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Exploring the Crochemore and Ziv-Lempel factorizations of some automatic sequences with the software Walnut

Marieh Jahannia, Manon Stipulanti

Abstract

We explore the Ziv-Lempel and Crochemore factorizations of some classical automatic sequences making an extensive use of the theorem prover Walnut.

Exploring the Crochemore and Ziv-Lempel factorizations of some automatic sequences with the software Walnut

Abstract

We explore the Ziv-Lempel and Crochemore factorizations of some classical automatic sequences making an extensive use of the theorem prover Walnut.
Paper Structure (10 sections, 10 theorems, 16 equations, 2 figures, 1 table)

This paper contains 10 sections, 10 theorems, 16 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. The Fibonacci word
  4. The $z$- and $c$- factorizations of some classical automatic sequences
  5. The Thue-Morse sequence
  6. The period-doubling sequence
  7. The Rudin-Shapiro sequence
  8. The paper-folding sequence
  9. The Mephisto-Waltz sequence
  10. Conclusion

Key Result

theorem 1

Let $z(\mathbf{t})=(z_0,z_1,\ldots)$ be the $z$-factorization of the Thue-Morse sequence $\mathbf{t}$. Then, for all $m\in\{0,\ldots,6\}$, $z_m$ is given in Table tab:z and c fact of our sequences and, for all $m\ge 7$, $z_m=\mathbf{t}[i..i+n-1]$ where

Figures (2)

  • Figure 1: DFAOs generating the automatic sequences of the paper.
  • Figure 2: An automaton accepting, among others, the base-$2$ representations of the pairs $(i,n)$ giving the position and length of factors of the $z$-factorization of the Fibonacci word.

Theorems & Definitions (21)

  • remark 1
  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • proof
  • theorem 4
  • proof
  • theorem 5
  • ...and 11 more