The Power of Depth 2 Circuits over Algebras
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 $\mathbb{F}$.
Polynomial identity testing
depth 3 circuits
matrix algebras
local rings
371-382
Regular Paper
Chandan
Saha
Chandan Saha
Ramprasad
Saptharishi
Ramprasad Saptharishi
Nitin
Saxena
Nitin Saxena
10.4230/LIPIcs.FSTTCS.2009.2333
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