Type inhabitation of atomic polymorphism is undecidable

Abstract

Abstract 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 $\mathbf{F}$.