As Large Language Models become more ubiquitous across domains, it becomes important to examine their inherent limitations critically. This work argues that hallucinations in language models are not just occasional errors but an inevitable feature of these systems. We demonstrate that hallucinations stem from the fundamental mathematical and logical structure of LLMs. It is, therefore, impossible to eliminate them through architectural improvements, dataset enhancements, or fact-checking mechanisms. Our analysis draws on computational theory and Godel's First Incompleteness Theorem, which references the undecidability of problems like the Halting, Emptiness, and Acceptance Problems. We demonstrate that every stage of the LLM process-from training data compilation to fact retrieval, intent classification, and text generation-will have a non-zero probability of producing hallucinations. This work introduces the concept of Structural Hallucination as an intrinsic nature of these systems. By establishing the mathematical certainty of hallucinations, we challenge the prevailing notion that they can be fully mitigated.
Then I’m confused what is your point on Halting Problem vis-a-vis hallucinations being un-mitigable qualities of LLMs? Did I misunderstood you proposed “return undecided (somehow magically, bypassing Halting Problem)” to be the solution?
First, there’s no “somehow magically” about it, the entire logic of the halting problem’s proof relies on being able to set up a contradiction. I’ll agree that returning undecidable doesn’t solve the problem as stated because the problem as stated only allows two responses.
My wider point is that the Halting problem as stated is a purely academic one that’s unlikely to ever cause a problem in any real world scenario. Indeed, the ability to say “I don’t know” to unsolvable questions is a hot topic of ongoing LLM research.
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Then I’m confused what is your point on Halting Problem vis-a-vis hallucinations being un-mitigable qualities of LLMs? Did I misunderstood you proposed “return undecided (somehow magically, bypassing Halting Problem)” to be the solution?
First, there’s no “somehow magically” about it, the entire logic of the halting problem’s proof relies on being able to set up a contradiction. I’ll agree that returning undecidable doesn’t solve the problem as stated because the problem as stated only allows two responses.
My wider point is that the Halting problem as stated is a purely academic one that’s unlikely to ever cause a problem in any real world scenario. Indeed, the ability to say “I don’t know” to unsolvable questions is a hot topic of ongoing LLM research.