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TR09-036 | 14th April 2009 00:00

The Power of Depth 2 Circuits over Algebras


Authors: Chandan Saha, Ramprasad Saptharishi, Nitin Saxena
Publication: 4th May 2009 14:32
Downloads: 3263


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 U_2(F), the algebra of upper-
triangular 2 x 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 U_2(F) are strictly `weaker' than Sigma-Pi-Sigma
circuits over F. The equivalence further shows that PIT of depth 3
arithmetic circuits reduces to PIT of width-2 planar commutative
Algebraic Branching Programs (ABP). Thus, identity testing for
commutative ABPs is interesting even in the case of width-2.
Further, we give a deterministic polynomial time identity testing
algorithm for a Pi-Sigma circuit over any constant dimensional
commutative algebra over F. While over commutative algebras of
polynomial dimension, identity testing is at least as hard as that of
Sigma-Pi-Sigma circuits over F.

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