On Model-Checking Quantum Pushdown Systems, Extensions and $omega$-PCTL$^*$ Characterization of Weak Bisimulation

2022-09-21

AI Summary

This paper addresses model checking for probabilistic and quantum ω-pushdown systems. It introduces probabilistic ω-pushdown automata (ω-PDAs) and their stateless subclass, ω-probabilistic basic process algebras (ω-pBPAs). It proves that model checking ω-pBPAs against ω-PCTL is undecidable—via a reduction from the Post Correspondence Problem. It establishes a sound and complete logical characterization of weak bisimulation for ω-PCTL*, and extends the polynomial-time algorithm for deciding weak bisimulation on probabilistic automata to the probabilistic labeled transition systems induced by ω-PDAs. The core contributions are: (i) identifying the inherent undecidability boundary for model checking in the ω-pushdown setting; (ii) providing the first exact ω-PCTL*-characterization of weak bisimulation; and (iii) enabling efficient (polynomial-time) decidability of weak bisimulation for ω-pushdown systems.

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.

Problem

• Extend probabilistic pushdown automata to ω-pushdown automata
• Study model-checking ω-pBPA against ω-PCTL undecidability
• Analyze weak bisimulation soundness for ω-PCTL* and algorithms

Innovation

• Extends probabilistic pushdown automaton to ω-pushdown automaton
• Analyzes ω-PCTL* soundness and completeness for weak bisimulation
• Develops polynomial-time algorithm for weak bisimulation checking