Languages that are Recognized by Simple Counter Automata are not necessarily Testable

Combinatorial property testing deals with the following relaxation of decision problems: Given a fixed property and an input f , one wants to decide whether f satisfies the property or is ‘far’ from satisfying the property. It has been shown that regular languages are testable, and that there exist context free language which are not testable. We show that there exists a language that is accepted by a deterministic counter automaton for which any 1/25-test requires Ω(poly(log n)) queries. Thus, proving that even if we restrict ourselves to, perhaps the simplest possible stack automaton model, we do not ensure testability. Electronic Colloquium on Computational Complexity, Report No. 152 (2005)