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Additive word complexity and Walnut

Pierre Popoli, Jeffrey Shallit, Manon Stipulanti

Abstract

In combinatorics on words, a classical topic of study is the number of specific patterns appearing in infinite sequences. For instance, many works have been dedicated to studying the so-called factor complexity of infinite sequences, which gives the number of different factors (contiguous subblocks of their symbols), as well as abelian complexity, which counts factors up to a permutation of letters. In this paper, we consider the relatively unexplored concept of additive complexity, which counts the number of factors up to additive equivalence. We say that two words are additively equivalent if they have the same length and the total weight of their letters is equal. Our contribution is to expand the general knowledge of additive complexity from a theoretical point of view and consider various famous examples. We show a particular case of an analog of the long-standing conjecture on the regularity of the abelian complexity of an automatic sequence. In particular, we use the formalism of logic, and the software Walnut, to decide related properties of automatic sequences. We compare the behaviors of additive and abelian complexities, and we also consider the notion of abelian and additive powers. Along the way, we present some open questions and conjectures for future work.

Additive word complexity and Walnut

Abstract

In combinatorics on words, a classical topic of study is the number of specific patterns appearing in infinite sequences. For instance, many works have been dedicated to studying the so-called factor complexity of infinite sequences, which gives the number of different factors (contiguous subblocks of their symbols), as well as abelian complexity, which counts factors up to a permutation of letters. In this paper, we consider the relatively unexplored concept of additive complexity, which counts the number of factors up to additive equivalence. We say that two words are additively equivalent if they have the same length and the total weight of their letters is equal. Our contribution is to expand the general knowledge of additive complexity from a theoretical point of view and consider various famous examples. We show a particular case of an analog of the long-standing conjecture on the regularity of the abelian complexity of an automatic sequence. In particular, we use the formalism of logic, and the software Walnut, to decide related properties of automatic sequences. We compare the behaviors of additive and abelian complexities, and we also consider the notion of abelian and additive powers. Along the way, we present some open questions and conjectures for future work.
Paper Structure (12 sections, 25 theorems, 8 equations, 6 figures, 3 tables)

This paper contains 12 sections, 25 theorems, 8 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General results
  4. Different behaviors and curiosities
  5. Bounded additive and abelian complexities
  6. The Tribonacci word
  7. The generalized Thue--Morse word on three letters
  8. Bounded additive and unbounded abelian complexities: a variant of the Thue--Morse word
  9. Unbounded additive and abelian complexities
  10. Equality between abelian and additive complexities
  11. Abelian and additive powers
  12. Semigroup trick

Key Result

Lemma 5

For all infinite words $\mathbf{x}$, we have $\rho^{\mathop{\mathrm{add}}\nolimits}_{\mathbf{x}}(n) \le \rho^{\mathop{\mathrm{ab}}\nolimits}_{\mathbf{x}}(n)$ for all $n\ge 0$.

Figures (6)

  • Figure 1: A four-state DFAO computing the additive complexity of the fixed point of $f \colon \{0,1,2\}^* \to \{0,1,2\}, 0\mapsto 012, 1\mapsto 112002, 2\mapsto \varepsilon$.
  • Figure 2: The first few values of the abelian and additive complexities for the Tribonacci word.
  • Figure 3: A DFAO computing the additive complexity of the $(1,2)$-Thue--Morse word.
  • Figure 4: The first few values of the abelian and additive complexities for the variant of the Thue--Morse word.
  • Figure 5: The first few values of the abelian and additive complexities as well as their difference for the fixed point of the morphism $0 \mapsto 03$, $1 \mapsto 43$, $3 \mapsto 1$, $4 \mapsto 01$.
  • ...and 1 more figures

Theorems & Definitions (49)

  • Definition 1
  • Example 2
  • Definition 3
  • Conjecture 4
  • Lemma 5
  • Lemma 6
  • Proposition 7
  • Proposition 8
  • proof
  • Theorem 9: Shallit-2021
  • ...and 39 more