Complexity and depth of formulas for symmetric Boolean functions

Abstract

A new approach for implementation of the counting function for a Boolean set is proposed. The approach is based on approximate calculation of sums. Using this approach, new upper bounds for the size and depth of symmetric functions over the basis B2 of all dyadic functions and over the standard basis B0 = {∧, ∨,- } were non-constructively obtained. In particular, the depth of multiplication of n-bit binary numbers is asymptotically estimated from above by 4.02 log2n relative to the basis B2 and by 5.14log2n relative to the basis B0.