2009/04/14 by Saha, Chandan, Saptharishi, Ramprasad, Saxena, Nitin
#Computational Complexity (cs.CC) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.0904.2058
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 U2(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 U2(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.