Vertical representation of $C^{\infty}$-words

TL;DR

This paper introduces a novel framework for representing $C^{ abla}$-words using frontiers and an infinite directed acyclic graph, providing new insights into their structure and related conjectures.

Contribution

It offers a compact, frontier-based representation of $C^{ abla}$-words and links their properties to an abstract graph structure, enabling recursive analysis without direct reference to the words.

Findings

Defined a new graph-based representation of $C^{ abla}$-words.

Connected conjectures on $C^{ abla}$-words to properties of the graph.

Provided a recursive map for analyzing the structure of $C^{ abla}$-words.

Abstract

We present a new framework for dealing with $C^{\infty}$-words, based on their left and right frontiers. This allows us to give a compact representation of them, and to describe the set of $C^{\infty}$-words through an infinite directed acyclic graph $G$. This graph is defined by a map acting on the frontiers of $C^{\infty}$-words. We show that this map can be defined recursively and with no explicit references to $C^{\infty}$-words. We then show that some important conjectures on $C^{\infty}$-words follow from analogous statements on the structure of the graph $G$.