Regular expressions over countable words

TL;DR

This paper extends classical regular language theory to countable words, establishing equivalences between logical, algebraic, and expression-based characterizations for these infinite structures.

Contribution

It introduces five classes of expressions for countable words and proves their equivalence with logical and algebraic characterizations, generalizing classical results.

Findings

Characterization of expression classes for countable words

Decidable algebraic properties of these classes

Completion of the logical-algebraic-expression triad for 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.