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 πk-formula of size n for which every sk-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 sk-formula of size exp o(n1/(k−1)). Thus our hierarchy theorem is the best possible.
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