TESTING MEMBERSHIP IN LANGUAGES THAT HAVE SMALL WIDTH BRANCHING PROGRAMS

Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and Ron in [J. ACM, 45 (1998), pp. 653–750] and inspired by Rubinfeld and Sudan [SIAM J. Comput., 25 (1996), pp. 252–271], 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 “far” from having the property. The main result here is that, if G = {gn : {0, 1}n → {0, 1}} is a family of Boolean functions which have oblivious read-once branching programs of width w, then, for every n and > 0, there is a randomized algorithm that always accepts every x ∈ {0, 1}n if gn(x) = 1 and rejects it with high probability if at least n bits of x should be modified in order for it to be in g−1 n (1). The algorithm makes ( 2 w )O(w) queries. In particular, for constant and w, the query complexity is O(1). This generalizes the results of Alon et al. [Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, 1999, pp. 645–655] asserting that regular languages are -testable for every > 0.

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