Testing Properties of Boolean Functions: Lower Bounds on Testing Fourier Degree

We consider the problem of deciding whether a given object has a given property or it is far from any object with the property, referred to as property testing. We focus on the case where the objects are Boolean functions, and we survey some of the previously known results about testing for properties such as the number of relevant variables and Fourier degree of a Boolean function. We present the recently developed technique for proving lower bounds in property testing, which is based on showing connections between property testing and communication complexity [13]. For the problem of testing whether a Boolean function has Fourier degree ≤ k or it is ǫ-far from any Boolean function with Fourier degree ≤ k, we improve the known lower bound of Ω(k) [13, 18], to Ω(k/ √ ǫ), using reductions from communication complexity. ∗Department of Computer Science, University of Chicago. Email:[email protected]