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LOWER BOUNDS FOR DEPTH-THREE ARITHMETIC CIRCUITS WITH SMALL BOTTOM FANIN

Kayal, Neeraj and Saha, Chandan (2016) LOWER BOUNDS FOR DEPTH-THREE ARITHMETIC CIRCUITS WITH SMALL BOTTOM FANIN. In: COMPUTATIONAL COMPLEXITY, 25 (2). pp. 419-454.

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Official URL: http://dx.doi.org/10.1007/s00037-016-0132-0

Abstract

Shpilka & Wigderson (IEEE conference on computational complexity, vol 87, 1999) had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth-three arithmetic circuits with bounded bottom fanin over a field F of characteristic zero. We resolve this problem by proving a N-Omega(d/tau) lower bound for (nonhomogeneous) depth-three arithmetic circuits with bottom fanin at most tau computing an explicit N-variate polynomial of degree d over F. Meanwhile, Nisan & Wigderson (Comp Complex 6(3):217-234, 1997) had posed the problem of proving super-polynomial lower bounds for homogeneous depth-five arithmetic circuits. Over fields of characteristic zero, we show a lower bound of N-Omega(root d) for homogeneous depth-five circuits (resp. also for depth-three circuits) with bottom fanin at most N-mu, for any fixed mu < 1. This resolves the problem posed by Nisan and Wigderson only partially because of the added restriction on the bottom fanin (a general homogeneous depth-five circuit has bottom fanin at most N).

Item Type: Journal Article
Additional Information: Copy right for this article belongs to the SPRINGER BASEL AG, PICASSOPLATZ 4, BASEL, 4052, SWITZERLAND
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Depositing User: Id for Latest eprints
Date Deposited: 22 Oct 2016 09:28
Last Modified: 31 Oct 2018 13:58
URI: http://eprints.iisc.ac.in/id/eprint/55035

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