{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7833","name":"Correlation Bounds Against Monotone NC^1","abstract":"This paper gives the first correlation bounds under product distributions, including the uniform distribution, against the class mNC^ of polynomial-size O(log(n))-depth monotone circuits. Our main theorem, proved using the pathset complexity framework introduced in [Rossmann,arXiv:1312.0355], shows that the average-case k-CYCLE problem (on Erd\u00f6s-Renyi random graphs with an appropriate edge density) is 1\/2 + 1\/poly(n) hard for mNC^1. Combining this result with O'Donnell's hardness amplification theorem [O'Donnell,2002], we obtain an explicit monotone function of n variables (in the class mSAC^1) which is 1\/2 + n^(-1\/2+epsilon) hard for mNC^1 under the uniform distribution for any desired constant epsilon > 0. This bound is nearly best possible, since every monotone function has agreement 1\/2 + Omega(log(n)\/sqrt(n)) with some function in mNC^1 [O'Donnell\/Wimmer,FOCS'09].\r\n\r\nOur correlation bounds against mNC^1 extend smoothly to non-monotone NC^1 circuits with a bounded number of negation gates. Using Holley's monotone coupling theorem [Holley,Comm. Math. Physics,1974], we prove the following lemma: with respect to any product distribution, if a balanced monotone function f is 1\/2 + delta hard for monotone circuits of a given size and depth, then f is 1\/2 + (2^(t+1)-1)*delta hard for (non-monotone) circuits of the same size and depth with at most t negation gates. We thus achieve a lower bound against NC^1 circuits with (1\/2-epsilon)*log(n) negation gates, improving the previous record of 1\/6*log(log(n)) [Amano\/Maruoka,SIAML J. Comp.,2005]. Our bound on negations is \"half\" optimal, since \\lceil log(n+1) \\rceil negation gates are known to be fully powerful for NC^1 [Ajtai\/Komlos\/Szemeredi,STOC'83; Fischer,GI'75].","keywords":["circuit complexity","average-case complexity"],"author":{"@type":"Person","name":"Rossman, Benjamin","givenName":"Benjamin","familyName":"Rossman"},"position":20,"pageStart":392,"pageEnd":411,"dateCreated":"2015-06-06","datePublished":"2015-06-06","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Rossman, Benjamin","givenName":"Benjamin","familyName":"Rossman"},"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.CCC.2015.392","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6236","volumeNumber":33,"name":"30th Conference on Computational Complexity (CCC 2015)","dateCreated":"2015-06-06","datePublished":"2015-06-06","editor":{"@type":"Person","name":"Zuckerman, David","givenName":"David","familyName":"Zuckerman"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7833","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":"#volume6236"}}}