Download Analysis of Boolean Functions by Ryan O'Donnell PDF

By Ryan O'Donnell

Boolean services are possibly the main simple items of research in theoretical computing device technology. in addition they come up in different parts of arithmetic, together with combinatorics, statistical physics, and mathematical social selection. the sector of research of Boolean capabilities seeks to appreciate them through their Fourier remodel and different analytic tools. this article provides an intensive assessment of the sector, starting with the main simple definitions and continuing to complicated themes reminiscent of hypercontractivity and isoperimetry. each one bankruptcy incorporates a "highlight program" reminiscent of Arrow's theorem from economics, the Goldreich-Levin set of rules from cryptography/learning thought, Håstad's NP-hardness of approximation effects, and "sharp threshold" theorems for random graph homes. The e-book contains approximately 450 workouts and will be used because the foundation of a one-semester graduate path. it may entice complicated undergraduates, graduate scholars, and researchers in desktop technological know-how concept and similar mathematical fields.

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W − (n − w) = 2w − n. The result follows. Rousseau (Rousseau, 1762) suggested that the ideal voting rule is one which maximizes the number of votes that agree with the outcome. 33. The unique maximizers of i=1 f (i) among all f : {−1, 1}n → {−1, 1} are the majority functions. In particular, √ √ I[f ] ≤ I[Majn ] = 2/π n + O(n−1/2 ) for all monotone f . Proof. 3), n f (i) = E[f (x)(x 1 + x 2 + · · · + x n )] ≤ E[|x 1 + x 2 + · · · + x n |], i=1 x x since f (x) ∈ {−1, 1} always. Equality holds if and only if f (x) = sgn(x1 + · · · + xn ) whenever x1 + · · · + xn = 0.

Show that Var[f 2 ] = 2 2 i=j f (i) f (j ) . 21 Prove that there are no functions f : {−1, 1}n → {−1, 1} with exactly 2 nonzero Fourier coefficients. What about exactly 3 nonzero Fourier coefficients? 26. 23 In this exercise you will prove some basic facts about “distances” between probability distributions. Let ϕ and ψ be probability densities on Fn2 . (a) Show that the total variation distance between ϕ and ψ, defined by dTV (ϕ, ψ) = maxn A⊆F2 is equal to 1 2 ϕ − ψ 1. Pr [ y ∈ A] − Pr [ y ∈ A] , y∼ϕ y∼ψ 22 1 Boolean Functions and the Fourier Expansion (b) Show that the collision probability of ϕ, defined to be Pr y, y ∼ϕ independently [ y = y ], is equal to ϕ 22 /2n .

How should we aggregate these preferences to produce a winning candidate? 42 2 Basic Concepts and Social Choice In his 1785 Essay on the Application of Analysis to the Probability of Majority Decisions (de Condorcet, 1785), Condorcet suggested using the voters’ preferences to conduct the three possible pairwise elections, a vs. b, b vs. c, and c vs. a. This calls for the use of a 2-candidate voting rule f : {−1, 1}n → {−1, 1}; Condorcet suggested f = Majn but we might consider any such rule. Thus a “3-candidate Condorcet election” using f is conducted as follows: Voters’ Preferences #1 #2 #3 ··· a (+1) vs.

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