Explainability of Probabilistic Bisimilarity Distances for Labelled Markov Chains

Abstract

AbstractProbabilistic bisimilarity distances measure the similarity of behaviour of states of a labelled Markov chain. The smaller the distance between two states, the more alike they behave. Their distance is zero if and only if they are probabilistic bisimilar. Recently, algorithms have been developed that can compute probabilistic bisimilarity distances for labelled Markov chains with thousands of states within seconds. However, say we compute that the distance of two states is 0.125. How does one explain that 0.125 captures the similarity of their behaviour?In this paper, we address this question by returning to the definition of probabilistic bisimilarity distances proposed by Desharnais, Gupta, Jagadeesan, and Panangaden more than two decades ago. We use a slight variation of their logic to construct for each pair of states a sequence of formulas that explains the probabilistic bisimilarity distance of the states. Furthermore, we present an algorithm that computes those formulas and we show that each formula can be computed in polynomial time.We also prove that our logic is minimal. That is, if we leave out any operator from the logic, then the resulting logic no longer provides a logical characterization of the probabilistic bisimilarity distances.