Classical results in computability theory, notably Rice's theorem, focus on the extensional content of programs, namely, on the partial recursive functions that programs compute. Later and more recent work investigated intensional generalisations of such results that take into account the way in which functions are computed, thus affected by the specific programs computing them. In this paper, we single out a novel class of program semantics based on abstract domains of program properties that are able to capture nonextensional aspects of program computations, such as their asymptotic complexity or logical invariants, and allow us to generalise some foundational computability results such as Rice's Theorem and Kleene's Second Recursion Theorem to these semantics. In particular, it turns out that for this class of abstract program semantics, any nontrivial abstract property is undecidable and every decidable overapproximation necessarily includes an infinite set of false positives which covers all values of the semantic abstract domain.

中文翻译：

---

抽象语义学的莱斯定理

可计算性理论的经典结果，特别是莱斯定理，关注程序的外延内容，即程序计算的部分递归函数。后来和最近的工作调查了这些结果的内涵概括，这些结果考虑了计算函数的方式，从而受到计算它们的特定程序的影响。在本文中，我们根据程序属性的抽象域挑选出一类新的程序语义，它们能够捕获程序计算的非扩展方面，例如渐近复杂性或逻辑不变量，并允许我们概括一些基本的可计算性结果，例如作为这些语义的莱斯定理和克莱恩第二递归定理。特别是，