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