License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2015.304
URN: urn:nbn:de:0030-drops-50548
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5054/
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Oliveira, Rafael ; Shpilka, Amir ; Volk, Ben Lee

Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

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Abstract

In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation
between black-box PIT and lower bounds we obtain lower bounds for these models.

For depth-3 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 2*delta/3))). This implies a lower bound of exp(~Omega(n^(1/2))) for depth-3 multilinear formulas, for some explicit polynomial.

For depth-4 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 4*delta/3)). This implies a lower bound of exp(~Omega(n^(1/4))) for depth-4 multilinear formulas, for some explicit polynomial.

A regular formula consists of alternating layers of +,* gates, where all gates at layer i have the same fan-in. We give a
hitting set of size (roughly) exp(n^(1-delta)), for regular depth-d multilinear formulas of size exp(n^delta), where delta = O(1/sqrt(5)^d)). This result implies a lower bound of roughly exp(~Omega(n^(1/sqrt(5)^d))) for such formulas.

We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of
a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known.

Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a
read-once algebraic branching program (from depth-3 formulas we go straight to read-once algebraic branching programs).

BibTeX - Entry

@InProceedings{oliveira_et_al:LIPIcs:2015:5054,
  author =	{Rafael Oliveira and Amir Shpilka and Ben Lee Volk},
  title =	{{Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{304--322},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{David Zuckerman},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5054},
  URN =		{urn:nbn:de:0030-drops-50548},
  doi =		{10.4230/LIPIcs.CCC.2015.304},
  annote =	{Keywords: Arithmetic Circuits, Derandomization, Polynomial Identity Testing}
}

Keywords: Arithmetic Circuits, Derandomization, Polynomial Identity Testing
Collection: 30th Conference on Computational Complexity (CCC 2015)
Issue Date: 2015
Date of publication: 06.06.2015

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