TY - GEN

T1 - Near-optimal pseudorandom generators for constant-depth read-once formulas

AU - Doron, Dean

AU - Hatami, Pooya

AU - Hoza, William M.

N1 - Funding Information:
Dean Doron: NSF Grant CCF-1705028. Pooya Hatami: Simons Investigator Award (#409864, David Zuckerman). William M. Hoza: NSF GRFP under Grant DGE-1610403 and a Harrington Fellowship from UT Austin.
Funding Information:
Funding Dean Doron: NSF Grant CCF-1705028. Pooya Hatami: Simons Investigator Award (#409864, David Zuckerman). William M. Hoza: NSF GRFP under Grant DGE-1610403 and a Harrington Fellowship from UT Austin.
Publisher Copyright:
© Dean Doron, Pooya Hatami, and William M. Hoza; licensed under Creative Commons License CC-BY 34th Computational Complexity Conference (CCC 2019).

PY - 2019/7/1

Y1 - 2019/7/1

N2 - 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.

AB - 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.

KW - Constant-depth formulas

KW - Explicit constructions

KW - Pseudorandom generators

UR - http://www.scopus.com/inward/record.url?scp=85070671257&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.CCC.2019.16

DO - 10.4230/LIPIcs.CCC.2019.16

M3 - Conference contribution

AN - SCOPUS:85070671257

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 34th Computational Complexity Conference, CCC 2019

A2 - Shpilka, Amir

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 34th Computational Complexity Conference, CCC 2019

Y2 - 18 July 2019 through 20 July 2019

ER -