Counting Subwords and Regular Languages

TL;DR

This paper investigates conditions under which languages defined by equal or ordered subword counts are regular, providing a precise criterion and an efficient method to verify it.

Contribution

It establishes a necessary and sufficient condition for the regularity of languages based on subword occurrence counts and offers an efficient checking procedure.

Findings

Characterizes when such subword-count languages are regular.

Provides an algorithm to verify the regularity condition.

Clarifies the structure of languages defined by subword occurrence constraints.

Abstract

Let $x$ and $y$ be words. We consider the languages whose words $z$ are those for which the numbers of occurrences of $x$ and $y$, as subwords of $z$, are the same (resp., the number of $x$'s is less than the number of $y$'s, resp., is less than or equal). We give a necessary and sufficient condition on $x$ and $y$ for these languages to be regular, and we show how to check this condition efficiently.