Typing Classes and Mixins with Intersection Types

TL;DR

This paper develops an intersection type system for lambda-calculus with records and merge operators, enabling natural typing of mixins and classes without recursive types, and adapts these concepts to Java and C#.

Contribution

It introduces a novel intersection type system that handles mixins and classes with recursive records without recursive or F-bounded types, and demonstrates adaptation to mainstream languages.

Findings

Types are preserved under subject reduction and expansion.

Mixins and classes can be meaningfully typed without recursive types.

Code snippets in Java and C# are shown to be typable within the system.

Abstract

We study an assignment system of intersection types for a lambda-calculus with records and a record-merge operator, where types are preserved both under subject reduction and expansion. The calculus is expressive enough to naturally represent mixins as functions over recursively defined classes, whose fixed points, the objects, are recursive records. In spite of the double recursion that is involved in their definition, classes and mixins can be meaningfully typed without resorting to neither recursive nor F-bounded polymorphic types. We then adapt mixin construct and composition to Java and C#, relying solely on existing features in such a way that the resulting code remains typable in the respective type systems. We exhibit some example code, and study its typings in the intersection type system via interpretation into the lambda-calculus with records we have proposed.