Well-Quasi-Orderings on Word Languages

TL;DR

This paper explores how different natural orderings on words, such as prefix, suffix, and infix, influence the well-quasi-ordering properties of languages, extending Higman's theorem beyond subword ordering.

Contribution

It characterizes languages that are well-quasi-ordered under prefix, suffix, and infix orderings and provides decision procedures for regular and context-free languages.

Findings

Characterization of well-quasi-ordered languages under prefix and suffix orderings.

Extension of Higman-like theorems to infix ordering with regularity assumptions.

Decision procedures for determining well-quasi-ordering in regular and context-free languages.

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.