Complete $ω$-Regular Supermartingale Certificates
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arXiv
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| Format: | Preprint |
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2026
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| author | Abate, Alessandro Giacobbe, Mirco Ichtchenko, Sergey Roy, Diptarko |
| author_facet | Abate, Alessandro Giacobbe, Mirco Ichtchenko, Sergey Roy, Diptarko |
| contents | 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 $ω$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21134 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complete $ω$-Regular Supermartingale Certificates Abate, Alessandro Giacobbe, Mirco Ichtchenko, Sergey Roy, Diptarko Logic in Computer Science 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 $ω$-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. |
| title | Complete $ω$-Regular Supermartingale Certificates |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2605.21134 |