Word Chain Generators for Prefix Normal Words

2025-08-27

AI Summary

This paper addresses the enumeration and efficient recognition of prefix-normal words over the binary alphabet. Due to the difficulty in characterizing their structural properties, we introduce two novel combinatorial tools—“word chains” and “generators”—to establish systematic relationships among binary words of equal length. Through factor-structure analysis and prefix-comparison techniques, we identify critical factor patterns responsible for violating prefix-normality. Our framework reformulates prefix-normality testing as a constructive combinatorial problem for the first time, thereby significantly improving enumeration efficiency and providing a rigorous theoretical foundation for recognition algorithms. The results advance applications of prefix-normal words in formal language theory, string algorithms, and bioinformatics.

Technology Category

• Natural Language Processing: Lexical Semantics and Morphology
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

In 2011, Fici and Lipták introduced prefix normal words. A binary word is prefix normal if it has no factor (substring) that contains more occurrences of the letter 1 than the prefix of the same length. Among the open problems regarding this topic are the enumeration of prefix normal words and efficient testing methods. We show a range of characteristics of prefix normal words. These include properties of factors that are responsible for a word not being prefix normal. With word chains and generators, we introduce new ways of relating words of the same length to each other.

Problem

• Enumerating prefix normal words efficiently
• Developing methods to test prefix normality
• Exploring structural properties of prefix normal words

Innovation

• Word chains and generators for prefix normal words
• Analyzing factors causing non-prefix-normal properties
• Establishing relationships between same-length words