# Chapter 4 notes

Mansour’s Conjecture is from [Man94]. Even the weaker version would imply that the Kushilevitz–Mansour algorithm learns the class of $\mathrm{poly}(n)$-size DNF with any constant error, using queries, in time $\mathrm{poly}(n)$. In fact, this learning result was subsequently obtained in a celebrated work of Jackson [Jac97], using a different method (which begins with Exercise 4). Nevertheless, the Mansour Conjecture remains important for learning theory since Gopalan, Kalai, and Klivans [GKK08] have shown that it implies the same learning result in the more challenging and realistic model of “agnostic learning”. Theorems 24 and 25 are also due to Mansour [Man95].

The method of random restrictions dates back to Subbotovskaya [Sub61]. Håstad’s Switching Lemma (from [Has87]) is the culmination of a line of work due to Furst, Saxe, and Sipser [FSS84], Ajtai [Ajt83], and Yao [Yao85]. Linial, Mansour, and Nisan proved Lemma 21, which allowed them to deduce the LMN Theorem and its consequences. [LMN89,LMN93] An additional cryptographic application of the LMN Theorem is found in [GR00a]. The strongest lower bound currently known for approximately computing parity in $\mathsf{AC}^0$ is due to Impagliazzo, Matthews, and Paturi [IMP12] and also independently to Håstad.

Theorem 20 and its generalization Theorem 30 are due to Boppana [Bop97]; Linial, Mansour, and Nisan had given the weaker bound $O(\log s)^d$. Exercise 10 is due to Amano [Ama11], and Exercise 15 is due to Talagrand [Tal96].