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)


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

  1. 1.Kyoto UniversityKyotoJapan
  2. 2.IIT MadrasChennaiIndia
  3. 3.Saarland UniversitySaarbrückenGermany

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