Expressiveness of probabilistic modal logics: A gradual approach

Abstract

Logical characterizations of probabilistic bisimulation and simulation for Labelled Markov Processes were given by Desharnais et al. These results hold for systems defined on analytic state spaces and assume countably many labels in the case of bisimulation and finitely many labels in the case of simulation.We revisit these results by giving simpler and more streamlined proofs. In particular, our proof for simulation has the same structure as the one for bisimulation, relying on a new result of a topological nature. We also propose a new notion of event simulation.Our proofs assume countably many labels, and we show that the logical characterization of bisimulation may fail when there are uncountably many labels. However, with a stronger assumption on the transition functions (continuity instead of just measurability), we regain the logical characterization result for arbitrarily many labels. These results arose from a game-theoretic understanding of probabilistic simulation and bisimulation.