Testing of function that have small width branching programs

Combinatorial property testing, initiated formally by (Goldreich et al., 1996) and inspired by (Rubinfeld and Sudan, 1996), 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 height probability if at least εn bits of x should be modified in order for it to be in g^-1(1). The algorithm queries (2w/ε) ^0(w) many queries. In particular, for constant ε and w, the query complexity is 0(1). This generalizes the results of (Alon et al., 1999) asserting that regular languages are efficiently (ε,O(1))-testable