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Hallucination is Inevitable: An Innate Limitation of Large Language Models

Written for the Research library (claude-opus-5-5) from the paper text, in our own words: the license "arXiv — non-exclusive license to distribute" does not allow redistributing the text itself. Original — arxiv.org/pdf/2401.11817

Summary

Large language models often produce fluent but false output. Most work on this "hallucination" is empirical, so it cannot settle whether the problem could ever be removed entirely. Xu, Jain and Kankanhalli address that question theoretically. Since nobody has yet settled how truth should be defined for the actual world, they build a simplified "formal world". In it, a computable ground truth function gives exactly one correct answer for every string. An LLM is treated purely as a total computable function that a training procedure updates step by step. It counts as hallucinating if, at every training stage, some input still receives a wrong answer. Using classical learning-theory results, the authors prove that no computable LLM can learn every computable function. This holds regardless of architecture, data, learning algorithm or prompting, so a general-purpose model must hallucinate. Because the formal world is part of the real one, the result also applies to deployed models. The authors also describe hallucination-prone tasks for polynomial-time models and test them empirically. They argue that mitigators can reduce errors but never abolish them, which makes external safeguards and human oversight essential.

Key points

  • Hallucination is formalised as follows (Definition 4): a model h hallucinates with respect to ground truth f if, at every training state h[i], some string s still yields hi ≠ f(s).
  • The core impossibility result relies on learning theory: computable LLMs cannot learn all computable functions. It also covers narrower classes such as polynomial-time-bounded LLMs, which form a proper subset of all LLMs.
  • The authors present this as the first formal definition and inevitability proof for hallucination. They contrast it with parallel work by Kalai and Vempala, which gives a statistical lower bound for calibrated LLMs and is therefore less general.
  • Practical implications:
    • LLMs should not be used autonomously in safety-critical decisions without guardrails, knowledge bases or human control.
    • An initial "upper bound" on LLM capabilities is sketched in Appendix E.
  • Stated limitations:
    • The theory does not explain errors on problems that lie within a model's computational reach.
    • It assumes a deterministic ground truth.
    • The empirical checks used existing LLMs without fine-tuning.
  • Reference: arXiv:2401.11817 (v2, February 2025).

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

Fields: Other

Cited by

148 papers of our library cite it (of the 140673 whose bibliographies we read); 4 of them have a page here, 144 not yet.

Cite

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.48550/arxiv.2401.11817 · arXiv 2401.11817

@misc{xu2024hallucination,
  title = {{Hallucination is Inevitable: An Innate Limitation of Large Language Models}},
  author = {Xu, Ziwei and Jain, Sanjay and Kankanhalli, Mohan},
  year = {2024},
  doi = {10.48550/arxiv.2401.11817},
  eprint = {2401.11817},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.2401.11817}
}

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