Criticality of AC^0 formulae
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Rossman [In Proc. 34th Comput. Complexity Conf., 2019] introduced the notion of criticality. The criticality of a Boolean function f : {0,1}^n →{0,1} is the minimum λ≥ 1 such that for all positive integers t, _ρ∼ℛ_p[DT_depth(f|_ρ) ≥ t] ≤ (pλ)^t. Hästad's celebrated switching lemma shows that the criticality of any k-DNF is at most O(k). Subsequent improvements to correlation bounds of AC^0-circuits against parity showed that the criticality of any AC^0-circuit of size S and depth d+1 is at most O(log S)^d and any regular AC^0-formula of size S and depth d+1 is at most O(1/d·log S)^d. We strengthen these results by showing that the criticality of any AC^0-formula (not necessarily regular) of size S and depth d+1 is at most O(1/d·log S)^d, resolving a conjecture due to Rossman. This result also implies Rossman's optimal lower bound on the size of any depth-d AC^0-formula computing parity [Comput. Complexity, 27(2):209–223, 2018.]. Our result implies tight correlation bounds against parity, tight Fourier concentration results and improved #SAT algorithm for AC^0-formulae.
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