Sequent Calculus in the Topos of Trees

Nakano's "later" modality, inspired by G\"{o}del-L\"{o}b 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 $\mathrm{KM}_{\mathrm{lin}}$. We give a sound and cut-free complete sequent calculus for $\mathrm{KM}_{\mathrm{lin}}$ 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.Comment: Extended version, with full proof details, of a paper accepted to FoSSaCS 2015 (this version edited to fix some minor typos

A fully labelled proof system for intuitionistic modal logics

Labelled proof theory has been famously successful for modal logics by mimicking their relational seman-tics within deductive systems. Simpson in particular designed a framework to study a variety of intuitionisticmodal logics integrating a binary relation symbol in the syntax. In this paper, we present a labelled sequentsystem for intuitionistic modal logics such that there is not only one, but two relation symbols appearingin sequents: one for the accessibility relation associated with the Kripke semantics for normal modal logicsand one for the preorder relation associated with the Kripke semantics for intuitionistic logic. This putsour system in close correspondence with the standard birelational Kripke semantics for intuitionistic modallogics. As a consequence it can be extended with arbitrary intuitionistic Scott-Lemmon axioms. We showsoundness and completeness, together with an internal cut elimination proof, encompassing a wider array ofintuitionistic modal logics than any existing labelled system

Countermodels from Sequent Calculi in Multi-Modal Logics

A novel countermodel-producing decision procedure that applies to several multi-modal logics, both intuitionistic and classical, is presented. Based on backwards search in labeled sequent calculi, the procedure employs a novel termination condition and countermodel construction. Using the procedure, it is argued that multi-modal variants of several classical and intuitionistic logics including K, T, K4, S4 and their combinations with D are decidable and have the finite model property. At least in the intuitionistic multi-modal case, the decidability results are new. It is further shown that the countermodels produced by the procedure, starting from a set of hypotheses and no goals, characterize the atomic formulas provable from the hypotheses.