A Decidable Subtyping Logic for Intersection and Union Types

Abstract

Using Curry-Howard isomorphism, we extend the typed lambda-calculus with intersection and union types, and its corresponding proof-functional logic, previously defined by the authors, with subtyping and explicit coercions. We show the extension of the lambda-calculus to be isomorphic to the Barbanera-Dezani-de'Liguoro type assignment system and we provide a sound interpretation of the proof-functional logic with the NJ(β) logic, using Mints' realizers. We finally present a sound and complete algorithm for subtyping in presence of intersection and union types. The algorithm is conceived to work for the (sub)type theory Ξ.

Empowering union and intersection types with integrated subtyping

“…Furthermore, essentially the same proof of the requirements applies to the variant of InOut in which L is allowed to be empty. This variant implements the additional extension ⊤ <: ⊤ → ⊤, which was postulated by Barbanera, Dezani-Ciancaglini, and de'Liguoro [1995], and which Liquori and Stolze [2017] recently proved decidable. Thus in all of the above variations on extensions, subtyping has already been proven decidable.…”

The Delta-framework

“…The ∆-framework Kinds In LF ∆ we can discuss also ad hoc polymorphism, while in the Implicit Calculus only structural polymorphism is encoded. Indeed, he cannot assign the type ((σ ∩ τ ) → σ) ∩ (ρ → ρ)) to the identity λx.x [28]. Kopylov [27] adds a dependent intersection type constructor x:A ∩ B[x] to NuPRL, allowing the resulting system to support dependent records (which are a very useful data structure to encode mathematics).…”

“…Intersection and union type disciplines started to be investigated in a explicitly typed programming language settings à la Church, much later by Reynolds and Pierce [41,38], Wells et al [48,49], Liquori et al [29,18], Frisch et al [21] and Dunfield [19]. From a logical point of view, there are many proposals to find a suitable logics to fit intersection: among them we cite [37,47,42,36,11,10,39], and previous papers by the authors [17,30]…”

Towards a Logical Framework with Intersection and Union Types

“…Of course, this process could require some modifications and corrections to the essence definition and/or rules of the typing system currently implemented. • To integrate in the ∆-framework the minimal (sub)type theory Ξ, as described in [2], to and to add an explicit coercion expression (τ )∆ of type τ if ⊢ ∆ : σ and σ ⩽ τ , as described in [19]; this will raise the ∆-framework to the full power of intersection and union types. • To extend the subtyping algorithm A, for intersection and union types, as introduced in [19], with a dependent type theory: this algorithm (presented in functional style) is conceived to work for the minimal type theory Ξ (i.e.…”

“…Adding a subtyping relation and a suitable subtype type theory is subject of future work and partially introduced in[19].…”