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Criticality of Regular Formulas Rossman, Benjamin AC^0 circuits formulas criticality We define the criticality of a boolean function f : {0,1}^n -> {0,1} as the minimum real number lambda >= 1 such that Pr [DT_{depth}(f|R_p) >= t] <= (p lambda)^t for all p in [0,1] and t in N, where R_p is the p-random restriction and DT_{depth} is decision-tree depth. Criticality is a useful parameter: it implies an O(2^((1- 1/(2 lambda))n)) bound on the decision-tree size of f, as well as a 2^{-Omega(k/lambda)} bound on Fourier weight of f on coefficients of size >= k. In an unpublished manuscript [Rossmann, 2018], the author showed that a combination of Håstad’s switching and multi-switching lemmas [Håstad, 1986; Håstad, 2014] implies that AC^0 circuits of depth d+1 and size s have criticality at most O(log s)^d. In the present paper, we establish a stronger O(1/d log s)^d bound for regular formulas: the class of AC^0 formulas in which all gates at any given depth have the same fan-in. This result is based on (i) a novel switching lemma for bounded size (unbounded width) DNF formulas, and (ii) an extension of (i) which analyzes a canonical decision tree associated with an entire depth-d formula. As corollaries of our criticality bound, we obtain an improved #SAT algorithm and tight Linial-Mansour-Nisan Theorem for regular formulas, strengthening previous results for AC^0 circuits due to Impagliazzo, Matthews, Paturi [Impagliazzo et al., 2012] and Tal [Tal, 2017]. As a further corollary, we increase from o(log n /(log log n)) to o(log n) the number of quantifier alternations for which the QBF-SAT (quantified boolean formula satisfiability) algorithm of Santhanam and Williams [Santhanam and Williams, 2014] beats exhaustive search. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Benjamin Rossman 2019 Is Part Of LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2019.1 urn:nbn:de:0030-drops-108230 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.1 eng https://creativecommons.org/licenses/by/3.0/legalcode