Revisiting Decidable Bounded Quantification, via Dinaturality

TL;DR

This paper introduces a semantic approach to unify different forms of bounded quantification in type systems, achieving decidable typechecking while maintaining expressiveness and minimal typing properties.

Contribution

It presents a semantic interpretation that combines two forms of bounded quantification, enabling a decidable and expressive type system with minimal typing.

Findings

Unified type system with both bounded quantification forms

Decidability of typechecking for the fragment over beta-normal terms

Equivalence of one fragment to Kernel Fsub and increased expressiveness over System Fsubtop

Abstract

We use a semantic interpretation to investigate the problem of defining an expressive but decidable type system with bounded quantification. Typechecking in the widely studied System Fsub is undecidable thanks to an undecidable subtyping relation, for which the culprit is the rule for subtyping bounded quantification. Weaker versions of this rule, allowing decidable subtyping, have been proposed. One of the resulting type systems (Kernel Fsub) lacks expressiveness, another (System Fsubtop) lacks the minimal typing property and thus has no evident typechecking algorithm. We consider these rules as defining distinct forms of bounded quantification, one for interpreting type variable abstraction, and the other for type instantiation. By giving a semantic interpretation for both in terms of unbounded quantification, using the dinaturality of type instantiation with respect to subsumption, we show that they can coexist within a single type system. This does have the minimal typing property and thus a simple typechecking procedure. We consider the fragments of this unified type system over types which contain only one form of bounded quantifier. One of these is equivalent to Kernel Fsub, while the other can type strictly more terms than System Fsubtop but the same set of beta-normal terms. We show decidability of typechecking for this fragment, and thus for System Fsubtop typechecking of beta-normal terms.