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

TL;DR

This paper formalizes the inevitability of hallucinations in large language models using computational boundaries and proposes two escape routes involving oracle models and continuous learning, along with a new security principle.

Contribution

It introduces a formal hierarchy of computational inevitability for LLM illusions and proposes novel theoretical frameworks for escape routes and AI security.

Findings

Illusions are proven inevitable on diagonalization, incomputability, and information theory boundaries.

RAGs can be modeled as oracle machines, providing a formal basis for their effectiveness.

A new principle, Computational Class Alignment, is proposed for AI security.

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 feasible new principle for artificial intelligence security - Computational Class Alignment (CCA), which requires strict matching between task complexity and the actual computing power of the system, providing theoretical support for the secure application of artificial intelligence.