{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article12501","name":"Criticality of Regular Formulas","abstract":"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= k. \r\nIn an unpublished manuscript [Rossmann, 2018], the author showed that a combination of H\u00e5stad\u2019s switching and multi-switching lemmas [H\u00e5stad, 1986; H\u00e5stad, 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 \r\n(i) a novel switching lemma for bounded size (unbounded width) DNF formulas, and \r\n(ii) an extension of (i) which analyzes a canonical decision tree associated with an entire depth-d formula. \r\nAs 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.","keywords":["AC^0 circuits","formulas","criticality"],"author":{"@type":"Person","name":"Rossman, Benjamin","givenName":"Benjamin","familyName":"Rossman","email":"mailto:ben.rossman@utoronto.ca","affiliation":"Departments of Mathematics and Computer Science, University of Toronto, Canada","funding":"NSERC and Sloan Research Fellowship"},"position":1,"pageStart":"1:1","pageEnd":"1:28","dateCreated":"2019-07-16","datePublished":"2019-07-16","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Rossman, Benjamin","givenName":"Benjamin","familyName":"Rossman","email":"mailto:ben.rossman@utoronto.ca","affiliation":"Departments of Mathematics and Computer Science, University of Toronto, Canada","funding":"NSERC and Sloan Research Fellowship"},"copyrightYear":"2019","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.CCC.2019.1","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/arxiv.org\/abs\/1703.10116","isPartOf":{"@type":"PublicationVolume","@id":"#volume6340","volumeNumber":137,"name":"34th Computational Complexity Conference (CCC 2019)","dateCreated":"2019-07-16","datePublished":"2019-07-16","editor":{"@type":"Person","name":"Shpilka, Amir","givenName":"Amir","familyName":"Shpilka","email":"mailto:shpilka@tauex.tau.ac.il","affiliation":"Tel Aviv University, Tel Aviv, 69978, Israel"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article12501","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6340"}}}