On the number of ANDs versus the number of ORs in monotone Boolean circuits

Abstract

Alon and Boppana showed that if a monotone Boolean function f of n variables can be computed by a monotone circuit containing k AND gates, where k ⪢ 1, then it can also be computed using a monotone circuit containing k AND gates and O(k(n + k)) OR gates. They note that their result is tight up to a logarithmic factor. Here we show that under the same assumption the function f can be computed using a monotone circuit containing k AND gates and O(k(n + k)log k) OR gates. This result is tight up to a constant factor. By duality the same result holds when the roles of the AND and OR gates are interchanged.