## Bacci, Giorgio and Bacci, Giovanni and Larsen, Kim G. and Mardare, Radu and Tang, Qiyi et al. - Computing Probabilistic Bisimilarity Distances for Probabilistic Automata

lmcs:5994 - Logical Methods in Computer Science, February 2, 2021, Volume 17, Issue 1
Computing Probabilistic Bisimilarity Distances for Probabilistic Automata

Authors: Bacci, Giorgio and Bacci, Giovanni and Larsen, Kim G. and Mardare, Radu and Tang, Qiyi and van Breugel, Franck

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 Mémoli. As an additional contribution, in this paper we show that Mémoli'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.

Volume: Volume 17, Issue 1
Published on: February 2, 2021
Submitted on: December 20, 2019
Keywords: Computer Science - Formal Languages and Automata Theory,Computer Science - Logic in Computer Science