Some Remarks on Almost Periodic Sequences and Languages

TL;DR

This paper investigates the properties of almost periodic sequences in formal language theory, providing negative answers to questions about their factor sets and discussing related topics in symbolic dynamics.

Contribution

It proves that almost periodic words cannot have context-free non-regular sets of factors, strengthening previous results and exploring related theoretical aspects.

Findings

Almost periodic words do not have context-free non-regular factor sets.

Provides a stronger negative answer to a question posed in prior research.

Discusses implications for symbolic dynamics and formal language theory.

Abstract

Almost periodicity has been considered in Formal Language Theory in connection with some topics in Symbolic Dynamics. In (P\u{a}un and Marcus, Bulletin of EATCS 53 (1994)) some problems concerning this property are raised. For instance it is asked whether there exists some almost periodic word $\alpha$ such that $Sub(\alpha)$, the set of its finite factors, is context-free non-regular. We answer negatively (even in a stronger form) this question, as well as discussing other related topics.