A canonical form for testing boolean function properties

Authors:
Dana Dachman-Soled;Rocco A. Servedio
Affiliations:
Columbia University, New York, NY;Columbia University, New York, NY
Venue:
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Year:
2011

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Abstract

In a well-known result Goldreich and Trevisan (2003) showed that every testable graph property has a "canonical" tester in which a set of vertices is selected at random and the edges queried are the complete graph over the selected vertices. We define a similar-inspirit canonical form for Boolean function testing algorithms, and show that under some mild conditions property testers for Boolean functions can be transformed into this canonical form. Our first main result shows, roughly speaking, that every "nice" family of Boolean functions that has low noise sensitivity and is testable by an "independent tester," has a canonical testing algorithm. Our second main result is similar but holds instead for families of Boolean functions that are closed under ID-negative minors. Taken together, these two results cover almost all of the constant-query Boolean function testing algorithms that we know of in the literature, and show that all of these testing algorithms can be automatically converted into a canonical form.