Certifying Almost-Sure Reachability in Unknown MDPs

Abstract

{\em Specification-Guided Reinforcement Learning (RL)} has largely concerned itself with \emph{learning} an optimal policy against a formal specification in an unknown {\em Markov Decision Process (MDP)}. We take up the dual question. Given a policy already in hand, can one \emph{verify} that it is correct against a formal specification in an unknown MDP, using sampling access alone? Such a question arises naturally in safety-critical systems. We investigate it for the {\em almost-sure reachability property}, whose correctness is witnessed by a {\em ranking supermartingale (RSM)} certificate. Our first finding is negative. In the absence of any further information about the MDP, one can neither learn such a certificate nor verify a candidate one with probabilistic guarantees. On the positive side, we show that, under a mild restriction, a known lower bound $p_{\min}$ on nonzero transition probabilities, a correct certificate \emph{can} be learned with PAC-style guarantees. To the best of our knowledge, this is the first study of learning and verifying certificates for almost-sure reachability in unknown MDPs from sampling access alone. Finally, our experiments show that the $p_{\min}$-based sufficient condition, while sound, is conservative: the learned certificate often remains valid on the true MDP even when the condition fails.