TR09-036 Authors: Chandan Saha, Ramprasad Saptharishi, Nitin Saxena

Publication: 4th May 2009 14:32

Downloads: 3263

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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.