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

翻译：在词组合学中，已被深入研究的序列因子复杂度函数 $\rho_{\bf x}$ 统计了有限字母表上序列 ${\bf x}$ 对于任意非负整数 $n$ 的不同长度为 $n$ 的因子的数量。本文引入 \emph{反射复杂度} 函数 $r_{\bf x}$，用于枚举序列 ${\bf x}$ 中出现的因子，这些因子在单词中符号顺序反转的意义下被视为相同。我们引入并证明了关于 $r_{\bf x}$ 的增长性质及其与其他复杂度函数关系的结果。我们证明，若 ${\bf x}$ 是 $k$-自动序列，则 $r_{\bf x}$ 是可计算 $k$-正则的，并利用软件 {\tt Walnut} 评估了自动序列（如 Thue--Morse 序列）的反射复杂度。我们证明了一个 Morse--Hedlund 型结果，该结果依据反射复杂度刻画了最终周期序列，并由此推导出 Sturmian 序列的一个特征。此外，我们还研究了 episturmian 序列、$(s+1)$ 维台球序列以及 Rote 序列的反射复杂度。关于这一度量仍存在许多未解决的问题。