Closure certificates

Abstract

A barrier certificate, defined over the states of a dynamical system, is a real-valued function whose zero level set characterizes an inductively verifiable state invariant separating reachable states from unsafe ones. When combined with powerful decision procedures — such as sum-of-squares programming (SOS) or satisfiability-modulo-theory solvers (SMT) — barrier certificates enable an automated deductive verification approach to safety. The barrier certificate approach has been extended to refute LTL and $\omega$-regular specifications by separating consecutive transitions of corresponding $\omega$-automata in the hope of denying all accepting runs. Unsurprisingly, such tactics are bound to be conservative as refutation of recurrence properties requires reasoning about ranking functions to prove liveness as well. This paper introduces the notion of closure certificates as a natural extension of barrier certificates from state invariants to transition invariants. We show how one may use ranking function arguments over such certificates to verify discrete-time dynamical systems against Linear Temporal logic formulae. We augment these definitions with SOS and SMT based characterization for automating the search of closure certificates and demonstrate their effectiveness over some case studies.