Spatio-temporal Logics for Continuous Dynamical Systems

We develop a multi-modal/temporal logic for the analysis of a general class of continuous dynamical systems over arbitrary time-lines. Our frames consist of a topological space together with a flow (or group action), which is a function from the product of space and time into space that satisfies the two flow laws; we further require this function to be continuous. Over Euclidean space and real time, flows include the solutions of ordinary differential equations. The syntax of the logic combines the Until and Since constructs of linear temporal logic with a modal box/diamond. The latter are interpreted by topological interior/closure, and are characterized by the axioms of S4, while the temporal operators are interpreted with respect to motion along the flow. For the temporal fragment, we adapt the axiomatization of Burgess for temporal logic over linear orders. Our axiomatization for continuous flow logic consists of the fusion of S4 and linear temporal logic together with an additional axiom scheme which characterizes the continuity of flows. The major result of the paper is that this axiomatization is sound and complete for the class of continuous flow frames. While the base logic consisting of the fusion of S4 and linear temporal logic is decidable, it is an open question whether this extension with the continuity scheme remains decidable.