On Computation and Communication with Small Bias

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 \emph{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 $\subsetneq$ UPP\cc. Our proof combines a query complexity separation due to Beigel with a technique of Razborov that translates the acceptance probability of \emph{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)