Automatic Sequences of Rank Two

TL;DR

This paper proves that it is decidable whether a $k$-automatic infinite word has rank two, meaning it can be constructed from a small set of words, expanding understanding of automatic sequences.

Contribution

It establishes the decidability of the rank-two property for all $k$-automatic sequences, a novel result in the study of automatic words.

Findings

Decidability of rank two for $k$-automatic words.

Characterization of automatic sequences with rank two.

Extension of rank concepts to automatic sequences.

Abstract

Given a right-infinite word $\bf x$ over a finite alphabet $A$, the rank of $\bf x$ is the size of the smallest set $S$ of words over $A$ such that $\bf x$ can be realized as an infinite concatenation of words in $S$. We show that the property of having rank two is decidable for the class of $k$-automatic words for each integer $k\ge 2$.