Abstract
Rossman [In $\textit{Proc. $34$th Comput. Complexity Conf.}$, 2019] introduced the notion of $\textit{criticality}$. The criticality of a Boolean function $f : {0,1}n \to {0,1}$ is the minimum $\lambda \geq 1$ such that for all positive integers $t$, [ \Pr{\rho \sim \mathcal{R}p}\left[\text{DT}{\text{depth}}(f|{\rho}) \geq t\right] \leq (p\lambda)t. ] H\"astad's celebrated switching lemma shows that the criticality of any $k$-DNF is at most $O(k)$. Subsequent improvements to correlation bounds of $\text{AC}0$-circuits against parity showed that the criticality of any $\text{AC}0$-$\textit{circuit}$ of size $S$ and depth $d+1$ is at most $O(\log S)d$ and any $\textit{regular}$ $\text{AC}0$-$\textit{formula}$ of size $S$ and depth $d+1$ is at most $O\left(\frac1d \cdot \log S\right)d$. We strengthen these results by showing that the criticality of $\textit{any}$ $\text{AC}0$-formula (not necessarily regular) of size $S$ and depth $d+1$ is at most $O\left(\frac1d\cdot {\log S}\right)d$, resolving a conjecture due to Rossman. This result also implies Rossman's optimal lower bound on the size of any depth-$d$ $\text{AC}0$-formula computing parity [$\textit{Comput. Complexity, 27(2):209--223, 2018.}$]. Our result implies tight correlation bounds against parity, tight Fourier concentration results and improved $#$SAT algorithm for $\text{AC}0$-formulae.
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