Emanuele Viola's On the Power of Small-Depth Computation (Foundations and PDF

By Emanuele Viola

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Additional resources for On the Power of Small-Depth Computation (Foundations and Trends in Theoretical Computer Science)

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M}. Consider the associated random degree-1 polynomials: Pi (x) := xi i∈Si for i = 0, . . , log m. First, we claim that every polynomial always computes ∨ correctly when the input is 0m . The proof of this claim is obvious. 8. If x = 0m then Pi (x) = 0 always and for all i. Second, we claim that on any input that is not 0m , with high probability there is some polynomial that computes ∨ correctly. 9. For all x ∈ {0, 1}m , x = 0m , with probability at least 1/6 over P0 , . . , Plog m there is i such that Pi (x) = 1.

5 with := /t to get polynomials p1 , . . , pt such that for every i ≤ t we have Prx [pi (x) = Ci (x)] ≥ 1 − . The degree of each pi is ≤ logO(d) (w/ ), since each circuit Ci has size ≤ w and depth ≤ d. Then, define: p(x) := i≤t t pi (x) − . 2 Note that the degree of p is equal to the maximum degree of any pi , which is logO(d) (w/ ) = logO(d) (w/ ) (because t ≤ w). Also note that, whenever we have pi (x) = Ci (x) for all i, then sign(p(x)) = Majority(C1 (x), . . , Ct (x)) = C(x). By a union bound, this happens with probability at least 1 − t · = 1 − .

1 0 0 0 ··· 0 1 Then, we define L(x) to be M (x) with the first column and last row removed. In the above example, L(x) is the (s − 1) × (s − 1) matrix   0 0 ··· 1 ··· 0 1 0 1 · · · 0 0      0 1 0 · · · 1 0     . .  . .  . .      0 ··· 1 0 We let m := s − 1 and note that L(x) is an m × m matrix. 1 Definitions and Main Result 59 Observe that the column number in the first row of L(x) tells us to which node we arrive after taking one step from the node Start.

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On the Power of Small-Depth Computation (Foundations and Trends in Theoretical Computer Science) by Emanuele Viola


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