Affine type systems are substructural type systems where copying of information is restricted, but discarding of information is permissible at all types. Such type systems are well-suited for describing quantum programming languages, because copying of quantum information violates the laws of quantum mechanics. In this paper, we consider a first-order affine type system with inductive data types and present a novel categorical semantics for it. The most challenging aspect of this interpretation comes from the requirement to construct appropriate discarding maps for our data types which might be defined by mutual/nested recursion. We show how to achieve this for all types by taking models of a first-order linear type system whose atomic types are discardable and then presenting an additional affine interpretation of types within the slice category of the model with the tensor unit. We present some concrete categorical models for the language ranging from classical to quantum. Finally, we discuss potential ways of dualising and extending our methods and using them for interpreting coalgebraic and lazy data types.

中文翻译：

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通过切片类别的一阶仿射归纳数据类型的语义

仿射类型系统是子结构类型系统，其中信息的复制受到限制，但在所有类型中都允许丢弃信息。这种类型系统非常适合描述量子编程语言，因为复制量子信息违反了量子力学定律。在本文中，我们考虑了具有归纳数据类型的一阶仿射类型系统，并为其提供了一种新颖的分类语义。这种解释最具挑战性的方面来自需要为我们的数据类型构建适当的丢弃映射，这些映射可能由相互/嵌套递归定义。我们展示了如何通过采用原子类型可丢弃的一阶线性类型系统的模型来展示如何为所有类型实现这一点，然后使用张量单元对模型的切片类别内的类型进行额外的仿射解释。我们为从经典到量子的语言提供了一些具体的分类模型。最后，我们讨论了对我们的方法进行二元化和扩展以及使用它们来解释代数和惰性数据类型的潜在方法。