By Ryan O'Donnell

Boolean services are might be the main simple gadgets 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 sphere of research of Boolean features seeks to appreciate them through their Fourier rework and different analytic tools. this article provides a radical evaluation of the sector, starting with the main simple definitions and continuing to complicated subject matters comparable to hypercontractivity and isoperimetry. every one bankruptcy features a "highlight software" equivalent 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 houses. The ebook comprises approximately 450 workouts 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 technology concept and similar mathematical fields.

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**Example text**

25 gives an alternate way of looking at this proof. 30. 48. Stabρ [f ] = f, Tρ f . 49. For f : {−1, 1}n → R, n ρ |S| f (S)2 = Stabρ [f ] = S⊆[n] ρ k · Wk [f ]. 6) n NSδ [f ] = (1 − (1 − 2δ)k ) · Wk [f ]. 7) k=0 Thus the noise stability of f at ρ is equal to the sum of its Fourier weights, attenuated by a factor which decreases exponentially with degree. 50. Let ρ ∈ (0, 1). If f : {−1, 1}n → {−1, 1} is unbiased, then Stabρ [f ] ≤ ρ, with equality if and only if f = ±χi for some i ∈ [n]. Proof. For unbiased f we have W0 [f ] = 0 and hence Stabρ [f ] = k k k k≥1 ρ W [f ].

Then we can determine whether g is isomorphic to h by checking whether can(g) = can(h). Here is one possible way to define a canonical form for f : 1. Set P0 = Sn . 2. For each k = 1, 2, 3, . . , n, 3. Define Pk to be the set of all π ∈ Pk−1 that make the sequence n (f π (S))|S|=k maximal in lexicographic order on R(k) . 4. Let can(f ) = f π for (any) π ∈ Pn . (b) Show that this is well-defined, meaning that can(f ) is the same function for any choice of π ∈ Pn . , it satisfies (i) and (ii) above.

N, 3. Define Pk to be the set of all π ∈ Pk−1 that make the sequence n (f π (S))|S|=k maximal in lexicographic order on R(k) . 4. Let can(f ) = f π for (any) π ∈ Pn . (b) Show that this is well-defined, meaning that can(f ) is the same function for any choice of π ∈ Pn . , it satisfies (i) and (ii) above. (d) Show that if f ({1}), . . , f ({n}) are distinct numbers then can(f ) can be computed in O(2n ) time. (e) We could more generally consider g, h : {−1, 1}n → {−1, 1} to be isomorphic if g(x) = h(±xπ (1) , .