Subtype

In programming language theory, subtyping (also subtype polymorphism or inclusion polymorphism) is a form of type polymorphism in which a subtype is a datatype that is related to another datatype (the supertype) by some notion of substitutability, meaning that program elements, typically subroutines or functions, written to operate on elements of the supertype can also operate on elements of the subtype. If S is a subtype of T, the subtyping relation is often written S <: T, to mean that any term of type S can be safely used in a context where a term of type T is expected. The precise semantics of subtyping crucially depends on the particulars of what "safely used in a context where" means in a given programming language. The type system of a programming language essentially defines its own subtyping relation, which may well be trivial.

Due to the subtyping relation, a term may belong to more than one type. Subtyping is therefore a form of type polymorphism. In object-oriented programming the term 'polymorphism' is commonly used to refer solely to this subtype polymorphism, while the techniques of parametric polymorphism would be considered generic programming.

Functional programming languages often allow the subtyping of records. Consequently, simply typed lambda calculus extended with record types is perhaps the simplest theoretical setting in which a useful notion of subtyping may be defined and studied. Because the resulting calculus allows terms to have more than one type, it is no longer a "simple" type theory. Since functional programming languages, by definition, support function literals, which can also be stored in records, records types with subtyping provide some of the features of object-oriented programming. Typically, functional programming languages also provide some, usually restricted, form of parametric polymorphism. In a theoretical setting, it is desirable to study the interaction of the two features; a common theoretical setting is system F_<:. Various calculi that attempt to capture the theoretical properties of object-oriented programming may be derived from system F_<:.

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