Correlation bounds against monotone NC1

Published

Conference Paper

© Benjamin Rossman; licensed under Creative Commons License CC-BY. This paper gives the first correlation bounds under product distributions, including the uniform distribution, against the class mNC1 of polynomial-size O(log n)-depth monotone circuits. Our main theorem, proved using the pathset complexity framework introduced in [56], shows that the average-case k-CYCLE problem (on Erdos-Rényi random graphs with an appropriate edge density) is 1/2+ 1/poly(n) hard for mNC1. Combining this result with O'Donnell's hardness amplification theorem [43], we obtain an explicit monotone function of n variables (in the class mSAC1) which is 1/2+n-1/2+∈ hard for mNC1 under the uniform distribution for any desired constant ∈ > 0. This bound is nearly best possible, since every monotone function has agreement 1/2 + Ω(log n√n) with some function in mNC1 [44]. Our correlation bounds against mNC1 extend smoothly to non-monotone NC1 circuits with a bounded number of negation gates. Using Holley's monotone coupling theorem [30], we prove the following lemma: with respect to any product distribution, if a balanced monotone function f is 1/2 + δ hard for monotone circuits of a given size and depth, then f is 1/2 + (2t+1 - 1)δ hard for (non-monotone) circuits of the same size and depth with at most t negation gates. We thus achieve a lower bound against NC1 circuits with (1/2 - ∈) log n negation gates, improving the previous record of 1/6 log log n [7]. Our bound on negations is "half" optimal, since dlog(n + 1)e negation gates are known to be fully powerful for NC1 [3, 21].

Full Text

Duke Authors

Cited Authors

  • Rossman, B

Published Date

  • June 1, 2015

Published In

Volume / Issue

  • 33 /

Start / End Page

  • 392 - 411

International Standard Serial Number (ISSN)

  • 1868-8969

International Standard Book Number 13 (ISBN-13)

  • 9783939897811

Digital Object Identifier (DOI)

  • 10.4230/LIPIcs.CCC.2015.392

Citation Source

  • Scopus