On Characterizations for Language Generation: Interplay of Hallucinations, Breadth, and Stability

We study language generation in the limit – introduced by Kleinberg and Mullainathan – building on classical works of Gold and Angluin. Kleinberg's and Mullainathan's main result is an algorithm for generating from any countable language collection in the limit. While their algorithm eventually generates unseen strings from the target language $K$, it sacrifices coverage or breadth, i.e., its ability to generate a rich set of strings.Recent work introduces different notions of breadth and explores when generation with breadth is possible, leaving a full characterization of these notions open. Our first set of results settles this by characterizing generation for existing notions of breadth and their natural combinations. Interestingly, our lower bounds are very flexible and extend to many performance metrics beyond breadth – for instance, showing that, in general, it is impossible to train generators that achieve a higher perplexity or lower hallucination rate for $K$ compared to other languages.Next, we study language generation with breadth with stable generators – which eventually stop changing after seeing an arbitrary but finite number of strings – and prove unconditional lower bounds for stable generators – strengthening the results of Kalavasis, Mehrotra, and Velegkas – and surprisingly demonstrating that generation with many existing notions of breadth becomes equally hard, when stability is required. This gives a separation for generation with approximate breadth, between stable and unstable generators, highlighting the rich interplay between breadth, stability, and consistency in language generation.