Neither computability nor sense data have any relevance. The prisoners are a literary device. As far as formalizing the lack of information, the elements of each equivalence class chosen need not depend on the particular arrangement of the prisoners--that is more than enough.
Would it help if I pointed out that the original poster and most of the commentators here are well aware that no human, or indeed, no finite thinking being, could do what the prisoners do? Because you're not telling us things we don't know.
Again, you're thinking of this far too literally.