A note on the Lie complexity and beyond

TL;DR

This paper explores the properties of Lie complexity and its extensions in automatic sequences, proving that certain related complexity functions are automatic and extending previous results in the area.

Contribution

It introduces extended and prefix Lie complexities, establishing their automaticity and extending prior work on Lie complexity in automatic sequences.

Findings

Extended Lie complexity shares properties with Lie complexity.

The extended Lie complexity function is automatic.

The first-order difference of prefix Lie complexity is automatic.

Abstract

In a recent paper, Jason P. Bell and Jeffrey Shallit introduced the notion of {\em Lie complexity} and proved that the Lie complexity function of an automatic sequence is automatic. In this note, we give more facts concerning Lie complexity and define the extended Lie complexity and the prefix Lie complexity. Further, we prove that some proprieties of Lie complexity also hold for the extended Lie complexity. Particularly, we prove that the extended Lie complexity function and the first-order difference sequence of the prefix Lie complexity function of an automatic sequence are both automatic.