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Any Boolean function (): {,} {,} can be uniquely extended (interpolated) to the real domain by a multilinear polynomial in , constructed by summing the truth table values multiplied by indicator polynomials: = {,} (): =: = For example, the extension of the binary XOR function is () + + + which equals + Some other examples are negation (), AND ...
Boolean function; Boolean-valued function; Boolean-valued model; Boolean satisfiability problem; Boolean differential calculus; Indicator function (also called the characteristic function, but that term is used in probability theory for a different concept) Espresso heuristic logic minimizer; Logical matrix; Logical value; Stone duality; Stone ...
Boolean algebra also deals with functions which have their values in the set {0,1}. A sequence of bits is a commonly used example of such a function. Another common example is the totality of subsets of a set E: to a subset F of E, one can define the indicator function that takes the value 1 on F, and 0 outside F.
For example, the truth value of the statement "for every number there is a prime larger than it" is the set of all programs that take as input a number , and output a prime larger than . In category theory , truth values appear as the elements of the subobject classifier .
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = {0, 1}), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information.
In computer science, a Boolean expression is an expression used in programming languages that produces a Boolean value when evaluated. A Boolean value is either true or false.A Boolean expression may be composed of a combination of the Boolean constants True/False or Yes/No, Boolean-typed variables, Boolean-valued operators, and Boolean-valued functions.
The functions studied are often, but not always, Boolean-valued, making them Boolean functions. The area has found many applications in combinatorics , social choice theory , random graphs , and theoretical computer science, especially in hardness of approximation , property testing , and PAC learning .
Balanced Boolean functions are used in cryptography, where being balanced is one of "the most important criteria for cryptographically strong Boolean functions". [3] If a function is not balanced, it will have a statistical bias , making it subject to cryptanalysis such as the correlation attack .