Intersection type

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Data type for values having two types

title: "Intersection type" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["type-theory", "type-systems", "data-types", "composite-data-types", "polymorphism-(computer-science)", "typescript"] description: "Data type for values having two types" topic_path: "technology/computing" source: "https://en.wikipedia.org/wiki/Intersection_type" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Data type for values having two types ::

In type theory, an intersection type can be allocated to values that can be assigned both the type \sigma and the type \tau. This value can be given the intersection type \sigma \cap \tau 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 implements both the and the interfaces. Therefore, an object of type can be safely passed to functions expecting an argument of type and to functions expecting an argument of type .

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 is the type of function taking a number as an argument and returning a number, and 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. Consequently, type inference for infinite-intersection types is undecidable, but it is decidable for all finite rank 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 , , and that each have a method returning an object of either type , , or . Additionally, the functions and require arguments of type and , respectively.

::code[lang=typescript] class Egg { private kind: "Egg"; }

class Milk { private kind: "Milk"; }

// "Produces" a product interface Producer { produce(): T; }

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

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

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

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

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

The following program code defines the ad hoc polymorphic function that invokes the member function of the given object . The function has two type annotations, namely and , connected via the intersection type constructor . Specifically, when applied to an argument of type returns an object of type , and when applied to an argument of type returns an object of type . Ideally, should not be applicable to any object having (possibly by chance) a method. ::code[lang=typescript] // given a chicken, produces an egg; given a cow, produces milk const animalToFood: ((animal: Chicken) => Egg) & ((animal: Cow) => Milk) = (animal: any) => { return animal.produce(); }; ::

Finally, the following program code demonstrates type safe use of the above definitions. ::code[lang=typescript] let chicken: Chicken = new Chicken(); let cow: Cow = new Cow(); let randomNumberGenerator: 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 , , and of their respective type.
  • Lines 5–7 print for the previously created objects the respective results (provided as comments) when invoking .
  • Line 9 (resp. 10) demonstrates type safe use of the method applied to (resp. ).
  • Line 11, if uncommented, would result in a type error at compile time. Although the implementation of could invoke the method of , the type annotation of disallows it. This is in accordance with the intended meaning of .
  • Line 13 (resp. 15) demonstrates that applying to (resp. ) results in an object of type (resp. ).
  • Line 14 (resp. 16) demonstrates that applying to (resp. ) does not result in an object of type (resp. ). 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 and from a base class . 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 that does not have a method;
  • a class that has a method returning ;
  • a class that has a method, which can be used only once, returning . 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 contains a method, the object should not be a valid argument for . The above example can be realized using duck typing, for instance by introducing a new field to the classes and signifying that objects of corresponding type are valid arguments for . However, this would not only increase the size of the respective classes (especially with the introduction of more methods similar to ), but is also a non-local approach with respect to .

Comparison to function overloading

The above example can be realized using function overloading, for instance by implementing two methods and . 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 method accepting an object implementing the interface (adding non-local boilerplate code). The function would be realized as the method of an implementation of . Unfortunately, the connection between the input type ( or ) and the result type ( or ) would be difficult to represent.

Comparison with parametric polymorphism

Parametric polymorphism is conceptually equivalent to infinite intersection types.

Intersection types have been promoted as being "compositional" in contrast with the let-bound polymorphism of ML (a restricted form of parametric polymorphism) because intersection types have principal typings while ML's type system does not (not to be confused with principal types, which ML does enjoy). The lack of principal typings in ML translates into the need to evaluate some expressions before others in an ML program; essentially resulting in a data dependency at type-inference level in ML, in particular in let expressions. Consequently, intersection types have been proposed as a way to improve incremental compilation and/or gradual typing.{{cite book | chapter = Gradual Typing with Union and Intersection Types | first1 = Giuseppe | last1= Castagna | first2 = Victor | last2= Lanvin | title = ICFP 2017 }}

On the other hand, intersection types have been criticized for not being compositional in another sense, namely that in a putative system that uses only intersection types but has no parametric polymorphism, inferred may types depend on the local features of a module, which may compose badly with other modules unless whole program compilation happens at source level. As a trivial example, an identity function that is exposed through a public interface, e.g. exported by a module, ideally is parametrically polymorphic, so that it can be used with future types that the writer (of that function) doesn't yet know about. However, in a system that only has intersection types, such a function would be inferred to intersect at best over types existing when it is compiled.

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

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. ::code[lang=Scala] 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. ::code[lang=Scala] object booleanWitness extends Witness { type T = Boolean val value = true } ::

Let \langle \textsf{x} : \sigma \rangle be the type (specifically, a record type) of objects having the member \textsf{x} of type \sigma. In the above example, the object booleanWitness can be assigned the dependent intersection type (x : \langle \textsf{T} : \text{Type} \rangle) \cap \langle \textsf{value} : x.\textsf{T} \rangle. 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 \langle \textsf{T} : \text{Type} \rangle. 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 \langle \textsf{value} : \textsf{booleanWitness.T} \rangle. Overall, the object booleanWitness has the intersection type \langle \textsf{T} : \text{Type} \rangle \cap \langle \textsf{value} : \textsf{booleanWitness.T} \rangle. Therefore, presenting self-reference as dependency, the object booleanWitness has the dependent intersection type (x : \langle \textsf{T} : \text{Type} \rangle) \cap \langle \textsf{value} : x.\textsf{T} \rangle.

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

Comparison of languages with intersection types

::data[format=table]

LanguageActively developedParadigm(s)StatusFeatures
C#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
F#
Flow
Forsythe
Java
PHP
Scala
TypeScript
Whiley
::

References

References

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  4. (1996). "Strong normalization and typability with intersection types". Notre Dame Journal of Formal Logic.
  5. (January 2004). "Principality and type inference for intersection types using expansion variables". Theoretical Computer Science.
  6. Giuseppe Castagna, Programming with Union, Intersection, and Negation Types, https://arxiv.org/pdf/2111.03354 p. 7 Also chapter 12 in "The French School of Programming", Springer 2024, ISBN 978-3-031-34517-3
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