In [ABM07], Abdulla et al. introduced the concept of decisiveness, an interesting tool for lifting good properties of finite Markov chains to denumerable ones. Later, this concept was extended to more general stochastic transition systems (STSs), allowing the design of various verification algorithms for large classes of (infinite) STSs. We further improve the understanding and utility of decisiveness in two ways. First, we provide a general criterion for proving decisiveness of general STSs. This criterion, which is very natural but whose proof is rather technical, (strictly) generalizes all known criteria from the literature. Second, we focus on stochastic hybrid systems (SHSs), a stochastic extension of hybrid systems. We establish the decisiveness of a large class of SHSs and, under a few classical hypotheses from mathematical logic, we show how to decide reachability problems in this class, even though they are undecidable for general SHSs. This provides a decidable stochastic extension of o-minimal hybrid systems. [ABM07] Parosh A. Abdulla, Noomene Ben Henda, and Richard Mayr. 2007. Decisive Markov Chains. Log. Methods Comput. Sci. 3, 4 (2007).

中文翻译：

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随机系统的决定性及其在混合模型中的应用

在 [ABM07] 中，Abdulla 等人。引入了决定性的概念，这是一个有趣的工具，用于将有限马尔可夫链的良好属性提升为可数链。后来，这个概念被扩展到更一般的随机转换系统 (STS)，允许为大类（无限）STS 设计各种验证算法。我们通过两种方式进一步提高了对果断性的理解和实用性。首先，我们提供了证明一般 STS 决定性的一般标准。这个标准非常自然，但其证明相当技术性，（严格地）概括了文献中所有已知的标准。其次，我们关注随机混合系统（SHS），这是混合系统的随机扩展。我们建立了一大类 SHS 的决定性，并且根据数理逻辑的一些经典假设，我们展示了如何在这个类中决定可达性问题，即使它们对于一般的 SHS 是不可判定的。这提供了 o 最小混合系统的可判定随机扩展。[ABM07] Parosh A. Abdulla、Noomene Ben Henda 和 Richard Mayr。2007. 决定性马尔可夫链。日志。方法计算。科学。3, 4 (2007)。