License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.ICALP.2016.35
URN: urn:nbn:de:0030-drops-63150
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6315/
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Cheraghchi, Mahdi ; Grigorescu, Elena ; Juba, Brendan ; Wimmer, Karl ; Xie, Ning

AC^0 o MOD_2 Lower Bounds for the Boolean Inner Product

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Abstract

AC^0 o MOD_2 circuits are AC^0 circuits augmented with a layer of parity gates just above the input layer. We study AC^0 o MOD2 circuit lower bounds for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have highlighted this problem as a frontier problem in circuit complexity that arose both as a first step towards solving natural special cases of the matrix rigidity problem and as a candidate for constructing pseudorandom generators of minimal complexity. We give the first superlinear lower bound for the Boolean Inner Product function against AC^0 o MOD2 of depth four or greater. Specifically, we prove a superlinear lower bound for circuits of arbitrary constant depth, and an ~Omega(n^2) lower bound for the special case of depth-4 AC^0 o MOD_2. Our proof of the depth-4 lower bound employs a new "moment-matching" inequality for bounded, nonnegative integer-valued random variables that may be of independent interest: we prove an optimal bound on the maximum difference between two discrete distributions’ values at 0, given that their first d moments match.

BibTeX - Entry

@InProceedings{cheraghchi_et_al:LIPIcs:2016:6315,
  author =	{Mahdi Cheraghchi and Elena Grigorescu and Brendan Juba and Karl Wimmer and Ning Xie},
  title =	{{AC^0 o MOD_2 Lower Bounds for the Boolean Inner Product}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{35:1--35:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6315},
  URN =		{urn:nbn:de:0030-drops-63150},
  doi =		{10.4230/LIPIcs.ICALP.2016.35},
  annote =	{Keywords: Boolean analysis, circuit complexity, lower bounds}
}

Keywords: Boolean analysis, circuit complexity, lower bounds
Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 23.08.2016

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