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Large language models have the same interface as a Markov chain - they're just predicting likelihoods of next tokens.

You generate text by sampling from that likelihood distribution.

No, an autoregressive language model is conditioned on all prior states, not the previous one.

Multiply out the states, "all prior states" is then the "previous one". Easy to model as Markov chain.

this is technically true but the Markov chain would be too big to store even with petabytes of storage.

Indeed. The argument boils down to: since it's finite, I can turn it into a FSA. Not only is that unhelpful, it doesn't tell you how to construct it, i.e. the learning process.

Also 'easy' to model as a lookup table containing all possible solutions.

Models like this are essentially extremely efficient markov chains. Calling them AI is disingenuous, like most things we call AI lately.