§6.3: Constructions of various pseudorandom functions

In this section we give some constructions of boolean functions with strong pseudorandomness properties.


§6.2: ${\mathbb F}_2$-polynomials

We began our study of boolean functions in Chapter 1.2 by considering their polynomial representations over the real field. In this section we take a brief look at their polynomial representations over the field ${\mathbb F}_2$, with $\mathsf{False}$, $\mathsf{True}$ being represented by $0, 1 \in {\mathbb F}_2$ as usual. Note that in the field ${\mathbb [...]

§6.1: Notions of pseudorandomness

The most obvious spectral property of a truly random function $\boldsymbol{f} : \{-1,1\}^n \to \{-1,1\}$ is that all of its Fourier coefficients are very small (as we saw in Exercise 5.8).


§5.5: Highlight: Peres’s Theorem

Theorem 14 says that if $f$ is an unbiased linear threshold function $f(x) = \mathrm{sgn}(a_1 x_1 + \cdots + a_n x_n)$ in which all $a_i$’s are “small” then the noise stability $\mathbf{Stab}_\rho[f]$ is at least (roughly) $\frac{2}{\pi} \arcsin \rho$. Rephrasing in terms of noise sensitivity, this means $\mathbf{NS}_\delta[f]$ is at most (roughly) $\tfrac{2}{\pi} \sqrt{\delta} [...]

§5.4: Degree-1 weight

In this section we prove two theorems about the degree-$1$ Fourier weight of boolean functions: \[ \mathbf{W}^{1}[f] = \sum_{i=1}^n \widehat{f}(i)^2. \]


§5.3: The Fourier coefficients of Majority

In this section we will analyze the Fourier coefficients of $\mathrm{Maj}_n$. In fact, we give an explicit formula for them in Theorem 16 below. But most of the time this formula is not too useful; instead, it’s better to understand the Fourier coefficients of $\mathrm{Maj}_n$ asymptotically as $n \to \infty$.


§5.2: Majority, and the Central Limit Theorem

Majority is one of the more important functions in boolean analysis and its study motivates the introduction of one of the more important tools: the Central Limit Theorem (CLT).


§5.1: Linear threshold functions and polynomial threshold functions

Recall from Chapter 2.1 that a linear threshold function (abbreviated LTF) is a boolean-valued function $f : \{-1,1\}^n \to \{-1,1\}$ that can be represented as \begin{equation} \label{eqn:generic-LTF} f(x) = \mathrm{sgn}(a_0 + a_1 x_1 + \cdots + a_n x_n) \end{equation} for some constants $a_0, a_1, \dots, a_n \in {\mathbb R}$.


§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.