Download e-book for kindle: Analysis of Boolean Functions by Ryan O'Donnell

By Ryan O'Donnell

ISBN-10: 1107038324

ISBN-13: 9781107038325

Boolean services are probably the main uncomplicated gadgets of research in theoretical desktop technological know-how. additionally they come up in different parts of arithmetic, together with combinatorics, statistical physics, and mathematical social selection. the sector of research of Boolean services seeks to appreciate them through their Fourier remodel and different analytic equipment. this article provides a radical assessment of the sector, starting with the main simple definitions and continuing to complex themes equivalent to hypercontractivity and isoperimetry. every one bankruptcy features a "highlight software" similar to Arrow's theorem from economics, the Goldreich-Levin set of rules from cryptography/learning conception, Håstad's NP-hardness of approximation effects, and "sharp threshold" theorems for random graph homes. The e-book comprises approximately 450 routines and will be used because the foundation of a one-semester graduate direction. it's going to entice complex undergraduates, graduate scholars, and researchers in desktop technological know-how conception and comparable mathematical fields.

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Symmetrically and ordering Fourier characters χS according to |S| rather than max(S). Bonami also obtained the first hypercontractivity result for the Boolean cube. This proved to be a crucial tool for analysis of Boolean functions; see Chapter 9. For an early survey on Walsh series, see Balashov and Rubinshtein (Balashov and Rubinshtein, 1973). Turning to Boolean functions and computer science, the idea of using Boolean logic to study “switching functions” (as engineers originally called Boolean functions) dates to the late 1930s and is usually credited to Nakashima (Nakashima, 1935), Shannon (Shannon, 1937), and Shestakov (Shestakov, 1938).

13. The variance of f : {−1, 1}n → R is Var[f ] = f − E[f ], f − E[f ] = E[f 2 ] − E[f ]2 = f (S)2 . 12. 14. For f : {−1, 1}n → {−1, 1}, Var[f ] = 1 − E[f ]2 = 4 Pr[f (x) = 1] Pr[f (x) = −1] ∈ [0, 1]. In particular, a Boolean-valued function f has variance 1 if it’s unbiased and variance 0 if it’s constant. More generally, the variance of a Boolean-valued function is proportional to its “distance from being constant”. 15. Let f : {−1, 1}n → {−1, 1}. Then 2 ≤ Var[f ] ≤ 4 , where = min{dist(f, 1), dist(f, −1)}.

Show that q(x) = 12 − 12 p(1 − 2x1 , . . , 1 − 2xn ). 10 Let f : {−1, 1}n → R be not identically 0. , max{|S| : f (S) = 0}. 20 1 Boolean Functions and the Fourier Expansion (a) Show that deg(f ) = deg(a + bf ) for any a, b ∈ R (assuming b = 0, a + bf = 0). (b) Show that deg(f ) ≤ k if and only if f is a real linear combination of functions g1 , . . , gs , each of which depends on at most k input coordinates. 1 have “nontrivial” degree? 11 Suppose that f : {−1, 1}n → {−1, 1} has deg(f ) = k ≥ 1.

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Analysis of Boolean Functions by Ryan O'Donnell

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