I was reading about Universal Turing Machines.
I see that the matching lemma states, that between two symbols X and Y, if there are only 1-s and blanks, then a TM exists, which can count the number of ones.
This is used to match strings of ones between two tapes.
Now I wonder, would there also be a TM or a method of implementation of a TM which can match two strings? To me it intuitively appears, that inifinitely many symbols implies infinitely many states for the machine, and an example workflow may look like: found symbol X, enter state stateX, exit stateX to state readnextchar if again X was found on the second tape.
Would that not mean an infinitely long transition table? or is there an way to represent a table in a smaller way?
if the symbols are represented via some kind of place value system, e.g. decimal, binary, would that shorten the transition table length?