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.

The definition implies that instead of the truth table, traditionally used to represent Boolean functions, one may use a 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.

• 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 equal to $k\mod m$ for a fixed k, m
• Parity functions: their value is 1 if the input vector has odd number of ones.

Research

Boolean functions have many applications, including design of electronic circuits and in cryptography.

In 1994 Johan Håstad was awarded Gödel Prize concerned his work on lower bounds on the size of constant-depth Boolean circuits for the parity function.