Unified Syntax with Iso-types

Abstract

Traditional designs for functional languages (such as Haskell or ML) have separate sorts of syntax for terms and types. In contrast, many dependently typed languages use a unified syntax that accounts for both terms and types. Unified syntax has some interesting advantages over separate syntax, including less duplication of concepts, and added expressiveness. However, integrating unrestricted general recursion in calculi with unified syntax is challenging when some level of type-level computation is present, as decidable type-checking is easily lost. This paper argues that the advantages of unified syntax also apply to traditional functional languages, and there is no need to give up decidable type-checking. We present a dependently typed calculus that uses unified syntax, supports general recursion and has decidable type-checking. The key to retain decidable type-checking is a generalization of iso-recursive types called iso-types. Iso-types replace the conversion rule typically used in dependently typed calculus, and make every computation explicit via cast operators. We study two variants of the calculus that differ on the reduction strategy employed by the cast operators, and give different trade-offs in terms of simplicity and expressiveness.