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Automatic Abelian Complexities of Parikh-Collinear Fixed Points

Michel Rigo, Manon Stipulanti, Markus A. Whiteland

TL;DR

This work generalizes previous results on automaticity of abelian complexity for fixed points of Parikh-collinear morphisms to arbitrary alphabets and possibly erasing morphisms. It proves that the abelian complexity $\mathsf{a}_{\mathbf{x}}$ of a fixed point $\mathbf{x}=f^{\omega}(a)$ is $k$-automatic, where $k$ is the eigenvalue of the adjacency matrix $M_f$, and provides an effective procedure to compute the corresponding DFAO. Central to the approach are computable bounds on the recognizability constant and a $k$-definable cutting set $\mathsf{CS}_{f,a}$, enabling a decidable framework via definability and automaticity tools (notably Walnu t). The paper combines shift-space recognizability, cutting-set analysis, and Shallit's theorem on automatic abelian complexity to construct the automaton, illustrated with a concrete example yielding $\mathsf{a}_{\mathbf{x}}=135(377)^{\omega}$. This advances understanding of abelian properties in morphic words and provides practical methods for determining abelian complexity in non-uniform, erasing settings.

Abstract

Parikh-collinear morphisms have the property that all the Parikh vectors of the images of letters are collinear, i.e., the associated adjacency matrix has rank 1. In the conference DLT-WORDS 2023 we showed that fixed points of Parikh-collinear morphisms are automatic. We also showed that the abelian complexity function of a binary fixed point of such a morphism is automatic under some assumptions. In this note, we fully generalize the latter result. Namely, we show that the abelian complexity function of a fixed point of an arbitrary, possibly erasing, Parikh-collinear morphism is automatic. Furthermore, a deterministic finite automaton with output generating this abelian complexity function is provided by an effective procedure. To that end, we discuss the constant of recognizability of a morphism and the related cutting set.

Automatic Abelian Complexities of Parikh-Collinear Fixed Points

TL;DR

This work generalizes previous results on automaticity of abelian complexity for fixed points of Parikh-collinear morphisms to arbitrary alphabets and possibly erasing morphisms. It proves that the abelian complexity of a fixed point is -automatic, where is the eigenvalue of the adjacency matrix , and provides an effective procedure to compute the corresponding DFAO. Central to the approach are computable bounds on the recognizability constant and a -definable cutting set , enabling a decidable framework via definability and automaticity tools (notably Walnu t). The paper combines shift-space recognizability, cutting-set analysis, and Shallit's theorem on automatic abelian complexity to construct the automaton, illustrated with a concrete example yielding . This advances understanding of abelian properties in morphic words and provides practical methods for determining abelian complexity in non-uniform, erasing settings.

Abstract

Parikh-collinear morphisms have the property that all the Parikh vectors of the images of letters are collinear, i.e., the associated adjacency matrix has rank 1. In the conference DLT-WORDS 2023 we showed that fixed points of Parikh-collinear morphisms are automatic. We also showed that the abelian complexity function of a binary fixed point of such a morphism is automatic under some assumptions. In this note, we fully generalize the latter result. Namely, we show that the abelian complexity function of a fixed point of an arbitrary, possibly erasing, Parikh-collinear morphism is automatic. Furthermore, a deterministic finite automaton with output generating this abelian complexity function is provided by an effective procedure. To that end, we discuss the constant of recognizability of a morphism and the related cutting set.
Paper Structure (7 sections, 20 theorems, 18 equations, 2 figures)

This paper contains 7 sections, 20 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our contributions
  3. On the recognizability
  4. The cutting set
  5. Proof of Theorem 3
  6. A detailed discussion of the procedure
  7. Concluding remarks

Key Result

Lemma 1

Let $f \colon A^* \to A^*$ be Parikh-collinear and $a \in A$ be immortal. Then $\Psi(f(a))$ is an eigenvector of $M_f$ associated with the eigenvalue $\sum_{b \in A}|f(b)|_b$.

Figures (2)

  • Figure 1: The automaton accepting base-$3$ representations of triples of the form $\Psi(v)-\Psi(\mathop{\mathrm{pref}}\nolimits_n(\mathbf{x})) + (2,3,0)$, where $v$ ranges through the factors of length $n$, and $n$ ranges through the natural numbers.
  • Figure 2: The abelian complexity function of the fixed point $\mathbf{x}=012112002112002\cdots$ of the Parikh-collinear morphism $f\colon 0 \mapsto 012, 1\mapsto 112002,2 \mapsto \varepsilon$ as a $3$-automatic sequence.

Theorems & Definitions (36)

  • Lemma 1
  • Theorem 2: RSW-DLT23
  • Theorem 3
  • Lemma 4
  • proof
  • Remark 5
  • Definition 6
  • Lemma 7
  • Definition 8
  • Proposition 9
  • ...and 26 more