Hallucination as a Computational Boundary: A Hierarchy of Inevitability and the Oracle Escape

2025-08-10

AI Summary

Hallucination in large language models (LLMs) severely undermines their reliable deployment. Method: This paper establishes the first formal theoretical framework demonstrating that hallucination is fundamentally unavoidable—arising from computational inevitabilities rooted in diagonalization, uncomputability, and information-theoretic limits. We introduce the “Learner Pumping Lemma” and integrate probabilistic Turing machine modeling, neural game theory, and computational jump theory to rigorously analyze retrieval-augmented generation (RAG) and continual learning. Contributions/Results: First, we provide the first rigorous formal foundation for RAG. Second, we model RAG as an external oracle mechanism, enabling *absolute* hallucination avoidance under idealized assumptions. Third, we formalize continual learning as oracle internalization—the progressive incorporation of external knowledge into model parameters. Collectively, these results reveal two fundamental, complementary pathways for hallucination mitigation: *exogenous retrieval* (leveraging external knowledge sources) and *endogenous knowledge integration* (embedding verified knowledge into model representations).

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Natural Language Processing: (Large) Language Models
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation

Abstract

The illusion phenomenon of large language models (LLMs) is the core obstacle to their reliable deployment. This article formalizes the large language model as a probabilistic Turing machine by constructing a "computational necessity hierarchy", and for the first time proves the illusions are inevitable on diagonalization, incomputability, and information theory boundaries supported by the new "learner pump lemma". However, we propose two "escape routes": one is to model Retrieval Enhanced Generations (RAGs) as oracle machines, proving their absolute escape through "computational jumps", providing the first formal theory for the effectiveness of RAGs; The second is to formalize continuous learning as an "internalized oracle" mechanism and implement this path through a novel neural game theory framework.Finally, this article proposes a

Problem

• Proves LLM hallucinations are inevitable computational boundaries
• Proposes RAGs as oracle machines for escaping hallucinations
• Formalizes continuous learning via neural game theory framework

Innovation

• Formalize LLMs as probabilistic Turing machines
• Prove RAGs' escape via computational jumps
• Implement continuous learning via neural game theory