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 (ΣΠΣ) arithmetic circuits over a field F is polynomial time equivalent to identity testing of depth 2 (ΠΣ) arithmetic circuits over U2(F), the algebra of upper-triangular 2 × 2 matrices with entries from F. Such a connection is a bit surprising since we also show that, as computational models, ΠΣ circuits over U2(F) are strictly ‘weaker’ than ΣΠΣ circuits over F. The equivalence further implies that PIT of ΣΠΣ circuits reduces to PIT of width-2 commutative Algebraic Branching Programs(ABP). Further, we give a deterministic polynomial time identity testing algorithm for a ΠΣ 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 ΠΣ circuits is at least as hard as that of ΣΠΣ circuits over F.

• Type: Conference
• Year: 2009
• Where: FSTTCS
• Authors: Chandan Saha, Ramprasad Saptharishi, Nitin Saxena
• Keywords: Arithmetic Circuits, FSTTCS 2009, Identity Testing, Theoretical Computer Science, ΣΠΣ Circuits