By Ryan O'Donnell

Boolean features are probably the main uncomplicated items of research in theoretical computing device technology. additionally they come up in different components of arithmetic, together with combinatorics, statistical physics, and mathematical social selection. the sphere of study of Boolean features seeks to appreciate them through their Fourier remodel and different analytic tools. this article supplies a radical evaluate of the sphere, starting with the main simple definitions and continuing to complex subject matters corresponding to hypercontractivity and isoperimetry. each one bankruptcy contains 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 publication contains approximately 450 workouts and will be used because the foundation of a one-semester graduate path. it may attract complicated undergraduates, graduate scholars, and researchers in laptop technological know-how conception and similar mathematical fields.

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For f : {−1, 1}n → R and i ∈ [n], Inf i [f ] = f (S)2 . S i In other words, the influence of coordinate i on f equals the sum of f ’s Fourier weights on sets containing i. This is another good example of being able to “read off” an interesting combinatorial property of a Boolean function from its Fourier expansion. In the special case that f : {−1, 1}n → {−1, 1} is monotone there is a much simpler way to read off its influences: they are the degree-1 Fourier coefficients. In what follows, we write f (i) in place of f ({i}).

13. The influence of coordinate i on f : {−1, 1}n → {−1, 1} is defined to be the probability that i is pivotal for a random input: Inf i [f ] = Pr x∼{−1,1}n [f (x) = f (x ⊕i )]. 14. For f : {−1, 1}n → {−1, 1}, the influence Inf i [f ] equals the fraction of dimension-i edges in the Hamming cube which are boundary edges. Here (x, y) is a dimension-i edge if y = x ⊕i ; it is a boundary edge if f (x) = f (y). 15. For the ith dictator function χi we have that coordinate i is pivotal for every input x; hence Inf i [χi ] = 1.

7. Exercises and Notes 23 to being affine. All four query inputs should have the uniform distribution on Fn2 (but of course need not be independent). (d) Give an alternate 4-query test for being affine in which three of the query inputs are uniformly distributed and the fourth is not random. 30 Permutations π ∈ Sn act on strings x ∈ {−1, 1}n in the natural way: (x π )i = xπ(i) . They also act on functions f : {−1, 1}n → R via f π (x) = f (x π ) for all x ∈ {−1, 1}n . We say that functions g, h : {−1, 1}n → {−1, 1} are (permutation-)isomorphic if g = hπ for some π ∈ Sn .