Varāhamihira

2014-01-28T11:32:25Z

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In [[computer science]], the '''Boolean''' or '''logical data type''' is a [[data type]], having two values (usually denoted '''true''' and '''false'''), intended to represent the [[truth value]]s of [[logic]] and [[Boolean algebra (logic)|Boolean algebra]]. It is named after [[George Boole]], who first defined an algebraic system of logic in the mid 19th century. The Boolean data type is the primary result of [[Conditional (programming)|conditional statements]], which allow different actions and change [[control flow]] depending on whether a programmer-specified boolean ''condition'' evaluates to true or false.

== Generalities ==
In [[programming language]]s that have a built-in Boolean data type, such as [[Pascal programming language|Pascal]] and [[Java programming language|Java]], the [[comparison operator]]s such as <code>></code> and <code>≠</code> are usually defined to return a Boolean value. [[if-then-else|Conditional]] and [[while loop|iterative command]]s may be defined to test Boolean-valued expressions.

Languages without an explicit Boolean data type, like [[ANSI C#C90|C90]] and [[Lisp (programming language)|Lisp]], may still represent truth values by some other data type. Lisp uses an empty list for false, and any other value for true. C uses an [[integer (computer science)|integer]] type, where relational expressions like <code>i > j</code> and logical expressions connected by <code>&&</code> and <code>||</code> are defined to have value 1 if true and 0 if false, whereas the test parts of <code>if</code>, <code>while</code>, <code>for</code>, etc., treat any non-zero value as true.<ref name="k&r1e">{{cite book | first1 = Brian W | last1 = Kernighan | authorlink1 = Brian Kernighan | first2 = Dennis M |last2 = Ritchie | authorlink2 = Dennis Ritchie | page = 41 | title = [[The C Programming Language (book)|The C Programming Language]] | edition = 1st | publisher = [[Prentice Hall]] | year= 1978 | location = [[Englewood Cliffs, NJ]] | ISBN = 0-13-110163-3 }}</ref><ref>{{ cite book | pages=86–93 | first1 = PJ |last1 = Plauger | authorlink1 = P. J. Plauger | first2 = Jim | last2 = Brodie | title = ANSI and ISO Standard C Programmer's reference | publisher = [[Microsoft Press]] | origyear = 1989 | year = 1992 | ISBN = 1-55615-359-7}}</ref> Indeed, a Boolean variable may be regarded (and be implemented) as a numerical variable with a single [[bit|binary digit (bit)]], which can store only two values. It is worth noting that the implementation of booleans in computers are most likely represented as a full [[Word (computer architecture)|word]], rather than a bit; this is usually due to the ways computers transfer blocks of information.

Most programming languages, even those that do not have an explicit Boolean type, have support for Boolean algebraic operations such as [[logical conjunction|conjunction]] (<code>AND</code>, <code>&</code>, <code>*</code>), [[logical disjunction|disjunction]] (<code>OR</code>, <code>|</code>, <code>+</code>), [[logical equivalence|equivalence]] (<code>EQV</code>, <code>=</code>, <code>==</code>), [[exclusive disjunction|exclusive or/non-equivalence]] (<code>XOR</code>, <code>NEQV</code>, <code>^</code>, <code>!=</code>), and [[negation]] (<code>NOT</code>, <code>~</code>, <code>!</code>).

In some languages, like [[Ruby (programming language)|Ruby]], [[Smalltalk]], and [[Alice (software)|Alice]] the "true" and "false" values belong to separate classes—e.g. <code>True</code> and <code>False</code>, resp.—so there is no single Boolean "type."

In [[SQL]], which uses a [[three-valued logic]] for explicit comparisons because of its special treatment of [[Null (SQL)|Null]]s, the Boolean data type (introduced in [[SQL:1999]]) is also defined to include more than two truth values, so that SQL "Booleans" can store all logical values resulting from the evaluation of predicates in SQL. A column of Boolean type can also be restricted to just <code>TRUE</code> and <code>FALSE</code> though.

In the [[lambda calculus]] model of computing, Boolean values can be represented as [[Church booleans]].

== ALGOL, Java, and C# ==
One of the earliest languages to provide an explicit '''Boolean''' data type was [[ALGOL 60]] (1960) with values '''true''' and '''false''' and logical operators denoted by symbols '<math>\wedge</math>' (and), '<math>\vee</math>' (or), '<math>\supset</math>' (implies), '<math>\equiv</math>' (equivalence), and '<math>\neg</math>' (not). Due to input device limitations of the time, however, most compilers used alternative representations for the latter, such as <code>AND</code> or <code>'AND'</code>.

This approach ("Boolean is a separate built-in [[primitive data type]]") was adopted by many later languages, such as [[ALGOL 68]] (1970),<ref>{{cite web| title=Report on the Algorithmic Language ALGOL 68, Section 10.2.2. | url=http://www.fh-jena.de/~kleine/history/languages/Algol68-Report.pdf |format=PDF| accessdate=April 2007 |date=August 1968 }}</ref> [[Java (programming language)|Java]], and [[C Sharp (programming language)|C#]].

== Fortran ==
The first version of [[Fortran|FORTRAN]] (1957) and its successor FORTRAN II (1958) did not have logical values or operations; even the conditional <code>IF</code> statement took an arithmetic expression and branched to one of three locations according to its sign; see [[arithmetic IF]]. FORTRAN IV (1962), however, followed the ALGOL 60 example by providing a Boolean data type (<code>LOGICAL</code>), truth literals (<code>.TRUE.</code> and <code>.FALSE.</code>), Boolean-valued numeric comparison operators (<code>.EQ.</code>, <code>.GT.</code>, etc.), and logical operators (<code>.NOT.</code>, <code>.AND.</code>, <code>.OR.</code>). In <code>FORMAT</code> statements, a specific control character ('<code>L</code>') was provided for the parsing or formatting of logical values.<ref>Digital Equipment Corporation, ''DECSystem10 FORTRAN IV Programmers Reference Manual''. Reprinted in ''Mathematical Languages Handbook''. [http://www.bitsavers.org/pdf/tymshare/tymcom-x/Tymcom-X_Reference_Series_Fortran_IV_Jan73.pdf Online version] accessed 2011-11-16.</ref> 

== Lisp and Scheme ==
The [[Lisp (programming language)|Lisp]] language (1958) never had a built-in Boolean data type. Instead, conditional constructs like <code>cond</code> assume that the logical value "false" is represented by the empty list <code>()</code>, which is defined to be the same as the special atom <code>nil</code> or <code>NIL</code>; whereas any other [[s-expression]] is interpreted as "true". For convenience, most modern dialects of Lisp predefine the atom <code>t</code> to have value <code>t</code>, so that one can use <code>t</code> as a mnemonic notation for "true".

This approach ("any value can be used as a Boolean value") was retained in most Lisp dialects ([[Common Lisp]], [[Scheme (programming language)|Scheme]], [[Emacs Lisp]]), and similar models were adopted by many [[scripting language]]s, even ones that do have a distinct Boolean type or Boolean values; although which values are interpreted as "false" and which are "true" vary from language to language. In Scheme, for example, the "false" value is an atom distinct from the empty list, so the latter is interpreted as "true".

== Pascal, Ada, and Haskell ==
The [[Pascal (programming language)|Pascal]] language (1978) introduced the concept of programmer-defined [[enumerated type]]s. A built-in <code>Boolean</code> data type was then provided as a predefined enumerated type with values <code>FALSE</code> and <code>TRUE</code>. By definition, all comparisons, logical operations, and conditional statements applied to and/or yielded <code>Boolean</code> values. Otherwise, the <code>Boolean</code> type had all the facilities which were available for enumerated types in general — such as ordering and use as indices. On the other hand, the conversion between <code>Boolean</code>s and integers (or any other types) still required explicit tests or function calls, as in ALGOL 60. This approach ("Boolean is an enumerated type") was adopted by most later languages which had enumerated types, such as [[Modula]], [[Ada (programming language)|Ada]] and [[Haskell (programming language)|Haskell]].

== C, C++, Objective-C, Awk, Perl ==
The initial implementations of the [[C (programming language)|C]] language (1972) provided no Boolean type, and to this day Boolean values are commonly represented by integers (<code>int</code>s) in C programs. The comparison operators ('<code>></code>', '<code>==</code>', etc.) are defined to return a signed integer (<code>int</code>) result, either zero (for false) or 1 (for true). Logical operators ('<code>&&</code>', '<code>||</code>', '<code>!</code>', etc.) and condition-testing statements ('<code>if</code>', '<code>while</code>') assume that zero is false and all other values are true. One problem with this approach is that the tests <code>if(t==TRUE){...}</code> and <code>if(t)</code> are not equivalent.

After enumerated types (<code>enum</code>s) were added to the [[ANSI C|ANSI version of C]] (1989), many C programmers got used to defining their own Boolean types as such, for readability reasons. However, enumerated types are equivalent to integers according to the language standards; so the effective identity between Booleans and integers is still valid for C programs.

Standard [[C programming language|C]] (since [[C99]]) and several dialects of C such as [[Objective-C]] provide definitions of a Boolean type as an integer type and macros for "false" and "true" as 0 and 1, respectively. Thus logical values can be stored in integer variables, and used anywhere integers would be valid, including in indexing, arithmetic, parsing, and formatting. This approach ("Boolean values are just integers") has been retained in all later versions of C.

[[C++]] has a separate Boolean data type (<code>'bool'</code>), but with automatic conversions from scalar and pointer values that are very similar to those of C. This approach was adopted also by many later languages, especially by some [[scripting language|scripting]] ones such as [[AWK]].

[[Objective-C]] also has a separate Boolean data type (<code>'BOOL'</code>), with possible values being <code>YES</code> or <code>NO</code>, equivalents of true and false respectively.<ref>http://developer.apple.com/library/ios/#documentation/cocoa/conceptual/ProgrammingWithObjectiveC/FoundationTypesandCollections/FoundationTypesandCollections.html</ref>

[[Perl]] has many values of <code>false</code>: the number zero, the strings "0" and "", the empty list "()", and the special value <code>undef</code>.<ref>{{cite web |url=http://perldoc.perl.org/perlsyn.html#Truth-and-Falsehood |title=perlsyn - Perl Syntax / Truth and Falsehood |accessdate=10 September 2013}}</ref> Everything else evaluates to <code>true</code>.

== Python, Ruby, and JavaScript ==
In [[Python (programming language)|Python]] from version 2.3 forward, there is a <code>bool</code> type which is a [[subclass (computer science)|subclass]] of <code>int</code>, the standard integer type.<ref>{{cite web | last = Van Rossum | first = Guido | authorlink = Guido van Rossum | title = PEP 285 -- Adding a bool type | date = 3 April 2002 | url = http://www.python.org/dev/peps/pep-0285/ | accessdate = 15 May 2013}}</ref> It has two possible values: <code>True</code> and <code>False</code>, which are "special versions" of 1 and 0 respectively and behave as such in arithmetic contexts. In addition, a numeric value of zero (integer or fractional), the null value (<code>None</code>), the empty [[String (computer science)|string]], and empty containers (i.e. [[list (computing)|lists]], [[set (computing)|sets]], etc.) are considered Boolean false; all other values are considered Boolean true by default.<ref>{{cite web |url= http://docs.python.org/3.3/reference/expressions.html |title= Expressions |website= Python v3.3.2 documentation | accessdate = 15 May 2013}}</ref> Classes can define how their instances are treated in a Boolean context through the special method <code>__nonzero__</code> (Python 2) or <code>__bool__</code> (Python 3). For containers, <code>__len__</code> (the special method for determining the length of containers) is used if the explicit Boolean conversion method is not defined.

In [[Ruby (programming language)|Ruby]], on the other hand, only <code>nil</code> (Ruby's null value) and a special <code>false</code> object are "false", everything else (including the integer 0 and empty arrays) is "true".

In [[JavaScript]], the empty string (<code>""</code>), <code>null</code>, <code>undefined</code>, <code>[[NaN]]</code>, +0, [[−0]] and <code>false</code><ref>{{cite web
|url=http://www.ecma-international.org/publications/files/ECMA-ST/ECMA-262.pdf |title=ECMAScript Language Specification |page=43}}</ref>
are sometimes called "falsy", and their [[complement (set theory)|complement]], "truthy", to distinguish between strictly [[type-checked]] and [[type coercion|coerced]] Booleans.<ref>{{cite web |url=http://javascript.crockford.com/style2.html |title=The Elements of JavaScript Style |publisher=Douglas Crockford |accessdate=5 March 2011}}</ref> Languages such as [[PHP]] also use this approach.

== SQL ==
The [[SQL:1999]] standard introduced a BOOLEAN data type as an optional feature (T031). When restricted by a <code>NOT NULL</code> constraint, a SQL BOOLEAN behaves like Booleans in other languages. In SQL however, the BOOLEAN type is [[nullable type|nullable]] by default like all other SQL data types, meaning it can have the special [[Null (SQL)|null]] value as well. Although the SQL standard defines three [[Literal (computer programming)|literals]] for the BOOLEAN type—TRUE, FALSE and UNKNOWN—, it also says that the NULL BOOLEAN and UNKNOWN "may be used interchangeably to mean exactly the same thing".<ref name="Date2011">{{cite book|author=C. Date|title=SQL and Relational Theory: How to Write Accurate SQL Code|url=http://books.google.com/books?id=Ew06OZtjuJEC&pg=PA83|year=2011|publisher=O'Reilly Media, Inc.|isbn=978-1-4493-1640-2|page=83}}</ref><ref>ISO/IEC 9075-2:2011 §4.5</ref> This has caused some controversy because the identification subjects UNKNOWN to the equality comparison rules for NULL. More precisely UNKNOWN = UNKNOWN is not TRUE but UNKNOWN/NULL.<ref name="Prigmore2007">{{cite book|author=Martyn Prigmore|title=Introduction to Databases With Web Applications|url=http://books.google.com/books?id=PKggKqIZnN0C&pg=PA197|year=2007|publisher=Pearson Education Canada|isbn=978-0-321-26359-9|page=197}}</ref> As of 2012 few major SQL systems implement the T031 feature.<ref>Troels Arvin, [http://troels.arvin.dk/db/rdbms/#data_types-boolean Survey of BOOLEAN data type implementation]</ref> [[PostgreSQL]] is a notable exception, although it does not implement the UNKNOWN literal; NULL can be used instead.<ref>http://www.postgresql.org/docs/current/static/datatype-boolean.html</ref> (PostgreSQL does implement the IS UNKNOWN operator, which is part of an orthogonal feature, F571.) In other SQL implementations various ad hoc solutions are used, like [[bit]], [[byte]], and [[character (computing)|character]] to simulate Boolean values.

==See also==
*[[true and false (Unix)|true and false]] commands for [[shell scripting]]
*[[Shannon's expansion]]
*[[stdbool.h]] — C99 definitions for boolean

==References==
{{Reflist|2}}

{{data types}}

{{DEFAULTSORT:Boolean Data Type}}
[[Category:Boolean algebra]]
[[Category:Data types]]
[[Category:Primitive types]]
[[Category:Articles with example ALGOL 68 code]]

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