Avoidance bases for formulas with reversal

TL;DR

This paper determines the minimal sets of formulas with reversal for each of the first three avoidability indices, identifying unique formulas with the highest avoidability index in each set.

Contribution

It finds the n-avoidance bases for formulas with reversal for n=1,2,3, and identifies unique formulas with the highest avoidability index in each base.

Findings

Identified 1-avoidance, 2-avoidance, and 3-avoidance bases for formulas with reversal.

Found unique formulas with highest avoidability index in each base.

Confirmed these formulas belong to an infinite family previously studied.

Abstract

In the interest of studying formulas with reversal of high avoidability index, we find $n$-avoidance bases for formulas with reversal for $n\in\{1,2,3\}$. We demonstrate that there is a unique formula with reversal in each of these three bases of highest avoidability index $n+2$; these formulas are $xx$, $xyx\cdot y^R$, and $xyzx\cdot y^R\cdot z^R$, which belong to an infinite family of formulas with reversal that has been the subject of recent study by the authors.