We consider the almost-sure (a.s.) termination problem for probabilistic programs, which are a stochastic extension of classical imperative programs. Lexicographic ranking functions provide a sound and practical approach for termination of non-probabilistic programs, and their extension to probabilistic programs is achieved via lexicographic ranking supermartingales (LexRSMs). However, LexRSMs introduced in the previous work have a limitation that impedes their automation: all of their components have to be non-negative in all reachable states. This might result in a LexRSM not existing even for simple terminating programs. Our contributions are twofold. First, we introduce a generalization of LexRSMs that allows for some components to be negative. This standard feature of non-probabilistic termination proofs was hitherto not known to be sound in the probabilistic setting, as the soundness proof requires a careful analysis of the underlying stochastic process. Second, we present polynomial-time algorithms using our generalized LexRSMs for proving a.s. termination in broad classes of linear-arithmetic programs.

我们考虑概率程序的几乎确定（as）终止问题，概率程序是经典命令式程序的随机扩展。词典排序函数为终止非概率程序提供了一种合理且实用的方法，并且它们对概率程序的扩展是通过词典排序超级鞅（LexRSM）实现的。然而，之前的工作中引入的 LexRSM 有一个限制，阻碍了它们的自动化：它们的所有组件在所有可达状态下都必须是非负的。这可能会导致 LexRSM 不存在，即使对于简单的终止程序也是如此。我们的贡献是双重的。首先，我们引入 LexRSM 的泛化，允许某些分量为负。迄今为止，非概率终止证明的这一标准特征在概率设置中并不健全，因为健全性证明需要仔细分析潜在的随机过程。其次，我们提出了使用广义 LexRSM 的多项式时间算法，用于在广泛的线性算术程序中证明终止。