Elaborating intersection and union types

Abstract

Designing and implementing typed programming languages is hard. Every new type system feature requires extending the metatheory and implementation, which are often complicated and fragile. To ease this process, we would like to provide general mechanisms that subsume many different features.In modern type systems, parametric polymorphism is fundamental, but intersection polymorphism has gained little traction in programming languages. Most practical intersection type systems have supported only refinement intersections, which increase the expressiveness of types (more precise properties can be checked) without altering the expressiveness of terms; refinement intersections can simply be erased during compilation. In contrast, unrestricted intersections increase the expressiveness of terms, and can be used to encode diverse language features, promising an economy of both theory and implementation.We describe a foundation for compiling unrestricted intersection and union types: an elaboration type system that generates ordinary λ-calculus terms. The key feature is a Forsythe-like merge construct. With this construct, not all reductions of the source program preserve types; however, we prove that ordinary call-byvalue evaluation of the elaborated program corresponds to a typepreserving evaluation of the source program.We also describe a prototype implementation and applications of unrestricted intersections and unions: records, operator overloading, and simulating dynamic typing.

Elaborating evaluation-order polymorphism

“…This idea, developed by Barendregt et al (1983), can be automated; see, for example, Dunfield (2014).…”

“…The D intersection in this paper mixes features of general intersection and refinement intersection: the V and N instantiations have close-to-identical structure, but cbv and cbn functions aren't refinements of some "order-agnostic" base type. Our approach is descended mainly from the system of Dunfield (2014), which elaborates (general) intersection and union types into ordinary product and sum types. We differ in not having a source-level 'merge' construct e 1 ,, e 2 , where the type system can select either e 1 or e 2 , ignoring the other component.…”

“…Existential quantification. By analogy to union types (Dunfield 2014), an existential quantifier would elaborate to a sum type. For example, the sum tag on a function of type ∃a.…”

“…Datatype defini- tions, expressed as recursive/sum types, can also be polymorphic in evaluation order; for example, operations on binary search trees can be written just once. Much of the theory in this paper follows smoothly from existing work on intersection types, particularly Dunfield (2014). However, since we only consider intersections equivalent to the quantified type Da.…”

“…A, our intersected types have parametric structure: they differ only in the evaluation orders decorating the connectives. This limitation, a cousin of the refinement restriction in datasort refinement systems (Freeman and Pfenning 1991;Davies 2005), avoids the need for a merge construct (Reynolds 1996;Dunfield 2014) and the issues that arise from it.…”