We give a formal treatment of simple type theories, such as the simply-typed $\lambda$-calculus, using the framework of abstract clones. Abstract clones traditionally describe first-order structures, but by equipping them with additional algebraic structure, one can further axiomatize second-order, variable-binding operators. This provides a syntax-independent representation of simple type theories. We describe multisorted second-order presentations, such as the presentation of the simply-typed $\lambda$-calculus, and their clone-theoretic algebras; free algebras on clones abstractly describe the syntax of simple type theories quotiented by equations such as $\beta$- and $\eta$-equality. We give a construction of free algebras and derive a corresponding induction principle, which facilitates syntax-independent proofs of properties such as adequacy and normalization for simple type theories. Working only with clones avoids some of the complexities inherent in presheaf-based frameworks for abstract syntax.

中文翻译：

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抽象语法的抽象克隆

我们使用抽象克隆的框架对简单类型理论（例如简单类型的$ \ lambda $-演算）进行形式化处理。传统上，抽象克隆描述的是一阶结构，但是通过为它们配备其他代数结构，可以进一步公理化二阶变量绑定算子。这提供了简单类型理论的与语法无关的表示。我们描述了多种二阶表示形式，例如简单类型的$ \ lambda $演算及其克隆理论代数的表示。克隆上的免费代数抽象地描述了由诸如$ \ beta $-和$ \ eta $ -equality等式所引用的简单类型理论的语法。我们给出自由代数的构造，并推导相应的归纳原理，这有助于简化与语法无关的属性证明，例如简单类型理论的充分性和规范化。仅使用克隆可以避免基于presheaf的抽象语法框架固有的复杂性。