Computing Probabilistic Bisimilarity Distances for Probabilistic Automata

TL;DR

This paper introduces a practical algorithm for computing probabilistic bisimilarity distances in probabilistic automata by characterizing them through stochastic games and proves its termination and effectiveness.

Contribution

It presents the first practical algorithm for bisimilarity distance computation using a novel game-theoretic characterization and extends theoretical bounds on property satisfaction probabilities.

Findings

Algorithm terminates for non-stopping games.

First practical solution for bisimilarity distance computation.

Provides bounds on probability differences for $ ext{LTL}$ properties.

Abstract

The probabilistic bisimilarity distance of Deng et al. has been proposed as a robust quantitative generalization of Segala and Lynch's probabilistic bisimilarity for probabilistic automata. In this paper, we present a characterization of the bisimilarity distance as the solution of a simple stochastic game. The characterization gives us an algorithm to compute the distances by applying Condon's simple policy iteration on these games. The correctness of Condon's approach, however, relies on the assumption that the games are stopping. Our games may be non-stopping in general, yet we are able to prove termination for this extended class of games. Already other algorithms have been proposed in the literature to compute these distances, with complexity in $\textbf{UP} \cap \textbf{coUP}$ and \textbf{PPAD}. Despite the theoretical relevance, these algorithms are inefficient in practice. To the best of our knowledge, our algorithm is the first practical solution. The characterization of the probabilistic bisimilarity distance mentioned above crucially uses a dual presentation of the Hausdorff distance due to M\'emoli. As an additional contribution, in this paper we show that M\'emoli's result can be used also to prove that the bisimilarity distance bounds the difference in the maximal (or minimal) probability of two states to satisfying arbitrary $\omega$-regular properties, expressed, eg., as LTL formulas.