nLab polymorphism

This page is about the concept of polymorphism of declarations in computer science. For polymorphism of type universes see instead at universe polymorphism. For the generalisation of morphisms introduced by Shinichi Mochizuki, see poly-morphism.

Idea

In computer science, polymorphism refers to situations either where the same name is used to refer to more than one function, or where the same function is used at more than one type. One usually distinguishes these two kinds of polymorphism as ad hoc polymorphism and parametric polymorphism, respectively.

Ad hoc polymorphism

In ad hoc polymorphism, one simply defines multiple functions with the same name and different types, relying on the compiler (or, in some cases, the run-time system) to determine the correct function to call based on the types of its arguments and return value. This is also called overloading. For instance, using a mathematical notation, one might define functions

$$add : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$

$$add : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$$

and then when $add(3,2)$ is invoked, the compiler knows to call the first function since $3$ and $2$ are natural numbers, whereas when $add(4.2,\pi)$ is invoked it calls the second function since $4.2$ and $\pi$ are real numbers.

Note that there is nothing which stipulates that the behavior of a class of ad-hocly polymorphic functions with the same name should be at all similar. Nothing prevents us from defining $add : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ to add its arguments but $add : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ to subtract its arguments. Of course, it is good programming practice to make overloaded functions similar in their behavior.

In the example above, there might even be a coercion function $c : \mathbb{N} \to \mathbb{R}$, to be invoked whenever a natural number appears where the compiler expects a real number, giving a commutative diagram

$$\array { \mathbb{N} \times \mathbb{N} & \overset{add}\to & \mathbb{N} \\ \mathllap{c \times c} \downarrow & & \downarrow \mathrlap{c} \\ \mathbb{R} \times \mathbb{R} & \underset{add}\to & \mathbb{R} }$$

But things don't always work out this way.

Parametric polymorphism

In parametric polymorphism or parametricity, one writes code to define a function once, which contains a “type variable” that can be instantiated at many different types to produce different functions. For instance, we can define a function

$$first : A\times A \to A$$

where $A$ is a type variable (or parameter), by

$$first(x,y) \coloneqq x.$$

Now the compiler automatically instantiates a copy of this function, with identical code, for any type at which it is called. Thus we can behave as if we had functions

$$first : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$

$$first : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$$

and so on, for any types we wish. In contrast to ad hoc polymorphism, in this case we do have a guarantee that all these same-named functions are doing “the same thing”, because they are all instantiated by the same original polymorphic code.

In a dependently typed programming language with a type of types, such as Coq or Agda, a parametrically polymorphic family of functions can simply be considered to be a single dependently typed function whose first argument is a type. Thus our function above would be typed as

$$first : \prod_{A:Type} A\times A \to A$$

However, parametric polymorphism makes sense and is very useful even in languages with less rich type systems, such as Haskell and Standard ML?.

Theorems for Free

In type systems with parametric polymorphism, such as System F, there is a family of proofs for every polymorphic type signature which each give an algebraic law for any function with that signature. For example, the type signature $A \times B \to A$ has the free theorem?

$$\alpha \circ f = f \circ (\alpha \times \beta)$$

for all functions $f$ with that signature (Wadler 89, figure 1).

Related concepts

• logical relations

• impredicative polymorphism

• parametric dependent type theory

• dependent type theory with type variables

References

The distinction between ad hoc and parametric polymorphism is originally due to Christopher Strachey:

• Christopher Strachey, Fundamental concepts in programming languages, Lecture Notes from August 1967 (International Summer School in Computer Programming, Copenhagen), later published in Higher-Order and Symbolic Computation, 13, 11-49, 2000. (pdf)

Other classic papers include:

• John C. Reynolds, Towards a Theory of Type Structure, Lecture Notes in Computer Science 19, 408-425, 1974. (pdf)

• Robin Milner, A Theory of Type Polymorphism in Programming, Journal of Computer and System Sciences 17, 348-375, 1978. (pdf)

• John C. Reynolds, Types, Abstraction, and Parametric Polymorphism, Information Processing 83, 1983. (pdf)

Theorems for free were explained in:

• Philip Wadler, Theorems for free!, in Proceedings of the fourth international conference on Functional programming languages and computer architecture (FPCA 1989). 1989. (pdf, doi:10.1145/99370.99404)

See also the references at parametric dependent type theory for the various notions of parametricity in various kinds of dependent type theories.