Analysis of Boolean Functions

Analysis of Boolean Functions

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Description

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, Hastad'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.show more

Product details

  • Electronic book text
  • CAMBRIDGE UNIVERSITY PRESS
  • Cambridge University Press (Virtual Publishing)
  • Cambridge, United Kingdom
  • 13 b/w illus. 450 exercises
  • 113995038X
  • 9781139950381

Review quote

'The applications of the ideas in this book are plentiful and diverse, and O'Donnell does an excellent job of leading the reader from one viewpoint to the next. I found it especially enjoyable to see theorems that I'm personally familiar with as a cryptographer, such as the Goldreich-Levin theorem, placed alongside other things I didn't know as well, like Arrow's theorem from social choice - with everything woven into a single, consistent story. I suspect other 'fresh readers' will similarly find parts of this book that they recognize, and others they don't. The relationships exposed between these ideas should be of interest to everyone. Altogether, I highly recommend that you take a glance at Analysis of Boolean Functions.' Daniel Apon, SIGACT News '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, MathSciNetshow more

Table of contents

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.show more

About Ryan O'Donnell

Ryan O'Donnell is an Associate Professor in the Computer Science Department at Carnegie Mellon University.show more

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