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DOI: 10.4230/LIPIcs.FSTTCS.2009.2333
URN: urn:nbn:de:0030-drops-23334
URL: http://drops.dagstuhl.de/opus/volltexte/2009/2333/
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Saha, Chandan ; Saptharishi, Ramprasad ; Saxena, Nitin

The Power of Depth 2 Circuits over Algebras

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Abstract

We study the problem of polynomial identity testing (PIT) for depth $2$ arithmetic circuits over matrix algebra. We show that identity testing of depth $3$ ($\Sigma \Pi \Sigma$) arithmetic circuits over a field $\F$ is polynomial time equivalent to identity testing of depth $2$ ($\Pi \Sigma$) arithmetic circuits over $\mathsf{U}_2(\mathbb{F})$, the algebra of upper-triangular $2\times 2$ matrices with entries from $\F$. Such a connection is a bit surprising since we also show that, as computational models, $\Pi \Sigma$ circuits over $\mathsf{U}_2(\mathbb{F})$ are strictly `weaker' than $\Sigma \Pi \Sigma$ circuits over $\mathbb{F}$. The equivalence further implies that PIT of $\Sigma \Pi \Sigma$ circuits reduces to PIT of width-$2$ commutative \emph{Algebraic Branching Programs}(ABP). Further, we give a deterministic polynomial time identity testing algorithm for a $\Pi \Sigma$ circuit of size $s$ over commutative algebras of dimension $O(\log s/\log\log s)$ over $\F$. Over commutative algebras of dimension $\poly(s)$, we show that identity testing of $\Pi \Sigma$ circuits is at least as hard as that of $\Sigma \Pi \Sigma$ circuits over $\mathbb{F}$.

BibTeX - Entry

@InProceedings{saha_et_al:LIPIcs:2009:2333,
  author =	{Chandan Saha and Ramprasad Saptharishi and Nitin Saxena},
  title =	{{The Power of Depth 2 Circuits over Algebras}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{371--382},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-13-2},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{4},
  editor =	{Ravi Kannan and K. Narayan Kumar},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/2333},
  URN =		{urn:nbn:de:0030-drops-23334},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2009.2333},
  annote =	{Keywords: Polynomial identity testing, depth 3 circuits, matrix algebras, local rings}
}

Keywords: Polynomial identity testing, depth 3 circuits, matrix algebras, local rings
Seminar: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science
Issue Date: 2009
Date of publication: 14.12.2009


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