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• W2114522915 abstract "We present two results for computational models that allow error probabilities close to 1/2. 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 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>cc</sup> , and a version with unrestricted bias called UPP <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>cc</sup> . 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 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>cc</sup> subne UPP <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>cc</sup> . 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. 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 average- case bias can be made single-exponentially small in the sign-degree (which we show to be close to optimal)." @default.
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• W2114522915 date "2007-06-01" @default.
• W2114522915 modified "2023-09-26" @default.
• W2114522915 title "On Computation and Communication with Small Bias" @default.
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• W2114522915 doi "https://doi.org/10.1109/ccc.2007.18" @default.
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