Deciding probabilistic bisimilarity distance one for probabilistic automata

Probabilistic bisimilarity, due to Segala and Lynch, is an equivalence relation that captures which states of a probabilistic automaton behave exactly the same. Deng et al. proposed a robust quantitative generalization of probabilistic bisimilarity. Their probabilistic bisimilarity distances of states of a probabilistic automaton capture 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. Although the complexity of computing probabilistic bisimilarity distances for probabilistic automata has already been studied, we are not aware of any practical algorithms to compute those distances. In this paper, we provide several key results towards algorithms to compute probabilistic bisimilarity distances for probabilistic automata. In particular, we present a polynomial time algorithm that decides distance one. Furthermore, we give an alternative characterization of the probabilistic bisimilarity distances as a basis for a policy iteration algorithm.