# Analysis of Boolean Functions by Ryan O'Donnell

By Ryan O'Donnell

Boolean features are probably the main uncomplicated items of analysis in theoretical machine technological know-how. additionally they come up in different components of arithmetic, together with combinatorics, statistical physics, and mathematical social selection. the sector of study of Boolean capabilities seeks to appreciate them through their Fourier remodel and different analytic tools. this article provides a radical evaluate of the sector, starting with the main easy definitions and continuing to complex themes equivalent to hypercontractivity and isoperimetry. each one bankruptcy features a "highlight software" comparable to 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 ebook contains approximately 450 routines and will be used because the foundation of a one-semester graduate path. it's going to attract complicated undergraduates, graduate scholars, and researchers in machine technology conception and similar mathematical fields.

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The work of Bellare et al. 7. 28. , 2010) for further slight improvement. , 2006); the hemi-icosahedron function was introduced by Kushilevitz (Nisan and Wigderson, 1995). 12 is due to Lechner (Lechner, 1963). 2 Basic Concepts and Social Choice In this chapter we introduce a number of important basic concepts including influences and noise stability. Many of these concepts are nicely motivated using the language of social choice. The chapter is concluded with Kalai’s Fourier-based proof of Arrow’s Theorem.

Since ρ < ρ for all k > 1, noise stability is maximized if all of f ’s Fourier weight is on degree 1. 19(a). For a fixed function f , it’s often interesting to see how Stabρ [f ] varies as a function of ρ. 49 we see that Stabρ [f ] is a (univariate) polynomial with nonnegative coefficients; in particular, it’s an increasing function of ρ on [0, 1]. 51. For f : {−1, 1}n → R, d Stabρ [f ] dρ ρ=0 d Stabρ [f ] dρ ρ=1 = W1 [f ], = I[f ]. For f : {−1, 1}n → {−1, 1} we have that NSδ [f ] is an increasing function of δ on [0, 1/2], and the second identity is equivalent to d NSδ [f ] dδ δ=0 = I[f ].

45. It’s good to think of the set J in this proposition as the “notable” coordinates for function f . , the parity function χ[n] has all n of its influences equal to 1). 5. 33). Unfortunately, as soon as there are 3 (or more) candidates the problem of social choice becomes much more difficult. For example, suppose we have candidates a, b, and c, and each of n voters has a ranking of them. 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.