Environmental bisimulations for probabilistic higher-order languages are studied. In contrast with applicative bisimulations, environmental bisimulations are known to be more robust and do not require sophisticated techniques such as Howe’s in the proofs of congruence. As representative calculi, call-by-name and call-by-value λ-calculus, and a (call-by-value) λ-calculus extended with references (i.e., a store) are considered. In each case, full abstraction results are derived for probabilistic environmental similarity and bisimilarity with respect to contextual preorder and contextual equivalence, respectively. Some possible enhancements of the (bi)simulations, as “up-to techniques,” are also presented. Probabilities force a number of modifications to the definition of environmental bisimulations in non-probabilistic languages. Some of these modifications are specific to probabilities, others may be seen as general refinements of environmental bisimulations, applicable also to non-probabilistic languages. Several examples are presented, to illustrate the modifications and the differences.

中文翻译：

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概率高阶语言的环境互模拟

研究了概率高阶语言的环境互模拟。与应用性双模拟相比，已知环境双模拟更健壮，并且在一致性证明中不需要像 Howe 那样的复杂技术。作为代表演算，考虑按名称调用和按值调用λ演算，以及用引用扩展的（按值调用）λ演算（即，存储）。在每种情况下，分别针对上下文预序和上下文等价的概率环境相似性和双相似性得出完整的抽象结果。还介绍了（双）模拟的一些可能的增强，作为“最新技术”。概率迫使对非概率语言中环境互模拟的定义进行一些修改。这些修改中的一些是特定于概率的，其他的可能被视为环境互模拟的一般改进，也适用于非概率语言。提供了几个示例，以说明修改和差异。