2015/01/14 by Ranald Clouston, Clouston, Ranald, Rajeev Goré +1
Computer Science · #FOS: Computer and information sciences #Logic in Computer Science (cs.LO) #cs.LO
paper · pdf · doi:10.48550/arxiv.1501.03293
Extended version, with full proof details, of a paper accepted to FoSSaCS 2015 (this version edited to fix some minor typos)
arxiv created 2015/04/17 · arxiv updated 2015/04/20
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 KMlin. We give a sound and cut-free complete sequent calculus for KMlin 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.