Hallucination Rates in Language Generation

2026-07-25

AI Summary

This study investigates whether language generation algorithms can transcend traditional capability limits to produce broader classes of languages when permitted a low-frequency form of “hallucination”—specifically, errors occurring infinitely often but on a set of measure zero. Within a limit-language generation framework, the authors introduce hallucination rate as a key parameter and, by integrating computational learning theory, measure theory, and set-theoretic methods, establish for the first time a rigorous dual hierarchy linking hallucination rate to generative expressiveness. The results demonstrate that even zero-measure hallucinations substantially enhance generative power, and that certain language classes are generable only when such hallucinations are allowed. This hierarchical structure remains valid even under non-repetitive generation conditions, revealing the fundamental role of hallucination in expanding the boundaries of language generation.

Abstract

Language generation in the limit is an elegant model introduced by Kleinberg and Mullainathan [KM24] to formally study language generation by an algorithm that learns solely based on example strings. In this model, an algorithm is said to correctly generate from a language if it never makes an error after some finite time. In contrast, even sophisticated language models are known to regularly hallucinate in practice. In this paper, we initiate the study of language generation in the limit with (infinite) hallucination, i.e., the algorithm may generate incorrect strings infinitely often, but the errors occur at a limited rate (possibly even with 0-measure). We first show that hallucination, even at rate 0, makes generation in the limit strictly more powerful: there are language collections that cannot be generated with finite error but can be generated with infinite error, even when errors occur on a 0-measure set of time-steps. Furthermore, while all countable collections are generatable with finite error, we show a strict hierarchy of (uncountable) language collections characterized by the hallucination rate. This hierarchy extends to breadth, the fraction of the target language generated. While all countable collections can attain the optimal breadth of 1/2 [KW26b], we show strict separation at every breadth and hallucination rate. Finally, we study generation in the limit without repetition, where the algorithm may not repeat strings. This lets us compare the sets of correct and incorrect strings generated, rather than the fractions of correct and incorrect time-steps. Once again, we demonstrate a strict hierarchy at every hallucination rate and breadth. Taken together, these results reveal rich structure in language collections generatable in the limit with hallucination and establish hallucination rate as an important parameter in the theoretical study of language generation.

Problem

• hallucination rate
• language generation
• generation in the limit
• breadth
• uncountable language collections

Innovation

• hallucination rate
• language generation in the limit
• zero-measure error
• generation breadth
• non-repetitive generation