Set-Theoretic Types for Polymorphic Variants

Set-Theoretic Types for Polymorphic Variants by Giuseppe Castagna, Tommaso Petrucciani, and Kim Nguyễn:

Polymorphic variants are a useful feature of the OCaml language whose current definition and implementation rely on kinding constraints to simulate a subtyping relation via unification. This yields an awkward formalization and results in a type system whose behaviour is in some cases unintuitive and/or unduly restrictive.

In this work, we present an alternative formalization of polymorphic variants, based on set-theoretic types and subtyping, that yields a cleaner and more streamlined system. Our formalization is more expressive than the current one (it types more programs while preserving type safety), it can internalize some meta-theoretic properties, and it removes some pathological cases of the current implementation resulting in a more intuitive and, thus, predictable type system. More generally, this work shows how to add full-fledged union types to functional languages of the ML family that usually rely on the Hindley-Milner type system. As an aside, our system also improves the theory of semantic subtyping, notably by proving completeness for the type reconstruction algorithm.

Looks like a nice result. They integrate union types and restricted intersection types for complete type inference, which prior work on CDuce could not do. The disadvantage is that it does not admit principal types, and so inference is non-deterministic (see section 5.3.2).

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Thanks. I read over (somewhat quickly) but didn't come away with a good intuition about the limitations of this approach or what it feels like. They gave three examples of Ocaml not working well in the introduction, and I was hoping to see more examples of their system working well at the end.

At some point, shouldn't we consider an intricate type system like this as part of a general theorem proving tool and not expect nice completeness results?