On Hybrid Systems and the Modal µ-calculus

We start from a basic and fruitful idea in current work on the formal analysis and verification of hybrid and real-time systems: the uniform representation of both sorts of state dynamics - both continuous evolution within a control mode, and the effect of discrete jumps between control modes - as abstract transition relations over a hybrid space X ⊆ Q × Rn, where Q is a finite set of control modes. The resulting "machine" or transition system model is currently analyzed using the resources of concurrent and reactive systems theory and temporal logic verification, abstracted from their original setting of finite state spaces and purely discrete transitions. One such resource is the propositional µ-calculus: a richly expressive formal logic of transition system models (of arbitrary cardinality), which subsumes virtually all temporal and modal l ogics. The key move here is to view the transition system models of hybrid automata not merely as some form of "discrete abstraction", but rather as a skeleton which can be fleshed out by imbuing the state space with topological, metric tolerance or other structure. Drawing on the resources of modal logics, we give explicit symbolic representation to such structure in polymodal logics extending the modal µ-calculus. The result is a logical formalism in which we can directly and simply express continuity properties of transition relations and metric tolerance properties such as "being within distance ∈" of a set. Moreover, the logics have sound and complete deductive proof systems, so assumptions of continuity or tolerance can be used as hypotheses in deductive verification. By also viewing transition relations in their equivalent form as set-valued functions, and drawing on the resources of set-valued analysis and dynamical systems theory, we open the way to a richer formal analysis of robustness and stability for hybrid automata and related classes of systems.