Witnessing subsystems have proven to be a useful concept in the analysis of probabilistic systems, for example as diagnostic information on why a given property holds or as input to refinement algorithms. This paper introduces witnessing subsystems for reachability problems in probabilistic timed automata (PTA). Using a new operation on difference bounds matrices, it is shown how Farkas certificates of finite-state bisimulation quotients of a PTA can be translated into witnessing subsystems. We present algorithms for the computation of minimal witnessing subsystems under three notions of minimality, which capture the timed behavior from different perspectives, and discuss their complexity.

中文翻译：

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概率定时自动机的最少见证

见证子系统已被证明是概率系统分析中的一个有用概念，例如作为关于给定属性为何成立的诊断信息或作为细化算法的输入。本文介绍了概率定时自动机 (PTA) 中可达性问题的见证子系统。使用差分边界矩阵上的新操作，展示了如何将 PTA 的有限状态互模拟商的 Farkas 证书转换为见证子系统。我们提出了在三种最小性概念下计算最小见证子系统的算法，这些算法从不同的角度捕捉定时行为，并讨论它们的复杂性。