The Word Problem of $\mathbb{Z}^n$ Is a Multiple Context-Free Language

TL;DR

This paper proves that the word problem for the free abelian group  is a multiple context-free language, extending previous results from  to all dimensions, revealing new connections between group theory and formal language classes.

Contribution

It generalizes Salvati's result by showing the word problem of  is a multiple context-free language for any n, broadening understanding of formal language classifications of group word problems.

Findings

The word problem of  is a multiple context-free language for all n.

Extends Salvati's result from  to arbitrary n.

Links group theory with formal language hierarchy.

Abstract

The \emph{word problem} of a group $G = \langle \Sigma \rangle$ can be defined as the set of formal words in $\Sigma^*$ that represent the identity in $G$. When viewed as formal languages, this gives a strong connection between classes of groups and classes of formal languages. For example, Anisimov showed that a group is finite if and only if its word problem is a regular language, and Muller and Schupp showed that a group is virtually-free if and only if its word problem is a context-free language. Above this, not much was known, until Salvati showed recently that the word problem of $\mathbb{Z}^2$ is a multiple context-free language, giving first such example. We generalize Salvati's result to show that the word problem of $\mathbb{Z}^n$ is a multiple context-free language for any $n$.