Upper bounds for the formula size of the majority function

TL;DR

This paper establishes upper bounds on the formula size needed to compute the majority function and related symmetric functions, demonstrating that these bounds are polynomial in the number of variables over different Boolean bases.

Contribution

It provides new polynomial upper bounds for the formula complexity of the majority and symmetric functions over various Boolean bases, improving understanding of their computational complexity.

Findings

Formula size for counting functions is O(n^3.06) over all 2-input functions.

Formula size for counting functions is O(n^4.54) over the standard basis.

Bounds for symmetric functions are O(n^3.23) and O(n^4.82).

Abstract

It is shown that the counting function of n Boolean variables can be implemented with the formulae of size O(n^3.06) over the basis of all 2-input Boolean functions and of size O(n^4.54) over the standard basis. The same bounds follow for the complexity of any threshold symmetric function of n variables and particularly for the majority function. Any bit of the product of binary numbers of length n can be computed by formulae of size O(n^4.06) or O(n^5.54) depending on basis. Incidentally the bounds O(n^3.23) and O(n^4.82) on the formula size of any symmetric function of n variables with respect to the basis are obtained.