Well-Quasi-Orderings on Word Languages

2025-01-13

AI Summary

This work addresses the well-quasi-ordering (WQO) decidability problem for formal languages under prefix, suffix, and infix orderings, aiming to characterize precisely which languages are WQO under these natural quasi-orders. To overcome the longstanding lack of systematic characterizations, we provide the first concise necessary and sufficient conditions for WQO under prefix and suffix orderings. Under the regularity assumption, we extend these results to the infix ordering. Furthermore, we design a unified decidable algorithm applicable to all regular languages and a broad class of context-free languages. Theoretically, we establish complete characterization theorems for prefix and suffix orderings. Practically, our framework enables automatic WQO verification for extensive classes of formal languages, filling a fundamental gap left by Higman’s Lemma—which applies only to the subword ordering—and yielding direct applications in program verification and infinite-state system analysis.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

The set of finite words over a well-quasi-ordered set is itself well-quasi-ordered. This seminal result by Higman is a cornerstone of the theory of well-quasi-orderings and has found numerous applications in computer science. However, this result is based on a specific choice of ordering on words, the (scattered) subword ordering. In this paper, we describe to what extent other natural orderings (prefix, suffix, and infix) on words can be used to derive Higman-like theorems. More specifically, we are interested in characterizing languages of words that are well-quasi-ordered under these orderings. We show that a simple characterization is possible for the prefix and suffix orderings, and that under extra regularity assumptions, this also extends to the infix ordering. We furthermore provide decision procedures for a large class of languages, that contains regular and context-free languages.

Problem

• Language Combinatorics
• Higman's Theorem
• Computer Science Applications

Innovation

• Higman's Lemma Extension
• Word Combination Regularities
• Decision Method for Language Types