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Complexity of Linear Subsequences of Fibonacci-Automatic Sequences

Delaram Moradi, Narad Rampersad, Jeffrey Shallit

Abstract

We construct automata with input(s) in Fibonacci representation (also known as Zeckendorf representation) recognizing some basic arithmetic relations and study their number of states. We also consider some basic operations on Fibonacci-automatic sequences and discuss their state complexity. Furthermore, as a consequence of our results, we improve a bound in a recent paper of Bosma and Don. We also discuss the state complexity and runtime complexity of using a reasonable interpretation of Büchi arithmetic to actually construct some of the studied automata recognizing relations.

Complexity of Linear Subsequences of Fibonacci-Automatic Sequences

Abstract

We construct automata with input(s) in Fibonacci representation (also known as Zeckendorf representation) recognizing some basic arithmetic relations and study their number of states. We also consider some basic operations on Fibonacci-automatic sequences and discuss their state complexity. Furthermore, as a consequence of our results, we improve a bound in a recent paper of Bosma and Don. We also discuss the state complexity and runtime complexity of using a reasonable interpretation of Büchi arithmetic to actually construct some of the studied automata recognizing relations.
Paper Structure (11 sections, 22 theorems, 36 equations, 3 figures)

This paper contains 11 sections, 22 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Finite Automata
  4. Fibonacci Representation
  5. Recognizing Relations
  6. Addition
  7. Multiplication
  8. Linear Subsequences
  9. Application to a Problem of Bosma and Don
  10. Construction by Büchi Arithmetic
  11. Conclusion

Key Result

Lemma 1

We have $\rho_{\bf x}(n) = O(n)$. Let ${\bf x}$ be Fibonacci-automatic sequence generated by an $m$-state DFAO with msd-first input. Then $\rho_{\bf x} (n) = O(n m^2)$ for all $n \geq 1$.

Figures (3)

  • Figure 1: Automaton for the Fibonacci word $\bf f$.
  • Figure 2: Multiplication by $2$ in Fibonacci representation.
  • Figure 3: A Fibonacci DFAO for ${\bf f}[2i]$.

Theorems & Definitions (47)

  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 37 more