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We tackle the question of whether an agent can, by suitable choice ofprompts, control an AI bot to any state. To that end, we first introduce aformal definition of ``meaning'' that is amenable to analysis. Then, wecharacterize ``meaningful data'' on which large language models (LLMs) areostensibly trained, and ``well-trained LLMs'' through conditions that arelargely met by today's LLMs. While a well-trained LLM constructs an embeddingspace of meanings that is Euclidean, meanings themselves do not form a vector(linear) subspace, but rather a quotient space within. We then characterize thesubset of meanings that can be reached by the state of the LLMs for some inputprompt, and show that a well-trained bot can reach any meaning albeit withsmall probability. We then introduce a stronger notion of controllability as{\em almost certain reachability}, and show that, when restricted to the spaceof meanings, an AI bot is controllable. We do so after introducing a functionalcharacterization of attentive AI bots, and finally derive necessary andsufficient conditions for controllability. The fact that AI bots arecontrollable means that an adversary could steer them towards any state.However, the sampling process can be designed to counteract adverse actions andavoid reaching undesirable regions of state space before their boundary iscrossed.

Taming AI Bots: Controllability of Neural States in Large Language Models