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Effective Computation of Generalized Abelian Complexity for Pisot Type Substitutive Sequences

Jean-Michel Couvreur, Martin Delacourt, Nicolas Ollinger, Pierre Popoli, Jeffrey Shallit, Manon Stipulanti

TL;DR

The paper addresses the computation of generalized abelian (and $k$-abelian) complexities for infinite sequences, focusing on fixed points of Pisot-type substitutions. It develops two effective methods: (i) a direct approach for uniformly factor-balanced, $ q$-automatic sequences using Dumont--Thomas numeration and Walnut to obtain $ q$-regular two-dimensional complexity and $ q$-automatic $k$-abelian complexities, and (ii) a Pisot-type subsequence approach leveraging sequence automata to prove automaticity in Dumont--Thomas (and Bertrand) numeration systems, yielding bounded abelian complexity and synchronized Parikh-prefix data. The Tribonacci sequence is analyzed in depth, yielding a tight uniform bound on factor-balancedness, a large 2D linear representation (dimension $264$), and $ ho^{k}_{f t}(n)$ being automatic in the Tribonacci numeration system; the Narayana sequence is used to illustrate $k$-abelian computations up to $k=10$. Collectively, these results provide effective, automata-theoretic tools to compute generalized abelian complexities for a broad class of automatic and substitutive sequences, with concrete new properties for classical sequences (Fibonacci, Pell, Tribonacci) and practical demonstrations via Walnut. The work advances both the theory and practice of abelian-type complexity computations in symbolic dynamics and combinatorics on words, enabling systematic analysis of a wide range of substitutions and their fixed points.

Abstract

Generalized abelian equivalence compares words by their factors up to a certain bounded length. The associated complexity function counts the equivalence classes for factors of a given size of an infinite sequence. How practical is this notion? When can these equivalence relations and complexity functions be computed efficiently? We study the fixed points of substitution of Pisot type. Each of their $k$-abelian complexities is bounded and the Parikh vectors of their length-$n$ prefixes form synchronized sequences in the associated Dumont--Thomas numeration system. Therefore, the $k$-abelian complexity of Pisot substitution fixed points is automatic in the same numeration system. Two effective generic construction approaches are investigated using the \texttt{Walnut} theorem prover and are applied to several examples. We obtain new properties of the Tribonacci sequence, such as a uniform bound for its factor balancedness together with a two-dimensional linear representation of its generalized abelian complexity functions.

Effective Computation of Generalized Abelian Complexity for Pisot Type Substitutive Sequences

TL;DR

The paper addresses the computation of generalized abelian (and -abelian) complexities for infinite sequences, focusing on fixed points of Pisot-type substitutions. It develops two effective methods: (i) a direct approach for uniformly factor-balanced, -automatic sequences using Dumont--Thomas numeration and Walnut to obtain -regular two-dimensional complexity and -automatic -abelian complexities, and (ii) a Pisot-type subsequence approach leveraging sequence automata to prove automaticity in Dumont--Thomas (and Bertrand) numeration systems, yielding bounded abelian complexity and synchronized Parikh-prefix data. The Tribonacci sequence is analyzed in depth, yielding a tight uniform bound on factor-balancedness, a large 2D linear representation (dimension ), and being automatic in the Tribonacci numeration system; the Narayana sequence is used to illustrate -abelian computations up to . Collectively, these results provide effective, automata-theoretic tools to compute generalized abelian complexities for a broad class of automatic and substitutive sequences, with concrete new properties for classical sequences (Fibonacci, Pell, Tribonacci) and practical demonstrations via Walnut. The work advances both the theory and practice of abelian-type complexity computations in symbolic dynamics and combinatorics on words, enabling systematic analysis of a wide range of substitutions and their fixed points.

Abstract

Generalized abelian equivalence compares words by their factors up to a certain bounded length. The associated complexity function counts the equivalence classes for factors of a given size of an infinite sequence. How practical is this notion? When can these equivalence relations and complexity functions be computed efficiently? We study the fixed points of substitution of Pisot type. Each of their -abelian complexities is bounded and the Parikh vectors of their length- prefixes form synchronized sequences in the associated Dumont--Thomas numeration system. Therefore, the -abelian complexity of Pisot substitution fixed points is automatic in the same numeration system. Two effective generic construction approaches are investigated using the \texttt{Walnut} theorem prover and are applied to several examples. We obtain new properties of the Tribonacci sequence, such as a uniform bound for its factor balancedness together with a two-dimensional linear representation of its generalized abelian complexity functions.

Paper Structure

This paper contains 21 sections, 37 theorems, 14 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and Notations
  3. General and abelian combinatorics on words
  4. Automatic, synchronized, and regular sequences
  5. The case of uniformly factor-balanced sequences
  6. Effective computation
  7. Checking the validity of the result
  8. Application to Sturmian sequences
  9. The Fibonacci sequence
  10. The Pell sequence
  11. Application to the Tribonacci sequence
  12. The case of Pisot substitutions
  13. Abelian complexity
  14. Generalized abelian complexity
  15. Application to the Narayana sequence
  16. ...and 6 more sections

Key Result

Lemma 1

Let $u,v\in A^*$ be two finite words and $k\ge 1$. The following statements are equivalent characterizations of $u\sim_k v$:

Figures (2)

  • Figure 1: The Tribonacci sequence has bounded discrete derivatives (first 4096 values, $n$ on the horizontal axis, $k$ on the vertical axis, one color per value, a different palette for each picture).
  • Figure 2: The converter between the Dumont--Thomas numeration systems associated with the Narayana substitution $\tau$ and the substitution behind the length-$3$ sliding-block code of its fixed point (here it computes the identity).

Theorems & Definitions (55)

  • Lemma 1
  • Lemma 2
  • proof
  • Remark 3
  • Lemma 4
  • Lemma 5: Karhumaki:2013
  • Lemma 6
  • proof
  • Lemma 7
  • Lemma 8
  • ...and 45 more