Testing of function that have small width branching programs

Combinatorial property testing, initiated formally by Goldreich, Goldwasser and Ron in [11], and inspired by Rubinfeld and Sudan, deals with the following relaxation of decision problems: Given a fixed property and an input x, one wants to decide whether x has the property or is being `far' from having the property. The main result here is that if G = {g : {0,1}ⁿ→{0, 1}} is a family of Boolean functions that have read-once branching programs of width w, then for every n and ε>0 there is a randomized algorithm that always accepts every x∈{0, 1}ⁿ if g(x) = 1, and rejects it with high probability if at least εn bits of x should be modified in order for it to be in g^-1 (1). The algorithm queries (2^w/ε)^O(w) many queries. In particular, for constant ε and w, the query complexity is O(1). This generalizes the results of Alon et. al. asserting that regular languages are efficiently (ε, O(1))-testable.