3-Key-Input: Exploring the Theoretical Minimum Keys for Text Entry

Human input reaches language models by typing or speaking, and each channel leaves a distinct signature: orthographic noise for keyboards; for voice, disfluency from conventional transcription and restructuring from AI-backed dictation tools. How do they impact an LLM's performance? In this paper we present HIVE (Human Input-Variation Engine), a suite of voice transcription perturbations and QWERTY keyboard perturbations. We use HIVE to evaluate how robust models are to these perturbations. We present seven findings. (i) Voice transcription perturbations lower accuracy across every instruction-tuned model we test, and it is the structure of the transcription rather than its fillers that carries the cost. (ii) QWERTY keyboard perturbations cost less, and a model absorbs a lot of them before accuracy falls away. (iii) Both trace back to one cause, how many of the question's tokens survive the perturbation: destroying a token is what hurts, while adding new ones alongside it costs little. (iv) The gap between the two channels appears only where the answer must be constructed or deduced; on multiple choice there is none. (v) The harm does not solely come from test-set contamination. (vi) It cannot be trained away with lightweight adaptation. (vii) A thinking budget recovers the keyboard channel almost entirely but leaves the spoken registers untouched, and compressed speech is worse with it.

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.

We introduce Telegraph English (TE), a prompt-compression protocol that rewrites natural language into a symbol-rich, formally-structured dialect. Where token-deletion methods such as LLMLingua-2 train a classifier to delete low-importance tokens at a fixed ratio, TE performs a full semantic rewrite: it decomposes the input into atomic fact lines, substitutes verbose phrases with $\sim$40 logical and relational symbols, and lets the compression ratio adapt to each document's information density. A consequence of the line-structure rule is that compression and semantic chunking become the same operation -- each output line is an independently addressable fact, so the compressed representation is simultaneously a semantic index. We evaluate TE on 4{,}081 question-answer pairs from LongBench-v2 across five OpenAI models and two difficulty levels. At roughly 50% token reduction, TE preserves 99.1% accuracy on key facts with GPT-4.1 and outperforms LLMLingua-2 at matched compression ratios on every model and task tested. The gap widens on smaller models -- up to 11 percentage points on fine-detail tasks -- suggesting that explicit relational structure compensates for limited model capacity. We release the grammar specification, compression prompt, benchmark data, and reference implementation.