Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Meta-automatic Sequences

John M. Campbell, Benoit Cloitre

Abstract

Nested (or meta-Fibonacci) recurrences, such as the recurrence used to define Hofstadter's $Q$-sequence, along with the digit-based recurrences that underlie automatic sequences are of interest from both number-theoretic and combinatorial points of view. In this direction, Allouche and Shallit showed how the frequency sequence of a variant of the $Q$-sequence is $2$-automatic. This inspires us to introduce what may be seen as a natural combination of the recurrences for meta-Fibonacci and automatic sequences, by introducing the concept of a meta-automatic sequence. We show how it is possible to construct a meta-automatic sequence that is not denestable in terms of not being reducible, in a specific sense formalized in this paper, to certain digit-based recurrences for automatic sequences. This motivates our study of the non-denestable, meta-automatic sequences $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ introduced in this paper. For each of these integer sequences, we prove explicit DFAO evaluations, together with $4$-uniform morphisms, and we also consider the factor complexities of these sequences.

Meta-automatic Sequences

Abstract

Nested (or meta-Fibonacci) recurrences, such as the recurrence used to define Hofstadter's -sequence, along with the digit-based recurrences that underlie automatic sequences are of interest from both number-theoretic and combinatorial points of view. In this direction, Allouche and Shallit showed how the frequency sequence of a variant of the -sequence is -automatic. This inspires us to introduce what may be seen as a natural combination of the recurrences for meta-Fibonacci and automatic sequences, by introducing the concept of a meta-automatic sequence. We show how it is possible to construct a meta-automatic sequence that is not denestable in terms of not being reducible, in a specific sense formalized in this paper, to certain digit-based recurrences for automatic sequences. This motivates our study of the non-denestable, meta-automatic sequences and introduced in this paper. For each of these integer sequences, we prove explicit DFAO evaluations, together with -uniform morphisms, and we also consider the factor complexities of these sequences.
Paper Structure (17 sections, 25 theorems, 59 equations, 3 figures, 1 table)

This paper contains 17 sections, 25 theorems, 59 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Outline
  3. Preliminaries
  4. Meta-automatic sequences
  5. Denestability
  6. A non-nested baseline case
  7. Meta(1)
  8. A 4-state DFAO
  9. Meta(2)
  10. Lifted state and affine closure
  11. An explicit digit formula
  12. Factor complexity
  13. Quarto
  14. Meta(1) and Meta(2)
  15. Concluding remarks
  16. ...and 2 more sections

Key Result

Theorem 3

(Allouche $\&$ Shallit, 2012) Let the terms of $(U(n))_{n \geq 0}$ be in a finite set $\mathcal{A}$, and let $q\geq 2$ denote an integer. Then $U$ is $q$-automatic if there are nonnegative integers $t$, $a$, $b$, and $n_0$ together with a family $\{ f_{j} : j = 0, 1, \ldots, m(q, t) \}$ of functions for all $n \geq n_0$ and for all $j \in \{ 0, 1, \ldots, m(q, t) \}$AlloucheShallit2012.

Figures (3)

  • Figure 1: DFAO for $\mathcal{Q}$ (MSB-first). State labels are state/output. The initial state is $A$.
  • Figure 2: DFAO for $\mathcal{M}_{1}$ (4 states, base 4, MSB-first). State labels are state/output. The initial state is $\mathsf{0}$.
  • Figure 3: DFAO for $\mathcal{M}_{2}$ (4 states, base 4, MSB-first). State labels are state/output. The initial state is $\mathsf{0}$.

Theorems & Definitions (69)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Definition 4
  • Example 5
  • Definition 6
  • Example 7
  • Definition 8
  • Example 9
  • Lemma 10
  • ...and 59 more