Counting Subwords in Circular Words and Their Parikh Matrices

TL;DR

This paper explores methods for counting subwords in circular words, extends Parikh matrices to this context, and investigates ambiguity in word identification, with implications for applications like splicing systems.

Contribution

It introduces two new methods for counting subwords in circular words and extends Parikh matrices to this setting, addressing ambiguity in word inference.

Findings

Developed two counting methods for subwords in circular words

Extended Parikh matrices to circular words

Identified rewriting rules generating ambiguous circular words

Abstract

The word inference problem is to determine languages such that the information on the number of occurrences of those subwords in the language can uniquely identify a word. A considerable amount of work has been done on this problem, but the same cannot be said for circular words despite growing interests on the latter due to their applications -- for example, in splicing systems. Meanwhile, Parikh matrices are useful tools and well established in the study of subword occurrences. In this work, we propose two ways of counting subword occurrences in circular words. We then extend the idea of Parikh matrices to the context of circular words and investigate this extension. Motivated by the word inference problem, we study ambiguity in the identification of a circular word by its Parikh matrix. Accordingly, two rewriting rules are developed to generate ternary circular words which share the same Parikh matrix.