Technology

You're correct that the formal definition of a Markov process does not exclude internal computation, and that it only requires the next state to depend solely on the current state. But what defines a classical Markov chain in practice is not just the formal dependency structure but how the transition function is structured and used. A traditional Markov chain has a discrete and enumerable state space with explicit, often simple transition probabilities between those states. LLMs do not operate this way.

The claim that an LLM is "just" a large compressed Markov chain assumes that its function is equivalent to a giant mapping of input sequences to output distributions. But this interpretation fails to account for the fundamental difference in how those distributions are generated. An LLM is not indexing a symbolic structure. It is computing results using recursive transformations across learned embeddings, where those embeddings reflect complex relationships between tokens, concepts, and tasks. That is not reducible to discrete symbolic transitions without losing the model’s generalization capabilities. You could record outputs for every sequence, but the moment you present a sequence that wasn't explicitly in that set, the Markov table breaks. The LLM does not.

Yes, you can say a table is just one implementation of a function, and from a purely mathematical perspective, any function can be implemented as a table given enough space. But the LLM’s function is general-purpose. It extrapolates. A precomputed table cannot do this unless those extrapolations are already baked in, in which case you are no longer talking about a classical Markov system. You are describing a model that encodes relationships far beyond discrete transitions.

The pi analogy applies to deterministic functions with fixed outputs, not to learned probabilistic functions that approximate conditional distributions over language. If you give an LLM a new input, it will return a meaningful distribution even if it has never seen anything like it. That behavior depends on internal structure, not retrieval. Just because a function is deterministic at temperature 0 does not mean it is a transition table. The fact that the same input yields the same output is true for any deterministic function. That does not collapse the distinction between generalization and enumeration.

So while yes, you can implement any deterministic function as a lookup table, the nature of LLMs lies in how they model relationships and extrapolate from partial information. That ability is not captured by any classical Markov model, no matter how large.