Representations of Circular Words

TL;DR

This paper introduces two novel representations for circular words—tuple-based and tree-based—that facilitate compact notation, processing conjugates, and visualizing periodic properties, along with an iterative encoding method for words over a two-letter alphabet.

Contribution

It presents two distinct representations of circular words and an iterative encoding method, enhancing analysis and visualization of their properties.

Findings

Tree representation enables easy processing of conjugates.

Tuple notation offers a compact way to represent circular words.

Iterative construction allows building any two-letter word from 'ab'.

Abstract

In this article we give two different ways of representations of circular words. Representations with tuples are intended as a compact notation, while representations with trees give a way to easily process all conjugates of a word. The latter form can also be used as a graphical representation of periodic properties of finite (in some cases, infinite) words. We also define iterative representations which can be seen as an encoding utilizing the flexible properties of circular words. Every word over the two letter alphabet can be constructed starting from ab by applying the fractional power and the cyclic shift operators one after the other, iteratively.