Testing Membership in Formal Languages Implicitly Represented by Boolean Functions

Combinatorial property testing, initiated formally by Goldreich, Goldwasser,and Ron in [Goldreich et al. (1998)] and inspired by Rubinfeld and Sudan in[Rubinfeld and Sudan 1996], deals with the relaxation of decision problems. Given aproperty P the aim is to decide whether a given input satisfies the property P or is farfrom having the property. A property P can be described as a language, i.e., a nonemptyfamily of binary words. The associated property to a family of boolean functionsf = (fn) is the set of 1-inputs of f. By an attempt to correlate the notion of testingto other notions of low complexity property testing has been considered in the contextof formal languages. Here, a brief summary of results on testing properties defined byformal languages and by languages implicitly represented by small restricted branchingprograms is provided.