Modern language models are internally -- and mathematically -- distributions over token strings rather than \emph{character} strings, posing numerous challenges for programmers building user applications on top of them. For example, if a prompt is specified as a character string, it must be tokenized before passing it to the token-level language model. Thus, the tokenizer and consequent analyses are very sensitive to the specification of the prompt (e.g., if the prompt ends with a space or not). This paper presents algorithms for converting token-level language models to character-level ones. We present both exact and approximate algorithms. In the empirical portion of the paper, we benchmark the practical runtime and approximation quality. We find that -- even with a small computation budget -- our method is able to accurately approximate the character-level distribution (less than 0.00021 excess bits / character) at reasonably fast speeds (46.3 characters / second) on the Llama 3.1 8B language model.
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