Although computational complexity is a fundamental aspect of program behavior, it is often at odds with common type theoretic principles such as function extensionality, which identifies all functions with the same $\textit{input-output}$ behavior. We present a computational type theory called $\mathbf{CATT}$ that has a primitive notion of cost (the number of evaluation steps). We introduce a new dependent function type "funtime" whose semantics can be viewed as a cost-aware version of function extensionality. We prove a collection of lemmas for $\mathbf{CATT}$, including a novel introduction rule for the new funtime type. $\mathbf{CATT}$ can be simultaneously viewed as a framework for analyzing computational complexity of programs and as the beginnings of a semantic foundation for characterizing feasible mathematical proofs.

中文翻译：

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成本意识类型理论

尽管计算复杂度是程序行为的一个基本方面，但它通常与常见的类型理论原则不一致，例如函数扩展性，后者标识所有具有相同 $\textit{input-output}$ 行为的函数。我们提出了一个名为 $\mathbf{CATT}$ 的计算类型理论，它具有成本的原始概念（评估步骤的数量）。我们引入了一种新的依赖函数类型“funtime”，其语义可以被视为函数扩展性的成本感知版本。我们证明了 $\mathbf{CATT}$ 的一系列引理，包括新的 funtime 类型的新颖介绍规则。$\mathbf{CATT}$ 可以同时被视为分析程序计算复杂性的框架和表征可行数学证明的语义基础的开端。