A Canonical Form for Testing Boolean Function Properties

In a well-known result on graph property testing, [GT03] showed that every testable graph property has a “canonical” testing algorithm in which a set of vertices is selected uniformly at random and the edges queried are the complete graph over the selected vertices. In this paper we define a similar-in-spirit canonical form for Boolean function testing algorithms, and show that under some mild conditions on the function class and testing algorithm, property testers for Boolean functions can be transformed into this canonical form. We establish two main results. The first 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. The second 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 constantquery 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. ∗Supported in part by an FFSEAS Presidential Fellowship. †Supported by NSF grants CCF-0347282, CCF-0523664 and CNS-0716245, and by DARPA award HR0011-08-1-0069.