LIPIcs.STACS.2009.1834.pdf
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A new class of languages of infinite words is introduced, called the \emph{max-regular languages}, extending the class of $\omega$-regular languages. The class has two equivalent descriptions: in terms of automata (a type of deterministic counter automaton), and in terms of logic (weak monadic second-order logic with a bounding quantifier). Effective translations between the logic and automata are given.
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