Quantitative Supermartingale Certificates
Alessandro Abate, Mirco Giacobbe, Diptarko Roy
TL;DR
This work addresses the problem of quantitative verification and control under shift-invariant specifications for probabilistic systems, including ω-regular and LTL properties, where standard finite-state methods are inadequate. It develops a general theory of quantitative supermartingale certificates by decomposing a specification into a stochastic invariant that controls the exit probability from a region and an almost-sure certificate conditional on the invariant, leveraging Lévy's 0-1 law to connect conditional and overall satisfaction. The authors introduce the first quantitative Streett supermartingale certificates, enabling lower and upper bounds on ω-regular satisfaction and providing an algorithmic framework that uses polynomial templates and Handelman representations to synthesize certificates and parametrised control policies for general state-space models. Their experimental results on infinite-state examples demonstrate tight bounds and effective control synthesis, unifying and extending prior martingale-based certificates for reachability, safety, cost bounds, and asymptotic objectives, and laying the groundwork for scalable quantitative model checking and policy learning with formal guarantees.
Abstract
We introduce a general methodology for quantitative model checking and control synthesis with supermartingale certificates. We show that every specification that is invariant to time shifts admits a stochastic invariant that bounds its probability from below; for systems with general state space, the stochastic invariant bounds this probability as closely as desired; for systems with finite state space, it quantifies it exactly. Our result enables the extension of every certificate for the almost-sure satisfaction of shift-invariant specifications to its quantitative counterpart, ensuring completeness up to an approximation in the general case and exactness in the finite-state case. This generalises and unifies existing supermartingale certificates for quantitative verification and control under reachability, safety, reach-avoidance, and stability specifications, as well as asymptotic bounds on accrued costs and rewards. Furthermore, our result provides the first supermartingale certificate for computing upper and lower bounds on the probability of satisfying $ω$-regular and linear temporal logic specifications. We present an algorithm for quantitative $ω$-regular verification and control synthesis based on our method and demonstrate its practical efficacy on several infinite-state examples.