On Σ ∧ Σ ∧ Σ Circuits: The Role of Middle Σ Fan-In, Homogeneity and Bottom Degree

We study polynomials computed by depth five Sigma boolean AND Sigma boolean AND Sigma arithmetic circuits where 'Sigma' and 'boolean AND' represent gates that compute sum and power of their inputs respectively. Such circuits compute polynomials of the form Sigma(t)(i=1) Q(i)(alpha i), where Q(i) = Sigma(ri)(j=1) l(ij)(dij) where l(ij) are linear forms and r(i), alpha(i), t > 0. These circuits are a natural generalization of the well known class of Sigma boolean AND Sigma circuits and received significant attention recently. We prove an exponential lower bound for the monomial x(1) ... x(n) against depth five Sigma boolean AND Sigma([<= n]) boolean AND([>= 21]) Sigma and Sigma boolean AND Sigma([<= 2 root n/1000]) boolean AND([>=root n]) Sigma arithmetic circuits where the bottom Sigma gate is homogeneous. Our results show that the fan-in of the middle Sigma gates, the degree of the bottom powering gates and the homogeneity at the bottom Sigma gates play a crucial role in the computational power of Sigma boolean AND Sigma boolean AND Sigma circuits.