Computing reachability probabilities is a fundamental problem in the analysis of randomized programs. This article aims at a comprehensive and comparative account of various martingale-based methods for over- and under-approximating reachability probabilities. Based on the existing works that stretch across different communities (formal verification, control theory, etc.), we offer a unifying account. In particular, we emphasize the role of order-theoretic fixed points—a classic topic in computer science—in the analysis of randomized programs. This leads us to two new martingale-based techniques, too. We also make an experimental comparison using our implementation of template-based synthesis algorithms for those martingales.

中文翻译：

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随机程序中可达性的排序和排斥超级鞅

计算可达性概率是随机程序分析中的一个基本问题。本文旨在对各种基于鞅的方法用于过高和过低的可达性概率。基于跨越不同社区的现有工作（形式验证、控制理论等），我们提供了一个统一的说明。特别是，我们强调了序论不动点（计算机科学中的一个经典主题）在随机程序分析中的作用。这也将我们引向了两种新的基于鞅的技术。我们还使用我们对这些鞅的基于模板的合成算法的实现进行了实验比较。