In programming languages and type theory, 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 and form the basis of generic programming. 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 where denotes the type of lists with elements of type a. We say that the type of append is parameterized by a for all values of a. For each place where append is applied, a value is decided for a. Following Christopher Strachey, parametric polymorphism may be contrasted with ad hoc polymorphism, in which a single polymorphic function can have a number of distinct and potentially heterogeneous implementations depending on the type of argument to which it is applied. Thus, ad hoc polymorphism can generally only support a limited number of such distinct types, since a separate implementation has to be provided for each type.
History
Parametric polymorphism was first introduced to programming languages in ML in 1975. Today it exists in Standard ML, OCaml, F#, Ada, Haskell, Mercury, Visual Prolog, Scala, Julia, and others. Java, C#, Visual Basic.NET and Delphi have each 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 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.
Higher-ranked polymorphism
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". 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 types, which are sometimes called monotypes. A consequence is that all types can be written in a form that places all quantifiers at the outermost position. For example, consider the append function described above, which has type 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 is predicative. This is because predicativity, together with other restrictions, makes the type system simple enough that full type inference is always possible. As a practical example, OCaml performs type inference and supports impredicative polymorphism, but in some cases when impredicative polymorphism is used, the system's type inference is incomplete unless some explicit type annotations are provided by the programmer.
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 k or more arrows. 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 and impredicativity
Predicative polymorphism
In a predicative parametric polymorphic system, a type containing a type variable may not be used in such a way that is instantiated to a polymorphic type. Predicative type theories include Martin-Löf Type Theory and NuPRL.
Impredicative polymorphism
Impredicative polymorphism is the most powerful form of parametric polymorphism. A definition is said to be impredicative if it is self-referential; in type theory this allows the instantiation of a variable in a type with any type, including polymorphic types, such as itself. An example of this is the System F with the type variable X in the type, where X could even refer toT itself. In type theory, the most frequently studied impredicative typed λ-calculi are based on those of the lambda cube, especially System F.
Bounded parametric polymorphism
In 1985, Luca Cardelli and Peter Wegner recognized 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 in Haskell. In most object-oriented programming languages that support parametric polymorphism, parameters can be constrained to be subtypes of a given type.
