Intersection type

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In type theory, an intersection type can be allocated to values that can be assigned both the type [math]\sigma[/math] and the type [math]\tau[/math]. This value can be given the intersection type [math]\sigma \cap \tau[/math] in an intersection type system.[1] 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.[2] 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.[3] Remarkably, program termination can be precisely characterized using intersection types.[4]

TypeScript example

TypeScript supports intersection types,[5] 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 function Template:TS-lang (resp. Template:TS-lang) requires an object of type Template:TS-lang (resp. Template:TS-lang) as argument.

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:

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:

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](x : \sigma) \cap \tau[/math], is a dependent type in which the type [math]\tau[/math] may depend on the term variable [math]x[/math].[6] In particular, if a term [math]M[/math] has the dependent intersection type [math](x : \sigma) \cap \tau[/math], then the term [math]M[/math] has both the type [math]\sigma[/math] and the type [math]\tau[x := M][/math], where [math]\tau[x := M][/math] is the type which results from replacing all occurrences of the term variable [math]x[/math] in [math]\tau[/math] by the term [math]M[/math].

Scala example

Scala supports type declarations [7] 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.[8] For example, the following program text defines a Scala trait Witness, which can be used to implement the singleton pattern.[9]

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 programm 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]\langle \textsf{x} : \sigma \rangle[/math] be the type (specifically, a record type) of objects having the member [math]\textsf{x}[/math] of type [math]\sigma[/math]. In the above example, the object booleanWitness can be assigned the dependent intersection type [math](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]\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]\langle \textsf{value} : \textsf{booleanWitness.T} \rangle[/math]. Overall, the object booleanWitness has the intersection type [math]\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](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.[10] In comparison to dependent intersection types, dependent record types constitute a strictly more specialized type theoretic concept.[6]

Intersection of a type family

An intersection of a type family, denoted [math]\bigcap_{x : \sigma} \tau[/math], is a dependent type in which the type [math]\tau[/math] may depend on the term variable [math]x[/math].[6] In particular, if a term [math]M[/math] has the type [math]\bigcap_{x : \sigma} \tau[/math], then for each term [math]N[/math] of type [math]\sigma[/math], the term [math]M[/math] has the type [math]\tau[x := N][/math]. This notion is also called implicit Pi type,[11] observing that the argument [math]N[/math] is not kept at term level.

Comparison of languages with intersection types

Language Actively developed Paradigm(s) Status Features
C# Yes[12] Under discussion[13] ?
Ceylon Yes[14] Supported[15]
  • Type refinement
  • Interface composition
  • Subtyping in width
F# Yes[16] Under discussion[17] ?
Flow Yes[18] Supported[19]
  • Type refinement
  • Interface composition
Forsythe No Supported[20]
  • Function type intersection
  • Distributive, co- and contravariant function type subtyping
Java Yes[21] Supported[22]
  • Type refinement
  • Interface composition
  • Subtyping in width
Scala Yes[23] Supported[24][25]
  • Type refinement
  • Trait composition
  • Subtyping in width
TypeScript Yes[26] Supported[5]
  • Arbitrary type intersection
  • Interface composition
  • Subtyping in width and depth
Whiley Yes[27] Supported[28] ?

References

  1. Barendregt, Henk; Coppo, Mario; Dezani-Ciancaglini, Mariangiola (1983). "A filter lambda model and the completeness of type assignment". Journal of Symbolic Logic 48 (4): 931–940. doi:10.2307/2273659. 
  2. Palsberg, Jens (2012). "Overloading is NP-Complete". Logic and Program Semantics. Lecture Notes in Computer Science. 7230. pp. 204–218. doi:10.1007/978-3-642-29485-3_13. ISBN 978-3-642-29484-6. 
  3. Henk Barendregt; Wil Dekkers; Richard Statman (20 June 2013). Lambda Calculus with Types. Cambridge University Press. pp. 1–. ISBN 978-0-521-76614-2. https://books.google.com/books?id=2UVasvrhXl8C&pg=PR1. 
  4. Ghilezan, Silvia (1996). "Strong normalization and typability with intersection types". Notre Dame Journal of Formal Logic 37 (1): 44–52. doi:10.1305/ndjfl/1040067315. 
  5. 5.0 5.1 "Intersection Types in TypeScript". https://www.typescriptlang.org/docs/handbook/advanced-types.html#intersection-types. Retrieved 2019-08-01. 
  6. 6.0 6.1 6.2 Kopylov, Alexei (2003). "Dependent intersection: A new way of defining records in type theory". LICS 2003. IEEE Computer Society. pp. 86–95. doi:10.1109/LICS.2003.1210048. 
  7. "Type declarations in Scala". https://www.scala-lang.org/files/archive/spec/2.12/04-basic-declarations-and-definitions.html#type-declarations-and-type-aliases. 
  8. Amin, Nada; Grütter, Samuel; Odersky, Martin; Rompf, Tiark; Stucki, Sandro (2016). "The essence of dependent object types". 9600. Springer. 249–272. doi:10.1007/978-3-319-30936-1_14. 
  9. "Singletons in the Scala shapeless library". https://github.com/milessabin/shapeless/blob/master/core/src/main/scala/shapeless/singletons.scala. 
  10. Pollack, Robert (2000). "Dependently typed records for representing mathematical structure". TPHOLs 2000. Springer. pp. 462–479. doi:10.1007/3-540-44659-1_29. 
  11. Stump, Aaron (2018). "From realizability to induction via dependent intersection". Annals of Pure and Applied Logic 169 (7): 637–655. doi:10.1016/j.apal.2018.03.002. 
  12. "C# Guide". http://csharp.net/. Retrieved 2019-08-08. 
  13. "Discussion: Union and Intersection types in C Sharp". https://github.com/dotnet/csharplang/issues/399. Retrieved 2019-08-08. 
  14. "Eclipse Ceylon: Welcom to Ceylon". https://ceylon-lang.org/. Retrieved 2019-08-08. 
  15. "Intersection Types in Ceylon". https://ceylon-lang.org/documentation/tour/types/#intersection_types. Retrieved 2019-08-08. 
  16. "F# Software Foundation". http://fsharp.org/. Retrieved 2019-08-08. 
  17. "Add Intersection Types to F Sharp". https://github.com/fsharp/fslang-suggestions/issues/600. Retrieved 2019-08-08. 
  18. "Flow: A Static Type Checker for JavaScript". https://flow.org/en/. Retrieved 2019-08-08. 
  19. "Intersection Type Syntax in Flow". https://flow.org/en/docs/types/intersections/#intersection-type-syntax. Retrieved 2019-08-08. 
  20. Reynolds, J. C. (1988). Preliminary design of the programming language Forsythe.
  21. "Java Software". https://www.oracle.com/java/. Retrieved 2019-08-08. 
  22. "IntersectionType (Java SE 12 & JDK 12)". https://docs.oracle.com/en/java/javase/12/docs/api/java.compiler/javax/lang/model/type/IntersectionType.html. Retrieved 2019-08-01. 
  23. "The Scala Programming Language". https://scala-lang.org/. Retrieved 2019-08-08. 
  24. "Compound Types in Scala". https://www.scala-lang.org/files/archive/spec/2.12/03-types.html#compound-types. Retrieved 2019-08-01. 
  25. "Intersection Types in Dotty". http://dotty.epfl.ch/docs/reference/new-types/intersection-types.html. Retrieved 2019-08-01. 
  26. "TypeScript - JavaScript that scales.". https://www.typescriptlang.org/. Retrieved 2019-08-01. 
  27. "Whiley: an Open Source Programming Language with Extended Static Checking". http://whiley.org/. Retrieved 2019-08-01. 
  28. "Whiley language specification". http://whiley.org/download/WhileyLanguageSpec.pdf. Retrieved 2019-08-01. 






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