On the Capability and Limitation of Hard Prompt

In this paper, we define the quantity of prompting complexity: for a fixed instruction-tuned language model, what is the shortest plausible prompt that makes deterministic decoding produce a target text? It is an LM-relative analogue of resource-bounded Kolmogorov complexity: the prompt is a program, the model interface is the interpreter, and information omitted from the prompt is supplied by the model's weights, training distribution, tokenizer, template, and decoding rule. Unlike classical Kolmogorov complexity, this measure is intentionally non-universal. In the finite-context setting it is computable by enumeration, but there is no model-independent invariance theorem; the same text may be cheap for one model and inaccessible or expensive for another. To keep the search space aligned with prompt engineering, we restrict programs to plausible human-readable texts rather than arbitrary token strings. We extend the exact definition to soft prompting complexity for approximate outputs, yielding a lossy notion of model-relative text compression and a formal target for prompt optimization. We also define prompting distance by comparing shortest generating prompts, and behavioral prompting complexity for reaching any output satisfying a specification. Based on these formulations, we define a research agenda for empirically studying which texts and behaviors are accessible from short plausible prompts under a fixed LM interface.

Large Language Models (LLMs) are frequently portrayed as general-purpose solvers capable of solving arbitrary tasks. We argue that this view overlooks a fundamental constraint: language is a compressed and capacity-limited interface for conveying task information. Modelling User--System interaction as a bilevel cheap-talk game, we analyse how latent tasks are encoded into prompts and reinterpreted under alignment and safety constraints. We introduce a conceptual decomposition separating task inference from execution and derive PAC-Bayes bounds that distinguish finite-sample estimation error from irreducible structural limitations. Our first main result establishes an expressivity floor: language acts as a capacity-limited communication channel, and whenever the informational complexity of a task family exceeds the capacity of that channel, distinct tasks become unavoidably indistinguishable to the Solver, inducing a strictly positive error floor that cannot be eliminated by additional data, optimisation, or model scaling alone. We then establish an objective-misalignment floor: when alignment constraints restrict the admissible output set, the User-ideal distribution may lie outside the feasible class, inducing an irreducible distortion. Together, these results yield a formal negative conclusion: prompt-conditioned LLMs are not universal problem solvers through prompting alone, as there exist task families for which correct behaviour is provably unattainable even in the infinite-data regime. More broadly, our analysis shows the limits of prompt-based generalisation arise from information-constrained communication and alignment-constrained objectives. This suggests that interfaces beyond natural language, including multimodal observations and, external memory, may reduce the inherent LLM limitations by increasing the task-relevant information available to the System.

Prompting is a widely used method for adapting general-purpose LLMs to domain-specific tasks. Effective prompts can be tedious to create manually, so prompt writing is increasingly handed to LLMs, with recent work training LLMs to write prompts for other models. These techniques either write prompts without observing how the task model behaves, blind to the failure modes the prompt should address, or must run alongside the task LLM at inference time, adding overhead to every query. Neither reflects how prompt engineering is often practiced. To close this gap, we introduce Reinforcement Learned Prompt Engineering (RLPE), which trains a model to analyze traces of a separate task model and write a prompt targeting the failures it observes. RLPE rewards the trained model strictly on prompt performance, so it learns the prompting strategies that improve the task model. The prompt is generated once and reused across the task, so inference does not require the trained model. Since RLPE needs only query access to the model it prompts, it is also compatible with closed-source models. We apply RLPE to Enigmata, BIG-Bench Extra Hard, and PlanBench. We train a 9B-parameter model that writes better prompts than state-of-the-art frontier models (gpt-5.6-sol and claude-opus-5) and prompt optimizers (GEPA and TextGrad) on 2/3 benchmarks. Although the model is never explicitly trained to iterate on prompts, this capability emerges at a level that matches or beats frontier models. The learned prompting capability also generalizes, with the trained model improving task models 6x larger than the one it trained with, including ones from entirely different model families, and matching claude-opus-5 on a fourth benchmark held out from training.