Document Type

Conference Proceeding


Mathematics (HMC)

Publication Date



We prove a hierarchy theorem for the representation of monotone Boolean functions by monotone Boolean functions by monotone formulae with restricted depth. Specifically, we show that there are functions with Πk-formulae 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.


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