research-article

Breaking the minsky-papert barrier for constant-depth circuits

Author:

Alexander A. SherstovAuthors Info & Claims

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

Pages 223 - 232

https://doi.org/10.1145/2591796.2591871

Published: 31 May 2014 Publication History

Abstract

The threshold degree of a Boolean function f is the minimum degree of a real polynomial p that represents f in sign: f(x) ≡ sgn p(x). In a seminal 1969 monograph, Minsky and Papert constructed a polynomial-size constant-depth {∧, ∨)-circuit in n variables with threshold degree Ω(n^1/3). This bound underlies some of today's strongest results on constant-depth circuits. It has been an open problem (O'Donnell and Servedio, STOC 2003) to improve Minsky and Papert's bound to n^Ω(1)+1/3.

We give a detailed solution to this problem. For any fixed k ≥ 1, we construct an {∧, ∨)-formula of size n and depth k with threshold degree Ω(n k-1/2k-1). This lower bound nearly matches a known O(√n) bound for arbitrary formulas, and is exactly tight for regular formulas. Our result proves a conjecture due to O'Donnell and Servedio (STOC 2003) and a different conjecture due to Bun and Thaler (2013). Applications to communication complexity and computational learning are given.

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Cited By

Bun MKothari RThaler JDiakonikolas IKempe DHenzinger M(2018)The polynomial method strikes back: tight quantum query bounds via dual polynomialsProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3188745.3188784(297-310)Online publication date: 20-Jun-2018
https://dl.acm.org/doi/10.1145/3188745.3188784
Bouland AChen LHolden DThaler JVasudevan P(2017)On the Power of Statistical Zero Knowledge2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2017.71(708-719)Online publication date: Oct-2017
https://doi.org/10.1109/FOCS.2017.71
Bun MThaler J(2017)A Nearly Optimal Lower Bound on the Approximate Degree of AC^02017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2017.10(1-12)Online publication date: Oct-2017
https://doi.org/10.1109/FOCS.2017.10
Show More Cited By

Index Terms

Breaking the minsky-papert barrier for constant-depth circuits
1. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes

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cover image ACM Conferences

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

May 2014

984 pages

ISBN:9781450327107

DOI:10.1145/2591796

Program Chair:
David Shmoys
Cornell University

Copyright © 2014 ACM.

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Publication History

Published: 31 May 2014

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STOC '14

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STOC '14: Symposium on Theory of Computing

May 31 - June 3, 2014

New York, New York

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STOC '14 Paper Acceptance Rate 91 of 319 submissions, 29%;

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

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57th Annual ACM Symposium on Theory of Computing (STOC 2025)

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Cited By

Bun MKothari RThaler JDiakonikolas IKempe DHenzinger M(2018)The polynomial method strikes back: tight quantum query bounds via dual polynomialsProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3188745.3188784(297-310)Online publication date: 20-Jun-2018
https://dl.acm.org/doi/10.1145/3188745.3188784
Bouland AChen LHolden DThaler JVasudevan P(2017)On the Power of Statistical Zero Knowledge2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2017.71(708-719)Online publication date: Oct-2017
https://doi.org/10.1109/FOCS.2017.71
Bun MThaler J(2017)A Nearly Optimal Lower Bound on the Approximate Degree of AC^02017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2017.10(1-12)Online publication date: Oct-2017
https://doi.org/10.1109/FOCS.2017.10
Göös MJayram T(2016)A composition theorem for conical juntasProceedings of the 31st Conference on Computational Complexity10.5555/2982445.2982450(1-16)Online publication date: 29-May-2016
https://dl.acm.org/doi/10.5555/2982445.2982450
Alman JChan TWilliams R(2016)Polynomial Representations of Threshold Functions and Algorithmic Applications2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2016.57(467-476)Online publication date: Oct-2016
https://doi.org/10.1109/FOCS.2016.57
Bun MThaler J(2015)Hardness Amplification and the Approximate Degree of Constant-Depth CircuitsAutomata, Languages, and Programming10.1007/978-3-662-47672-7_22(268-280)Online publication date: 20-Jun-2015
https://doi.org/10.1007/978-3-662-47672-7_22

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