research-article

Quantified derandomization of linear threshold circuits

Author:

Roei TellAuthors Info & Claims

STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

Pages 855 - 865

https://doi.org/10.1145/3188745.3188822

Published: 20 June 2018 Publication History

Abstract

One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for TC⁰, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for TC⁰. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of TC⁰ circuits of depth d>2.

Our first main result is a quantified derandomization algorithm for TC⁰ circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a TC⁰ circuit C over n input bits with depth d and n^1+exp(−d) wires, runs in almost-polynomial-time, and distinguishes between the case that C rejects at most 2^{n^1−1/5d} inputs and the case that C accepts at most 2^{n^1−1/5d} inputs. In fact, our algorithm works even when the circuit C is a linear threshold circuit, rather than just a TC⁰ circuit (i.e., C is a circuit with linear threshold gates, which are stronger than majority gates).

Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of TC⁰, and would consequently imply that NEXP⊈TC⁰. Specifically, if there exists a quantified derandomization algorithm that gets as input a TC⁰ circuit with depth d and n^1+O(1/d) wires (rather than n^1+exp(−d) wires), runs in time at most 2^{n^exp(−d)}, and distinguishes between the case that C rejects at most 2^{n^1−1/5d} inputs and the case that C accepts at most 2^{n^1−1/5d} inputs, then there exists an algorithm with running time 2^{n^1−Ω(1)} for standard derandomization of TC⁰.

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Hatami PHoza WTal ATell RSaha BServedio R(2023)Depth-𝑑 Threshold Circuits vs. Depth-(𝑑+1) AND-OR TreesProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585216(895-904)Online publication date: 2-Jun-2023
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Index Terms

Quantified derandomization of linear threshold circuits
1. Theory of computation
  1. Computational complexity and cryptography
    1. Circuit complexity
  2. Randomness, geometry and discrete structures
    1. Pseudorandomness and derandomization

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cover image ACM Conferences

STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

June 2018

1332 pages

ISBN:9781450355599

DOI:10.1145/3188745

General Chairs:
Ilias Diakonikolas
University of Southern California, USA
,
David Kempe
University of Southern California, USA
,
Program Chair:
Monika Henzinger
University of Vienna, Austria

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Published: 20 June 2018

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https://doi.org/10.1109/FOCS61266.2024.00140
Hatami PHoza WTal ATell RSaha BServedio R(2023)Depth-𝑑 Threshold Circuits vs. Depth-(𝑑+1) AND-OR TreesProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585216(895-904)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585216
Doron DMoshkovitz DOh JZuckerman D(2022)Nearly Optimal Pseudorandomness from HardnessJournal of the ACM10.1145/355530769:6(1-55)Online publication date: 17-Nov-2022
https://dl.acm.org/doi/10.1145/3555307
Fan ZLi JYang TLeonardi SGupta A(2022)The exact complexity of pseudorandom functions and the black-box natural proof barrier for bootstrapping results in computational complexityProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520010(962-975)Online publication date: 9-Jun-2022
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Hatami PHoza WTal ATell R(2022)Fooling Constant-Depth Threshold Circuits (Extended Abstract)2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00019(104-115)Online publication date: Feb-2022
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Chen LTell RKhuller SVassilevska Williams V(2021)Simple and fast derandomization from very hard functions: eliminating randomness at almost no costProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451059(283-291)Online publication date: 15-Jun-2021
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Chen LJin CWilliams RMakarychev KMakarychev YTulsiani MKamath GChuzhoy J(2020)Sharp threshold results for computational complexityProceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3357713.3384283(1335-1348)Online publication date: 22-Jun-2020
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