Logical, Metric, and Algorithmic Characterisations of Probabilistic Bisimulation

TL;DR

This paper unifies various probabilistic bisimulation concepts through a common lifting operation linked to the Kantorovich metric, providing logical, metric, and algorithmic characterisations, including an efficient on-the-fly checking algorithm.

Contribution

It introduces a unified framework for probabilistic bisimulation based on a lifting operation related to the Kantorovich metric, with new logical and algorithmic characterisations.

Findings

Lifting operation corresponds to the Kantorovich metric.

Extended modal logics are adequate and expressive for probabilistic bisimilarity.

An efficient on-the-fly algorithm for bisimulation checking is developed.

Abstract

Many behavioural equivalences or preorders for probabilistic processes involve a lifting operation that turns a relation on states into a relation on distributions of states. We show that several existing proposals for lifting relations can be reconciled to be different presentations of essentially the same lifting operation. More interestingly, this lifting operation nicely corresponds to the Kantorovich metric, a fundamental concept used in mathematics to lift a metric on states to a metric on distributions of states, besides the fact the lifting operation is related to the maximum flow problem in optimisation theory. The lifting operation yields a neat notion of probabilistic bisimulation, for which we provide logical, metric, and algorithmic characterisations. Specifically, we extend the Hennessy-Milner logic and the modal mu-calculus with a new modality, resulting in an adequate and an expressive logic for probabilistic bisimilarity, respectively. The correspondence of the lifting operation and the Kantorovich metric leads to a natural characterisation of bisimulations as pseudometrics which are post-fixed points of a monotone function. We also present an "on the fly" algorithm to check if two states in a finitary system are related by probabilistic bisimilarity, exploiting the close relationship between the lifting operation and the maximum flow problem.