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I don't know how to prove or show that:

$ L_1 = \{xx|x \in \Sigma^\ast\} $ (that can be accepted by queue automaton)

If it would be possible, show by deterministic queue automaton, if not, by non-deterministic queue automaton.

If it's not that hard (and if I can), please give me a hint. Thanks in advance.

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  • $\begingroup$ We generally prefer that you ask only one question per post. We also prefer that you show us what progress you've made. Perhaps it is worth spending a little more time on $L_1$: can you at least find a nondeterministic queue automaton? $\endgroup$
    – D.W.
    Jan 21 at 23:23
  • $\begingroup$ @D.W. Sorry, you're right. I changed the question. Actually no, could you please give me some explanation/hint? then I try and come back $\endgroup$ Jan 21 at 23:26
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Working with a queue automaton for me is intuitively equivalent to working with a circular tape.

Read the input $w$ from the automaton, and put it on the tape. Add two markers $A$ and $B$, so we obtain $Aw_1Bw_2$. Now we can one by one eat a letter after $A$ and one after $B$.

For the deterministic variant it suffices to put $B$ exactly in the middle of the input string. That position can be found using a method similar to this: How to find middle element of linked list in one pass?

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