Finite acceptance of infinite words
I. Litovsky, L. Staiger
In this paper we consider the following two types of finite
acceptance of infinite words by finite automata: An infinite
word $\xi$ is accepted if and only if there is a run on input
$\xi$ for which an accepting state is visited at least once
or at least once but only finitely often. The resulting
classes of regular $\omega$-languages are characterized by
language-theoretic means, and they are positioned into the
known hierarchies of regular $\omega$-languages.