Testing basic boolean formulae

Abstract

We consider the problem of determining whether a given function f : {0, 1}n → {0, 1} belongs to a certain class of Boolean functions ℱ or whether it is far from the class. More precisely, given query access to the function f and given a distance parameter ε, we would like to decide whether f ε ℱ or whether it differs from every g ε ℱ on more than an ε-fraction of the domain elements. The classes of functions we consider are singleton ("dictatorship") functions, monomials, and monotone disjunctive normal form functions with a bounded number of terms. In all cases we provide algorithms whose query complexity is independent of n (the number of function variables), and linear in 1/ε.