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.

行为等效被引入作为一种简单而优雅的证明方法，用于确定外部观察者是否无法区分两个过程的行为。观察者的知识通常取决于他们对过程行为的观察。此外，并发系统中不确定性和概率的组合导致对过程行为的几种解释。显然，不同类型的观察以及不同的解释会导致不同类型的行为关系，例如（双向）模拟，跟踪和测试。如果仅将注意力集中在线性特性上，则可以确定三种主要的跟踪和测试方法语义：迹线分布，逐迹线和极值概率方法。在本文中，我们提出了基于上述三种经典方法的行为度量的新颖概念，这些概念可用于度量与跟踪和测试语义相关的过程线性行为的差异。我们研究了这些度量的属性，例如组成性（以非扩展性表示），并比较了它们的表达能力。更准确地说，我们还将它们与（bi）模拟指标进行比较，从而获得第一个指标线性时间-分支时间谱。