Sequent Calculus in the Topos of Trees [article]

Ranald Clouston, Rajeev Goré
2015 arXiv   pre-print
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
more » ... ogic 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.
arXiv:1501.03293v2 fatcat:gjav5oyn3vbbnbo43xzx46o6yi