We study a cost-aware programming language for higher-order recursion dubbed $\textbf{PCF}_\mathsf{cost}$ in the setting of synthetic domain theory (SDT). Our main contribution relates the denotational cost semantics of $\textbf{PCF}_\mathsf{cost}$ to its computational cost semantics, a new kind of dynamic semantics for program execution that serves as a mathematically natural alternative to operational semantics in SDT. In particular we prove an internal, cost-sensitive version of Plotkin's computational adequacy theorem, giving a precise correspondence between the denotational and computational semantics for complete programs at base type. The constructions and proofs of this paper take place in the internal dependent type theory of an SDT topos extended by a phase distinction in the sense of Sterling and Harper. By controlling the interpretation of cost structure via the phase distinction in the denotational semantics, we show that $\textbf{PCF}_\mathsf{cost}$ programs also evince a noninterference property of cost and behavior. We verify the axioms of the type theory by means of a model construction based on relative sheaf models of SDT.

中文翻译：

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合成域理论中高阶递归的成本敏感计算充分性

我们在合成域理论（SDT）的背景下研究了一种用于高阶递归的成本感知编程语言，称为 $\textbf{PCF}_\mathsf{cost}$。我们的主要贡献将 $\textbf{PCF}_\mathsf{cost}$ 的指称成本语义与其计算成本语义联系起来，这是一种用于程序执行的新型动态语义，可作为 SDT 中操作语义的数学自然替代方案。特别是，我们证明了 Plotkin 计算充分性定理的内部、成本敏感版本，给出了基本类型的完整程序的指称语义和计算语义之间的精确对应。本文的构造和证明发生在 SDT 拓扑的内部依赖类型理论中，该拓扑通过 Sterling 和 Harper 意义上的相位区分进行扩展。通过指称语义中的阶段区别来控制成本结构的解释，我们表明 $\textbf{PCF}_\mathsf{cost}$ 程序也表现出成本和行为的互不干扰属性。我们通过基于SDT相关层模型的模型构建来验证类型论的公理。