{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article1418","name":"On Probabilistic Time versus Alternating Time","abstract":"Sipser and G\u00c3\u0192\u00c2\u00a1cs, and independently Lautemann, proved in '83 that probabilistic polynomial time is contained in the second level of the polynomial-time hierarchy, i.e. BPP is in Sigma_2 P. This is essentially the only non-trivial upper bound that we have on the power of probabilistic computation. More precisely, the Sipser-G\u00c3\u0192\u00c2\u00a1cs-Lautemann simulation shows that probabilistic time can be simulated deterministically, using two quantifiers, **with a quadratic blow-up in the running time**. That is, BPTime(t) is contained in Sigma_2 Time(t^2).\r\n\r\nIn this talk we discuss whether this quadratic blow-up in the running time is necessary. We show that the quadratic blow-up is indeed necessary for black-box simulations that use two quantifiers, such as those of Sipser, G\u00c3\u0192\u00c2\u00a1cs, and Lautemann. To obtain this result, we prove a new circuit lower bound for computing **approximate majority**, i.e. computing the majority of a given bit-string whose fraction of 1's is bounded away from 1\/2 (by a constant): We show that small depth-3 circuits for approximate majority must have bottom fan-in Omega(log n).\r\n\r\nOn the positive side, we obtain that probabilistic time can be simulated deterministically, using three quantifiers, in quasilinear time. That is, BPTime(t) is contained in Sigma_3 Time(t polylog t). Along the way, we show that approximate majority can be computed by uniform polynomial-size depth-3 circuits. This is a uniform version of a striking result by Ajtai that gives *non-uniform* polynomial-size depth-3 circuits for approximate majority.\r\n\r\nIf time permits, we will discuss some applications of our results to proving lower bounds on randomized Turing machines.","keywords":["Probabilistic time","alternating time","polynomial-time hierarchy","approximate majority","constant-depth circuit"],"author":{"@type":"Person","name":"Viola, Emanuele","givenName":"Emanuele","familyName":"Viola"},"position":11,"pageStart":1,"pageEnd":0,"dateCreated":"2006-11-30","datePublished":"2006-11-30","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Viola, Emanuele","givenName":"Emanuele","familyName":"Viola"},"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06111.11","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume601","volumeNumber":6111,"name":"Dagstuhl Seminar Proceedings, Volume 6111","dateCreated":"2006-10-09","datePublished":"2006-10-09","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article1418","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume601"}}}