<mathematics, logic> 1. Boolean algebra. <programming> 2. (bool) The type of an expression with two possible values, "true" and "false". Also, a variable of Boolean type or a function with Boolean arguments or result. The most common Boolean functions are AND, OR and NOT. (1997-12-01)

Any kind of logic, function, expression, or theory based on the work of George Boole is considered Boolean.

Boolean grammars, introduced by , are a class of formal grammars studied in formal language theory. They extend the basic type of grammars, the context-free grammars, with conjunction and negation operations.

In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible input length.

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network.

<mathematics, logic> (After the logician George Boole) 1. Commonly, and especially in computer science and digital electronics, this term is used to mean two-valued logic. 2. This is in stark contrast with the definition used by pure mathematicians who in the 1960s introduced "Boolean-valued models" into logic precisely because a "Boolean-valued model" is an interpretation of a theory that allows more than two possible truth values! Strangely, a Boolean algebra (in the mathematical sense) is not strictly an algebra, but is in fact a lattice. A Boolean algebra is sometimes defined as a "complemented distributive lattice". Boole's work which inspired the mathematical definition concerned algebras of sets, involving the operations of intersection, union and complement on sets. Such algebras obey the following identities where the operators ^, V, - and constants 1 and 0 can be thought of either as set intersection, union, complement, universal, empty; or as two-valued logic AND, OR, NOT, TRUE, FALSE; or any other conforming system. a ^ b = b ^ a a V b = b V a (commutative laws) (a ^ b) ^ c = a ^ (b ^ c) (a V b) V c = a V (b V c) (associative laws) a ^ (b V c) = (a ^ b) V (a ^ c) a V (b ^ c) = (a V b) ^ (a V c) (distributive laws) a ^ a = a a V a = a (idempotence laws) --a = a -(a ^ b) = (-a) V (-b) -(a V b) = (-a) ^ (-b) (de Morgan's laws) a ^ -a = 0 a V -a = 1 a ^ 1 = a a V 0 = a a ^ 0 = 0 a V 1 = 1 -1 = 0 -0 = 1 There are several common alternative notations for the "-" or logical complement operator. If a and b are elements of a Boolean algebra, we define a <= b to mean that a ^ b = a, or equivalently a V b = b. Thus, for example, if ^, V and - denote set intersection, union and complement then <= is the inclusive subset relation. The relation <= is a partial ordering, though it is not necessarily a linear ordering since some Boolean algebras contain incomparable values. Note that these laws only refer explicitly to the two distinguished constants 1 and 0 (sometimes written as LaTeX op and ot), and in two-valued logic there are no others, but according to the more general mathematical definition, in some systems variables a, b and c may take on other values as well. (1997-02-27)

<information science> (Or "Boolean query") A query using the Boolean operators, AND, OR, and NOT, and parentheses to construct a complex condition from simpler criteria. A typical example is searching for combinatons of keywords on a World-Wide Web search engine. Examples: car or automobile "New York" and not "New York state" The term is sometimes stretched to include searches using other operators, e.g. "near". Not to be confused with binary search. See also: weighted search. (1999-10-23)

In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as {0, 1}, or \mathbb{B}.

<mathematics> A logic based on Boolean algebra. (1995-03-25)