This paper studies the linear time-branching time spectrum of equivalences for a subclass of interactive Markov chains (IMCs) that are not subject to any further synchronization. This subclass of IMCs is known as closed IMCs. We define several variants of trace equivalence by performing button pushing experiments on stochastic trace machines. We establish the relation between these equivalences and also compare them with bisimulation for closed IMCs. Next, we define several variants of stutter trace equivalence for closed IMCs. We perform button pushing experiments with stutter insensitive stochastic trace machines to obtain these equivalences. We investigate the relationship among these stutter trace equivalences and also compare them with weak bisimulation for closed IMCs. We discuss the relation between several strong and weak trace equivalences defined in this paper. Next, we use these results to sketch the linear time-branching time spectrum of equivalences for IMCs. Finally, we prove that maximum and minimum probabilities of satisfying properties specified using metric temporal logic (MTL) formulas are preserved under some of these trace and stutter trace equivalences.

本文研究了交互式马尔可夫链（IMC）的子类的等价线性时间分支时间谱，该子类无需进一步同步。IMC的此子类称为封闭IMC。通过在随机跟踪计算机上执行按钮按下实验，我们定义了跟踪等效项的几种变体。我们建立了这些等价关系，并将它们与封闭IMC的双仿真进行比较。接下来，我们为封闭的IMC定义了几种等效的口吃痕迹。我们使用口吃不敏感的随机跟踪机器执行按钮按下实验，以获取这些等效项。我们研究了这些口吃痕迹等效项之间的关系，并将它们与封闭IMC的弱双仿真进行了比较。我们讨论了本文定义的几种强和弱迹线当量之间的关系。接下来，我们将使用这些结果来绘制IMC等效项的线性时间分支时间谱。最后，我们证明了使用度量时间逻辑（MTL）公式指定的满足属性的最大和最小概率在这些迹线和口吃迹线等效项中得以保留。