What's Decidable about (Atomic) Polymorphism

TL;DR

This paper explores the decidability of type-checking and related properties in the weak fragment of System F called atomic System F, revealing decidability of type-checking but undecidability of contextual equivalence.

Contribution

It introduces atomic System F, proves type-checking is decidable in this fragment, and analyzes the decidability of free theorems and contextual equivalence.

Findings

Type-checking in atomic System F is decidable.

Contextual equivalence is undecidable in atomic System F.

The source of undecidability in full System F is better understood through this fragment.

Abstract

Due to the undecidability of most type-related properties of System F like type inhabitation or type checking, restricted polymorphic systems have been widely investigated (the most well-known being ML-polymorphism). In this paper we investigate System Fat, or atomic System F, a very weak predicative fragment of System F whose typable terms coincide with the simply typable ones. We show that the type-checking problem for Fat is decidable and we propose an algorithm which sheds some new light on the source of undecidability in full System F. Moreover, we investigate free theorems and contextual equivalence in this fragment, and we show that the latter, unlike in the simply typed lambda-calculus, is undecidable.