Symmetric Boolean function
In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the permutation of its input bits, i.e., it depends only on the number of ones in the input.
From the definition follows, that there are 2n+1 symmetric n-ary Boolean functions. It implies that 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 -vector, whose i-th entry is the value of the function on an input vector with i ones.A number of special cases are recognized.
- 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 functions: their value is 1 if the input vector has odd number of ones.
