Deciding Probabilistic Bisimilarity Distance One for Labelled Markov Chains

Qiyi Tang,Franck Van Breugel

doi:10.1007/978-3-319-96145-3_39

Abstract

Probabilistic bisimilarity is an equivalence relation that captures which states of a labelled Markov chain behave the same. Since this behavioural equivalence only identifies states that transition to states that behave exactly the same with exactly the same probability, this notion of equivalence is not robust. Probabilistic bisimilarity distances provide a quantitative generalization of probabilistic bisimilarity. The distance of states captures the similarity of their behaviour. The smaller the distance, the more alike the states behave. In particular, states are probabilistic bisimilar if and only if their distance is zero. This quantitative notion is robust in that small changes in the transition probabilities result in small changes in the distances.

Highlights

A behavioural equivalence captures which states of a model give rise to the same behaviour
The probabilistic bisimilarity distances that we study in this paper were first defined by Desharnais et al in [11]
To compare this algorithm with our new algorithm consisting of the components D0, D1 and simple policy iteration (SPI), we implemented all the components in Java and ran both implementations on several labelled Markov chains

Summary

Introduction

A behavioural equivalence captures which states of a model give rise to the same behaviour. The state of the art algorithm to compute the probabilistic bisimilarity distances consists of two components: D0 and SPI To compare this algorithm with our new algorithm consisting of the components D0 , D1 and SPI, we implemented all the components in Java and ran both implementations on several labelled Markov chains. To determine the number of non-trivial distances of a labelled Markov chain, we use the following algorithm. We compare the running time of our new algorithm with the state of the art algorithm, that combines algorithms due to Derisavi et al and due to Bacci et al The results are shown in the table below In this protocol, the number of non-trivial distances is zero. To compute the non-trivial distances smaller than a positive number, ε, we use the following algorithm

Compute the query set

Findings

Conclusion