On Model-Checking Probabilistic $ω$-Pushdown Systems, and $ω$-PCTL$^*$ Characterization of Weak Bisimulation

Abstract: In this paper, we obtain the following equally important new results: We first extend the notion of {\em probabilistic pushdown automaton} to {\em probabilistic $\omega$-pushdown automaton} for the first time and study the model-checking question of {\em stateless probabilistic $\omega$-pushdown system ($\omega$-pBPA)} against $\omega$-PCTL (defined by Chatterjee, Sen, and Henzinger in \cite{CSH08}), showing that model-checking of {\em stateless probabilistic $\omega$-pushdown systems ($\omega$-pBPA)} against $\omega$-PCTL is generally undecidable. Our approach is to construct $\omega$-PCTL formulas encoding the {\em Post Correspondence Problem}. We study and analyze the soundness and completeness of {\em weak bisimulation} for {\em $\omega$ probabilistic computational tree logic ($\omega$-PCTL$<sup>*$)}, showing that it is sound and complete. Our models are probabilistic labelled transition systems induced by probabilistic $\omega$-pushdown automata defined in this paper.