Generalization of automatic sequences for numeration systems on a regular language

TL;DR

This paper extends the concept of automatic sequences to numeration systems based on regular languages, exploring their properties and connections with numeration systems.

Contribution

It introduces a generalized framework for automatic sequences derived from regular languages and analyzes their fundamental properties and relationships with numeration systems.

Findings

Sequences generalize k-automatic sequences for regular languages

Properties of these sequences are characterized

Connections with numeration systems are established

Abstract

Let L be an infinite regular language on a totally ordered alphabet (A,<). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the output alphabet of the automaton. This process generalizes the concept of k-automatic sequence for abstract numeration systems on a regular language (instead of systems in base k). Here, I study the first properties of these sequences and their relations with numeration systems.