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

Clemens Heuberger, Daniel Krenn, Gabriel F. Lipnik

TL;DR

It is shown that the minimisation algorithm for recognisable series can also be used to minimise linear representations of $q$-regular sequences.

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.

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

TL;DR

It is shown that the minimisation algorithm for recognisable series can also be used to minimise linear representations of -regular sequences.

Abstract

In this note, we precisely elaborate the connection between recognisable series (in the sense of Berstel and Reutenauer) and -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 -regular sequences.
Paper Structure (8 sections, 5 theorems, 20 equations)

This paper contains 8 sections, 5 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Recognisable Series and Regular Sequences
  4. Minimisation Failing?
  5. Why this Note is Needed
  6. Structure of this Note
  7. Recognisable Series
  8. Regular Sequences

Key Result

Lemma 2.2

Let $\mathcal{A}$ be a finite set, $x\in K^{\mathcal{A}^\star}$ be a recognisable series and $(u, M, w)$ be a minimal linear representation of $x$ of dimension $D$. Then $\mathop{\mathrm{span}}\nolimits(\setm{u M(b)}{b\in\mathcal{A}^{\star}})=K^{1\times D}$.

Theorems & Definitions (21)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Example 1.4: Continuation of Example \ref{['ex:minimzation-gone-wrong']}
  • Definition 2.1
  • Lemma 2.2: Berstel--Reutenauer Berstel-Reutenauer:2011:noncommutative-rational-series
  • proof : Proof of Lemma \ref{['lemma:necessary-condition-minimality']}
  • Proposition 2.3
  • proof
  • Definition 2.4
  • ...and 11 more