The Gowers norm in the testing of Boolean functions

A property tester is a fast, randomized algorithm that reads only a few entries of the input, and based on the values of these entries, it distinguishes whether the input has a certain property or is "different" from any input having this property. Furthermore, we say that a property tester has completeness c and soundness s if it accepts all inputs having the property with probability at least c and accepts "different" inputs with probability at most s + o(1). In this thesis we present two property testers for boolean functions on the boolean cube f0; 1gn. We summarize our contribution as follows. We present a new dictatorship test that determines whether the function is a dictator (of the form f(x) = xi for some coordinate i), or a function that is an "anti-dictator." Our test is "adaptive," makes q queries, has completeness 1, and soundness O(q3) 2??q. Previously, a dictatorship test that has soundness (q + 1) . 2-q is achieved by Samorodnitsky and Trevisan, but their test has completeness strictly less than 1. Furthermore, the previously best known dictatorship test from the PCP literature with completeness 1 has soundness ... . Our contribution lies in achieving perfect completeness and low sound- ness simultaneously. We consider properties of functions that are invariant under linear transformations of the boolean cube. Previous works, such as linearity testing and low-degree testing, have focused on linear properties.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2009

Author dc:creator

• Chen, Victor Yen-Wen

Advisor dc:contributor.advisor

• Madhu Sudan.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/49880

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/49880