Logical characterization of branching metrics for nondeterministic probabilistic transition systems

Abstract

In this paper we propose a logical characterization of (bi)simulation metrics obtained by a probabilistic variant of Hennessy-Milner logic enriched with variables, whose semantics is defined following the equational μ-calculus approach. Our characterization is based on the novel notions of mimicking formulae and syntactical distance on formulae. The former ones are the quantitative analogous to characteristic formulae. The latter is a 1-bounded pseudometric on formulae measuring their syntactical disparities. The characterization is obtained by showing that the (bi)simulation distance between processes corresponds to the syntactical distance between their mimicking formulae.We also discuss the expressive power of mimicking formulae with respect to probabilistic (bi)simulations. We show that two processes are bisimilar if and only if their mimicking formulae are syntactically equivalent. Moreover, we obtain that mimicking formulae of processes coincide with their characteristic formulae for ready simulation and that negation free mimicking formulae coincide with the characteristic formulae for simulation.