The Value of a Prompt: An LLM-Relative Kolmogorov-Complexity Approach

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.

Artificial Intelligence (cs.AI); Computational Complexity (cs.CC); Information Theory (cs.IT)

arXiv:2608.16438 [cs.AI]

(or arXiv:2608.16438v1 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2608.16438 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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