Upper bounds for the formula size of the majority function

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.