Given a regular language $L$ over a unary alphabet $\Sigma = \{ a \}$.
How to decide whether there are two words $w,w' \in L$ such that the length of $w$ is relatively prime to the length of $w'$ ?
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1Why are you interested in such a contrived question? – Yuval Filmus Sep 03 '13 at 18:54
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1This should help: What are the possible sets of word lengths in a regular language? – Gilles 'SO- stop being evil' Sep 03 '13 at 21:22
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Consider a minimal DFA for $L$. Each state in the DFA has one outgoing edge, so the DFA looks like a path entering a cycle. That means that $W = \{ n : a^n \in L \}$ is eventually periodic: for some $k,l$, for all $n \geq k$ it holds that $n \in W$ iff $n + l \in W$. We are now left with a problem in number theory, which I leave for you to ponder. (Try a few examples to see when an eventually periodic $W$ can fail to satisfy the property you're looking for.)
Yuval Filmus
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1Thank you for your hint. Our regular language has additional property: for any words $w \in L$ we have also $w^n \in L$, for all $n \geq 1$. Actually we want to decide existence of $m$ such that all words longer than $m$ are in $L$. Due to your helpful comment we now see that it is equivalent to having a loop in DFA containing only accepting states. – Barbara Sep 04 '13 at 11:28