What's the Magic Word? A Control Theory of LLM Prompting

Prompt engineering is effective and important in the deployment of LLMs but is poorly understood mathematically. Here, we formalize prompt engineering as an optimal control problem on LLMs -- where the prompt is considered a control variable for modulating the output distribution of the LLM. Within this framework, we ask a simple question: given a sequence of tokens, does there always exist a prompt we can prepend that will steer the LLM toward accurately predicting the final token? We call such an optimal prompt the magic word since prepending the prompt causes the LLM to output the correct answer. If magic words exist, can we find them? If so, what are their properties? We offer analytic analysis on the controllability of the self-attention head where we prove a bound on controllability as a function of the singular values of its weight matrices. We take inspiration from control theory to propose a metric called $k-\epsilon$ controllability to characterize LLM steerability. We compute the $k-\epsilon$ controllability of a panel of large language models, including Falcon-7b, Llama-7b, and Falcon-40b on 5000 WikiText causal language modeling tasks. Remarkably, we find that magic words of 10 tokens or less exist for over 97% of WikiText instances surveyed for each model.