programming language A programming language is a system of notation for writing computer programs. Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually def ...

s and

type theory In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of ...

, a product of ''types'' is another, compounded, type in a structure. The "operands" of the product are

type Type may refer to: Science and technology Computing * Typing, producing text via a keyboard, typewriter, etc. * Data type, collection of values used for computations. * File type * TYPE (DOS command), a command to display contents of a file. * ...

s, and the structure of a product type is determined by the fixed order of the operands in the product. An instance of a product type retains the fixed order, but otherwise may contain all possible instances of its

primitive data type In computer science, primitive data types are a set of basic data types from which all other data types are constructed. Specifically it often refers to the limited set of data representations in use by a particular processor, which all compiled ...

s. The expression of an instance of a product type will be a

tuple In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...

, and is called a "tuple type" of expression. A product of types is a direct product of two or more types. If there are only two component types, it can be called a "pair type". For example, if two component types A and B are the set of all possible values of that type, the product type written A × B contains elements that are pairs (a,b), where "a" and "b" are instances of A and B respectively. The pair type is a special case of the dependent pair type, where the type B may depend on the instance picked from A. In many languages, product types take the form of a record type, for which the components of a tuple can be accessed by

label A label (as distinct from signage) is a piece of paper, plastic film, cloth, metal, or other material affixed to a container or product. Labels are most often affixed to packaging and containers using an adhesive, or sewing when affix ...

. In languages that have

algebraic data type In computer programming, especially functional programming and type theory, an algebraic data type (ADT) is a kind of composite data type, i.e., a data type formed by combining other types. Two common classes of algebraic types are product ty ...

s, as in most

functional programming In computer science, functional programming is a programming paradigm where programs are constructed by Function application, applying and Function composition (computer science), composing Function (computer science), functions. It is a declarat ...

languages, algebraic data types with one constructor are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

to a product type. In the

Curry–Howard correspondence In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs. It is also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-p ...

, product types are associated with

logical conjunction In logic, mathematics and linguistics, ''and'' (\wedge) is the Truth function, truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or ...

(AND) in

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

. The notion directly extends to the product of an arbitrary finite number of types (an ''n''-ary product type), and in this case, it characterizes the expressions that behave as tuples of expressions of the corresponding types. A degenerate form of product type is the

unit type In the area of mathematical logic and computer science known as type theory, a unit type is a type that allows only one value (and thus can hold no information). The carrier (underlying set) associated with a unit type can be any singleton set. ...

: it is the product of no types. In

call-by-value In a programming language, an evaluation strategy is a set of rules for evaluating expressions. The term is often used to refer to the more specific notion of a ''parameter-passing strategy'' that defines the kind of value that is passed to the ...

programming languages, a product type can be interpreted as a set of pairs whose first component is a value in the first type and whose second component is a value in the second type. In short, it is a

cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

and it corresponds to a product in the category of types. Most

languages have a

primitive notion In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to Intuition (knowledge), intuition or taken ...

of product type. For instance, the product of ''type₁'', ..., ''type_n'' is written ''type₁''*...*''type_n'' in ML and (''type₁'',...,''type_n'') in

Haskell Haskell () is a general-purpose, statically typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research, and industrial applications, Haskell pioneered several programming language ...

. In both these languages, tuples are written (''v₁'',...,''v_n'') and the components of a tuple are extracted by pattern-matching. Additionally, many functional programming languages provide more general

s, which extend both product and sum types. Product types are the dual of sum types.

References

*
''Homotopy Type Theory: Univalent Foundations of Mathematics'', The Univalent Foundations Program, Institute for Advanced Study
''See section 1.5''. {{DEFAULTSORT:Product Type Data types Type theory Composite data types

See also

References