Dependence Logics in Temporal Settings

Abstract: Many forms of dependence manifest themselves over time, with behavior of variables in dynamical systems as a paradigmatic example. This paper studies temporal dependence in dynamical systems from a logical perspective, by enriching a minimal modal base logic of static functional dependencies. We first introduce a logic for dynamical systems featuring temporalized variables, provide a complete axiomatic proof calculus, and show that its satisfiability problem is decidable. Then, to capture explicit reasoning about dynamic transition functions, we enhance the framework with function symbols and term identity. Next we combine temporalized variables with a modality for next-time truth from standard temporal logic, where modal correspondence analysis reveals the principles needed for a complete and decidable logic of timed dynamical systems supporting reductions between the two ways of referring to time. Our final result is an axiomatization of a general decidable logic of dependencies in arbitrary dynamical systems. We conclude with a brief outlook on how the systems introduced here mesh with richer temporal logics of system behavior, and with dynamic topological logic.