图灵机元素区别问题

Question

So the language is as follows:

E = {#x1#x2...#xi where alphabet is {0,1}* and no string can be a duplicate of another string }

I am trying to create the state diagram for this, but even before that I was coming up with the algorithm to solve it, but the issue I was encountering is whenever I compare the first two strings, I have to mark each character with an 'x' so how would I restore the first string? Like first I compare x1 and x2, by the time I'm done, in x2 and all characters in x1 would be marked with 'x', so when I move on to x3, x1 has nothing to compare.

Answer 1

Instead of marking considered symbols with an x, mark them with special symbols corresponding to the symbols being marked. So, instead of writing x for 0 and x for 1, write a for 0 and b for 1. In fact, go ahead and use symbols c and d also to replace values in "the earliest thing I need to check" so you can check all pairs. A high-level description of a Turing machine using this strategy is the following:

1. begin reading the first input, replacing 0 with c and 1 with d
2. go to the second input and if the second input is a match so far, write a for 0 and b for 1, then continue. If it's not a match, we know that these inputs don't match and we can begin comparing other pairs. Change the input you're checking to a and b only and reset the first input to 0 and 1 only.
3. repeat this process skipping over all a and b already there to check all pairs involving the first term.
4. once you've checked all pairs involving the first term, cross it out (using x maybe) and then repeat the whole process on the remaining input

This will check all pairs and work as expected. The key is, as you correctly surmised, being able to reconstruct parts of the input, meaning you need extra symbols in your tape alphabet. Never hesitate to introduce tape symbols - they're free and can never hurt.