The Spectrum of Small DeMorgan Formulas.
Anat Ganor, Ilan Komargodski, Ran Raz · Electronic colloquium on computational complexity · 2012
We show a connection between the deMorgan formula size of a Boolean function and the noise stability of the function. Using this connection, we show that the Fourier spectrum of any balanced Boolean function computed by a deMorgan formula of size s is concentrated on coefficients of degree up to O( √ s). These results have several applications that apply to any function f that can be computed by a deMorgan formula of size s. First, we get that f can be approximated (in L2-norm) with constant error by a polynomial of degree O( √ s). Second, we show an upper bound of O( √ s) on the average sensitivity of f . Our main result stems from a generalization of Khrapchenko’s bound [Khr71], that might be of independent interest, and some Fourier analysis on the Boolean cube. Previous works prove that any function f : {0, 1}n → {0, 1} that can be computed by a deMorgan formula of size s, can be approximated point-wise by a polynomial of degree O(s/2+o(1)) with constant point-wise error. We note that this result can be easily extended to have a polynomial of degree O(t ·s/2+o(1)) that approximates f with point-wise error 2−t, for any t > 0. This was shown in a long line of results in quantum complexity, including [BBC+01] and [FGG08, ACR+07, RS08, Rei09].