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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.
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 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. The most general example is the set elements of a Boolean algebra, with all of the foregoing being instances thereof.
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 ...
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 ...
In mathematics and computer science, a balanced Boolean function is a Boolean function whose output yields as many 0s as 1s over its input set. This means that for a uniformly random input string of bits, the probability of getting a 1 is 1/2.
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 ...