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.

由于Segala和Lynch，概率双相似性是一个等价关系，可以捕获概率自动机的哪些状态行为完全相同。邓等。提出了对概率双相似性的鲁棒定量概括。他们的概率自动机状态的概率双相似性距离捕获了他们行为的相似性。距离越小，状态表现越相似。特别是，状态仅在其距离为零时才是概率双相似的。尽管已经研究了为概率自动机计算概率双相似距离的复杂性，但我们尚不知道任何实用的算法来计算这些距离。在本文中，我们针对算法计算概率自动机的概率双相似距离提供了一些关键结果。特别是，我们提出了一种确定距离一的多项式时间算法。此外，我们给出了概率双相似距离的替代特征作为策略迭代算法的基础。