The reflection complexity of sequences over finite alphabets

2024-06-13

arXiv.org

AI Summary

This paper introduces *reflexive complexity*, a novel sequence complexity measure defined as the number of distinct equivalence classes of length-$n$ factors under reversal. It systematically investigates the growth order, computability, and structural characterization power of this measure for infinite sequences over finite alphabets. Method: The analysis integrates combinatorics on words, $k$-regular sequence theory, and the Walnut formal verification tool. Contribution/Results: The work establishes the first Morse–Hedlund-type criterion, precisely characterizing eventually periodic sequences via reflexive complexity. It fully determines the asymptotic behavior of reflexive complexity for classical families—including Sturmian, Rote, and rich sequences. Moreover, it proves that the reflexive complexity of any $k$-automatic sequence is a computable $k$-regular function, and explicitly computes it for canonical examples such as the Thue–Morse sequence. These results provide a new theoretical lens and practical tool for structural analysis of sequences.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Evaluation and Analysis

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• Web Mining and Content Analysis: Web measurements
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

In combinatorics on words, the well-studied factor complexity function $ ho_{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 exttt{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.

Problem

• Introduces reflection complexity to count sequence factors up to symbol reversal
• Characterizes periodic and Sturmian sequences via reflection complexity properties
• Analyzes reflection complexity of automatic sequences using computable regularity

Innovation

• Introduces reflection complexity function r_x
• Proves Morse-Hedlund-type result for periodicity
• Links k-automatic sequences to k-regularity