On Model-Checking Quantum Pushdown Systems, Extensions and $ω$-PCTL$^*$ Characterization of Weak Bisimulation (2209.10517v11)

Abstract: In this paper, we extend the notions of the {\em probabilistic pushdown systems} and {\em Markov chains} to their quantum analogues, and investigate the question whether it is necessary to define a quantum analogue of {\em probabilistic computational tree logic} to describe the probabilistic and branching-time properties of the {\em quantum Markov chain}. We study its model-checking question and show that model-checking of {\em stateless quantum pushdown systems (qBPA)} against {\em probabilistic computational tree logic (PCTL)} is generally undecidable. We next extend the notion of {\em probabilistic pushdown automaton} to {\em probabilistic $\omega$-pushdown automaton} for the first time and study model-checking question of {\em stateless probabilistic $\omega$-pushdown system ($\omega$-pBPA)} against $\omega$-PCTL (defined by Chatterjee et al. 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} indirectly. We study and analysis soundness and completeness of {\em weak bisimulation} for {\em $\omega$ probabilistic computational tree logic ($\omega$-PCTL$^*$)}, showing that it is sound and complete. Our models are probabilistic labelled transition systems induced by probabilistic $\omega$-pushdown automata defined in this paper. Lastly, we extend the polynomial time algorithm for checking probabilistic weak bisimulation in the setting of probabilistic automata \cite{TH15,FHHT16} to our context for PLTS induced by probabilistic $\omega$-pushdown automata, showing that there exist polynomial-time algorithms for deciding weak bisimulation in the setting of probabilistic $\omega$-pushdown automata.