On $\varSigma \wedge \varSigma \wedge \varSigma$ Circuits: The Role of Middle $\varSigma$ Fan-In, Homogeneity and Bottom Degree

Abstract

We study polynomials computed by depth five $\varSigma \wedge \varSigma \wedge \varSigma$ arithmetic circuits where ‘$\varSigma$’ and ‘$\wedge$’ represent gates that compute sum and power of their inputs respectively. Such circuits compute polynomials of the form $\sum _{i=1}^t Q_i^{\alpha _{i}}$, where $Q_i = \sum _{j=1}^{r_i}\ell _{ij}^{d_{ij}}$ where $\ell _{ij}$ are linear forms and $r_i$, $\alpha _{i}$, $t>0$. These circuits are a natural generalization of the well known class of $\varSigma \wedge \varSigma$ circuits and received significant attention recently. We prove an exponential lower bound for the monomial $x_1\cdots x_n$ against depth five $\varSigma \wedge \varSigma ^{[\le n]}\wedge ^{[\ge 21]}\varSigma$ and $\varSigma \wedge \varSigma ^{[\le 2^{\sqrt{n}/1000}]}\wedge ^{[\ge \sqrt{n}]}\varSigma$ arithmetic circuits where the bottom $\varSigma$ gate is homogeneous.

Our results show that the fan-in of the middle $\varSigma$ gates, the degree of the bottom powering gates and the homogeneity at the bottom $\varSigma$ gates play a crucial role in the computational power of $\varSigma \wedge \varSigma \wedge \varSigma$ circuits.

K. Sreenivasaiah—This work was done while the author was working at Max Planck Institute for Informatics, Saarbrücken supported by IMPECS post doctoral fellowship.