In the Turing machine formalism that was taught at my university(1), the set of symbols that can be stored in a cell must be finite, but can otherwise be anythi

In the Turing machine formalism that was taught at my university(1), the set of symbols that can be stored in a cell must be finite, but can otherwise be anything needed to solve the problem at hand. For each state in the associated state machine, there’s effectively a lookup table that maps the current tape symbol to some other, and an instruction about how to move the tape head for the next step. This means, in particular, that you can annotate symbols on the tape by having different variants of each input symbol, so long as the number of variants is strictly bounded. (1) There are many equally-powerful variants, so there’s no guarantee that others will be working with the same definition.