The reflection complexity of sequences over finite alphabets

TL;DR

This paper introduces the reflection complexity function for sequences over finite alphabets, exploring its properties, relationships with other complexities, and applications to automatic and well-known sequences like Thue–Morse.

Contribution

It defines and analyzes the reflection complexity, providing characterizations of periodicity and Sturmian sequences, and connects it with automatic sequences using computational tools.

Findings

Reflection complexity characterizes periodic and Sturmian sequences.

For k-automatic sequences, reflection complexity is computably k-regular.

Software Walnut can evaluate reflection complexity of automatic sequences.

Abstract

In combinatorics on words, the well-studied factor complexity function $\rho_{\infw{x}}$ 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.