The metric linear-time branching-time spectrum on nondeterministic probabilistic processes

Behavioral equivalences were introduced as a simple and elegant proof methodology for establishing whether the behavior of two processes cannot be distinguished by an external observer. The knowledge of observers usually depends on the observations that they can make on process behavior. Furthermore, the combination of nondeterminism and probability in concurrent systems leads to several interpretations of process behavior. Clearly, different kinds of observations as well as different interpretations lead to different kinds of behavioral relations, such as (bi)simulations, traces and testing. If we restrict our attention to linear properties only, we can identify three main approaches to trace and testing semantics: the trace distributions, the trace-by-trace and the extremal probabilities approaches. In this paper, we propose novel notions of behavioral metrics that are based on the three classic approaches above, and that can be used to measure the disparities in the linear behavior of processes with respect to trace and testing semantics. We study the properties of these metrics, like compositionality (expressed in terms of the non-expansiveness property), and we compare their expressive powers. More precisely, we compare them also to (bi)simulation metrics, thus obtaining the first metric linear time – branching time spectrum.