Counterexample Guided Abstraction Refinement for Polyhedral Probabilistic Hybrid Systems

Abstract

We consider the problem of safety analysis of probabilistic hybrid systems, which capture discrete, continuous and probabilistic behaviors. We present a novel counterexample guided abstraction refinement (CEGAR) algorithm for a subclass of probabilistic hybrid systems, called polyhedral probabilistic hybrid systems (PHS), where the continuous dynamics is specified using a polyhedral set within which the derivatives of the continuous executions lie. Developing a CEGAR algorithm for PHS is complex owing to the branching behavior due to the probabilistic transitions, and the infinite state space due to the real-valued variables. We present a practical algorithm by choosing a succinct representation for counterexamples, an efficient validation algorithm and a constructive method for refinement that ensures progress towards the elimination of a spurious abstract counterexample. The technical details for refinement are non-trivial since there are no clear disjoint sets for separation. We have implemented our algorithm in a Python toolbox called Procegar; our experimental analysis demonstrates the benefits of our method in terms of successful verification results, as well as bug finding.