Bisimulations Meet PCTL Equivalences for Probabilistic Automata

Probabilistic automata (PA) [20] have been successfully applied in the formal verification of concurrent and stochastic systems. Efficient model checking algorithms have been studied, where the most often used logics for expressing properties are based on PCTL [11] and its extension PCTL [4]. Various behavioral equivalences are proposed for PAs, as a powerful tool for abstraction and compositional minimization for PAs. Unfortunately, the behavioral equivalences are well-known to be strictly stronger than the logical equivalences induced by PCTL or PCTL. This paper introduces novel notions of strong bisimulation relations, which characterizes PCTL and PCTL exactly. We also extend weak bisimulations characterizing PCTL and PCTL without next operator, respectively. Thus, our paper bridges the gap between logical and behavioral equivalences in this setting.