| Theories and practice of polymorphism |
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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 several fundamentally different kinds of polymorphism, two of which were 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 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, subtype polymorphism or 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.[1] 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. Alternatively, inclusion polymorphism may be achieved through type coercion, also known as type casting.
Interaction between parametric polymorphism and subtyping leads to the concepts of bounded quantification and covariance and contravariance (or polarity) of type constructors.
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Chris Strachey[2] 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 allows a function or a data type to be written generically, so that it can handle values identically without depending on their type.[3] 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 is often simply referred to as "polymorphism". The following example shows 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.
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:
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()); }
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