[Paper Review] Envelope Words and the Reflexivity of the Return Word Sequences in the Period-doubling Sequence

Yuke Huang, Zhi-Ying Wen|arXiv (Cornell University)|Mar 21, 2017

semigroups and automata theory12 references3 citations

TL;DR

This paper introduces the concept of envelope words to overcome the failure of kernel word techniques in the period-doubling sequence $Δ$. It proves that return word sequences of any factor are either $Θ_1$ or $Θ_2$, both substitutive sequences, establishing a reflexivity property. This allows full characterization of return word sequences and enables solving combinatorial problems like square and cube detection via a new spectrum-based method.

ABSTRACT

We consider the infinite one-sided sequence over alphabet $\{a,b\}$ generated by the period-doubling substitution $σ(a)=ab$ and $σ(b)=aa$, denoted by $\mathbb{D}$. Let $r_p(ω)$ be the $p$-th return word of factor $ω$. The main result of this paper is twofold. (1) For any factor $ω$ in $\mathbb{D}$, the return word sequence $\{r_p(ω)\}_{p\geq1}$ is $Θ_1$ or $Θ_2$. Both of them are substitutive sequences and determined completely in this paper. (2) For any factor $ω$ in $Θ_1$ (resp. $Θ_2$), the return word sequence $\{r_p(ω)\}_{p\geq1}$ is still $Θ_1$ or $Θ_2$. We call it the reflexivity property of the return word sequence. As an application, we introduce a notion of spectrum for studying some typical combinatorial properties, such as separated, adjacent and overlapped.

Motivation & Objective

• To address the lack of a kernel word structure in the period-doubling sequence, which prevents standard return word analysis.
• To develop a new framework using envelope words to characterize return word sequences in $Δ$.
• To establish the reflexivity property: return word sequences of any factor in $Δ$ are themselves $Θ_1$ or $Θ_2$.
• To apply this structure to solve combinatorial problems such as detecting adjacent, separated, or overlapped occurrences of factors.
• To introduce a spectrum-based method for analyzing the positions of factor occurrences, enabling solutions to Q2 (position-specific properties) in addition to Q1 (existence of properties).

Proposed method

• Define envelope words $E^1_m = A_m\delta_m^{-1}$ and $E^2_m = B_m B_{m-1} \delta_m^{-1}$, which are odd-length factors capturing the structure of return words.
• Prove the uniqueness of envelope extension, replacing the failed kernel decomposition in $Δ$.
• Define two sequences $\Theta_1 = \tau_1(\u0394)$ and $\Theta_2 = \tau_2(\u0394)$ via substitutions $\tau_1(a)=a, \tau_1(b)=bb$ and $\tau_2(a)=ab, \tau_2(b)=acac$, which are shown to be the return word sequences.
• Use the structure of $\Theta_1$ and $\Theta_2$ to determine return word lengths: $|r_p(\omega)|$ depends on the symbol at position $p$ in $\Theta_1$ or $\Theta_2$.
• Introduce the spectrum of a property $\mathcal{P}$ to analyze both the factor $\omega$ and its occurrence position $p$, enabling solution of Q2 (position-specific properties).

Experimental results

Research questions

• RQ1 What are the return word sequences of any factor $\omega$ in the period-doubling sequence $\u0394$?
• RQ2 Can the return word sequence of a factor $\omega$ in $\u0394$ be characterized as a substitutive sequence?
• RQ3 Does the return word sequence of a factor $\omega$ in $\u0394$ preserve the same structural type (i.e., $\Theta_1$ or $\Theta_2$), establishing reflexivity?
• RQ4 How can the return word sequence structure be used to determine whether two occurrences of a factor are adjacent, separated, or overlapped?
• RQ5 Can a spectrum-based method be constructed to analyze both the factor and its occurrence position, solving Q2 in combinatorics on words?

Key findings

• For any factor $\omega$ in $\u0394$, the return word sequence $\{r_p(\omega)\}_{p \geq 1}$ is either $\Theta_1$ or $\Theta_2$, both of which are substitutive sequences defined by explicit substitutions.
• The return word sequence of any factor $\omega$ in $\Theta_1$ or $\Theta_2$ is again $\Theta_1$ or $\Theta_2$, establishing the reflexivity property of return word sequences.
• The length of the $p$-th return word $|r_p(\omega)|$ is determined by the symbol at position $p$ in $\Theta_1$ or $\Theta_2$: $|r_p(\omega)| = 2^m$ if $\Theta_1[p] = a$, $2^{m-1}$ if $\Theta_1[p] = b$, and so on for $\Theta_2$.
• The spectrum of the separated, adjacent, and overlapped properties is fully characterized: $(\omega,p) \in \mathcal{P}_1$ (separated) if $\text{Env}(\omega) = E^1_m$ and $\Theta_1[p] = a$, or $\text{Env}(\omega) = E^1_m$, $\Theta_1[p] = b$, and $|\omega| < 2^{m-1}$, or $\text{Env}(\omega) = E^2_m$ and $\Theta_2[p] \neq a$.
• The spectrum of adjacent occurrences $(\omega,p) \in \mathcal{P}_2$ holds if and only if $\text{Env}(\omega) = E^1_m$ and $|\omega| = 2^{m-1}$.
• The spectrum of overlapped occurrences $(\omega,p) \in \mathcal{P}_3$ holds if $\text{Env}(\omega) = E^1_m$ and $|\omega| > 2^{m-1}$, or $\text{Env}(\omega) = E^2_m$ and $\Theta_2[p] = a$.