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Characterizing Language Generation in the Limit: Finite Witnesses and a Separation-Width Hierarchy

arXiv · AI, language, vision and robotics · article · Sep 9, 2026 · UTC

Language generation in the limit asks for valid unseen elements from every exhaustive positive presentation of an unknown infinite language. We characterize this task for arbitrary families over a countable universe. Generation is possible exactly when each target can be assigned a finite positive witness so that the targets activated by any finite sample have an infinite common intersection. The necessary direction follows from a universal normalization: a search through unconfirmed histories converts any successful generator into one depending only on the observed set. We then ask how large

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