A Logical Analysis of Dynamic Dependence

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 extending a minimal modal base logic of static functional dependencies. We define a logic for dynamical systems with single time steps, provide a complete axiomatic proof calculus, and show the decidability of the satisfiability problem for a substantial fragment. The system comes in two guises: modal and first-order, that naturally complement each other. Next, we consider a timed semantics for our logic, as an intermediate between state spaces and temporal universes for the unfoldings of a dynamical system. We prove completeness and decidability by combining techniques from dynamic-epistemic logic and modal logic of functional dependencies with complex terms for objects. Also, we extend these results to the timed logic with functional symbols and term identity. Finally, we conclude with a brief outlook on how the system proposed here connects with richer temporal logics of system behavior, and with dynamic topological logic.