Proclaiming Dictators and Juntas or Testing Boolean Formulae

We consider the problem of determining whether a given function belongs to a certain class of Boolean functions or whether it is far from the class. More precisely, given query access to the function and given a distance parameter , we would like to decide whether or whether it differs from every on more than an -fraction of the domain elements. The classes of functions we consider are singleton (“dictatorship”) functions, monomials, and monotone DNF functions with a bounded number of terms. In all cases we provide algorithms whose query complexity is independent of (the number of function variables), and polynomial in the other relevant parameters.