A long-standing open problem in the semantics of programming languages supporting probabilistic choice is to find a commutative monad for probability on the category DCPO. In this paper we present three such monads and a general construction for finding even more. We show how to use these monads to provide a sound and adequate denotational semantics for the Probabilistic FixPoint Calculus (PFPC) -- a call-by-value simply-typed lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. We also show that in the special case where we consider continuous dcpo's, then all three monads coincide with the valuations monad of Jones and we fully characterise the induced Eilenberg-Moore categories by showing that they are all isomorphic to the category of continuous Kegelspitzen of Keimel and Plotkin.

中文翻译：

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概率编程语言的可交换单子

支持概率选择的编程语言语义中一个长期存在的开放问题是找到类别DCPO上的概率的可交换单子。在本文中，我们介绍了三个这样的单子，以及用于发现更多单子的一般构造。我们将展示如何使用这些单子为概率定点积分（PFPC）提供合理且适当的指称语义-概率混合的递归类型，项递归和概率选择的按值调用简单型lambda演算。我们还表明，在考虑连续dcpo的特殊情况下，所有三个单子都与Jones的估值单子重合，并且通过证明它们都是与Keimel的连续Kegelspitzen的同构而完全归纳出诱导的Eilenberg-Moore类别。和普洛特金。