Sufficient and Necessary Barrier-like Conditions for Safety and Reach-avoid Verification of Stochastic Discrete-time Systems

Abstract: In this paper, we examine necessary and sufficient barrier-like conditions for infinite-horizon safety verification and reach-avoid verification of stochastic discrete-time systems, derived through a relaxation of Bellman equations. Unlike previous methods focused on barrier-like conditions that primarily address sufficiency, our work rigorously integrates both necessity and sufficiency for properties pertaining to infinite time. Safety verification aims to certify the satisfaction of the safety property, which stipulates that the probability of the system, starting from a specified initial state, remaining within a safe set always is greater than or equal to a specified lower bound. A necessary and sufficient barrier-like condition is formulated for safety verification. In contrast, reach-avoid verification extends beyond safety to include reachability, seeking to certify the satisfaction of the reach-avoid property. It requires that the probability of the system, starting from a specified initial state, reaching a target set eventually while remaining within a safe set until the first hit of the target, is greater than or equal to a specified lower bound. Two necessary and sufficient barrier-like conditions are formulated under certain assumptions.