On intuitionistic modal and tense logics and their classical companion logics: Topological semantics and bisimulations

We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order (or partial-order) with the modal accessibility relation generalise to the requirement that the relation and its inverse be lower semi-continuous with respect to the topology. We then investigate the notion of topological bisimulation relations between topological Kripke frames, as introduced by Aiello and van Benthem, and show that their topology-preserving conditions are equivalent to the properties that the inverse-relation and the relation are lower semi-continuous with respect to the topologies on the two models. Our first main result is that this notion of topological bisimulation yields semantic preservation w.r.t. topological Kripke models for both intuitionistic tense logics, and for their classical companion multi-modal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbert-style axiomatizations of the Fischer Servi logic and its classical companion logic, we use the canonical model in a second main result which characterizes a Hennessy-Milner class of topological models between any pair of which there is a maximal topological bisimulation that preserve the intuitionistic semantics. The Hennessy-Milner class we identify includes transition system representations of hybrid automata over a product state space whose factors are a Euclidean space and a finite discrete space equipped with an Alexandrov topology determined by a pre-order. ∗Partially supported by Australian Research Council grants DP0208553 and LX0242359. A preliminary version of parts of this paper appeared as ‘Topological semantics and bisimulations for intuitionistic modal logics and their classical companion logics’, in Logical Foundations of Computer Science (LFCS’07), Springer-Verlag, 2007 (LNCS volume 4514, pp. 162-179).