Exponential Lower Bounds for Some Restricted Depth Five Powering Circuits

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Suryajith Chillara, Christian Engels, B. V. Raghavendra Rao, Ramprasad Saptharishi and Karteek Sreenivasiah

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  • Journal: Chicago Journal of Theoretical Computer Science (accepted)


Depth five powering circuits are arithmetic circuits of the form $\Sigma\wedge\Sigma\wedge\Sigma$ where $\Sigma$ 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 each $Q_i$ is a sum of powers of linear polynomials. These circuits are a natural generalization of the well known class of depth three powering circuits ($\Sigma\wedge\Sigma$ circuits). In this paper, we study the complexity of some restricted classes of depth five powering circuits that compute the monomial $x_1x_2\cdots x_n$. The restrictions we study are on the fan-ins of the $\Sigma$ and $\wedge$ gates (denoted by $\Sigma^{[a]}$ and $\wedge^{[d]}$) while the bottom $\Sigma$ gates compute homogeneous linear polynomials.

We prove size lower bound of $2^{\Omega(n)}$ against the following classes of depth five powering circuits that compute the monomial $x_1x_2\cdots x_n$.

  • The $\Sigma\wedge\Sigma^{[m]}\wedge^{[\ge d]}\Sigma_{\text{Hom}}$ circuits where $d\geq 8$ and $m< \frac{0.29n}{d^2}2^{0.955d}$.
  • The $\Sigma\wedge\Sigma^{{r}}\wedge^{[= d]}\Sigma_{\text{Hom}}$ circuits where $r < \epsilon\cdot d$ is the rank of the linear forms under that sum gate, $\epsilon >0$ is a constant and $d\geq 8$.

From this, we infer that the monomial can be computed by a $\Sigma\wedge\Sigma^{[=2^{\sqrt{n}}]}\wedge^{[=\sqrt{n}]}\Sigma_{\text{Hom}}$ (or a $\Sigma\wedge\Sigma^{{=\sqrt{n}}}\wedge^{[=\sqrt{n}]}\Sigma_{\text{Hom}}$) circuit of size $2^{O(\sqrt{n})}$ but any $\Sigma\wedge\Sigma^{[\leq 2^{0.955\sqrt{n}}]}\wedge^{[\ge \sqrt{n}]}\Sigma_{\text{Hom}}$ (or a $\Sigma\wedge\Sigma^{{\leq \epsilon\sqrt{n}}}\wedge^{[= \sqrt{n}]}\Sigma_{\text{Hom}}$) circuit computing it must be of size $2^{\Omega(n)}$.

Our results show how 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 $\Sigma\wedge\Sigma\wedge\Sigma$ circuits.