k-Inductive Barrier Certificates for Stochastic Systems

Barrier certificates are inductive invariants that provide guarantees on the safety and reachability behaviors of continuous dynamical systems. For stochastic dynamical systems, barrier certificates take the form of inductive "expectation" invariants. In this context, a barrier certificate is a non-negative real-valued function over the state space of the system satisfying a strong supermartingale condition: it decreases in expectation as the system evolves The existence of barrier certificates, then, provides lower bounds on the probability of satisfaction of safety or reachability specifications over unbounded-time horizons. Unfortunately, establishing supermartingale conditions on barrier certificates can often be restrictive. In practice, we strive to overcome this challenge by utilizing a weaker condition called c-martingale that permits a bounded increment in expectation at every time step; unfortunately this only guarantees the property of interest for a bounded time horizon. The idea of k-inductive invariants, often utilized in software verification, relaxes the need for the invariant to be inductive with every transition of the system to requiring that the invariant holds in the next step if it holds for the last k steps. This paper synthesizes the idea of k-inductive invariants with barrier certificates. These refinements that we dub as k-inductive barrier certificates relax the supermartingale requirements at each time step to supermartingale requirements in k-steps with potential c-martingale requirements at each step, while still providing unbounded-time horizon probabilistic guarantees. We characterize a notion of k-inductive barrier certificates for safety and two distinct notions of k-inductive barrier certificates for reachability. Correspondingly, utilizing such k-inductive barrier certificates, we obtain probabilistic lower bounds on the satisfaction of safety and reachability specifications, respectively. We present a computational method based on sum-of-squares (SOS) programming to synthesize suitable k-inductive barrier certificates and, demonstrate the effectiveness of the proposed methods via some case studies.