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On monotone formulae with restricted depth

Authors:

Wolfgang J. Paul,

Nicholas Pippenger,

Mihalis YannakakisAuthors Info & Claims

STOC '84: Proceedings of the sixteenth annual ACM symposium on Theory of computing

Pages 480 - 487

https://doi.org/10.1145/800057.808717

Published: 01 December 1984 Publication History

Abstract

We prove a hierarchy theorem for the representation of monotone Boolean functions by monotone formulae with restricted depth. Specifically, we show that there are functions with π_k-formula of size n for which every σ_k-formula has size exp ω(n^1/(k−1)). A similar lower bound applies to concrete functions such as transitive closure and clique. We also show that any function with a formula of size n (and any depth) has a σ_k-formula of size exp o(n^1/(k−1)). Thus our hierarchy theorem is the best possible.

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Cited By

Emdin GKulikov AMihajlin ISlezkin N(2024)CNF Encodings of Symmetric FunctionsTheory of Computing Systems10.1007/s00224-024-10168-wOnline publication date: 26-Mar-2024
https://doi.org/10.1007/s00224-024-10168-w
Lecomte VRamakrishnan PTan L(2022)The composition complexity of majorityProceedings of the 37th Computational Complexity Conference10.4230/LIPIcs.CCC.2022.19(1-26)Online publication date: 20-Jul-2022
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2022.19
de Rezende SGöös MRobere R(2022)Guest ColumnACM SIGACT News10.1145/3532737.353274653:1(59-82)Online publication date: 20-Apr-2022
https://dl.acm.org/doi/10.1145/3532737.3532746
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Index Terms

On monotone formulae with restricted depth
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Paths and connectivity problems
2. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes

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cover image ACM Conferences

STOC '84: Proceedings of the sixteenth annual ACM symposium on Theory of computing

December 1984

547 pages

ISBN:0897911334

DOI:10.1145/800057

Chairman:
Richard DeMillo

Copyright © 1984 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 01 December 1984

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Cited By

Emdin GKulikov AMihajlin ISlezkin N(2024)CNF Encodings of Symmetric FunctionsTheory of Computing Systems10.1007/s00224-024-10168-wOnline publication date: 26-Mar-2024
https://doi.org/10.1007/s00224-024-10168-w
Lecomte VRamakrishnan PTan L(2022)The composition complexity of majorityProceedings of the 37th Computational Complexity Conference10.4230/LIPIcs.CCC.2022.19(1-26)Online publication date: 20-Jul-2022
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2022.19
de Rezende SGöös MRobere R(2022)Guest ColumnACM SIGACT News10.1145/3532737.353274653:1(59-82)Online publication date: 20-Apr-2022
https://dl.acm.org/doi/10.1145/3532737.3532746
Charikar MMa WTan LMiller ACensor-Hillel K(2021)Brief AnnouncementProceedings of the 2021 ACM Symposium on Principles of Distributed Computing10.1145/3465084.3467951(431-433)Online publication date: 21-Jul-2021
https://dl.acm.org/doi/10.1145/3465084.3467951
Yehudayoff A(2020)Pointer chasing via triangular discriminationCombinatorics, Probability and Computing10.1017/S0963548320000085(1-10)Online publication date: 15-May-2020
https://doi.org/10.1017/S0963548320000085
Limaye NSreenivasaiah KSrinivasan STripathi UVenkitesh SCharikar MCohen E(2019)A fixed-depth size-hierarchy theorem for AC0[⊕] via the coin problemProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing10.1145/3313276.3316339(442-453)Online publication date: 23-Jun-2019
https://dl.acm.org/doi/10.1145/3313276.3316339
Meir OWigderson A(2019)Prediction from Partial Information and Hindsight, with Application to Circuit Lower BoundsComputational Complexity10.1007/s00037-019-00177-428:2(145-183)Online publication date: 1-Jun-2019
https://dl.acm.org/doi/10.1007/s00037-019-00177-4
Håstad JRossman BServedio RTan L(2017)An Average-Case Depth Hierarchy Theorem for Boolean CircuitsJournal of the ACM10.1145/309579964:5(1-27)Online publication date: 29-Aug-2017
https://dl.acm.org/doi/10.1145/3095799
Chen XOliveira IServedio RTan LWichs DMansour Y(2016)Near-optimal small-depth lower bounds for small distance connectivityProceedings of the forty-eighth annual ACM symposium on Theory of Computing10.1145/2897518.2897534(612-625)Online publication date: 19-Jun-2016
https://dl.acm.org/doi/10.1145/2897518.2897534
de Rezende SNordstrom JVinyals M(2016)How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity)2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2016.40(295-304)Online publication date: Oct-2016
https://doi.org/10.1109/FOCS.2016.40
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