Criticality of AC

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 and all p ∈ [0,1], [EQUATION] where R p refers to the distribution of p -random restrictions. Håstad'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 [EQUATION]. 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 [EQUATION], 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.