Saha, Chandan ;
Saptharishi, Ramprasad ;
Saxena, Nitin
The Power of Depth 2 Circuits over Algebras
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 uppertriangular $2\times
2$ 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 = {371382},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897132},
ISSN = {18688969},
year = {2009},
volume = {4},
editor = {Ravi Kannan and K. Narayan Kumar},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2009/2333},
URN = {urn:nbn:de:0030drops23334},
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: 

2009 