Programming with Union, Intersection, and Negation Types

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 á 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.