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 combinations, 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.

Polymorphism in (early-bound) strongly-typed languages

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Parametric polymorphism is a way to make a language more expressive, while still maintaining full static type-safety. Using parametric polymorphism, a function or a data type can be written generically so that it can handle values identically without depending on their type. Such functions and data types are called generic functions and generic datatypes respectively.

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For example, a function append that joins two lists can be constructed so that it does not care about the type of elements: it can append lists of integers, lists of real numbers, lists of strings, and so on. Let the type variable a denote the type of elements in the lists. Then append can be typed [a] × [a] → [a], where [a] denotes a list of elements of type a. We say that the type of append is parameterized by a for all values of a. (Note that since there is only one type variable, the function cannot be applied to just any pair of lists: the pair, as well as the result list, must consist of the same type of elements.) For each place where append is applied, a value is decided for a.

Parametric polymorphism was first introduced to programming languages in ML in 1976. Today it exists in Standard ML, OCaml, Ada, Haskell, Visual Prolog and others. Java, C#, Visual Basic .NET and Delphi (CodeGear) have each recently introduced "generics" for parametric polymorphism. Some implementations of type polymorphism are superficially similar to parametric polymorphism while also introducing ad-hoc aspects. One example is C++ template specialization.

The most general form of polymorphism is "higher-rank impredicative polymorphism". Two popular restrictions of this form are restricted rank polymorphism (for example, rank-1 or prenex polymorphism) and predicative polymorphism. Together, these restrictions give "predicative prenex polymorphism", which is essentially the form of polymorphism found in ML and early versions of Haskell.

Rank restrictions

Rank-1 (prenex) polymorphism

In a prenex polymorphic system, type variables may not be instantiated with polymorphic types. This is very similar to what is called "ML-style" or "Let-polymorphism" (technically ML's Let-polymorphism has a few other syntactic restrictions).

This restriction makes the distinction between polymorphic and non-polymorphic types very important; thus in predicative systems polymorphic types are sometimes referred to as type schemas to distinguish them from ordinary (monomorphic) types, which are sometimes called monotypes. A consequence is that all types can be written in a form which places all quantifiers at the outermost (prenex) position.

For example, consider the append function described above, which has type [a] × [a] → [a]; in order to apply this function to a pair of lists, a type must be substituted for the variable a in the type of the function such that the type of the arguments matches up with the resulting function type. In an impredicative system, the type being substituted may be any type whatsoever, including a type that is itself polymorphic; thus append can be applied to pairs of lists with elements of any type—even to lists of polymorphic functions such as append itself.

Polymorphism in the language ML and its close relatives is predicative. This is because predicativity, together with other restrictions, makes the type system simple enough that type inference is possible. In languages where explicit type annotations are necessary when applying a polymorphic function, the predicativity restriction is less important; thus these languages are generally impredicative. Haskell manages to achieve type inference without predicativity but with a few complications.

Rank-k polymorphism

For some fixed value k, rank-k polymorphism is a system in which a quantifier may not appear to the left of more than k arrows (when the type is drawn as a tree).

Type inference for rank-2 polymorphism is decidable, but reconstruction for rank-3 and above is not.

Rank-n ("higher-rank") polymorphism

Rank-n polymorphism is polymorphism in which quantifiers may appear to the left of arbitrarily many arrows.

Predicativity restrictions

Predicative polymorphism

In a predicative parametric polymorphic system, a type $\tau$ containing a type variable $\alpha$ may not be used in such a way that $\alpha$ is instantiated to a polymorphic type.

Impredicative polymorphism ("first class" polymorphism)

Also called first-class polymorphism. Impredicative polymorphism allows the instantiation of a variable in a type $\tau$ with any type, including polymorphic types, such as $\tau$ itself.

In type theory, the most frequently studied impredicative typed λ-calculi are based on those of the lambda cube, especially System F. Predicative type theories include Martin-Löf Type Theory and NuPRL.

Bounded parametric polymorphism

Cardelli and Wegner recognized in 1985 the advantages of allowing bounds on the type parameters. Many operations require some knowledge of the data types but can otherwise work parametrically. For example, to check whether an item is included in a list, we need to compare the items for equality. In Standard ML, type parameters of the form ’’a are restricted so that the equality operation is available, thus the function would have the type ’’a × ’’a list → bool and ’’a can only be a type with defined equality. In Haskell, bounding is achieved by requiring types to belong to a type class; thus the same function has the type ${\scriptstyle Eq\,\alpha \,\Rightarrow \alpha \,\rightarrow \left[\alpha \right]\rightarrow Bool}$ in Haskell. In most object-oriented programming languages that support parametric polymorphism, parameters can be constrained to be subtypes of a given type (see Subtyping polymorphism below and the article on Generic programming).

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.

Ad-hoc polymorphism for early bound languages

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). 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.

Ad-hoc polymorphism is a dispatch mechanism: control moving through one named function is dispatched to various other functions without having to specify the exact function being called. Overloading allows multiple functions taking different types to be defined with the same name; the compiler or interpreter automatically calls the right one. This way, functions appending lists of integers, lists of strings, lists of real numbers, and so on could be written, and all be called append—and the right append function would be called based on the type of lists being appended. This differs from parametric polymorphism, in which the function would need to be written generically, to work with any kind of list. Using overloading, it is possible to have a function perform two completely different things based on the type of input passed to it; this is not possible with parametric polymorphism. Another way to look at overloading is that a routine is uniquely identified not by its name, but by the combination of its name and the number, order and types of its parameters.

This type of polymorphism is common in object-oriented programming languages, many of which allow operators to be overloaded in a manner similar to functions (see operator overloading). Some languages which are not dynamically typed and lack ad-hoc polymorphism (including type classes) have longer function names such as print_int, print_string, etc. This can be seen as advantage (more descriptive) or a disadvantage (overly verbose) depending on one's point of view.

An advantage that is sometimes gained from overloading is the appearance of specialization, e.g., a function with the same name can be implemented in multiple different ways, each optimized for the particular data types that it operates on. This can provide a convenient interface for code that needs to be specialized to multiple situations for performance reasons.

Since overloading is done at compile time, it is not a substitute for late binding as found in subtyping polymorphism.

Ad-hoc polymorphism for late bound languages

The previous section notwithstanding, there are other ways in which ad-hoc polymorphism can work out. Consider for example the Smalltalk language. In Smalltalk, the overloading is done at run time because the methods ("function implementation") for each overloaded message ("overloaded function") are resolved when they are about to be executed. This happens at run time, after the program is compiled. Therefore, polymorphism is given by subtyping polymorphism as in other languages, and it is also extended in functionality by ad-hoc polymorphism at run time.

A closer look will also reveal that Smalltalk provides a slightly different variety of ad-hoc polymorphism. Since Smalltalk has a late bound execution model, and since it provides objects the ability to handle messages which are not understood, it is possible to go ahead and implement functionality using polymorphism without explicitly overloading a particular message. This may not be generally recommended practice for everyday programming, but it can be quite useful when implementing proxies.

Also, while in general terms common class method and constructor overloading is not considered polymorphism, there are more uniform languages in which classes are regular objects. In Smalltalk, for instance, classes are regular objects. In turn, this means messages sent to classes can be overloaded, and it is also possible to create objects that behave like classes without their classes inheriting from the hierarchy of classes. These are effective techniques which can be used to take advantage of Smalltalk's powerful reflection capabilities. Similar arrangements are also possible in languages such as Self and Newspeak.

Example

Imagine an operator + that may be used in the following ways:

1. 1 + 2 = 3
2. 3.14 + 0.0015 = 3.1415
3. 1 + 3.7 = 4.7
4. [1, 2, 3] + [4, 5, 6] = [1, 2, 3, 4, 5, 6]
5. [true, false] + [false, true] = [true, false, false, true]
6. "sat" + "ish" = "satish"

Overloading

To handle these six function calls, four different pieces of code are needed—or three, if strings are considered to be lists of characters:

• In the first case, integer addition must be invoked.
• In the second and third cases, floating-point addition must be invoked (with type promotion, or type coercion, in the third case).
• In the fourth and fifth cases, list concatenation must be invoked.
• In the last case, string concatenation must be invoked, unless this too is handled as list concatenation (e.g., Haskell).

Thus, the name + actually refers to three or four completely different functions. This is an example of overloading.

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)