Article

Separating AC⁰ from depth-2 majority circuits

Author:

Alexander A. SherstovAuthors Info & Claims

STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing

Pages 294 - 301

https://doi.org/10.1145/1250790.1250834

Published: 11 June 2007 Publication History

Abstract

We prove that AC⁰ cannot be efficiently simulated by MAJºMAJ circuits. Namely, we construct an AC⁰ circuit of depth 3 that requires MAJºMAJ circuits of size 2^{Ω(n^1/5)}. This matches Allender's classic result that AC⁰ can be simulated by MAJºMAJºMAJ circuits of quasipolynomial size.

Our proof is based on communication complexity. To obtain the above result, we develop a novel technique for communication lower bounds, the Degree/Discrepancy Theorem. This technique is a separate contribution of our paper. It translates lower bounds on the threshold degree of a Boolean function into upper bounds on the discrepancy of a related function. Upper bounds on the discrepancy, in turn, immediately imply communication lower bounds as well as lower bounds against threshold circuits.

As part of our proof, we use the Degree/Discrepancy Theorem to obtain an explicit AC⁰ circuit of depth 3 that has discrepancy 2^{-Ω(n^1/5)}, under an explicit distribution. This yields the first known AC⁰ function with exponentially small discrepancy. Finally, we apply our work to learning theory, showing that polynomial-size DNF and CNF formulas have margin complexity 2^{Ω(n^1/5)}.

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Kane DRao SServedio R(2018)A PRG for boolean PTF of degree 2 with seed length subpolynomial in ϵ and logarithmic in nProceedings of the 33rd Computational Complexity Conference10.5555/3235586.3235588(1-24)Online publication date: 22-Jun-2018
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Index Terms

Separating AC⁰ from depth-2 majority circuits
1. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes

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cover image ACM Conferences

STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing

June 2007

734 pages

ISBN:9781595936318

DOI:10.1145/1250790

General Chair:
David Johnson
AT&T Labs - Research
,
Program Chair:
Uriel Feige
Microsoft Research and Weizmann Institute

Copyright © 2007 ACM.

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Published: 11 June 2007

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June 11 - 13, 2007

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Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Kane DRao SServedio R(2018)A PRG for boolean PTF of degree 2 with seed length subpolynomial in ϵ and logarithmic in nProceedings of the 33rd Computational Complexity Conference10.5555/3235586.3235588(1-24)Online publication date: 22-Jun-2018
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Amaru LAmaru L(2016)Majority Normal Form Representation and SatisfiabilityNew Data Structures and Algorithms for Logic Synthesis and Verification10.1007/978-3-319-43174-1_5(121-133)Online publication date: 3-Aug-2016
https://doi.org/10.1007/978-3-319-43174-1_5
Lee TZhang S(2010)Composition theorems in communication complexityProceedings of the 37th international colloquium conference on Automata, languages and programming10.5555/1880918.1880971(475-489)Online publication date: 6-Jul-2010
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Подольский ВPodolskii VШерстов АSherstov A(2010)Небольшое уменьшение степени многочлена с заданной знаковой функцией может экспоненциально увеличить его вес и длинуA Small Decrease in the Degree of a Polynomial with a Given Sign Function Can Exponentially Increase Its Weight and LengthМатематические заметкиMatematicheskie Zametki10.4213/mzm873487:6(885-899)Online publication date: 2010
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Chattopadhyay APitassi T(2010)The story of set disjointnessACM SIGACT News10.1145/1855118.185513341:3(59-85)Online publication date: 3-Sep-2010
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Beame PHuynh TPitassi TMitzenmacher MSchulman L(2010)Hardness amplification in proof complexityProceedings of the forty-second ACM symposium on Theory of computing10.1145/1806689.1806703(87-96)Online publication date: 5-Jun-2010
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Beame PHuynh-Ngoc D(2009)Multiparty Communication Complexity and Threshold Circuit Size of AC^02009 50th Annual IEEE Symposium on Foundations of Computer Science10.1109/FOCS.2009.12(53-62)Online publication date: Oct-2009
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Jayram TKopparty SRaghavendra P(2009)On the Communication Complexity of Read-Once AC^0 FormulaeProceedings of the 2009 24th Annual IEEE Conference on Computational Complexity10.1109/CCC.2009.39(329-340)Online publication date: 15-Jul-2009
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Lee TSchechtman GShraibman A(2009)Lower Bounds on Quantum Multiparty Communication ComplexityProceedings of the 2009 24th Annual IEEE Conference on Computational Complexity10.1109/CCC.2009.24(254-262)Online publication date: 15-Jul-2009
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