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

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 $n$.