One of the major advantages of model checking over other formal methods is its ability to generate a counterexample when a model does not satisfy is its specification. A counterexample is an error trace that helps to locate the source of the error. Therefore, the counterexample represents a valuable tool for debugging. In Probabilistic Model Checking (PMC), the task of counterexample generation has a quantitative aspect. Unlike the previous methods proposed for conventional model checking that generate the counterexample as a single path ending with a bad state representing the failure, the task in PMC is completely different. A counterexample in PMC is a set of evidences or diagnostic paths that satisfy a path formula, whose probability mass violates the probability threshold.

Counterexample generation is not sufficient for finding the exact source of the error. Therefore, in conventional model checking, many debugging techniques have been proposed to act on the counterexamples generated to locate the source of the error. In PMC, debugging counterexamples is more challenging, since the probabilistic counterexample consists of multiple paths and it is probabilistic. In this article, we propose a debugging technique based on stochastic games to analyze probabilistic counterexamples generated for probabilistic models described as Markov chains in PRISM language. The technique is based mainly on the idea of considering the modules composing the system as players of a reachability game, whose actions contribute to the evolution of the game. Through many case studies, we will show that our technique is very effective for systems employing multiple components. The results are also validated by introducing a debugging tool called GEPCX (Game Explainer of Probabilistic Counterexamples).

与其他形式化方法相比，模型检查的主要优点之一是它能够在模型不满足规范时生成反例。一个反例是有助于定位错误源的错误跟踪。因此，反例代表了一个有价值的调试工具。在概率模型检查 (PMC) 中，反例生成任务具有定量方面。与之前为传统模型检查提出的方法不同，这些方法将反例生成为以表示失败的不良状态结束的单个路径，PMC 中的任务完全不同。PMC 中的一个反例是一组满足路径公式的证据或诊断路径，其概率质量违反概率阈值。

反例生成不足以找到错误的确切来源。因此，在传统的模型检查中，已经提出了许多调试技术来作用于生成的反例以定位错误的根源。在 PMC 中，调试反例更具挑战性，因为概率反例由多条路径组成，并且是概率性的。在本文中，我们提出了一种基于随机博弈的调试技术，以分析为 PRISM 语言中描述为马尔可夫链的概率模型生成的概率反例。该技术主要基于将构成系统的模块视为可达性游戏的玩家的想法，其行为有助于游戏的发展。通过大量案例研究，我们将证明我们的技术对于使用多个组件的系统非常有效。还通过引入称为 GEPCX（概率反例的游戏解释器）的调试工具来验证结果。