Analysis of Boolean Functions

This graduate-level text gives a thorough overview of the analysis of Boolean functions, beginning with the most basic definitions and proceeding to advanced topics.

Ryan O'Donnell (Author)

9781107038325, Cambridge University Press

Hardback, published 5 June 2014

444 pages, 13 b/w illus. 450 exercises
22.9 x 15.2 x 2.5 cm, 0.76 kg

'This 423-page book is a rich source of material presented in an attractive form. Each chapter highlights one main result which provides a focus and incentive for the reader to go to the end of the chapter.' Martin C. Cooper, MathSciNet

Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a 'highlight application' such as Arrow's theorem from economics, the Goldreich–Levin algorithm from cryptography/learning theory, Håstad's NP-hardness of approximation results, and 'sharp threshold' theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students and researchers in computer science theory and related mathematical fields.

1. Boolean functions and the Fourier expansion
2. Basic concepts and social choice
3. Spectral structure and learning
4. DNF formulas and small-depth circuits
5. Majority and threshold functions
6. Pseudorandomness and F2-polynomials
7. Property testing, PCPPs, and CSPs
8. Generalized domains
9. Basics of hypercontractivity
10. Advanced hypercontractivity
11. Gaussian space and invariance principles.

Subject Areas: Maths for computer scientists [UYAM], Computer science [UY], Statistical physics [PHS], Combinatorics & graph theory [PBV], Mathematics [PB]