G-automata, counter languages and the Chomsky hierarchy

TL;DR

This paper explores the relationship between G-automata languages and formal language classes, showing that counter languages are context-sensitive and linking their complexity to the word problem of groups like a5^n.

Contribution

It establishes a connection between the word problem of groups and the classification of G-automata languages within the Chomsky hierarchy.

Findings

Counter languages are context-sensitive.

Counter languages are indexed iff the word problem for a5^n is indexed.

G-automata languages are contained within the class accepted by the group's word problem.

Abstract

We consider how the languages of $G$-automata compare with other formal language classes. We prove that if the word problem of a group $G$ is accepted by a machine in the class $\mathcal M$ then the language of any $G$-automaton is in the class $\mathcal M$. It follows that the so called {\emph counter languages} (languages of $\mathbb Z^n$-automata) are context-sensitive, and further that counter languages are indexed if and only if the word problem for $\mathbb Z^n$ is indexed.