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As in, given a regular language $V$, does there exist a deterministic Büchi automaton $\mathcal{A}$, or equivalently a regular language $W$ such that $\mathcal{L}(\mathcal{A})=\vec{W}=V^\omega$?

For clarity, if $L$ is a finite-word language, then $L^\omega$ and $\vec{L}$ are resp. the $\omega$-closure and the limit of $L$; see here.

Edit: My guess is that $V^\omega=\vec{V^\ast}$, but I'm unable to (dis)prove it. see comments.

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  • $\begingroup$ Regarding your conjecture: it probably is not simple as that. In a paper by Choueka, Theories of automata on $\omega$-tapes, we find the following Lemma 5.2: For every regular $V \subseteq X^*$ one can effectively find some regular $\widetilde V \subseteq X^*$ such that $V^\omega = V^* ( \lim \widetilde V)$. $\endgroup$ Commented Apr 28, 2022 at 16:34
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    $\begingroup$ Quoting from the same paper, just above the lemma: Certainly one cannot say, in general, that $V^\omega = \lim V$ or $V^\omega = \lim V^*$: the set $V = \{1\}\{0\}^*$ is a counterexample to both assertions, since $10^\omega$ is not in $V^\omega$ although it is in $\lim V$ and in $\lim V^*$. $\endgroup$ Commented Apr 28, 2022 at 16:39

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