Characteristic sets for polynomial grammatical inference

When concerned about efficient grammatical inference two issues are relevant: the first one is to determine the quality of the result, and the second is to try to use polynomial time and space. A typical idea to deal with the first point is to say that an algorithm performs well if it infers in the limit the correct language. The second point has led to debate about how to define polynomial time: the main definitions of polynomial inference have been proposed by Pitt and Angluin. We return in this paper to a definition proposed by Gold that requires a characteristic set of strings to exist for each grammar, and this set to be polynomial in the size of the grammar or automaton that is to be learned, where the size of the sample is the sum of the lengths of all strings it includes. The learning algorithm must also infer correctly as soon as the characteristic set is included in the data. We first show that this definition corresponds to a notion of teachability as defined by Goldman and Mathias. By adapting their teacher/learner model to grammatical inference we prove that languages given by context-free grammars, simple deterministic grammars, linear grammars and nondeterministic finite automata are not identifiable in the limit from polynomial time and data.