Two different paths exist when one is interested in validating an idea for a learning algorithm. On the one hand, the practical approach consists in using the available data to test the quality of the learning algorithm (for instance the widely used cross-validation technique). On the other hand, a theoretical approach is possible by using a learning paradigm, which is an attempt to formalize what learning means. Such models provide a framework to study the behavior of learning algorithms and

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... formally establish their soundness. The most widely used learning paradigm in Grammatical Inference is the one known as identification in the limit. The original definition has been found lacking because no efficiency bound is required. It remains an open problem how to best incorporate a notion of efficiency and tractability in this framework. This chapter surveys the different refinements that have been developed and studied, and the challenges they face. Main results for each formalisation, along with comparisons, are provided.