Harmonic
Analysis of Boolean Functions 

in

 Computer Science

 

The course is given by Guy Kindler, and will cover topics concerning analysis of Boolean functions, and their relations to computer science.

Course Note

          Notes for most classes of the course, written by Nick Weininger. (PDF, TEX)

Homework

       Ex 1,     Ex 2,     Ex 3,     Ex 4,

     Ex 5

Additional Sources

1.     Testing Juntas:    A Powerpoint Presentation, and a paper.

2.     Learning Juntas: a paper.

3.   Beckner’s Thesis:           a thesis.

Procedural Details

  1. Time and Place:
    Spring Semester 2003/4: Tuesdays and Thursdays, 14:50-16:10, Hill 423.

2.     Requirements:
Basic knowledge in Linear Algebra and Normed Spaces.

3.     Contact:

    1. By email:
      “g” and then “kindler” and then the at sign and then dimacs.rutgers.edu
    2. By phone:
      732 445 4577 (office), 609 430 1278 (home)

Program of the course:

1.           Introduction:
Influence of variables, norms, Fourier representation. Voting systems: dictatorship, juntas, monotonicity.

2.           Preliminaries:
The space of Boolean functions, the Fourier-Walsh representation, and some formulas connecting them.

3.           Testing linear codes:
Testing the Hadamard code and the Long-code, and relations with Noise-Sensitivity. 

4.           Testing proofs:
How to check mathematical proofs without reading them.

5.           Testing juntas:
Testing whether a given Boolean function is a J-junta, in time polynomial in J.

6.           Learning Juntas.

7.           Average sensitivity and juntas:
The Bonami-Beckner Inequality, a theorem of Friedgut, extensions to the biased case.

8.           Monotonicity and symmetry:
A lemma of Margulis, A theorem of Kahn, Kalai and Linial, threshold phenomena for graph-properties, first passage percolation.

9.           Linear and almost-linear functions:
Almost linear functions are dictatorships. The linear weight is bounded by the expectation.

10.      Rationality and social choice:
The only rational preference selection is a dictatorship

11.      Noise-resistant boolean functions are juntas.

12.      Low-degree bounded functions are juntas.

13.      Vertex-Cover and Max-Cut.