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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10472)

## 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.

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© Springer-Verlag GmbH Germany 2017

## Authors and Affiliations

• Christian Engels
• 1
• B. V. Raghavendra Rao
• 2
Email author
• Karteek Sreenivasaiah
• 3
1. 1.Kyoto UniversityKyotoJapan