Symbolic Control for Stochastic Systems via Parity Games

We consider the problem of computing the maximal probability of satisfying an $\omega$-regular specification for stochastic, continuous-state, nonlinear systems evolving in discrete time. The problem reduces, after automata-theoretic constructions, to finding the maximal probability of satisfying a parity condition on a (possibly hybrid) state space. While characterizing the exact satisfaction probability is open, we show that a lower bound on this probability can be obtained by (I) computing an under-approximation of the qualitative winning region, i.e., states from which the parity condition can be enforced almost surely, and (II) computing the maximal probability of reaching this qualitative winning region. The heart of our approach is a technique to symbolically compute the under-approximation of the qualitative winning region in step (I) via a finite-state abstraction of the original system as a $2\frac{1}{2}$-player parity game. Our abstraction procedure uses only the support of the probabilistic evolution; it does not use precise numerical transition probabilities. We prove that the winning set in the abstract $2\frac{1}{2}$-player game induces an under-approximation of the qualitative winning region in the original synthesis problem, along with a policy to solve it. By combining these contributions with (a) existing symbolic fixpoint algorithms to solve $2\frac{1}{2}$-player games and (b) existing techniques for reachability policy synthesis in stochastic nonlinear systems, we get an abstraction-based symbolic algorithm for finding a lower bound on the maximal satisfaction probability. We have implemented our approach and evaluated it on the nonlinear model of the perturbed Dubins vehicle.