§4.5: Highlight: LMN’s work on constant-depth circuits

Having derived strong results about the Fourier spectrum of small DNFs and CNFs, we will now extend to the case of constant-depth circuits. We begin by describing how Håstad applied his Switching Lemma to constant-depth circuits. We then describe some Fourier-theoretic consequences coming from a very early (1989) work in analysis of boolean functions [...]

§4.4: Håstad’s Switching Lemma and the spectrum of DNFs

Let’s further investigate how random restrictions can simplify DNF formulas.