On Probabilistic $ω$-Pushdown Systems, and $ω$-Probabilistic Computational Tree Logic

In this paper, we define the notion of {\em probabilistic $ω$-pushdown automaton} and study its model-checking problem against the logic of $ω$-probabilistic computational tree logic ($ω$-PCTL) and its bounded version from a computational complexity view. Specifically, we obtain the following equally important new results: (1) We define {\em probabilistic $ω$-pushdown automaton} for the first time and study the model-checking question of {\em stateless probabilistic $ω$-pushdown system ($ω$-pBPA)} against $ω$-PCTL (defined by Chatterjee, Sen, and Henzinger in \cite{CSH08}), showing that model-checking of {\em stateless probabilistic $ω$-pushdown systems ($ω$-pBPA)} against $ω$-PCTL is generally undecidable. Our approach is to construct $ω$-PCTL formulas encoding the {\em Post Correspondence Problem}. (2) We then study in which case there exists an algorithm for model-checking {\it stateless probabilistic $ω$-pushdown systems} and show that the problem of model-checking {\it stateless probabilistic $ω$-pushdown systems} against $ω$-{\it bounded probabilistic computational tree logic} ($ω$-bPCTL) is decidable, and further show that this problem is $\mathit{NP}$-hard.