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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.
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.
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.
Examples of balanced Boolean functions are the majority function, [1] the "dictatorship function" that copies the first bit of its input to the output, [1] and the parity check function that produces the exclusive or of the input bits.
The following formula shows that a 4-ary function is bent when its nonlinearity is 6: = = In the mathematical field of combinatorics, a bent function is a Boolean function that is maximally non-linear; it is as different as possible from the set of all linear and affine functions when measured by Hamming distance between truth tables.
The example of the Boolean function given by S(x, y, z) = z if x = y and S(x, y, z) = x otherwise shows that this condition is strictly weaker than functional completeness. [ 5 ] [ 6 ] [ 7 ] Characterization of functional completeness