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In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the order of its input bits, i.e., it depends only on the number of ones (or zeros) in the input. [1] For this reason they are also known as Boolean counting functions. [2] There are 2 n+1 symmetric n-ary Boolean functions.
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {-1,1}). [1] [2] Alternative names are switching function, used especially in older computer science literature, [3] [4] and truth function (or logical function), used in logic.
Now the completed features contain every Boolean function on Boolean variables, with each one exactly once. Viewing these Boolean functions as polynomials in k {\displaystyle k} variables over GF(2), segregate the functions into pairs ( f , g ) {\displaystyle (f,g)} where f {\displaystyle f} contains the i {\displaystyle i} -th coordinate as a ...
Parity only depends on the number of ones and is therefore a symmetric Boolean function.. The n-variable parity function and its negation are the only Boolean functions for which all disjunctive normal forms have the maximal number of 2 n − 1 monomials of length n and all conjunctive normal forms have the maximal number of 2 n − 1 clauses of length n.
The Boolean function is said to be linearly separable provided these two sets of points are linearly separable. The number of distinct Boolean functions is where n is the number of variables passed into the function. [3] Such functions are also called linear threshold logic, or perceptrons.
Another way to see why the free Boolean algebra on an n-element set has elements is to note that each element is a function from n bits to one. There are 2 n {\displaystyle 2^{n}} possible inputs to such a function and the function will choose 0 or 1 to output for each input, so there are 2 2 n {\displaystyle 2^{2^{n}}} possible functions.
A random Boolean network (RBN) is one that is randomly selected from the set of all possible Boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior ...
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 .