research-article

Nonuniform ACC Circuit Lower Bounds

Author:

Ryan WilliamsAuthors Info & Claims

Journal of the ACM (JACM), Volume 61, Issue 1

Article No.: 2, Pages 1 - 32

https://doi.org/10.1145/2559903

Published: 01 January 2014 Publication History

Abstract

The class ACC consists of circuit families with constant depth over unbounded fan-in AND, OR, NOT, and MOD_m gates, where m > 1 is an arbitrary constant. We prove the following.

---NEXP, the class of languages accepted in nondeterministic exponential time, does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynomials and other less natural functions.

---E^NP, the class of languages recognized in 2^O(n) time with an NP oracle, doesn’t have nonuniform ACC circuits of 2ⁿ^o(1) size. The lower bound gives an exponential size-depth tradeoff: for every d, m there is a δ > 0 such that E^NP doesn’t have depth-d ACC circuits of size 2ⁿ^δ with MOD_m gates.

Previously, it was not known whether EXP^NP had depth-3 polynomial-size circuits made out of only MOD₆ gates. The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail these lower bounds. The algorithms combine known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a strengthening of the author’s prior work.

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Index Terms

Nonuniform ACC Circuit Lower Bounds
1. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes

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cover image Journal of the ACM

Journal of the ACM Volume 61, Issue 1

January 2014

222 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2578041

Editor:
Victor Vianu
University of California, San Diego

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Publication History

Published: 01 January 2014

Accepted: 01 May 2013

Revised: 01 July 2012

Received: 01 May 2011

Published in JACM Volume 61, Issue 1

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