The critical complexity of all (monotone) boolean functions and monotone graph properties
CREW-PRAM's build a powerful model of parallel computers. Cook/Dwork/Reischuk proved that the CREW-PRAM complexity of Boolean functions is bounded below by logbc(f) where b ≈ 4.79 and c(f) is the critical complexity of f. This lower bound is often even tight. For a class of functions F the critical complexity c(F), the minimum of all c(f) where f ∈ F, is the best general lower bound on the critical complexity of all f ∈ F. We determine the critical complexity of the set of all nondegenerate Boolean functions and all monotone nondegenerate Boolean functions up to a small additive term. And we compute exactly the critical complexity of the class of all monotone graph properties proving partially a conjecture of Turán.
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