A Class of Automatic Sequences

TL;DR

This paper proves that the characteristic sequences of regular languages over a k-letter alphabet are k-automatic, extending to multiplicatively dependent alphabet sizes, thus linking regular languages and automatic sequences.

Contribution

It establishes a new connection between regular languages and automatic sequences for different alphabet sizes, including multiplicatively dependent cases.

Findings

Characteristic sequence of a regular language over a k-letter alphabet is k-automatic.

Extension to t-letter alphabets where t and k are multiplicatively dependent.

Provides a broader understanding of automaticity in regular languages.

Abstract

Let $k\ge 2$. We prove that the characteristic sequence of a regular language over a $k$-letter alphabet is $k$-automatic. More generally, if $t\ge 2$ and $t,k$ are multiplicatively dependent, we show that the characteristic sequence of a regular language over a $t$-letter alphabet is $k$-automatic.