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The Sign-Rank of AC^O

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2 Author(s)
Razborov, A.A. ; Inst. for Adv. Study, Princeton, NJ ; Sherstov, A.A.

The sign-rank of a matrix A = [Aij] with plusmn1 entries is the least rank of a real matrix B = [Bij] with AijBij > 0 for all i, j. We obtain the first exponential lower bound on the sign-rank of a function in AC0. Namely, let f(x, y) = Lambdai=1 m Lambdaj=1 m 2(xij Lambda yij). We show that the matrix [f(x, y)]x, y has sign-rank 2Omega(m). This in particular implies that Sigma2 ccnsubeUPPcc, which solves a long-standing open problem posed by Babai, Frankl, and Simon (1986). Our result additionally implies a lower bound in learning theory. Specifically, let Phi1,..., Phir : {0, 1}n rarrRopf be functions such that every DNF formula f : {0, 1}n rarr {-1, +1} of polynomial size has the representation f equiv sign(a1Phi1 + hellip + arPhir) for some reals a1,..., ar. We prove that then r ges 2Omega(n 1/3 ), which essentially matches an upper bound of 2Otilde(n 1/3 ) due to Klivans and Servedio (2001). Finally, our work yields the first exponential lower bound on the size of threshold-of-majority circuits computing a function in AC0. This substantially generalizes and strengthens the results of Krause and Pudlak (1997).

Published in:

Foundations of Computer Science, 2008. FOCS '08. IEEE 49th Annual IEEE Symposium on

Date of Conference:

25-28 Oct. 2008