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 2n+1 symmetric n-ary Boolean functions. Instead of the truth table, traditionally used to represent Boolean functions, one may use a more compact representation for an n-variable symmetric Boolean function: the (n + 1)-vector, whose i-th entry (i = 0, ..., n) is the value of the function on an input vector with i ones. Mathematically, the symmetric Boolean functions correspond one-to-one with the functions that map n+1 elements to two elements, .
Symmetric Boolean functions are used to classify Boolean satisfiability problems.
^Ingo Wegener, "The Complexity of Symmetric Boolean Functions", in: Computation Theory and Logic, Lecture Notes in Computer Science, vol. 270, 1987, pp. 433–442
^"BooleanCountingFunction—Wolfram Language Documentation". reference.wolfram.com. Retrieved 2021-05-25.
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