Intersection type

Short description: Concept in type theory

In type theory, an intersection type can be allocated to values that can be assigned both the type [math]\displaystyle{ \sigma }[/math] and the type [math]\displaystyle{ \tau }[/math]. This value can be given the intersection type [math]\displaystyle{ \sigma \cap \tau }[/math] in an intersection type system. Generally, if the ranges of values of two types overlap, then a value belonging to the intersection of the two ranges can be assigned the intersection type of these two types. Such a value can be safely passed as argument to functions expecting either of the two types. For example, in Java the class Boolean implements both the Serializable and the Comparable interfaces. Therefore, an object of type Boolean can be safely passed to functions expecting an argument of type Serializable and to functions expecting an argument of type Comparable.

Intersection types are composite data types. Similar to product types, they are used to assign several types to an object. However, product types are assigned to tuples, so that each tuple element is assigned a particular product type component. In comparison, underlying objects of intersection types are not necessarily composite. A restricted form of intersection types are refinement types.

Intersection types are useful for describing overloaded functions. For example, if Template:TS-lang is the type of function taking a number as an argument and returning a number, and Template:TS-lang is the type of function taking a string as an argument and returning a string, then the intersection of these two types can be used to describe (overloaded) functions that do one or the other, based on what type of input they are given.

Contemporary programming languages, including Ceylon, Flow, Java, Scala, TypeScript, and Whiley (see comparison of languages with intersection types), use intersection types to combine interface specifications and to express ad hoc polymorphism. Complementing parametric polymorphism, intersection types may be used to avoid class hierarchy pollution from cross-cutting concerns and reduce boilerplate code, as shown in the TypeScript example below.

The type theoretic study of intersection types is referred to as the intersection type discipline. Remarkably, program termination can be precisely characterized using intersection types.

TypeScript example

TypeScript supports intersection types, improving expressiveness of the type system and reducing potential class hierarchy size, demonstrated as follows.

The following program code defines the classes Template:TS-lang, Template:TS-lang, and Template:TS-lang that each have a method Template:TS-lang returning an object of either type Template:TS-lang, Template:TS-lang, or Template:TS-lang. Additionally, the functions Template:TS-lang and Template:TS-lang require arguments of type Template:TS-lang and Template:TS-lang, respectively.

class Egg { private kind: "Egg" }
class Milk { private kind: "Milk" }

// produces eggs
class Chicken { produce() { return new Egg(); } }

// produces milk
class Cow { produce() { return new Milk(); } }

// produces a random number
class RandomNumberGenerator { produce() { return Math.random(); } }

// requires an egg
function eatEgg(egg: Egg) {
    return "I ate an egg.";
}

// requires milk
function drinkMilk(milk: Milk) {
    return "I drank some milk.";
}

The following program code defines the ad hoc polymorphic function Template:TS-lang that invokes the member function Template:TS-lang of the given object Template:TS-lang. The function Template:TS-lang has two type annotations, namely Template:TS-lang and Template:TS-lang, connected via the intersection type constructor Template:TS-lang. Specifically, Template:TS-lang when applied to an argument of type Template:TS-lang returns an object of type type Template:TS-lang, and when applied to an argument of type Template:TS-lang returns an object of type type Template:TS-lang. Ideally, Template:TS-lang should not be applicable to any object having (possibly by chance) a Template:TS-lang method.

// given a chicken, produces an egg; given a cow, produces milk
let animalToFood: ((_: Chicken) => Egg) & ((_: Cow) => Milk) =
    function (animal: any) {
        return animal.produce();
    };

Finally, the following program code demonstrates type safe use of the above definitions.

var chicken = new Chicken();
var cow = new Cow();
var randomNumberGenerator = new RandomNumberGenerator();

console.log(chicken.produce()); // Egg { }
console.log(cow.produce()); // Milk { }
console.log(randomNumberGenerator.produce()); //0.2626353555444987

console.log(animalToFood(chicken)); // Egg { }
console.log(animalToFood(cow)); // Milk { }
//console.log(animalToFood(randomNumberGenerator)); // ERROR: Argument of type 'RandomNumberGenerator' is not assignable to parameter of type 'Cow'

console.log(eatEgg(animalToFood(chicken))); // I ate an egg.
//console.log(eatEgg(animalToFood(cow))); // ERROR: Argument of type 'Milk' is not assignable to parameter of type 'Egg'
console.log(drinkMilk(animalToFood(cow))); // I drank some milk.
//console.log(drinkMilk(animalToFood(chicken))); // ERROR: Argument of type 'Egg' is not assignable to parameter of type 'Milk'

The above program code has the following properties:

• Lines 1–3 create objects Template:TS-lang, Template:TS-lang, and Template:TS-lang of their respective type.
• Lines 5–7 print for the previously created objects the respective results (provided as comments) when invoking Template:TS-lang.
• Line 9 (resp. 10) demonstrates type safe use of the method Template:TS-lang applied to Template:TS-lang (resp. Template:TS-lang).
• Line 11, if uncommented, would result in a type error at compile time. Although the implementation of Template:TS-lang could invoke the Template:TS-lang method of Template:TS-lang, the type annotation of Template:TS-lang disallows it. This is in accordance with the intended meaning of Template:TS-lang.
• Line 13 (resp. 15) demonstrates that applying Template:TS-lang to Template:TS-lang (resp. Template:TS-lang) results in an object of type Template:TS-lang (resp. Template:TS-lang).
• Line 14 (resp. 16) demonstrates that applying Template:TS-lang to Template:TS-lang (resp. Template:TS-lang) does not result in an object of type Template:TS-lang (resp. Template:TS-lang). Therefore, if uncommented, line 14 (resp. 16) would result in a type error at compile time.

Comparison to inheritance

The above minimalist example can be realized using inheritance, for instance by deriving the classes Template:TS-lang and Template:TS-lang from a base class Template:TS-lang. However, in a larger setting, this could be disadvantageous. Introducing new classes into a class hierarchy is not necessarily justified for cross-cutting concerns, or maybe outright impossible, for example when using an external library. Imaginably, the above example could be extended with the following classes:

• a class Template:TS-lang that does not have a Template:TS-lang method;
• a class Template:TS-lang that has a Template:TS-lang method returning Template:TS-lang;
• a class Template:TS-lang that has a Template:TS-lang method, which can be used only once, returning Template:TS-lang.

This may require additional classes (or interfaces) specifying whether a produce method is available, whether the produce method returns food, and whether the produce method can be used repeatedly. Overall, this may pollute the class hierarchy.

Comparison to duck typing

The above minimalist example already shows that duck typing is less suited to realize the given scenario. While the class Template:TS-lang contains a Template:TS-lang method, the object Template:TS-lang should not be a valid argument for Template:TS-lang. The above example can be realized using duck typing, for instance by introducing a new field Template:TS-lang to the classes Template:TS-lang and Template:TS-lang signifying that objects of corresponding type are valid arguments for Template:TS-lang. However, this would not only increase the size of the respective classes (especially with the introduction of more methods similar to Template:TS-lang), but is also a non-local approach with respect to Template:TS-lang.

Comparison to function overloading

The above example can be realized using function overloading, for instance by implementing two methods Template:TS-lang and Template:TS-lang. In TypeScript, such a solution is almost identical to the provided example. Other programming languages, such as Java, require distinct implementations of the overloaded method. This may lead to either code duplication or boilerplate code.

Comparison to the visitor pattern

The above example can be realized using the visitor pattern. It would require each animal class to implement an Template:TS-lang method accepting an object implementing the interface Template:TS-lang (adding non-local boilerplate code). The function Template:TS-lang would be realized as the Template:TS-lang method of an implementation of Template:TS-lang. Unfortunately, the connection between the input type (Template:TS-lang or Template:TS-lang) and the result type (Template:TS-lang or Template:TS-lang) would be difficult to represent.

Limitations

On the one hand, intersection types can be used to locally annotate different types to a function without introducing new classes (or interfaces) to the class hierarchy. On the other hand, this approach requires all possible argument types and result types to be specified explicitly. If the behavior of a function can be specified precisely by either a unified interface, parametric polymorphism, or duck typing, then the verbose nature of intersection types is unfavorable. Therefore, intersection types should be considered complementary to existing specification methods.

Dependent intersection type

A dependent intersection type, denoted [math]\displaystyle{ (x : \sigma) \cap \tau }[/math], is a dependent type in which the type [math]\displaystyle{ \tau }[/math] may depend on the term variable [math]\displaystyle{ x }[/math]. In particular, if a term [math]\displaystyle{ M }[/math] has the dependent intersection type [math]\displaystyle{ (x : \sigma) \cap \tau }[/math], then the term [math]\displaystyle{ M }[/math] has both the type [math]\displaystyle{ \sigma }[/math] and the type [math]\displaystyle{ \tau[x := M] }[/math], where [math]\displaystyle{ \tau[x := M] }[/math] is the type which results from replacing all occurrences of the term variable [math]\displaystyle{ x }[/math] in [math]\displaystyle{ \tau }[/math] by the term [math]\displaystyle{ M }[/math].

Scala example

Scala supports type declarations as object members. This allows a type of an object member to depend on the value of another member, which is called a path-dependent type. For example, the following program text defines a Scala trait Witness, which can be used to implement the singleton pattern.

trait Witness {
  type T
  val value: T {}
}

The above trait Witness declares the member T, which can be assigned a type as its value, and the member value, which can be assigned a value of type T. The following program text defines an object booleanWitness as instance of the above trait Witness. The object booleanWitness defines the type T as Boolean and the value value as true. For example, executing System.out.println(booleanWitness.value) prints true on the console.

object booleanWitness extends Witness {
  type T = Boolean
  val value = true
}

Let [math]\displaystyle{ \langle \textsf{x} : \sigma \rangle }[/math] be the type (specifically, a record type) of objects having the member [math]\displaystyle{ \textsf{x} }[/math] of type [math]\displaystyle{ \sigma }[/math]. In the above example, the object booleanWitness can be assigned the dependent intersection type [math]\displaystyle{ (x : \langle \textsf{T} : \text{Type} \rangle) \cap \langle \textsf{value} : x.\textsf{T} \rangle }[/math]. The reasoning is as follows. The object booleanWitness has the member T that is assigned the type Boolean as its value. Since Boolean is a type, the object booleanWitness has the type [math]\displaystyle{ \langle \textsf{T} : \text{Type} \rangle }[/math]. Additionally, the object booleanWitness has the member value that is assigned the value true of type Boolean. Since the value of booleanWitness.T is Boolean, the object booleanWitness has the type [math]\displaystyle{ \langle \textsf{value} : \textsf{booleanWitness.T} \rangle }[/math]. Overall, the object booleanWitness has the intersection type [math]\displaystyle{ \langle \textsf{T} : \text{Type} \rangle \cap \langle \textsf{value} : \textsf{booleanWitness.T} \rangle }[/math]. Therefore, presenting self-reference as dependency, the object booleanWitness has the dependent intersection type [math]\displaystyle{ (x : \langle \textsf{T} : \text{Type} \rangle) \cap \langle \textsf{value} : x.\textsf{T} \rangle }[/math].

Alternatively, the above minimalistic example can be described using dependent record types. In comparison to dependent intersection types, dependent record types constitute a strictly more specialized type theoretic concept.

Intersection of a type family

An intersection of a type family, denoted [math]\displaystyle{ \bigcap_{x : \sigma} \tau }[/math], is a dependent type in which the type [math]\displaystyle{ \tau }[/math] may depend on the term variable [math]\displaystyle{ x }[/math]. In particular, if a term [math]\displaystyle{ M }[/math] has the type [math]\displaystyle{ \bigcap_{x : \sigma} \tau }[/math], then for each term [math]\displaystyle{ N }[/math] of type [math]\displaystyle{ \sigma }[/math], the term [math]\displaystyle{ M }[/math] has the type [math]\displaystyle{ \tau[x := N] }[/math]. This notion is also called implicit Pi type, observing that the argument [math]\displaystyle{ N }[/math] is not kept at term level.

Comparison of languages with intersection types

Language	Actively developed	Paradigm(s)	Status	Features
C#	Yes	Functional Imperative Object-oriented	Under discussion	Additionally, generic type parameters can have constraints that require their (monomorphized) type-arguments to implement multiple interfaces, whereupon the runtime type represented by the generic type parameter becomes an intersection-type of all listed interfaces.
Ceylon	No	Object-oriented	Supported	Type refinement Interface composition Subtyping in width
F#	Yes	Functional Imperative Object-oriented	Under discussion	?
Flow	Yes	Functional Imperative Object-oriented Scripting	Supported	Type refinement Interface composition
Forsythe	No	Imperative	Supported	Function type intersection Distributive, co- and contravariant function type subtyping
Java	Yes	Imperative Object-oriented	Supported	Type refinement Interface composition Subtyping in width
PHP	Yes	Functional Imperative Object-oriented Scripting	Supported	Type refinement Interface composition
Scala	Yes	Functional Imperative Object-oriented	Supported	Type refinement Trait composition Subtyping in width
TypeScript	Yes	Functional Imperative Object-oriented Scripting	Supported	Arbitrary type intersection Interface composition Subtyping in width and depth
Whiley	Yes	Functional Imperative	Supported	?