The Semantics of Miranda's Algebraic Types

Abstract

Miranda has two interesting features in its typing system: implicit polymorphism (also known as ML-style polymorphism) and algebraic types. Algebraic types create new types from old and can operate on arbitrary types. This paper argues that functions of types, or type constructors, best represent the meaning of algebraic types. Building upon this idea, we develop a denotational semantics for algebraic types. We first define a typed lambda calculus that specifies type constructors. A semantic model of type constructors is them built, using the ideal model as a basis. (The ideal model gives the most natural semantics for Miranda's implicit polymorphism.) The model is shown to be sound with respect to this lambda calculus. FInally, we demonstrate how to use the model to interpret algebraic types, and prove that the translation produces elements in the model.