Pseudorandomness for Read-Once, Constant-Depth Circuits

TL;DR

This paper introduces an improved explicit pseudorandom generator for read-once, constant-depth circuits, utilizing Fourier analysis to achieve a seed length of O(log^{D+1} n), which is an advancement over previous bounds.

Contribution

It provides a new Fourier growth bound for read-once circuits and constructs a more efficient pseudorandom generator with reduced seed length.

Findings

New Fourier growth bound for read-once circuits

Explicit pseudorandom generator with seed length O(log^{D+1} n)

Improved derandomization for read-once, depth-D circuits

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 $AC^0$. 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 $\mathbb{Z}^n_2$.