Buechi automata are used to recognize languages of infinite words. Such languages have been introduced to describe the behavior of real time systems or infinite games. The question of inferring them from infinite examples has already been studied, but it may seem more reasonable to believe that the data from which we want to learn is a set of finite words, namely the prefixes of accepted or rejected infinite words. We describe the problems of identification in the limit and polynomial identification in the limit from given data associated to different interpretations of these prefixes: a positive prefix is universal (respectively existential) when all the infinite words of which it is a prefix are in the language (respectively when at least one is); the same applies to the negative prefixes. We prove that the classes of regular ω-languages (those recognized by Buechi automata) and of deterministic ω-languages (those recognized by deterministic Buechi automata) are not identifiable in the limit, whichever interpretation for the prefixes is taken. We give a polynomial algorithm that identifies the class of safe languages from positive existential prefixes and negative universal prefixes. We show that this class is maximal for polynomial identification in the limit from given data, in the sense that no superclass can even be identified in the limit.
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