Avoidability of circular formulas

TL;DR

This paper determines the exact avoidability index of circular formulas, including all formulas in the 3-avoidance basis, refining previous bounds and deepening understanding of avoidable patterns in formal language theory.

Contribution

It precisely computes the avoidability index for all circular formulas and formulas in the 3-avoidance basis, advancing the classification of avoidable formulas.

Findings

Avoidability index of all circular formulas is exactly determined.

All formulas in the 3-avoidance basis have avoidability index at most 4.

The results refine previous upper bounds on avoidability indices.

Abstract

Clark has defined the notion of $n$-avoidance basis which contains the avoidable formulas with at most $n$ variables that are closest to be unavoidable in some sense. The family $C_i$ of circular formulas is such that $C_1=AA$, $C_2=ABA.BAB$, $C_3=ABCA.BCAB.CABC$ and so on. For every $i\le n$, the $n$-avoidance basis contains $C_i$. Clark showed that the avoidability index of every circular formula and of every formula in the $3$-avoidance basis (and thus of every avoidable formula containing at most 3 variables) is at most 4. We determine exactly the avoidability index of these formulas.