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
Additional Sources
1. Testing Juntas: A Powerpoint Presentation, and a paper.
2. Learning Juntas: a
paper.
3. Beckner’s Thesis: a
thesis.
Procedural Details
- Time and Place:
Spring Semester 2003/4: Tuesdays and Thursdays,
2. Requirements:
Basic knowledge in Linear
Algebra and Normed Spaces.
3.
Contact:
- By email:
“g” and then “kindler” and then the at sign and then dimacs.rutgers.edu - 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.