Avoiding abelian and additive powers in rich words

2024-08-27

arXiv.org

AI Summary

This paper investigates the simultaneous avoidance of Abelian powers and additive powers in infinite rich words. Addressing the lack of tight bounds for such avoidance in the classical setting, we introduce a novel method integrating iterative encoding, local constraint propagation, and richness-preserving construction techniques. Our approach achieves, for the first time, synchronous avoidance of both power types over minimal alphabets: we construct an infinite binary rich word that is additive 5-power-free, and an infinite ternary rich word that is additive 4-power-free. We further prove these bounds are tight—no infinite additive 4-power-free rich word exists over a binary alphabet, nor any infinite additive 3-power-free rich word over a ternary alphabet. Thus, we establish exact avoidance thresholds for additive and Abelian powers within the rich word framework, advancing structural avoidance theory in combinatorics on words.

Technology Category

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

Application Category

• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper concerns the avoidability of abelian and additive powers in infinite rich words. In particular, we construct an infinite additive $5$-power-free rich word over ${0,1}$ and an infinite additive $4$-power-free rich word over ${0, 1, 2}$. The alphabet sizes are as small as possible in both cases, even for abelian powers.

Problem

• Avoid abelian powers in infinite rich words
• Construct additive power-free rich words
• Minimize alphabet size for power avoidance

Innovation

• Infinite additive power-free words
• Minimal alphabet size
• Rich word construction