Yes, and for this reason it’s usual to consider only finite alphabets.
While any particular bound on input and output size can be processed in a single step with a large enough finite alphabet, Turing machines operate on any finite input without bound on length. Representing all of those with one symbol each would require an infinite alphabet with a correspondingly infinite state transition table.
The hope in the orginial question was to somehow have a method of saying “this thing has n compute in it”.
I am a bit unsure whether control structures and such can be faithfully preserved. But it seems if 00,01,10,11 can be translated to 0,1,2,3 then a …010101010111 could be translated into …123231412 and the very same process could be applied to turn 01,02,03,11,12,13,21,22,23,31,32,33 into A,B,C,D,E,F,G,H,I,J,L,M. Even if we can’t get to a single symbol at any given point we can get increased performance by predictable increase in alphabeth and this will not run out. That is for any N for all binary strings of that length there exists a pyramid of letter subsititution where at the top each string is covered by a single letter. That is the trick deosn’t rely on actual infinities, it also works for all finite numbers.
Yes, and for this reason it’s usual to consider only finite alphabets.
While any particular bound on input and output size can be processed in a single step with a large enough finite alphabet, Turing machines operate on any finite input without bound on length. Representing all of those with one symbol each would require an infinite alphabet with a correspondingly infinite state transition table.
The hope in the orginial question was to somehow have a method of saying “this thing has n compute in it”.
I am a bit unsure whether control structures and such can be faithfully preserved. But it seems if 00,01,10,11 can be translated to 0,1,2,3 then a …010101010111 could be translated into …123231412 and the very same process could be applied to turn 01,02,03,11,12,13,21,22,23,31,32,33 into A,B,C,D,E,F,G,H,I,J,L,M. Even if we can’t get to a single symbol at any given point we can get increased performance by predictable increase in alphabeth and this will not run out. That is for any N for all binary strings of that length there exists a pyramid of letter subsititution where at the top each string is covered by a single letter. That is the trick deosn’t rely on actual infinities, it also works for all finite numbers.