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By unique words we understand the such word $w$ that every character in the word occur at most once, for example: $\Sigma = \{a,b,c\}$ $w = abc$, $v = abca$

$w$ is unique, $v$ is not.

Now, we have a problem:

We have given an alphabet $\Sigma$. We have a given length of word $|w| = n$. How to recognize using $NFA$ whether $w$ is unique?

$NFA$ is expected to has polynomial number of states regard to $n$.

Please hint me.

Carol
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  • We get asked this kind of question a lot, so we have written a general reference on how to prove that a particular language is regular (or, equivalently, that it can be recognized by a NFA). I suggest studying the material in our reference question, and try applying those methods to your problem. If you're still stuck, edit the question to show what progress you've made and where you got stuck, and see if you can ask a more specific question about the problem you're solving. (continued) – D.W. Aug 07 '17 at 22:08
  • Also, tell us whether you want a NFA that recognizes the language of unique words, or one that recognizes the language of words that aren't unique. We're happy to help you understand the concepts but just solving exercises for you is unlikely to achieve that. You might find this page helpful in improving your question. – D.W. Aug 07 '17 at 22:08
  • @D.W., it is not problem whether language is regular or not. It is finite so it is regular. The problem is to construct $NFA$ with polynomial number of states. – Carol Aug 08 '17 at 06:37
  • I suggest you edit the question to clarify which language you want the NFA to recognize, and to show us what progress you've made (what's the smallest NFA you've been able to find so far? how large is it?). Also, it wouldn't hurt to tell us how you know that polynomial-size NFA exists. If this is an exercise, I encourage you to credit the source of the exercise and where you got the idea that a polynomial-size NFA exists. – D.W. Aug 08 '17 at 06:41

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