Bimodal Logics for Reasoning About Continuous Dynamics

We study a propositional bimodal logic consisting of two S4 modalities and [a], together with the interaction axiom scheme 〈a〉 φ → 〈a〉φ. In the intended semantics, the plain is given the McKinsey-Tarski interpretation as the interior operator of a topology, while the labelled [a] is given the standard Kripke semantics using a reflexive and transitive binary relation Ra. The interaction axiom expresses the property that the Ra relation is lower semi-continuous with respect to the topology. The class of topological Kripke frames axiomatised by the logic includes all frames over Euclidean space where Ra is the positive flow relation of a differential equation. We establish the completeness of the axiomatisation with respect to the intended class of topological Kripke frames, and investigate tableau calculi for the logic, although decidability is still an open question.