Symmetric Boolean function
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.
Special cases
A number of special cases are recognized:[1]
- Majority function: their value is 1 on input vectors with more than n/2 ones
- Threshold functions: their value is 1 on input vectors with k or more ones for a fixed k
- Exact-value functions: their value is 1 on input vectors with k ones for a fixed k
- Counting functions: their value is 1 on input vectors with the number of ones congruent to k mod m for fixed k, m
- Parity function: their value is 1 if the input vector has odd number of ones
See also
References
- ^ a b 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.