Complete Supermartingale Certificates for $ω$-Regular Properties

2026-05-20

AI Summary

This work addresses the problem of almost-sure and quantitative probabilistic verification of ω-regular properties—such as liveness or reactivity—for time-homogeneous Markov chains over countably infinite state spaces. By decomposing ω-regular specifications into obligations that almost surely terminate in absorbing regions, and leveraging the existence of such regions, the paper presents the first complete supermartingale certificate for almost-sure satisfaction of ω-regular properties. For the quantitative setting, it further develops an ε-complete supermartingale-based verification framework. Integrating supermartingale theory, Markov chain analysis, and reactive decomposition techniques, this approach achieves, for the first time, complete (or ε-complete) probabilistic verification of ω-regular properties in this class of infinite-state systems.

Abstract

We introduce a general methodology for the construction of sound and complete proof rules for the almost-sure and quantitative acceptance of reactivity properties on time-homogeneous Markov chains with general state spaces. Reactivity captures $ω$-regular properties and subsumes linear temporal logic. Our core technical result establishes that every reactivity property admits decomposition into multiple obligations of almost-sure termination into absorbing regions, and that appropriate absorbing regions always exist on general state spaces. This enables the extension of every complete proof rule for almost-sure termination into a proof rule for reactivity that is complete in the almost-sure case, and complete up to an arbitrarily small $\varepsilon$-approximation in the quantitative case. We apply our new methodology to recent results on sound and complete supermartingale certificates for almost-sure termination in the special case of countably infinite state spaces, alongside standard results on quantitative safety. As a result, we obtain the first sound and complete supermartingale certificates for almost-sure $ω$-regular properties and the first sound and $\varepsilon$-complete supermartingale certificates for quantitative $ω$-regular properties on time-homogeneous Markov chains with countably infinite state spaces.

Problem

• supermartingale certificates
• ω-regular properties
• Markov chains
• almost-sure termination
• quantitative verification

Innovation

• supermartingale certificates
• ω-regular properties
• almost-sure termination
• Markov chains
• reactivity