Complete ω-Regular Supermartingale Certificates

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.