Efficient Verification for Stochastic Mixed Monotone Systems

We present an efficient computational procedure to perform model checking on discrete-time, mixed monotone stochastic systems subject to an affine random disturbance. Specifically, we exploit the structure of such systems in order to efficiently compute a finite-state Interval-valued Markov Chain (IMC) that over-approximates the system's behavior. To that end, we first make the assumption that the disturbance is unimodal, symmetric, and independent on each coordinate of the domain. Next, given a rectangular partition of the state-space, we compute bounds on the probability of transition between all the states in the partition. The ease of computing the one-step reachable set of rectangular states under mixed monotone dynamics renders the computation of these transition bounds highly efficient. We furthermore investigate a method for over-approximating the IMC of mixed monotone systems when the disturbance is only approximately unimodal symmetric, and we discuss state-space refinement heuristics. Lastly, we present two verification case studies.