Supermartingale Certificates for Quantitative Omega-regular Verification and Control

TL;DR

We address verification and control of discrete-time infinite-state stochastic systems against quantitative $\omega$-regular specifications. Our approach builds limit-deterministic Büchi supermartingales (LDBSM) on the product of the system and an LDBA, combining safety and liveness certificates in a hierarchical, sound framework. We provide template-based automated synthesis for polynomial dynamics via SMT-based constraint solving, enabling both verification and controller synthesis with provable probabilistic guarantees. Experiments on a 1D random walk demonstrate high success probabilities (e.g., $p$ near 1) and capability beyond prior supermartingale-based methods. The work broadens certificate-based verification to arbitrary quantitative $\omega$-regular properties and supports probabilistic programs and general stochastic dynamics.

Abstract

We present the first supermartingale certificate for quantitative $ω$-regular properties of discrete-time infinite-state stochastic systems. Our certificate is defined on the product of the stochastic system and a limit-deterministic Büchi automaton that specifies the property of interest; hence we call it a limit-deterministic Büchi supermartingale (LDBSM). Previously known supermartingale certificates applied only to quantitative reachability, safety, or reach-avoid properties, and to qualitative (i.e., probability 1) $ω$-regular properties. We also present fully automated algorithms for the template-based synthesis of LDBSMs, for the case when the stochastic system dynamics and the controller can be represented in terms of polynomial inequalities. Our experiments demonstrate the ability of our method to solve verification and control tasks for stochastic systems that were beyond the reach of previous supermartingale-based approaches.