Pseudorandomness for Read-Once, Constant-Depth Circuits
(1504.04675)Abstract
For Boolean functions computed by read-once, depth-$D$ circuits with unbounded fan-in over the de Morgan basis, we present an explicit pseudorandom generator with seed length $\tilde{O}(\log{D+1} n)$. The previous best seed length known for this model was $\tilde{O}(\log{D+4} n)$, obtained by Trevisan and Xue (CCC 13) for all of $AC^0$ (not just read-once). Our work makes use of Fourier analytic techniques for pseudorandomness introduced by Reingold, Steinke, and Vadhan (RANDOM
13) to show that the generator of Gopalan et al. (FOCS `12) fools read-once $AC0$. To this end, we prove a new Fourier growth bound for read-once circuits, namely that for every $F: {0,1}n\to{0,1}$ computed by a read-once, depth-$D$ circuit, \begin{equation}\sum_{s\subseteq[n], |s|=k}|\hat{F}[s]|\le O(\log{D-1}n)k,\end{equation} where $\hat{F}$ denotes the Fourier transform of $F$ over $\mathbb{Z}n_2$.
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