When quoting this document, please refer to the following
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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### 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},
`