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The Power of Depth 2 Circuits over Algebras

Authors Chandan Saha, Ramprasad Saptharishi, Nitin Saxena



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Chandan Saha
Ramprasad Saptharishi
Nitin Saxena

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Chandan Saha, Ramprasad Saptharishi, and Nitin Saxena. The Power of Depth 2 Circuits over Algebras. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 371-382, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009) https://doi.org/10.4230/LIPIcs.FSTTCS.2009.2333

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 $\mathbb{F}$.

Subject Classification

Keywords
  • Polynomial identity testing
  • depth 3 circuits
  • matrix algebras
  • local rings

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