Decidability of Approximate Skolem Problem and Applications to Logical Verification of Dynamical Properties of Markov Chains

Abstract

When studying probabilistic dynamical systems, temporal logic has typically been used to analyze path properties. Recently, there has been some interest in analyzing the dynamical evolution of state probabilities of these systems. In this article, we show that verifying linear temporal properties concerning the state evolution induced by a Markov chain is equivalent to the decidability of the Skolem problem -- a long-standing open problem in Number Theory. However, from a practical point of view, usually it is enough to check properties up to some acceptable error bound ϵ. We show that an approximate version of the Skolem problem is decidable, and that it can be applied to verify, up to arbitrarily small ϵ, linear temporal properties of the state evolution induced by a Markov chain.