Programming with union, intersection, and negation types

TL;DR

This paper advocates for set-theoretic types, including union, intersection, and negation, highlighting their necessity and advantages in programming language design and type safety.

Contribution

It introduces a theoretical framework for set-theoretic types, extends semantic subtyping with polymorphism, and proposes practical language restrictions for implementation.

Findings

Set-theoretic types are essential for precise typing of programming patterns.

Three effective language restrictions enable practical implementation of set-theoretic types.

The framework supports advanced features like pattern matching, gradual typing, and denotational semantics.

Abstract

In this essay, I present the advantages and, I dare say, the beauty of programming in a language with set-theoretic types, that is, types that include union, intersection, and negation type connectives. I show by several examples how set-theoretic types are necessary to type some common programming patterns, but also how they play a key role in typing several language constructs-from branching and pattern matching to function overloading and type-cases-very precisely. I start by presenting the theory of types known as semantic subtyping and extend it to include polymorphic types. Next, I discuss the design of languages that use these types. I start by defining a theoretical framework that covers all the examples given in the first part of the presentation. Since the system of the framework cannot be effectively implemented, I then describe three effective restrictions of this system: (i) a polymorphic language with explicitly-typed functions, (ii) an implicitly-typed polymorphic language \`a la Hindley-Milner, and (iii) a monomorphic language that, by implementing classic union-elimination, precisely reconstructs intersection types for functions and implements a very general form of occurrence typing. I conclude the presentation with a short overview of other aspects of these languages, such as pattern matching, gradual typing, and denotational semantics.