Verification of Parametric Properties of Linear Discrete-time Stochastic Systems

Abstract

We study the problem of verifying parametric properties of linear discrete-time stochastic systems (linear DTSSs) over bounded and unbounded time horizons. A linear DTSS captures random processes, where the one-step transition relation between the current random vector X and the next-step random vector X’ is linear and is given by X’ = AX + W, where A is an n x n matrix and W is a random noise vector. We assume that the initial and noise random vectors are multivariate normal, each of which is expressed as a pair of two parameters, namely, mean and covariance. We are interested in verifying whether the system always satisfies a given parametric property specified as a subset of the parameters’ space over bounded or unbounded time horizons. For bounded time steps, we reduce the problem to the satisfiability of a semidefinite programming problem. For the unbounded time steps, we propose a novel abstraction procedure to reduce the verification problem to that of a finite graph, wherein, the nodes of the graph correspond to the regions of a partition of the random vector space, in contrast to existing works that partition the state-space. More precisely, we partition the parameter space of normal random vectors, namely, the space of means and covariance matrices, and apply semi-definite programming to compute the edges. We show that our abstraction procedure is sound. Our experiments with varying number of vehicle dynamics and different dimensions of random models show that our methods are reasonably scalable.