Weak Mso with the Unbounding Quantifier

Abstract: A new class of languages of infinite words is introduced, called the 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.