Correlation bounds against monotone NC1
© 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 , 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 , 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 . 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 , 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 . Our bound on negations is "half" optimal, since dlog(n + 1)e negation gates are known to be fully powerful for NC1 [3, 21].
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