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DOI: 10.4230/LIPIcs.ITCS.2019.13

Torus Polynomials: An Algebraic Approach to ACC Lower Bounds

Authors: Abhishek Bhrushundi, Kaave Hosseini, Shachar Lovett, and Sankeerth Rao

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract

We propose an algebraic approach to proving circuit lower bounds for ACC^0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC^0 and ACC^0 can be reformulated in this framework, implying that ACC^0 can be approximated by low-degree torus polynomials. Furthermore, as a step towards proving ACC^0 lower bounds for the majority function via our approach, we show that MAJORITY cannot be approximated by low-degree symmetric torus polynomials. We also pose several open problems related to our framework.

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Abhishek Bhrushundi, Kaave Hosseini, Shachar Lovett, and Sankeerth Rao. Torus Polynomials: An Algebraic Approach to ACC Lower Bounds. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bhrushundi_et_al:LIPIcs.ITCS.2019.13,
  author =	{Bhrushundi, Abhishek and Hosseini, Kaave and Lovett, Shachar and Rao, Sankeerth},
  title =	{{Torus Polynomials: An Algebraic Approach to ACC Lower Bounds}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.13},
  URN =		{urn:nbn:de:0030-drops-101066},
  doi =		{10.4230/LIPIcs.ITCS.2019.13},
  annote =	{Keywords: Circuit complexity, ACC, lower bounds, polynomials}
}

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DOI: 10.4230/LIPIcs.CCC.2018.2

A PRG for Boolean PTF of Degree 2 with Seed Length Subpolynomial in epsilon and Logarithmic in n

Authors: Daniel Kane and Sankeerth Rao

Published in: LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

Abstract

We construct and analyze a pseudorandom generator for degree 2 boolean polynomial threshold functions. Random constructions achieve the optimal seed length of O(log n + log 1/epsilon), however the best known explicit construction of [Ilias Diakonikolas, 2010] uses a seed length of O(log n * epsilon^{-8}). In this work we give an explicit construction that uses a seed length of O(log n + (1/epsilon)^{o(1)}). Note that this improves the seed length substantially and that the dependence on the error epsilon is additive and only grows subpolynomially as opposed to the previously known multiplicative polynomial dependence. Our generator uses dimensionality reduction on a Nisan-Wigderson based pseudorandom generator given by Lu, Kabanets [Kabanets and Lu, 2018].

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Daniel Kane and Sankeerth Rao. A PRG for Boolean PTF of Degree 2 with Seed Length Subpolynomial in epsilon and Logarithmic in n. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 2:1-2:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kane_et_al:LIPIcs.CCC.2018.2,
  author =	{Kane, Daniel and Rao, Sankeerth},
  title =	{{A PRG for Boolean PTF of Degree 2 with Seed Length Subpolynomial in epsilon and Logarithmic in n}},
  booktitle =	{33rd Computational Complexity Conference (CCC 2018)},
  pages =	{2:1--2:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-069-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{102},
  editor =	{Servedio, Rocco A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.2},
  URN =		{urn:nbn:de:0030-drops-88861},
  doi =		{10.4230/LIPIcs.CCC.2018.2},
  annote =	{Keywords: Pseudorandomness, Polynomial Threshold Functions}
}

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Documents authored by Rao, Sankeerth

Torus Polynomials: An Algebraic Approach to ACC Lower Bounds

Abstract

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A PRG for Boolean PTF of Degree 2 with Seed Length Subpolynomial in epsilon and Logarithmic in n

Abstract

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