We introduce a method for proving almost sure termination in the context of lambda calculus with continuous random sampling and explicit recursion, based on ranking supermartingales. This result is extended in three ways. Antitone ranking functions have weaker restrictions on how fast they must decrease, and are applicable to a wider range of programs. Sparse ranking functions take values only at a subset of the program's reachable states, so they are simpler to define and more flexible. Ranking functions with respect to alternative reduction strategies give yet more flexibility, and significantly increase the applicability of the ranking supermartingale approach to proving almost sure termination, thanks to a novel (restricted) confluence result which is of independent interest. The notion of antitone ranking function was inspired by similar work by McIver, Morgan, Kaminski and Katoen in the setting of a first-order imperative language, but adapted to a higher-order functional language. The sparse ranking function and confluent semantics extensions are unique to the higher-order setting. Our methods can be used to prove almost sure termination of programs that are beyond the reach of methods in the literature, including higher-order and non-affine recursion.

中文翻译：

---

超级市场，排名函数和概率Lambda演算

我们介绍了一种基于连续超级抽样和连续随机抽样以及显式递归来证明lambda演算中几乎确定终止的方法。此结果以三种方式扩展。反音等级功能对必须降低的速度有较弱的限制，并且适用于更广泛的程序。稀疏排序函数仅在程序可访问状态的子集中获取值，因此它们更易于定义且更灵活。相对于替代减少策略的排名功能还提供了更大的灵活性，并且由于具有独特利益的新颖（受限）融合结果，可以大大提高排名超级马丁方法证明几乎确定的终止的适用性。反音阶排序功能的概念是由McIver，Morgan，Kaminski和Katoen在设置一阶命令式语言时所做的类似工作启发而来的，但后来又适应了更高阶的功能语言。稀疏排序功能和融合语义扩展对于高阶设置是唯一的。我们的方法可以用来证明几乎可以肯定地终止程序，而这些程序是文献中方法所不能及的，包括高阶和非仿射递归。