Regular expressions over countable words

Abstract: We investigate the expressive power of regular expressions for languages of countable words and establish their expressive equivalence with logical and algebraic characterizations. Our goal is to extend the classical theory of regular languages - defined over finite words and characterized by automata, monadic second-order logic, and regular expressions - to the setting of countable words. In this paper, we introduce and study five classes of expressions: marked star-free expressions, marked expressions, power-free expressions, scatter-free expressions, and scatter expressions. We show that these expression classes characterize natural fragments of logic over countable words and possess decidable algebraic characterizations. As part of our algebraic analysis, we provide a precise description of the relevant classes in terms of their J-class structure. These results complete a triad of equivalences - between logic, algebra, and expressions - in this richer setting, thereby generalizing foundational results from classical formal language theory.