Bisimulations Meet PCTL Equivalences for Probabilistic Automata

TL;DR

This paper introduces new strong bisimulation relations for probabilistic automata that exactly characterize PCTL and PCTL*, bridging the gap between logical and behavioral equivalences.

Contribution

The paper proposes novel strong bisimulation notions that precisely match PCTL and PCTL* semantics, extending to weak bisimulations and simulation preorders.

Findings

New strong bisimulation relations characterize PCTL and PCTL* exactly.

Extended weak bisimulations for PCTL and PCTL* without next operator.

Framework connects logical and behavioral equivalences in probabilistic automata.

Abstract

Probabilistic automata (PAs) have been successfully applied in 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 probabilistic computation tree logic (PCTL) and its extension PCTL^*. Various behavioral equivalences are proposed, as a powerful tool for abstraction and compositional minimization for PAs. Unfortunately, the equivalences are well-known to be sound, but not complete with respect to the logical equivalences induced by PCTL or PCTL*. The desire of a both sound and complete behavioral equivalence has been pointed out by Segala in 1995, but remains open throughout the years. In this paper we introduce novel notions of strong bisimulation relations, which characterize PCTL and PCTL* exactly. We extend weak bisimulations that characterize PCTL and PCTL* without next operator, respectively. Further, we also extend the framework to simulation preorders. Thus, our paper bridges the gap between logical and behavioral equivalences and preorders in this setting.