Thoughts on Intersection Types

Facebook’s Flow is a static type checker, designed to find type errors in JavaScript programs. Flow’s type system is similar to that of TypeScript in that it is a gradual type system and relies heavily on type inference to find type errors.

One interesting feature of Flow that is missing in TypeScript is intersection types. For example, in Flow you can represent a function which takes either Bar or Foo with intersection types.

/* @flow */
class Foo {}
class Bar {}
declare var f: ((x: Foo) => void) & ((x: Bar) => void);
f(new Foo());
f(true); // Flow will error here.

The function f has type ((x: Foo) => void) & ((x: Bar) => void). The notation A & B represents the intersection type of A and B, meaning a value of A & B belongs to both A and B. We can capture this intuition with the following sub-typing rules:

[1] A & B <: A
[2] A & B <: B
[3] S <: A, S <: B ~> S <: A & B

In Flow, intersection types are primarily used to support finitary overloading. func takes either number or string and returns nothing. This can be represented as intersection of two function types as in the following:

type A = (t: number) => void & (t: string) => void
var func : A;

However, TypeScript doesn’t need intersection types to support finitary overloading because it directly supports function overloading.

interface A {
  (t: number): void;
  (t: string): void;
}
var func: A

Or in TypeScript 1.4, you can represent the same function more succinctly with union types.

interface A {
  (t: number | string): void;
}
var func: A

This is not a coincidence! Intersection types are the formal dual of union types. When an arrow is distributed over a union, the union changes to an intersection.

(A -> T) & (B -> T)  <:  (A | B) -> T

So it means the following two types are equivalent:

  • (t: number | string): void
  • (t: number) => void & (t: string) => void

So far intersection types seem redundant once a language have union types (or vice versa). However, there are some real world use cases where intersection type are actually required.

Let’s check how ES6’s new function Object.assign is typed in TypeScript (lib.core.es6.d.ts).

interface ObjectConstructor {
    /**
      * Copy the values of all of the enumerable own properties from one or more source objects to a 
      * target object. Returns the target object.
      * @param target The target object to copy to.
      * @param sources One or more source objects to copy properties from.
      */
    assign(target: any, ...sources: any[]): any;
    ...
}

Currently, both the argument types and the return type of Object.assign is any because the type system of TypeScript can’t represent the type correctly. Let’s try to type this function correctly. First, we can make this function polymorphic by assigning A to target and B[] to sources.

Object.assign<A,B>(target: A, ...sources: B[]): ?;

Okay. We are now stuck with the return type. The return value has all the properties of both A and B. It means with structural typing, the return value is a subtype of both ‘A’ and ‘B’ (belongs to both A and B). Yeah! We need intersection types to represent this value. So the correct signature of this function is:

Object.assign<A, B>(target: A, ...sources: B[]): A & B;

The same reasoning also applies to mixins.

mixins<A,B>(base: { new() :A }, b: B}) : { new(): A & B} 

TypeScript developers are still discussing on adding intersection types. Many people think that there are not enough compelling use cases. Also intersecting two primitive types like string & number makes no sense, which makes intersection types less desirable.

I think adding intersection types to TypeScript makes the language more consistent because these two concepts are the dual of each other and can’t really be separable. But I also think that the language design must be conservative. So let’s just wait until we have more compelling reasons to add intersection types.