The reflection complexity of sequences over finite alphabets

TL;DR

This paper introduces reflection complexity $r_{\infw{x}}$, which counts length-$n$ factors of a sequence $\infw{x}$ up to reversal, and explores its interplay with the classical factor complexity $\rho_{\infw{x}}$, palindrome counts, and unreflected factors. By establishing fundamental inequalities, a graph-theoretic framework, and a Morse–Hedlund-type dichotomy, the authors characterize eventual periodicity and Sturmian sequences, and extend the analysis to quasi-Sturmian, episturmian, billiard, Rote, and rich sequences. They further show that for $k$-automatic sequences, $r_{\infw{x}}$ is computably $k$-regular and provide concrete evaluations for classic automatic sequences via the Walnut tool. The work reveals a rich structure for reflection complexity, including a robust method to compute it in many important cases and numerous avenues for future investigation.

Abstract

In combinatorics on words, the well-studied factor complexity function $ρ_{\infw{x}}$ of a sequence $\infw{x}$ over a finite alphabet counts, for every nonnegative integer $n$, the number of distinct length-$n$ factors of $\infw{x}$. In this paper, we introduce the \emph{reflection complexity} function $r_{\infw{x}}$ to enumerate the factors occurring in a sequence $\infw{x}$, up to reversing the order of symbols in a word. We prove a number of results about the growth properties of $r_{\infw{x}}$ and its relationship with other complexity functions. We also prove a Morse--Hedlund-type result characterizing eventually periodic sequences in terms of their reflection complexity, and we deduce a characterization of Sturmian sequences. We investigate the reflection complexity of quasi-Sturmian, episturmian, $(s+1)$-dimensional billiard, complementation-symmetric Rote, and rich sequences. Furthermore, we prove that if $\infw{x}$ is $k$-automatic, then $r_{\infw{x}}$ is computably $k$-regular, and we use the software \texttt{Walnut} to evaluate the reflection complexity of some automatic sequences, such as the Thue--Morse sequence. We note that there are still many unanswered questions about this reflection measure.

Figures (1)

• Figure 1: The consequence of Lemma \ref{['lem:main']} in the graph $\Lambda_{\mathbf{x}}(n)$, when $u = v^R$ but $u \not= v$.

Theorems & Definitions (89)

• Definition 1
• Example 2
• Definition 3
• Example 4
• Theorem 5: MorseHedlund1938
• Remark 6
• Example 7
• Lemma 8
• proof
• Theorem 9