On Behavioral Metric for Probabilistic Systems: Definition and Approximation Algorithm

In this paper, we consider the behavioral pseu- dometrics for probabilistic systems. The model we are interested in is probabilistic automata, which are based on state transition systems and make a clear distinction between probabilistic and nondeterministic choices. The pseudometrics are defined as the greatest fixpoint of a monotonic functional on the complete lat- tice of state metrics. A distinguished characteristic of this pseudometric lies in that it does not discount the future, which addresses some algorithmic challenges to compute the distance of two states in the model. We solve this problem by providing an approximation al- gorithm: up to any desired degree of accuracy , the distance can be approximated to within in time ex- ponential in the size of the model and logarithmic in 1 . A key ingredient of our algorithm is to express a pseudometric being a post-fixpoint as the elementary sentence over real closed fields, which allows us to ex- ploit Tarski's decision procedure, together with binary search to approximate the behavioral distance.