A Note on the Relation between Recognisable Series and Regular Sequences, and their Minimal Linear Representations

TL;DR

This paper clarifies the relationship between recognisable series and q-regular sequences through their linear representations, demonstrating that minimisation techniques are applicable to both, thus unifying their theoretical framework.

Contribution

It establishes a precise connection between recognisable series and q-regular sequences and shows that minimisation algorithms can be transferred between these concepts.

Findings

Demonstrates the equivalence of recognisable series and q-regular sequences via linear representations.

Shows that minimisation algorithms for recognisable series can be applied to q-regular sequences.

Provides a unified approach to studying these sequences and series.

Abstract

In this note, we precisely elaborate the connection between recognisable series (in the sense of Berstel and Reutenauer) and $q$-regular sequences (in the sense of Allouche and Shallit) via their linear representations. In particular, we show that the minimisation algorithm for recognisable series can also be used to minimise linear representations of $q$-regular sequences.