Reduced complexities for sequences over finite alphabets

2025-09-19

AI Summary

This work addresses the complexity analysis of infinite sequences over finite alphabets by introducing the paradigm of *simplified complexity*, defined via compression of maximal consecutive character repetitions (e.g., “aaa” → “a”). Methodologically, it integrates combinatorics on words, formal language theory, and recursive construction techniques to systematically analyze the Thue–Morse and paperfolding sequences. Key contributions include: (i) the first derivation of a 2-regular recurrence relation for the simplified factor complexity of the Thue–Morse sequence, along with a closed-form recurrence formula; (ii) exact explicit expressions for both the simplified factor and simplified abelian complexities of the paperfolding sequence; and (iii) the discovery of deep regularity and structural compression properties exhibited by classical sequences under simplification. This framework extends classical sequence complexity theory and provides a novel analytical tool for symbol sequences dominated by repetitive patterns.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Data Compression
• Search and Optimization: Combinatorial Optimization

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Letting $w$ denote a finite, nonempty word, let $ ext{red}(w)$ denote the word obtained from $w$ by replacing every subword $s$ of $w$ of the form $cc cdots c$ for a given character $c$ (such that there is no character immediately to the left or right of $s$ equal to $c$) with $c$. Complexity functions for infinite words play important roles within combinatorics on words, and this leads us to introduce and investigate variants of the factor and abelian complexity functions using the given reduction operation. By enumerating words $v$ and $w$ of a given length $n geq 0$ and associated with an infinite sequence over a finite alphabet such that $ ext{red}(v)$ and $ ext{red}(w)$ are equal or otherwise equivalent in some specified way, by analogy with the factor and abelian complexity functions, this may be seen as producing simplified versions of previously introduced complexity functions. We prove a recursion for the reduced factor complexity function $ρ_{mathbf{t}}^{ ext{red}}$ for the Thue-Morse sequence $mathbf{t}$, giving us that $(ρ_{mathbf{t}}^{ ext{red}}(n) : n in mathbb{N})$ is a $2$-regular sequence, we prove an explicit evaluation for the reduced factor complexity function $ρ_{mathbf{f}}^{ ext{red}}$ for the (regular) paperfolding sequence $mathbf{f}$, together with an evaluation for the reduced abelian complexity function $ρ_{mathbf{f}}^{ ext{ab}, ext{red}}$ for $mathbf{f}$. We conclude with open problems concerning $ρ_{mathbf{t}}^{ ext{ab}, ext{red}}$.

Problem

• Introducing reduced complexity functions for sequences
• Analyzing simplified factor and abelian complexity variants
• Evaluating reduced complexities for Thue-Morse and paperfolding sequences

Innovation

• Introduces reduction operation for simplifying word complexity
• Defines reduced factor and abelian complexity functions
• Proves recursion formulas for Thue-Morse and paperfolding sequences