Near-optimal pseudorandom generators for constant-depth read-once formulas

Dean Doron, Pooya Hatami, William M. Hoza

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations

Abstract

We give an explicit pseudorandom generator (PRG) for read-once AC0, i.e., constant-depth read-once formulas over the basis {∧, ∨, ¬} with unbounded fan-in. The seed length of our PRG is Oe(log(n/ε)). Previously, PRGs with near-optimal seed length were known only for the depth-2 case [22]. For a constant depth d > 2, the best prior PRG is a recent construction by Forbes and Kelley with seed length Oe(log2 n + log n log(1/ε)) for the more general model of constant-width read-once branching programs with arbitrary variable order [17]. Looking beyond read-once AC0, we also show that our PRG fools read-once AC0[⊕] with seed length Oe(t + log(n/ε)), where t is the number of parity gates in the formula. Our construction follows Ajtai and Wigderson’s approach of iterated pseudorandom restrictions [1]. We assume by recursion that we already have a PRG for depth-d AC0 formulas. To fool depth-(d+ 1) AC0 formulas, we use the given PRG, combined with a small-bias distribution and almost k-wise independence, to sample a pseudorandom restriction. The analysis of Forbes and Kelley [17] shows that our restriction approximately preserves the expectation of the formula. The crux of our work is showing that after poly(log log n) independent applications of our pseudorandom restriction, the formula simplifies in the sense that every gate other than the output has only polylog n remaining children. Finally, as the last step, we use a recent PRG by Meka, Reingold, and Tal [32] to fool this simpler formula.

Original languageEnglish
Title of host publication34th Computational Complexity Conference, CCC 2019
EditorsAmir Shpilka
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771160
DOIs
StatePublished - 1 Jul 2019
Externally publishedYes
Event34th Computational Complexity Conference, CCC 2019 - New Brunswick, United States
Duration: 18 Jul 201920 Jul 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume137
ISSN (Print)1868-8969

Conference

Conference34th Computational Complexity Conference, CCC 2019
Country/TerritoryUnited States
CityNew Brunswick
Period18/07/1920/07/19

Keywords

  • Constant-depth formulas
  • Explicit constructions
  • Pseudorandom generators

ASJC Scopus subject areas

  • Software

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