, DBiKt, which uses deep inference, allowing one to apply inference rules at an arbitrary depth in a nested sequent. We prove the equivalence between LBiKt and DBiKt and outline a proof search strategy for DBiKt. We also give a Kripke semantics and prove that LBiKt is sound with respect to the semantics, but completeness is still an open problem. We then discuss various extensions of LBiKt.

    Original languageEnglish
    Title of host publicationAdvances in Modal Logic 2010
    Pages156-177
    Number of pages22
    Publication statusPublished - 2010
    Event8th International Conference on Advances in Modal Logic, AiML-2010 - Moscow, Russian Federation
    Duration: 24 Aug 201027 Aug 2010

    Publication series

    NameAdvances in Modal Logic 2006
    Volume8

    Conference

    Conference8th International Conference on Advances in Modal Logic, AiML-2010
    Country/TerritoryRussian Federation
    CityMoscow
    Period24/08/1027/08/10

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