Hallucination has been widely recognized as a significant drawback for large language models (LLMs). Many efforts have been made to reduce hallucination in these models, yet the fundamental question remains: can it ever be completely eliminated? This article explores the formal limitations of LLMs and demonstrates that hallucination is an unavoidable aspect of their operation.

By defining a formal world where hallucination is represented as inconsistencies between a computable LLM and a computable ground truth function, it is possible to show that LLMs cannot learn all computable functions. As a result, these models will inevitably generate hallucinations when used as general problem solvers. Since the real world is even more complex than this formal world, hallucinations in real-world LLMs are unavoidable.

Additionally, for real-world LLMs constrained by provable time complexity, hallucination-prone tasks are identified and empirically validated. Finally, using this formal framework, it is possible to explore existing hallucination mitigation techniques and discuss their practical implications on the safe deployment of LLMs.

The Essence of Large Language Models

At the core of LLMs lies a deceptively simple principle: the prediction of linguistic patterns. These models operate by predicting the most probable next word in a sequence, rather than possessing true understanding. Their capabilities, while impressive, are fundamentally limited by their training data and the probabilistic nature of their outputs.

This raises an important question: can LLMs ever truly understand language without experiencing the world? More critically, can hallucinations—instances where models generate plausible but incorrect information—be fully eliminated, or must they be managed as an intrinsic limitation?

Hallucinations in LLMs: A Structural Problem

Hallucinations in LLMs are not merely errors that can be fixed by improving datasets or refining model architectures. Instead, they are a direct result of the mathematical and logical foundations upon which these models are built. A key concept introduced in this discussion is Structural Hallucinations—hallucinations that arise due to the fundamental nature of LLMs.

Using concepts from computational theory and Gödel’s First Incompleteness Theorem, it is demonstrated that every stage of an LLM’s process—from training data compilation to fact retrieval, intent classification, and text generation—carries a non-zero probability of hallucinations. This means that even with extensive fact-checking mechanisms and improved training, hallucinations will always exist to some degree.

Needle in a Haystack: The Challenge of Accurate Information Retrieval

One of the primary sources of hallucination in LLMs is their inability to retrieve specific information reliably. When faced with vast amounts of data, LLMs often struggle to locate the most relevant and accurate piece of information. This problem, sometimes referred to as the “Needle in a Haystack” issue, is a major cause of factual inaccuracies.

Even if a model is trained on an extensive dataset, it does not guarantee that it will retrieve the correct fact when queried. Instead, it may blur multiple pieces of information together or fabricate missing details to produce a seemingly coherent response. This occurs because LLMs rely on statistical approximations rather than precise knowledge retrieval.

Addressing Hallucinations: Mitigation Strategies

While hallucinations cannot be completely eliminated, several strategies have been developed to reduce their frequency and impact:

1. Chain-of-Thought (CoT) Prompting

Encouraging LLMs to articulate their reasoning process can sometimes reduce logical inconsistencies and hallucinations. This method improves transparency in decision-making but does not eliminate errors entirely.

2. Self-Consistency

By generating multiple responses and selecting the most consistent one, models can improve their reliability. This approach assumes that correct answers are more likely to be arrived at consistently.

3. Retrieval-Augmented Generation (RAG)

Rather than relying solely on an internal dataset, RAG-based models fetch real-time, domain-specific data to supplement their knowledge. This reduces the risk of outdated or fabricated responses.

4. Uncertainty Quantification

Identifying when a model is uncertain about an answer can help flag potentially hallucinated responses. Various probabilistic methods are used to quantify this uncertainty.

5. Faithful Explanation Generation

Providing explanations for how an answer was derived allows users to verify the reliability of a model’s response. Techniques like Shapley values help measure the contribution of different factors in a model’s decision-making.

Why Hallucinations Can Never Be Fully Eliminated

Despite advances in model architecture, information retrieval, and fact-checking, hallucinations persist due to several fundamental reasons:

  1. Incomplete Training Data
    No dataset can contain all possible facts. Even with the largest datasets, there will always be gaps in knowledge.

  2. Undecidability of Accurate Retrieval
    Retrieving correct information from a vast dataset is inherently undecidable in computational theory. There will always be uncertainty in whether the retrieved data is accurate.

  3. Probabilistic Nature of LLMs
    Language models generate outputs based on statistical probabilities, not definitive logic. This means there is always a nonzero probability of generating an incorrect statement.

  4. Lack of True Understanding
    Unlike human cognition, LLMs do not “understand” concepts in a meaningful way. They generate text based on learned patterns rather than genuine comprehension.

  5. Logical Limitations (Gödel’s Incompleteness Theorem)
    Some truths cannot be derived within a formal system. This limitation applies to LLMs, ensuring that some forms of hallucination are inevitable.

Conclusion

Hallucinations in LLMs are not anomalies that can be fully removed; rather, they are a fundamental aspect of how these models function. While mitigation strategies can reduce their frequency, hallucinations will always exist due to the probabilistic nature of language models and the inherent incompleteness of training data.

Moving forward, the focus should not be on eliminating hallucinations but on managing them effectively. Strategies like Retrieval-Augmented Generation, uncertainty quantification, and structured reasoning can help improve reliability. However, it remains essential for users to approach LLM-generated content critically and verify information when accuracy is crucial.

Understanding and accepting the limitations of LLMs will be key in integrating them responsibly across various domains, from research to everyday applications.