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.

Parity function

The parity function is notable for its role in theoretical investigation of circuit complexity of Boolean functions.

In early 1980s Merrick Furst, James Saxe and Michael Sipser and independently Miklós Ajtai established super-polynomial lower bounds on the size of constant-depth Boolean circuits for the parity function,i.e., they have shown that polynomial-size constant-depth circuits cannot compute the parity function. Similar results were also established for the majority, multiplication and transitive closure functions, by reduction to the parity function problem. Until this time only linear lower bounds were known for various naturally arising functions.

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