On monotone formulae with restricted depth
Maria M. Klawe, Wolfgang J. Paul, Nicholas J. Pippenger, Mihalis Yannakakis · 1984
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 πk-formula of size n for which every σk-formula has size exp ω(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 σk-formula of size exp o(n1/(k−1)). Thus our hierarchy theorem is the best possible.