Property testing and the branching program size of boolean functions

Abstract

Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and Ron (1998) and inspired by Rubinfeld and Sudan (1996), deals with the relaxation of decision problems. Given a property P the aim is to decide whether a given input satisfies the property P or is far from having the property. For a family of boolean functions f = (fn) the associated property is the set of 1-inputs of f. Newman (2002) has proved that properties characterized by oblivious read-once branching programs of constant width are testable, i.e., a number of queries that is independent of the input size is sufficient. We show that Newman's result cannot be generalized to oblivious read-once branching programs of almost linear size. Moreover, we present a property identified by restricted oblivious read-twice branching programs of constant width and by CNFs with a linear number of clauses, where almost all clauses have constant length, but for which the query complexity is Ω(n1/4). © Springer-Verlag Berlin Heidelberg 2005.