# Correlation bounds against monotone NC1

Conference Paper

This paper gives the first correlation bounds under product distributions, including the uniform distribution, against the class mNC 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 mNC . Combining this result with O'Donnell's hardness amplification theorem [43], we obtain an explicit monotone function of n variables (in the class mSAC ) which is 1/2+n hard for mNC 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 mNC [44]. Our correlation bounds against mNC extend smoothly to non-monotone NC 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 + (2 - 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 NC 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 NC [3, 21]. 1 1 1 -1/2+∈ 1 1 1 1 t+1 1 1

• Rossman, B

• June 1, 2015

• 33 /

• 392 - 411

• 1868-8969

### International Standard Book Number 13 (ISBN-13)

• 9783939897811

### Digital Object Identifier (DOI)

• 10.4230/LIPIcs.CCC.2015.392

• Scopus