Verifying Asymptotic Temporal Properties of Continuous-State Probabilistic Systems

We describe a theory of probabilistic Morse decomposition for continuous-state, probabilistic dynamical systems. Morse decompositions, studied in the topological theory of dynamical systems, allow reasoning about asymptotic behaviors of dynamical systems. We generalize notions of attractors, repellers, and invariant sets to the probabilistic context, and show how these can be used to describe the topological structure of the probabilistic dynamics. Additionally, we show a Lyapunov function characterization for probabilistic Morse decompositions. Our probabilistic Morse decompositions enable an abstraction-based verification methodology for asymptotic specifications such as “the trajectories asymptotically converge to a set with positive probability,” which are not expressible in usual linear temporal logics. We describe computational approaches to computing probabilistic Morse decompositions, using state-space gridding as well as Lyapunov functions. Interestingly, the construction of Morse decompositions is crucial: we show that existing abstraction-based techniques based on gridding the state space are not sufficiently powerful to verify asymptotic specifications.