Revisiting Decidable Bounded Quantification, via Dinaturality

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.