{"title":"A non-linear time lower bound for Boolean branching programs","authors":"M. Ajtai","doi":"10.1109/SFFCS.1999.814578","DOIUrl":null,"url":null,"abstract":"We prove that for all positive integer k and for all sufficiently small /spl epsiv/>0 if n is sufficiently large then there is no Boolean (or 2-way) branching program of size less than 2/sup em/ which for all inputs X/spl sube/{0, 1, ..., n-1} computes in time kn the parity of the number of elements of the set of all pairs (x,y) with the property x/spl isin/X, y/spl isin/X, x<y, x+y/spl isin/X. For the proof of this fact we show that if A=(/spl alpha//sub i,j/)/sub i=0, j=0//sup n/ is a random n by n matrix over the field with 2 elements with the condition that \"/spl forall/, j, k, l/spl isin/{0, 1, ..., n-1}, i+j=k+l implies /spl alpha//sub i,j/=/spl alpha//sub k,l/\" then with a high probability the rank of each /spl delta/n by /spl delta/n submatrix of A is at least c/spl delta/|log /spl delta/|/sup -2/n, where c>0 is an absolute constant and n is sufficiently large with respect to /spl delta/.","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"103","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/SFFCS.1999.814578","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 103
Abstract
We prove that for all positive integer k and for all sufficiently small /spl epsiv/>0 if n is sufficiently large then there is no Boolean (or 2-way) branching program of size less than 2/sup em/ which for all inputs X/spl sube/{0, 1, ..., n-1} computes in time kn the parity of the number of elements of the set of all pairs (x,y) with the property x/spl isin/X, y/spl isin/X, x0 is an absolute constant and n is sufficiently large with respect to /spl delta/.
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