By Yaser S. Abu-Mostafa

The capacity and ends of knowledge thought and computational complexity have grown considerably nearer over the last decade. universal analytic instruments, resembling combinatorial arithmetic and knowledge circulation arguments, were the cornerstone of VLSl complexity and cooperative computation. the elemental assumption of constrained computing assets is the basis for cryptography, the place the excellence is made among on hand info and available info. a variety of different examples of universal pursuits and instruments among the 2 disciplines have formed a brand new examine classification of 'information and complexity theory'. This quantity is meant to show to the examine neighborhood a number of the fresh major issues alongside this subject matter. The contributions chosen listed here are all very uncomplicated, shortly energetic, quite well-established, and stimulating for immense follow-ups. this isn't an encyclopedia at the topic, it truly is involved purely with well timed contributions of enough coherence and promise. The sorts of the six chapters conceal a large spectrum from particular mathematical effects to surveys of enormous components. it's was hoping that the technical content material and topic of this quantity can help determine this normal learn zone. i want to thank the authors of the chapters for contributing to this quantity. I additionally wish to thank Ed Posner for his initiative to deal with this topic systematically, and Andy Fyfe and Ruth Erlanson for proofreading a few of the chapters.

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**Example text**

P D Thus, The inequality proved in Lemma 4 clearly holds if we replace E~(f) by the worst-case error E~(f). -j{6) is small. In the next lemma and Theorem 12 we do so for most functions. In Lemma 6 and Corollary 7, we prove the same results for the inner-product function. l} have >'j(6)<~. Proof: Fix a rectangle in {O,l}{O.... l} R of area IRI in {O, .. ,n-1} x{O, .. ,n-1}. The number offunctions with impurity pj(R) < ~ - 6 is < Therefore, for any A :S n 2 , the fraction of functions that have pj(R) rectangle R of area IRI ~ A, is at most fo· 2_2~:: .

Constant rectangle is of size n = 2N. , 0) ~ N = log n. -constant rectangle. First, modify the inner-product function to have the range ±I by letting: . p. (x,y)=1 -1 .. (x,y)=D. We will see in Lemma 6 that for every rectangle R L ~ {O, .. ,n-l} x {O, .. *(x,y)l::; JnlRI . -constant rectangle R must satisfy: IRI = I Hence, is tight IRI :::; n. *(x, y)1 :::; JnlRI . -constant rectangle {OJ X {O, .. ,n-l} proves that the bound 0 Let {O,l} {O, .. ,n-l}x{O, .. ,n-l} denote the set of boolean functions defined on {O, ..

N-I} for S in Theorem 2, and letting A, ~ A;o, .. ,n-I}X{O, .. ,n-I} denote the size of the largest f-constant rectangle in {D, .. ,n-I} x {O, .. ,n-I}, yields: Largest f-Constant Rectangle Argument D Theorem 4 [Yao 79] Example 2 Let n ~ 2N for some integer N, and Sx = Sy = {D,I}N. For X = Xl, ... , XN and Y = YI, ... ) denotes exclusive or. -constant rectangle is of size n = 2N. , 0) ~ N = log n. -constant rectangle. First, modify the inner-product function to have the range ±I by letting: . p.