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Opacity complexity of automatic sequences. The general case

J. -P. Allouche, J. -Y. Yao

TL;DR

The paper proposes opacity complexity as a measure for $k$-automatic sequences, formalized via finite $k$-automata with output and the intrinsic automaton concept. It shows that opacity can be computed via the intrinsic automaton and that for the standard prefix metric the opacity constant is $M_k=1/2$, enabling concrete calculations. The authors compute opacity properties for classical sequences (e.g., Thue-Morse, period-doubling, Golay-Shapiro-Rudin, paperfolding, Baum-Sweet, Tower of Hanoi) and show a spectrum of opacity outcomes. This framework provides a tool to classify automatic sequences by external visibility and lays groundwork for further study under alternative comparison metrics.

Abstract

In this work we introduce a new notion called opacity complexity to measure the complexity of automatic sequences. We study basic properties of this notion, and exhibit an algorithm to compute it. As applications, we compute the opacity complexity of some well-known automatic sequences, including in particular constant sequences, purely periodic sequences, the Thue-Morse sequence, the period-doubling sequence, the Golay-Shapiro(-Rudin) sequence, the paperfolding sequence, the Baum-Sweet sequence, the Tower of Hanoi sequence, and so on.

Opacity complexity of automatic sequences. The general case

TL;DR

The paper proposes opacity complexity as a measure for -automatic sequences, formalized via finite -automata with output and the intrinsic automaton concept. It shows that opacity can be computed via the intrinsic automaton and that for the standard prefix metric the opacity constant is , enabling concrete calculations. The authors compute opacity properties for classical sequences (e.g., Thue-Morse, period-doubling, Golay-Shapiro-Rudin, paperfolding, Baum-Sweet, Tower of Hanoi) and show a spectrum of opacity outcomes. This framework provides a tool to classify automatic sequences by external visibility and lays groundwork for further study under alternative comparison metrics.

Abstract

In this work we introduce a new notion called opacity complexity to measure the complexity of automatic sequences. We study basic properties of this notion, and exhibit an algorithm to compute it. As applications, we compute the opacity complexity of some well-known automatic sequences, including in particular constant sequences, purely periodic sequences, the Thue-Morse sequence, the period-doubling sequence, the Golay-Shapiro(-Rudin) sequence, the paperfolding sequence, the Baum-Sweet sequence, the Tower of Hanoi sequence, and so on.
Paper Structure (5 sections, 5 theorems, 26 equations, 8 figures)

This paper contains 5 sections, 5 theorems, 26 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Basic properties of opacity complexity
  3. Computation of opacity complexity
  4. Some examples
  5. Further study

Key Result

Proposition 1

For each finite $k$-automaton with output $(\mathscr{A},o)$, there exists a minimal $k$-automaton $(\mathscr{A}^{\prime },o^{\prime })$ (unique up to isomorphism) such that $\mathscr{A}^{\prime }$ is a factor of $\mathscr{A}$, and $(\mathscr{A},o)\approx (\mathscr{A}^{\prime },o^{\prime })$.

Figures (8)

  • Figure 1: Identity automaton $\mathscr{A}_{id}$
  • Figure 2: Thue-Morse Automaton $\mathscr{A}_{tm}$
  • Figure 3: Period-doubling automaton $\mathscr{A}_{pd}$
  • Figure 4: Golay-Shapiro(-Rudin) automaton $\mathscr{A}_{gsr}$
  • Figure 5: Paperfolding automaton $\mathscr{A}_{pf}$
  • ...and 3 more figures

Theorems & Definitions (22)

  • Example 1
  • Example 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • proof
  • Corollary 1
  • ...and 12 more