Supermartingale Certificates for Quantitative Omega-regular Verification and Control

TL;DR

This paper introduces a novel supermartingale certificate called LDBSM for verifying and controlling discrete-time stochastic systems against quantitative omega-regular properties, extending previous methods to more complex properties.

Contribution

It presents the first supermartingale certificate for quantitative omega-regular properties and automated synthesis algorithms for systems described by polynomial inequalities.

Findings

Successfully verified complex stochastic systems with the new method.

Automated synthesis of certificates for systems with polynomial dynamics.

Outperformed previous approaches on benchmark verification tasks.

Abstract

We present the first supermartingale certificate for quantitative $\omega$-regular properties of discrete-time infinite-state stochastic systems. Our certificate is defined on the product of the stochastic system and a limit-deterministic B\"uchi automaton that specifies the property of interest; hence we call it a limit-deterministic B\"uchi supermartingale (LDBSM). Previously known supermartingale certificates applied only to quantitative reachability, safety, or reach-avoid properties, and to qualitative (i.e., probability 1) $\omega$-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.