{
  "abstract": "In a world where valuable artifacts are increasingly created, completed, or processed by LLMs, the central economic question is not only what the LLM can produce, but what \\emph{value} remains in the inputs (i.e., the prompts) we provide to it. Given a prompt, hint, critique, problem statement, or partial solution that helps an LLM produce an artifact $z$---a proof, program, design, or scientific hypothesis---how should we measure the value of that input? Intuitively, an input is valuable when it makes the target artifact easier for the model to generate: either by increasing its sampling probability, or by reducing the thinking time needed to find it. We propose a computational Levin--Kolmogorov complexity approach to this problem, by appropriately replacing the universal Turing machine in the classical definitions by the LLM itself. Concretely, we introduce an LLM-relative notion of \\emph{probabilistic Levin--Kolmogorov complexity} $pKt$---treating the model's thinking as the random tape of the program, and charging logarithmically for it in Levin's manner---and define prompt value as algorithmic mutual information with respect to $pKt$. This captures the intuition above: a prompt having $b$ bits of value for an artifact $z$ makes $z$ $2^b$ times ``easier to obtain'', by multiplying the success probability by $2^b$, by dividing the required computation by $2^b$, or by any corresponding tradeoff between probability and computation. In contrast to the classical notion of algorithmic mutual information, ours is efficiently estimable. We additionally show that, under a natural reproduction experiment, a prompt value of \\(b\\) bits means that reproducing \\(z\\) without the prompt has median token cost \\(2^b\\) times that of reproducing it with the prompt.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.16438",
  "author_total": 1,
  "authors": [
    {
      "name": "Rafael Pass"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/893vhwzfbx5gn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.16438",
  "fulltext_url": null,
  "id": "893vhwzfbx5gn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.16438",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1703|Computability, Logic, AI Algorithms"
    ]
  ],
  "openalex": "W7203680732",
  "pdf_url": "https://arxiv.org/pdf/2608.16438",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/893vhwzfbx5gn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2608.16438",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.16438"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "The Value of a Prompt: An LLM-Relative Kolmogorov-Complexity Approach",
  "title_text": "The Value of a Prompt: An LLM-Relative Kolmogorov-Complexity Approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/893vhwzfbx5gn",
  "venue": "arXiv",
  "versions": []
}