Complete $\omega$-Regular Supermartingale Certificates

Alessandro Abate; Mirco Giacobbe; Sergey Ichtchenko; Diptarko Roy

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arXiv · 2605.21134

Complete $\omega$-Regular Supermartingale Certificates

Alessandro Abate·Mirco Giacobbe·Sergey Ichtchenko·Diptarko Roy

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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 the $\omega$-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 $\omega$-regular properties and the first sound and $\varepsilon$-complete supermartingale certificates for quantitative $\omega$-regular properties on time-homogeneous Markov chains with countably infinite state spaces.

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Alessandro Abate, Mirco Giacobbe, Sergey Ichtchenko, Diptarko Roy. 2026-05-20. Complete $\omega$-Regular Supermartingale Certificates. https://arxiv.org/abs/2605.21134

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