Intersection type

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 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 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.

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.

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 type, and when applied to an argument of type  returns an object of type type. Ideally, should not be applicable to any object having (possibly by chance) a  method.

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

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.

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, which can be used to implement the singleton pattern. The above trait  declares the member , which can be assigned a type as its value, and the member  , which can be assigned a value of type. The following program text defines an object  as instance of the above trait. The object  defines the type   as   and the value   as. For example, executing  prints   on the console.

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  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  has the member   that is assigned the type   as its value. Since  is a type, the object   has the type $$\langle \textsf{T} : \text{Type} \rangle$$. Additionally, the object  has the member   that is assigned the value   of type. Since the value of  is , the object   has the type $$\langle \textsf{value} : \textsf{booleanWitness.T} \rangle$$. Overall, the object  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  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.