Polymorphism (computer science)

This article is about the programming language theory concepts with direct application to functional programming languages. For a gentler introduction of these notions as commonly implemented in object-oriented programming, see Polymorphism in object-oriented programming.

In computer science, polymorphism is a programming language feature that allows values of different data types to be handled using a uniform interface. The concept of parametric polymorphism applies to both data types and functions. A function that can evaluate to or be applied to values of different types is known as a polymorphic function. A data type that can appear to be of a generalized type (e.g., a list with elements of arbitrary type) is designated polymorphic data type like the generalized type from which such specializations are made.

There are two fundamentally different kinds of polymorphism, originally informally described by Christopher Strachey in 1967. If the function denotes different and potentially heterogeneous implementations depending on a limited range of individually specified types and combination, it is called ad-hoc polymorphism. Ad-hoc polymorphism is supported in many languages using function and method overloading.

If all code is written without mention of any specific type and thus can be used transparently with any number of new types, it is called parametric polymorphism. John C. Reynolds (and later Jean-Yves Girard) formally developed this notion of polymorphism as an extension to the lambda calculus (called the polymorphic lambda calculus, or System F). Parametric polymorphism is widely supported in statically typed functional programming languages. In the object-oriented programming community, programming using parametric polymorphism is often called generic programming.

In object-oriented programming, inclusion polymorphism is a concept in type theory wherein a name may denote instances of many different classes as long as they are related by some common super class. Inclusion polymorphism is generally supported through subtyping, i.e., objects of different types are entirely substitutable for objects of another type (their base type(s)) and thus can be handled via a common interface. Alternately, inclusion polymorphism may be achieved through type coercion, also known as type casting.

Forms of polymorphism

Ad-hoc polymorphism

Main article: Ad-hoc polymorphism

Chris Strachey chose the term ad-hoc polymorphism to refer to polymorphic functions which can be applied to arguments of different types, but which behave differently depending on the type of the argument to which they are applied (also known as function overloading or operator overloading). The term "ad hoc" in this context is not intended to be pejorative; it refers simply to the fact that this type of polymorphism is not a fundamental feature of the type system. In the example below, the Add functions seems to work generically over various types when looking at the invocations, but are considered to be two entirely distinct functions by the compiler for all intents and purposes:

program Adhoc;

function Add( x, y : Integer ) : Integer;
begin
    Add := x + y
end;

function Add( s, t : String ) : String;
begin
    Add := Concat( s, t )
end;

begin
    Writeln(Add(1, 2));
    Writeln(Add('Hello, ', 'World!'));
end.

In dynamically typed languages the situation can be more complex as the correct function that needs to be invoked might only be determinable at run time.

Parametric polymorphism

Main article: Parametric polymorphism

Parametric polymorphism allows a function or a data type to be written generically, so that it can handle values identically without depending on their type. Parametric polymorphism is a way to make a language more expressive, while still maintaining full static type-safety.

Parametric polymorphism is ubiquitous in functional programming, where it often simply referred to as "polymorphism". The follow example show a parametrized list data type and two parametrically polymorphic functions on them:

data List a = Nil | Cons a (List a)

length :: List a -> Integer
length Nil         = 0
length (Cons x xs) = 1 + length xs

map :: (a -> b) -> List a -> List b
map f Nil         = Nil
map f (Cons x xs) = Cons (f x) (map f xs)

Parametric polymorphism is also available in several object-oriented languages, where it goes under the name "generics":

class List<T> {
    class Node<T> {
        T       elem;
        Node<T> next;
    }
    Node<T> head;
    int length() { ... }
}

List<B> map(Func<A,B> f, List<A> xs) {
    ...
}

Any parametrically polymorph function is necessarily restricted in what it can do, working on the shape of the data instead of its value, leading to the concept of parametricity.

Subtyping polymorphism (or inclusion polymorphism)

Main article: Subtype polymorphism

Some languages employ the idea of subtypes to restrict the range of types that can be used in a particular case of parametric polymorphism. In these languages, subtyping polymorphism (sometimes referred to as dynamic polymorphism) allows a function to be written to take an object of a certain type T, but also work correctly if passed an object that belongs to a type S that is a subtype of T (according to the Liskov substitution principle). This type relation is sometimes written S <: T. Conversely, T is said to be a supertype of S—written T :> S.

For example, if Number, Rational, and Integer are types such that Number :> Rational and Number :> Integer, a function written to take a Number will work equally well when passed an Integer or Rational as when passed a Number. The actual type of the object can be hidden from clients into a black box, and accessed via object identity. In fact, if the Number type is abstract, it may not even be possible to get your hands on an object whose most-derived type is Number (see abstract data type, abstract class). This particular kind of type hierarchy is known—especially in the context of the Scheme programming language—as a numerical tower, and usually contains many more types.

Object-oriented programming languages offer subtyping polymorphism using subclassing (also known as inheritance). In typical implementations, each class contains what is called a virtual table—a table of functions that implement the polymorphic part of the class interface—and each object contains a pointer to the "vtable" of its class, which is then consulted whenever a polymorphic method is called. This mechanism is an example of:

• late binding, because virtual function calls are not bound until the time of invocation, and
• single dispatch (i.e., single-argument polymorphism), because virtual function calls are bound simply by looking through the vtable provided by the first argument (the this object), so the runtime types of the other arguments are completely irrelevant.

The same goes for most other popular object systems. Some, however, such as CLOS, provide multiple dispatch, under which method calls are polymorphic in all arguments.

In the following example we make cats and dogs subtypes of animals. The procedure write accepts an animal, but will also work correctly if a subtype is passed to it:

abstract class Animal {
    String talk();
}

class Cat extends Animal {
    String talk() { return "Meow!"; }
}

class Dog extends Animal {
    String talk() { return "Woof!"; }
}

static void write(Animal a) {
    System.out.println(a.talk());
}

static void main() {
    write(new Cat());
    write(new Dog());
}

See also

• Polymorphism in object-oriented programming
• Duck typing for polymorphism without (static) types
• Polymorphic code (Computer virus terminology)
• System F for a lambda calculus with parametric polymorphism.
• Virtual inheritance

References

• Luca Cardelli, Peter Wegner. On Understanding Types, Data Abstraction, and Polymorphism, from Computing Surveys, (December, 1985)
• Philip Wadler, Stephen Blott. How to make ad-hoc polymorphism less ad hoc, from Proc. 16th ACM Symposium on Principles of Programming Languages, (January, 1989)
• Christopher Strachey. Fundamental Concepts in Programming Languages, from Higher-Order and Symbolic Computation, (April, 2000; original paper is dated 1967)
• Paul Hudak, John Peterson, Joseph Fasel. A Gentle Introduction to Haskell Version 98.
• Booch, et All. Object-Oriented Analysis and Design with Application.

External links

• C++ examples of polymorphism
• Objects and Polymorphism (Visual Prolog)