In

computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

and computer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...

, a data type (or simply type) is a set of possible values and a set of allowed operations on it. A data type tells the compiler
In computing, a compiler is a computer program that Translator (computing), translates computer code written in one programming language (the ''source'' language) into another language (the ''target'' language). The name "compiler" is primarily ...

or interpreter how the programmer intends to use the data. Most programming languages support basic data types of integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

numbers (of varying sizes), floating-point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an Integer (computer science), integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. ...

numbers (which approximate real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s), characters and Booleans. A data type constrains the possible values that an expression, such as a variable or a function, might take. This data type defines the operations that can be done on the data, the meaning of the data, and the way values of that type can be stored.
Concept

A data type is a collection or grouping of data values. Such a grouping may be defined for many reasons: similarity, convenience, or to focus the attention. It is frequently a matter of good organization that aids the understanding of complex definitions. Almost all programming languages explicitly include the notion of data type, though the possible data types are often restricted by considerations of simplicity, computability, or regularity. An explicit data type declaration typically allows the compiler to choose an efficient machine representation, but the conceptual organization offered by data types should not be discounted. Different languages may use different data types or similar types with different semantics. For example, in thePython programming language
Python is a high-level programming language, high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of off-side rule, significant indentation.
Python is type_system#DYNAMIC, dynamically- ...

, `int`

represents an arbitrary-precision integer which has the traditional numeric operations such as addition, subtraction, and multiplication. However in the Java programming language
Java is a High-level programming language, high-level, class-based programming, class-based, object-oriented programming, object-oriented programming language that is designed to have as few implementation Dependency (computer science), depende ...

, the type `int`

represents the set of 32-bit
In computer architecture
In computer engineering, computer architecture is a description of the structure of a computer system made from component parts. It can sometimes be a high-level description that ignores details of the implementation. A ...

integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

ranging in value from −2,147,483,648 to 2,147,483,647, with arithmetic operations that wrap on overflow. In Rust
Rust is an iron oxide, a usually reddish-brown oxide formed by the reaction of iron and oxygen in the catalytic presence of water or air moisture. Rust consists of hydrous ferric oxides, hydrous iron(III) oxides (Fe2O3·nH2O) and iron(III) oxi ...

this 32-bit integer type is denoted `i32`

and panics on overflow in debug mode.
Most programming languages also allow the programmer to define additional data types, usually by combining multiple elements of other types and defining the valid operations of the new data type. For example, a programmer might create a new data type named "complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

" that would include real and imaginary parts, or a color data type represented by three byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable un ...

s denoting the amounts each of red, green, and blue, and a string representing the color's name.
Data types are used within type system
In computer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programmin ...

s, which offer various ways of defining, implementing, and using them. In a type system, a data type represents a constraint placed upon the interpretation of data, describing representation, interpretation and structure of values or objects stored in computer memory. The type system uses data type information to check correctness of computer programs
In theoretical computer science, an algorithm is correct with respect to a program specification, specification if it behaves as specified. Best explored is ''functional'' correctness, which refers to the input-output behavior of the algorithm (i. ...

that access or manipulate the data. A compiler
In computing, a compiler is a computer program that Translator (computing), translates computer code written in one programming language (the ''source'' language) into another language (the ''target'' language). The name "compiler" is primarily ...

may use the static type of a value to optimize the storage it needs and the choice of algorithms for operations on the value. In many C compilers the data type, for example, is represented in 32 bits, in accord with the IEEE specification for single-precision floating point numbers. They will thus use floating-point-specific microprocessor operations on those values (floating-point addition, multiplication, etc.).
Most data types in statistics have comparable types in computer programming, and vice versa, as shown in the following table:
Definition

identified five definitions of a "type" that were used—sometimes implicitly—in the literature: ; Syntactic: A type is a purelysyntactic
In linguistics
Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature an ...

label associated with a variable when it is declared. Although useful for advanced type systems such as substructural type systems, such definitions provide no intuitive meaning of the types.
; Representation: A type is defined in terms of its composition of more primitive types—often machine types.
; Representation and behaviour: A type is defined as its representation and a set of operators manipulating these representations.
; Value space: A type is a set of possible values which a variable can possess. Such definitions make it possible to speak about ( disjoint) unions or Cartesian product
In mathematics, specifically set theory, the Cartesian product of two Set (mathematics), sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notatio ...

s of types.
; Value space and behaviour: A type is a set of values which a variable can possess and a set of functions that one can apply to these values.
The definition in terms of a representation was often done in imperative languages such as ALGOL
Algol , designated Beta Persei (β Persei, abbreviated Beta Per, β Per), known colloquially as the Demon Star, is a bright multiple star in the constellation of Perseus (constellation), Perseus and one of the fir ...

and Pascal, while the definition in terms of a value space and behaviour was used in higher-level languages such as Simula
Simula is the name of two Simulation language, simulation programming languages, Simula I and Simula 67, developed in the 1960s at the Norwegian Computing Center in Oslo, by Ole-Johan Dahl and Kristen Nygaard. Syntax (programming languages), Syn ...

and CLU. Types including behavior align more closely with object-oriented
Object-oriented programming (OOP) is a programming paradigm based on the concept of "Object (computer science), objects", which can contain data and Computer program, code. The data is in the form of Field (computer science), fields (often kno ...

models, whereas a structured programming
Structured programming is a programming paradigm
Programming paradigms are a way to classify programming languages based on their features. Languages can be classified into multiple paradigms.
Some paradigms are concerned mainly with impl ...

model would tend to not include code, and are called plain old data structures.
Classification

Data types may be categorized according to several factors: * ''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 central processing unit, process ...

s'' or ''built-in data types'' are types that are built-in to a language implementation. ''User-defined data types'' are non-primitive types. For example, Java's numeric types are primitive, while classes are user-defined.
* A value of an ''atomic type'' is a single data item that cannot be broken into component parts. A value of a '' composite type'' or ''aggregate type'' is a collection of data items that can be accessed individually. For example, an integer is generally considered atomic, although it consists of a sequence of bits, while an array of integers is certainly composite.
* ''Basic data types'' or ''fundamental data types'' are defined axiomatically from fundamental notions or by enumeration of their elements. ''Generated data types'' or ''derived data types'' are specified, and partly defined, in terms of other data types. All basic types are atomic. For example, integers are a basic type defined in mathematics, while an array of integers is the result of applying an array type generator to the integer type.
The terminology varies - in the literature, primitive, built-in, basic, atomic, and fundamental may be used interchangeably.
Examples

Machine data types

All data in computers based on digital electronics is represented as bits (alternatives 0 and 1) on the lowest level. The smallest addressable unit of data is usually a group of bits called abyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable un ...

(usually an octet
Octet may refer to:
Music
* Octet (music), ensemble consisting of eight instruments or voices, or composition written for such an ensemble
** String octet, a piece of music written for eight string instruments
*** Octet (Mendelssohn), 1825 com ...

, which is 8 bits). The unit processed by machine code
In computer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programm ...

instructions is called a word
A word is a basic element of language
Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, ...

(, typically 32 or 64 bits).
Machine data types ''expose'' or make available fine-grained control over hardware, but this can also expose implementation details that make code less portable. Hence machine types are mainly used in systems programming or low-level programming language
A low-level programming language is a programming language that provides little or no Abstraction (computer science), abstraction from a computer's instruction set architecture—commands or functions in the language map that are structurally sim ...

s. In higher-level languages most data types are ''abstracted'' in that they do not have a language-defined machine representation. The C programming language
''The C Programming Language'' (sometimes termed ''K&R'', after its authors' initials) is a computer programming book written by Brian Kernighan and Dennis Ritchie, the latter of whom originally designed and implemented the language, as well as ...

, for instance, supplies types such as booleans, integers, floating-point numbers, etc., but the precise bit representations of these types are implementation-defined. The only C type with a precise machine representation is the `char`

type that represents a byte.
Boolean type

The Boolean type represents the valuestrue
True most commonly refers to truth, the state of being in congruence with fact or reality.
True may also refer to:
Places
* True, West Virginia, an unincorporated community in the United States
* True, Wisconsin, a town in the United States
* Tr ...

and false. Although only two values are possible, they are more often represented as a word rather as a single bit as it requires more machine instructions to store and retrieve an individual bit. Many programming languages do not have an explicit Boolean type, instead using an integer type and interpreting (for instance) 0 as false and other values as true.
Boolean data refers to the logical structure of how the language is interpreted to the machine language. In this case a Boolean 0 refers to the logic False. True is always a non zero, especially a one which is known as Boolean 1.
Numeric types

Almost all programming languages supply one or moreinteger
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

data types. They may either supply a small number of predefined subtypes restricted to certain ranges (such as `short`

and `long`

and their corresponding `unsigned`

variants in C/C++); or allow users to freely define subranges such as 1..12 (e.g. Pascal/ Ada). If a corresponding native type does not exist on the target platform, the compiler will break them down into code using types that do exist. For instance, if a 32-bit integer is requested on a 16 bit platform, the compiler will tacitly treat it as an array of two 16 bit integers.
Floating point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an Integer (computer science), integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. ...

data types represent certain fractional values (rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

, mathematically). Although they have predefined limits on both their maximum values and their precision, they are sometimes misleadingly called reals (evocative of mathematical real numbers
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

). They are typically stored internally in the form (where and are integers), but displayed in familiar decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

form.
Fixed point data types are convenient for representing monetary values. They are often implemented internally as integers, leading to predefined limits.
For independence from architecture details, a Bignum or arbitrary precision `numeric`

type might be supplied. This represents an integer or rational to a precision limited only by the available memory and computational resources on the system. Bignum implementations of arithmetic operations on machine-sized values are significantly slower than the corresponding machine operations.
Enumerations

Theenumerated type
In computer programming, an enumerated type (also called enumeration, enum, or factor in the R (programming language), R programming language, and a categorical variable in statistics) is a data type consisting of a set of named value (computer sci ...

has distinct values, which can be compared and assigned, but which do not necessarily have any particular concrete representation in the computer's memory; compilers and interpreters can represent them arbitrarily. For example, the four suits in a deck of playing cards may be four enumerators named ''CLUB'', ''DIAMOND'', ''HEART'', ''SPADE'', belonging to an enumerated type named ''suit''. If a variable ''V'' is declared having ''suit'' as its data type, one can assign any of those four values to it. Some implementations allow programmers to assign integer values to the enumeration values, or even treat them as type-equivalent to integers.
String and text types

String
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* Strings (1991 film), ''Strings'' (1991 fi ...

s are a sequence of characters used to store words or plain text
In computing, plain text is a loose term for data (e.g. file contents) that represent only character (computing), characters of readable material but not its graphical representation nor other objects (Floating point numbers, floating-point numb ...

, most often textual markup languages
Markup language refers to a Encoding, text-encoding system consisting of a set of symbols inserted in a Text file, text document to control its structure, formatting, or the relationship between its parts. Markup is often used to control the dis ...

representing formatted text. Characters may be a letter of some alphabet
An alphabet is a standardized set of basic written graphemes (called letter (alphabet), letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character ...

, a digit, a blank space, a punctuation mark, etc. Characters are drawn from a character set such as ASCII
ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...

. Character and string types can have different subtypes according to the character encoding. The original 7-bit wide ASCII was found to be limited, and superseded by 8, 16 and 32-bit sets, which can encode a wide variety of non-Latin alphabets (such as Hebrew
Hebrew (; ; ) is a Northwest Semitic languages, Northwest Semitic language of the Afroasiatic languages, Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, ...

and Chinese) and other symbols. Strings may be of either variable length or fixed length, and some programming languages have both types. They may also be subtyped by their maximum size.
Since most character sets include the digits, it is possible to have a numeric string, such as `"1234"`

. These numeric strings are usually considered distinct from numeric values such as `1234`

, although some languages automatically convert between them.
Union types

A union type definition will specify which of a number of permitted subtypes may be stored in its instances, e.g. "float or long integer". In contrast with a record, which could be defined to contain a float ''and'' an integer, a union may only contain one subtype at a time. A tagged union (also called a variant, variant record, discriminated union, or disjoint union) contains an additional field indicating its current type for enhanced type safety.Algebraic data types

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

(ADT) is a possibly recursive sum type of product type
In programming languages and type theory, a product of ''types'' is another, compounded, type in a structure. The "operands" of the product are types, and the structure of a product type is determined by the fixed order of the operands in the pr ...

s. A value of an ADT consists of a constructor tag together with zero or more field values, with the number and type of the field values fixed by the constructor. The set of all possible values of an ADT is the set-theoretic disjoint union (sum), of the sets of all possible values of its variants (product of fields). Values of algebraic types are analyzed with pattern matching, which identifies a value's constructor and extracts the fields it contains.
If there is only one constructor, then the ADT corresponds to a product type similar to a tuple or record. A constructor with no fields corresponds to the empty product (unit type). If all constructors have no fields then the ADT corresponds to an enumerated type
In computer programming, an enumerated type (also called enumeration, enum, or factor in the R (programming language), R programming language, and a categorical variable in statistics) is a data type consisting of a set of named value (computer sci ...

.
One common ADT is the option type, defined in Haskell as .
Data structures

Some types are very useful for storing and retrieving data and are calleddata structure
In computer science, a data structure is a data organization, management, and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the rel ...

s. Common data structures include:
* An array (also called vector, list
A ''list'' is any set of items in a row. List or lists may also refer to:
People
* List (surname)
Organizations
* List College, an undergraduate division of the Jewish Theological Seminary of America
* SC Germania List, German rugby uni ...

, or sequence) stores a number of elements and provides random access
Random access (more precisely and more generally called direct access) is the ability to access an arbitrary element of a sequence in equal time or any datum from a population of Address space, addressable elements roughly as easily and efficient ...

to individual elements. The elements of an array are typically (but not in all contexts) required to be of the same type. Arrays may be fixed-length or expandable. Indices into an array are typically required to be integers (if not, one may stress this relaxation by speaking about an associative array
In computer science, an associative array, map, symbol table, or dictionary is an abstract data type that stores a collection (abstract data type), collection of attribute–value pair, (key, value) pairs, such that each possible key appears at ...

) from a specific range (if not all indices in that range correspond to elements, it may be a sparse array).
* Record (also called tuple or struct) Records are among the simplest data structure
In computer science, a data structure is a data organization, management, and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the rel ...

s. A record is a value that contains other values, typically in fixed number and sequence and typically indexed by names. The elements of records are usually called ''fields'' or ''members''.
* An object contains a number of data fields, like a record, and also offers a number of subroutines for accessing or modifying them, called methods
Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a prescribed process for completing a task. It may refer to:
*Scien ...

.
* the singly linked list, which can be used to implement a queue __NOTOC__
Queue () may refer to:
* Queue area, or queue, a line or area where people wait for goods or services
Arts, entertainment, and media
*''ACM Queue'', a computer magazine
* The Queue (Sorokin novel), ''The Queue'' (Sorokin novel), a 198 ...

and is defined in Haskell as the ADT , and
* the binary tree
In computer science, a binary tree is a N-ary tree, k-ary k = 2 Tree (data structure), tree data structure in which each node has at most two child node, children, which are referred to as the ' and the '. A recursive definition using just set ...

, which allows fast searching, and can be defined in Haskell as the ADT
Abstract data types

Anabstract data type
In computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied scie ...

is a data type that does not specify the concrete representation of the data. Instead, a formal ''specification'' based on the data type's operations is used to describe it. Any ''implementation'' of a specification must fulfill the rules given. For example, a stack has push/pop operations that follow a Last-In-First-Out rule, and can be concretely implemented using either a list or an array. Another example is a set which stores values, without any particular order, and no repeated values. Values themselves are not retrieved from sets, rather one tests a value for membership to obtain a boolean "in" or "not in".
Abstract data types are used in formal semantics
Semantics (from grc, wikt:σημαντικός, σημαντικός ''sēmantikós'', "significant") is the study of reference, Meaning (philosophy), meaning, or truth. The term can be used to refer to subfields of several distinct discipline ...

and program verification and, less strictly, in design
A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process or the result of that plan or specification in the form of a prototype, product, or process. The verb ''to design'' ...

. Beyond verification, a specification might immediately be turned into an implementation. The OBJ family of programming languages for instance bases on this option using equation
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s for specification and rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a well-formed formula, formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewr ...

to run them. Algebraic specification was an important subject of research in CS around 1980 and almost a synonym for abstract data types at that time. It has a mathematical foundation in Universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular Group (mathematics), groups as ...

. The specification language can be made more expressive by allowing other formulas than only equations.
A more involved example is the Boom hierarchy of the binary tree
In computer science, a binary tree is a N-ary tree, k-ary k = 2 Tree (data structure), tree data structure in which each node has at most two child node, children, which are referred to as the ' and the '. A recursive definition using just set ...

, list
A ''list'' is any set of items in a row. List or lists may also refer to:
People
* List (surname)
Organizations
* List College, an undergraduate division of the Jewish Theological Seminary of America
* SC Germania List, German rugby uni ...

, bag and set abstract data types. All these data types can be declared by three operations: ''null'', which constructs the empty container, ''single'', which constructs a container from a single element and ''append'', which combines two containers of the same type. The complete specification for the four data types can then be given by successively adding the following rules over these operations:
Access to the data can be specified by pattern-matching over the three operations, e.g. a ''member'' function for these containers by:
Care must be taken to ensure that the function is invariant under the relevant rules for the data type. Within each of the equivalence classes implied by the choosen subset of equations, it has to yield the same result for all of its members.
Pointers and references

The main non-composite, derived type is the pointer, a data type whose value refers directly to (or "points to") another value stored elsewhere in thecomputer memory
In computing, memory is a device or system that is used to store information for immediate use in a computer or related computer hardware and Digital data, digital Electronics, electronic devices. The term ''memory'' is often synonymous with th ...

using its address
An address is a collection of information, presented in a mostly fixed format, used to give the location of a building, apartment, or other structure or a plot of land, generally using border, political boundaries and street names as references, ...

. It is a primitive kind of reference
Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a ''name'' ...

. (In everyday terms, a page number in a book could be considered a piece of data that refers to another one). Pointers are often stored in a format similar to an integer; however, attempting to dereference or "look up" a pointer whose value was never a valid memory address would cause a program to crash. To ameliorate this potential problem, pointers are considered a separate type to the type of data they point to, even if the underlying representation is the same.
Function types

Functional programming
In computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied sc ...

languages treat functions as a distinct datatype and allow values of this type to be stored in variables and passed to functions. Some multi-paradigm languages such as JavaScript
JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web
The World Wide Web (WWW), commonly known as the Web, is an information system enabling documents and oth ...

also have mechanisms for treating functions as data. Most contemporary type systems go beyond JavaScript's simple type "function object" and have a family of function types differentiated by argument and return types, such as the type `Int -> Bool`

denoting functions taking an integer and returning a boolean. In C, a function is not a first-class data type but function pointer
A function pointer, also called a subroutine pointer or procedure pointer, is a pointer (computer programming), pointer that points to a function. As opposed to referencing a data value, a function pointer points to executable code within memory. ...

s can be manipulated by the program. Java and C++ originally did not have function values but have added them in C++11 and Java 8.
In mathematical logic, first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses Quantifica ...

does not allow the application of quantifiers on function or predicate names, although second-order logic
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of Propositional calculus, propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
First-ord ...

does.
Type constructors

A type constructor builds new types from old ones, and can be thought of as an operator taking zero or more types as arguments and producing a type. Product types, function types, power types and list types can be made into type constructors.Quantified types

Universally-quantified and existentially-quantified types are based onpredicate logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses Quantifica ...

. Universal quantification is written as $\backslash forall\; x.f(x)$ or `forall x. f x`

and is the intersection over all types `x`

of the body `f x`

, i.e. the value is of type `f x`

for every `x`

. Existential quantification written as $\backslash exists\; x.f(x)$ or `exists x. f x`

and is the union over all types `x`

of the body `f x`

, i.e. the value is of type `f x`

for some `x`

.
In Haskell, universal quantification is commonly used, but existential types must be encoded by transforming `exists a. f a`

to `forall r. (forall a. f a -> r) -> r`

or a similar type.
Refinement types

A refinement type is a type endowed with a predicate which is assumed to hold for any element of the refined type. For instance, the type of natural numbers greater than 5 may be written as $\backslash $Dependent types

A dependent type is a type whose definition depends on a value. Two common examples of dependent types are dependent functions and dependent pairs. The return type of a dependent function may depend on the value (not just type) of one of its arguments. A dependent pair may have a second value of which the type depends on the first value.Meta types

Some programming languages represent the type information as data, enabling type introspection and reflection. In contrast, higher order type systems, while allowing types to be constructed from other types and passed to functions as values, typically avoid basing computational decisions on them.Convenience types

For convenience, high-level languages and databases may supply ready-made "real world" data types, for instance times, dates, and monetary values (currency). These may be built-in to the language or implemented as composite types in a library.See also

* C data types *Data dictionary
A data dictionary, or metadata repository, as defined in the ''IBM Dictionary of Computing'', is a "centralized repository of information about data such as meaning, relationships to other data, origin, usage, and format". ''Oracle Corporation, O ...

* Functional programming
In computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied sc ...

* Kind
* Type (model theory)
* Type theory
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

for the mathematical models of types
* Type system
In computer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programmin ...

for different choices in programming language typing
* Type conversion
In computer science, type conversion, type casting, type coercion, and type juggling are different ways of changing an Expression (computer science), expression from one data type to another. An example would be the conversion of an integer (c ...

* ISO/IEC 11404, General Purpose Datatypes
References

Further reading

* * *External links

* {{DEFAULTSORT:Data Type Programming language concepts