A Gradual Interpretation of Union Types

Union types allow to capture the possibility of a term to be of several possibly unrelated types. Traditional static approaches to union types are untagged and tagged unions, which present dual advantages in their use. Inspired by recent work on using abstract interpretation to understand gradual typing, we present a novel design for union types, called gradual union types. Gradual union types combine the advantages of tagged and untagged union types, backed by dynamic checks. Seen as a gradual typing discipline, gradual union types are restricted imprecise types that denote a finite number of static types. We apply the Abstracting Gradual Typing (AGT) methodology of Garcia et al. to derive the static and dynamic semantics of a language that supports both gradual unions and the traditional, totally-unknown type. We uncover that gradual unions interact with the unknown type in a way that mandates a stratified approach to AGT, relying on a composition of two distinct abstract interpretations in order to retain optimality. Thanks to the abstract interpretation framework, the resulting language is type safe and satisfies the refined criteria for gradual languages. We also show how to compile such a language to a threesome cast calculus, and prove that the compilation preserves the semantics and properties of the language.