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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2012.36
URN: urn:nbn:de:0030-drops-38467
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2012/3846/
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Kopparty, Swastik ; Srinivasan, Srikanth

Certifying polynomials for AC^0(parity) circuits, with applications

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Abstract

In this paper, we introduce and develop the method of certifying polynomials for proving AC^0 circuit lower bounds.

We use this method to show that Approximate Majority cannot be computed by AC^0(parity) circuits of size n^{1 + o(1)}. This implies a separation between the power of AC^0(parity) circuits of
near-linear size and uniform AC^0(parity) (and even AC^0) circuits of polynomial size.
This also implies a separation between randomized AC^0(parity) circuits of linear size and deterministic AC^0(parity) circuits of near-linear size.

Our proof using certifying polynomials extends the deterministic restrictions technique of Chaudhuri and Radhakrishnan, who showed that Approximate Majority cannot be computed by AC^0 circuits of size n^{1+o(1)}.
At the technical level, we show that for every ACP circuit C of near-linear size, there is a low degree variety V over F_2 such that the restriction of C to V is constant.

We also prove other results exploring various aspects of the power of certifying polynomials. In the process, we show an essentially optimal lower bound of Omega\left(\log^{\Theta(d)} s \cdot \log \frac{1}{\epsilon} \right) on the degree of \epsilon-approximating polynomials for AC^0(parity) circuits of size s.

BibTeX - Entry

@InProceedings{kopparty_et_al:LIPIcs:2012:3846,
  author =	{Swastik Kopparty and Srikanth Srinivasan},
  title =	{{Certifying polynomials for AC^0(parity) circuits, with applications}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012) },
  pages =	{36--47},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-47-7},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{18},
  editor =	{Deepak D'Souza and Telikepalli Kavitha and Jaikumar Radhakrishnan},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2012/3846},
  URN =		{urn:nbn:de:0030-drops-38467},
  doi =		{10.4230/LIPIcs.FSTTCS.2012.36},
  annote =	{Keywords: Constant-depth Boolean circuits, Polynomials over finite fields, Size hierarchies}
}

Keywords: Constant-depth Boolean circuits, Polynomials over finite fields, Size hierarchies
Collection: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)
Issue Date: 2012
Date of publication: 14.12.2012

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