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Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07) (2007)
San Diego, California
June 13, 2007 to Mar. 16, 2007
ISSN: 1093-0159
ISBN: 0-7695-2780-9
pp: 24-32
Harry Buhrman , CWI Amsterdamn and University of Amsterdam, Netherlands
Nikolay Vereshchagin , Moscow State University, Russia
Ronald de Wolf , CWI Amsterdam, Netherlands
We present two results for computational models that allow error probabilities close to 1/2. <p>First, most computational complexity classes have an analogous class in communication complexity. The class PP in fact has two, a version with weakly restricted bias called PP^cc, and a version with unrestricted bias called UPP^cc. Ever since their introduction by Babai, Frankl, and Simon in 1986, it has been open whether these classes are the same. We show that PP^cc \varsubsetneq UPP^cc. Our proof combines a query complexity separation due to Beigel with a technique of Razborov that translates the acceptance probability of quantum protocols to polynomials.</p> <p>Second, we study how small the bias of minimal-degree polynomials that sign-represent Boolean functions needs to be. We show that the worst-case bias is at worst double-exponentially small in the sign-degree (which was very recently shown to be optimal by Podolski), while the averagecase bias can be made single-exponentially small in the sign-degree (which we show to be close to optimal).</p>

R. de Wolf, N. Vereshchagin and H. Buhrman, "On Computation and Communication with Small Bias," Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07)(CCC), San Diego, California, 2007, pp. 24-32.
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