Revised: August 1, 2022
Published: July 29, 2024
Abstract: [Plain Text Version]
We show how to distinguish circuits with $\log k$ negations (a.k.a. $k$-monotone functions) from uniformly random functions in $\exp\big({\tilde{O}}\big(n^{1/3}k^{2/3}\big)\big)$ time using random samples. The previous best distinguisher, due to the learning algorithm by Blais, Canonne, Oliveira, Servedio, and Tan (RANDOM'15), requires $\exp\big({\tilde{O}}(n^{1/2} k)\big)$ time.
Our distinguishers are based on Fourier analysis on slices of the Boolean cube. We show that some “middle” slices of negation-limited circuits have strong low-degree Fourier concentration and then we apply a variation of the classic Linial, Mansour, and Nisan “Low-Degree algorithm” (JACM'93) on slices. Our techniques also lead to a slightly improved weak learner for negation-limited circuits under the uniform distribution.
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