Atomic polymorphism |$\mathbf{F_{at}}$| is a restriction of Girard and Reynold’s system |$\mathbf{F} $|(or |$\lambda 2$|⁠) which was first introduced in Ferreira [ 2] in the context of a philosophical commentary on predicativism. |$\lambda 2$| is a well-known and powerful formal tool for studying polymorphic functional programming languages and formal methods in program specification and development, but its computational power far exceeds the recursive level of interest in applications. Hence, the interest of studying subsystems of |$\lambda 2$| with weaker computational power. |$\mathbf{F_{at}}$| is defined by restricting instantiation to atomic variables only. It turns out that the type system is still sufficiently powerful to possess embeddings of full intuitionistic propositional calculus [ 3, 4], and since the calculus has fewer connectives and strong normalizability is simple to prove [ 3], this result allows us to circumvent many of the extra computational complexities present when dealing with the proof theory of IPC. It is natural to inquire whether type inhabitation, i.e. provability in the corresponding fragment of second-order intuitionistic propositional logic, is decidable or not and in general to see whether the negative results involving the undecidability of type inhabitation, typability and type-checking for |$\mathbf{F}$| still hold in this fragment. A further theme would be to study the result of adding type constructors, recursors or even dependent types to |$\mathbf{F_{at}}$|⁠. In this paper, we show that type inhabitation for |$\mathbf{F_{at}}$| is undecidable by codifying within it an undecidable fragment of first-order intuitionistic predicate calculus, adapting and modifying the technique of Urzyczyn’s [ 1, 7] purely syntactic proof of the undecidability of type inhabitation for |$\mathbf{F}$|⁠.

中文翻译：

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原子多态性的类型居住是不确定的

原子多态性| $ \ mathbf {F_ {at}} $ | 是Girard和Reynold系统的限制| $ \ mathbf {F} $ | （或| $ \ lambda 2 $ |⁠），最早是在Ferreira [[2]在关于掠夺主义的哲学评论的背景下。| $ \ lambda 2 $ | 是一种众所周知的，功能强大的形式化工具，用于研究程序规范和开发中的多态函数式编程语言和形式化方法，但是其计算能力远远超出了应用程序对递归的关注程度。因此，研究| $ \ lambda 2 $ |的子系统的兴趣。计算能力较弱。| $ \ mathbf {F_ {at}} $ | 通过仅将实例化限制为原子变量来定义。事实证明，类型系统仍然足够强大，可以拥有完整的直觉命题演算的嵌入物[3， [4]，并且由于演算的结缔结较少且具有很强的可标准化性，因此很容易证明[ 3]，这个结果使我们能够避免在处理IPC的证明理论时出现的许多额外的计算复杂性。是很自然的查询类型居留问题，在第二阶直觉命题逻辑的相应片段即可证是否是可判定与否并在总体上看到涉及类型居住，typability和不可判定的阴性结果是否类型检查为| $ \ mathbf {F} $ | 仍然保留在这个片段中。另一个主题是研究将类型构造函数，递归甚至依赖类型添加到| $ \ mathbf {F_ {at}} $ |⁠中的结果。在本文中，我们显示了| $ \ mathbf {F_ {at}} $ |的类型居住。 通过将一阶直觉谓词演算的不确定片段编入其中，对Urzyczyn's [ 1， 7]纯粹的语法证明| $ \ mathbf {F} $ |⁠的类型居住不确定性。