My point is that your initial argument was missing several key pieces & if you specify the entire state space you will see that it's not as simple as you thought initially. I'm not saying it can't be done but that it's actually much more complicated than simply saying just take an incomplete board state s & uniform transitions between s, s' for valid solutions s' that are compatible with s. In fact, now that I spelled out the issues I still don't think this is a formal extensional equivalence. Prolog has interactive transitions between the states & it tracks choice points so compiling a sudoku solver to a Markov chain requires more than just tracking the board state in the context.

> My point is that your initial argument was missing several key pieces

My initial example was a response to "If you think it is possible then I'd like to see an implementation of a sudoku puzzle solver as Markov chain", describing how a Sudoku solver could be implemented as a Markov chain. I don't think there's anything missing from it - it solves all proper Sudokus, and I only left open the choice of how to handle improper Sudokus because that was unspecified (but trivial regardless of what's wanted).

> I'm not saying it can't be done but that it's actually much more complicated

If that's the case, then I did misinterpret your comments as saying it can't be done. But, I don't think it's really complicated regardless of whatever "ok but now it must encode choice points in its state" are thrown at it - it's just a state-to-state transition look-up table.

> so compiling a sudoku solver to a Markov chain requires more than just tracking the board state in the context.

As noted, you can keep all the same variables as a regular Sudoku solver as part of the Markov chain's state and transition instruction-by-instruction, if that's what you want.

If you mean inputs from a user, the same is true of LLMs which are typically ran interactively. Either model the whole universe including the user as part of state transition table (maybe impossible, depending on your beliefs about the universe), or have user interaction take the current state, modify it, and use it as initial state for a new run of the Markov chain.

> As noted, you can keep all the same variables as a regular Sudoku solver

What are those variables exactly?

For a depth-first solution (backtracking), I'd assume mostly just the partial solutions and a few small counters/indices/masks - like for tracking the cell we're up to and which cells were prefilled. Specifics will depend on the solver, but can be made part of Markov chain's state regardless.