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Analysis of Boolean Functions

Book 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 GoldreichÐLevin algorithm from cryptography/learning theory, HŒstad'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.

Table of Contents

  1. Cover
  2. Half title
  3. Title
  4. Copyright
  5. Dedication
  6. Table of Content
  7. Preface
    1. Acknowledgments
  8. List of Notation
  9. 1 Boolean Functions and the Fourier Expansion
    1. 1.1 On Analysis of Boolean Functions
    2. 1.2 The “Fourier Expansion”: Functions as Multilinear Polynomials
    3. 1.3 The Orthonormal Basis of Parity Functions
    4. 1.4 Basic Fourier Formulas
    5. 1.5 Probability Densities and Convolution
    6. 1.6 Highlight: Almost Linear Functions and the BLR Test
    7. 1.7 Exercises and Notes
      1. Notes
  10. 2 Basic Concepts and Social Choice
    1. 2.1 Social Choice Functions
    2. 2.2 Influences and Derivatives
    3. 2.3 Total Influence
    4. 2.4 Noise Stability
    5. 2.5 Highlight: Arrow’s Theorem
    6. 2.6 Exercises and Notes
      1. Notes
  11. 3 Spectral Structure and Learning
    1. 3.1 Low-Degree Spectral Concentration
    2. 3.2 Subspaces and Decision Trees
    3. 3.3 Restrictions
    4. 3.4 Learning Theory
    5. 3.5 Highlight: The Goldreich-Levin Algorithm
    6. 3.6 Exercises and Notes
      1. Notes
  12. 4 DNF Formulas and Small-Depth Circuits
    1. 4.1 DNF Formulas
    2. 4.2 Tribes
    3. 4.3 Random Restrictions
    4. 4.4 Håstad’s Switching Lemma and the Spectrum of DNFs
    5. 4.5 Highlight: LMN’s Work on Constant-Depth Circuits
    6. 4.6 Exercises and Notes
      1. Notes
  13. 5 Majority and Threshold Functions
    1. 5.1 Linear Threshold Functions and Polynomial Threshold Functions
    2. 5.2 Majority, and the Central Limit Theorem
    3. 5.3 The Fourier Coefficients of Majority
    4. 5.4 Degree-1 Weight
    5. 5.5 Highlight: Peres’s Theorem and Uniform Noise Stability
    6. 5.6 Exercises and Notes
      1. Notes
  14. 6 Pseudorandomness and F[sub(2)]-Polynomials
    1. 6.1 Notions of Pseudorandomness
    2. 6.2 F[sub(2)]-Polynomials
    3. 6.3 Constructions of Various Pseudorandom Functions
    4. 6.4 Applications in Learning and Testing
    5. 6.5 Highlight: Fooling F[sub(2)]-Polynomials
    6. 6.6 Exercises and Notes
      1. Notes
  15. 7 Property Testing, PCPPs, and CSPs
    1. 7.1 Dictator Testing
    2. 7.2 Probabilistically Checkable Proofs of Proximity
    3. 7.3 CSPs and Computational Complexity
    4. 7.4 Highlight: Håstad’s Hardness Theorems
    5. 7.5 Exercises and Notes
      1. Notes
  16. 8 Generalized Domains
    1. 8.1 Fourier Bases for Product Spaces
    2. 8.2 Generalized Fourier Formulas
    3. 8.3 Orthogonal Decomposition
    4. 8.4 p-Biased Analysis
    5. 8.5 Abelian Groups
    6. 8.6 Highlight: Randomized Decision Tree Complexity
    7. 8.7 Exercises and Notes
      1. Notes
  17. 9 Basics of Hypercontractivity
    1. 9.1 Low-Degree Polynomials Are Reasonable
    2. 9.2 Small Subsets of the Hypercube Are Noise-Sensitive
    3. 9.3 (2, q)- and (p, 2)-Hypercontractivity for a Single Bit
    4. 9.4 Two-Function Hypercontractivity and Induction
    5. 9.5 Applications of Hypercontractivity
    6. 9.6 Highlight: The Kahn–Kalai–Linial Theorem
    7. 9.7 Exercises and Notes
      1. Notes
  18. 10 Advanced Hypercontractivity
    1. 10.1 The Hypercontractivity Theorem for Uniform ±1 Bits
    2. 10.2 Hypercontractivity of General Random Variables
    3. 10.3 Applications of General Hypercontractivity
    4. 10.4 More on Randomization/Symmetrization
    5. 10.5 Highlight: General Sharp Threshold Theorems
    6. 10.6 Exercises and Notes
      1. Notes
  19. 11 Gaussian Space and Invariance Principles
    1. 11.1 Gaussian Space and the Gaussian Noise Operator
    2. 11.2 Hermite Polynomials
    3. 11.3 Borell’s Isoperimetric Theorem
    4. 11.4 Gaussian Surface Area and Bobkov’s Inequality
    5. 11.5 The Berry–Esseen Theorem
    6. 11.6 The Invariance Principle
    7. 11.7 Highlight: Majority Is Stablest Theorem
    8. 11.8 Exercises and Notes
      1. Notes
  20. Some Tips
  21. Bibliography
  22. Index