The Behavior of Probabilistic Systems: From Equivalences to Behavioral Distances

In this paper we synthesize our recent work on behavioral distances for probabilistic systems and present an overview of the current state of the art in the field. We mainly focus on behavioral distances for Markov chains, Markov decision processes, and Segala systems. We illustrate three different methods used for the definition of such metrics: logical, order theoretic, and measure-testing; and we discuss the relationships between them and provide the main arguments in support of each of them. We also overview the problem of computing such distances, both from a theoretical and a practical view point, including the exact and the ap-proximated methods.