On monotone formulae with restricted depth (preliminary version)

Abstract

We prove a hierarchy theorem for the representation of monotone Boolean functions by monotone formulae with restricted depth. Specifically, we show that there are functions with ∏k-formulae of size n for which every ∑k-formula has size exp ?(n1/(k-1)). A similar lower bound applies to concrete functions such as transitive closure and clique. We also show that any function with a formula of size n (and any depth) has a ∑k-formula of size exp (n1/(k-1)). Thus our hierarchy theorem is the best possible.