Visibly Pushdown Languages in Groups

TL;DR

This paper investigates the relationship between Visibly Pushdown Languages and finitely generated groups, establishing key characterizations and limitations of $ extbf{VPL}$ in group contexts.

Contribution

It proves that a group's word problem is $ extbf{VPL}$ only if the group is finite and explores the structure of $ extbf{VPL}$-related sets in groups.

Findings

Word problem of a finitely generated group is $ extbf{VPL}$ iff the group is finite.

Free reduction does not preserve $ extbf{VPL}$.

Solving equations in free groups with $ extbf{VPL}$ constraints is undecidable.

Abstract

In this paper we explore the connections between the class of Visibly Pushdown Languages ($\mathbf{VPL}$) and the natural sets of words one can associate to a finitely generated group. We show that the word problem of a finitely generated group is $\mathbf{VPL}$ exactly when the group is finite. We also show that free reduction does not preserve $\mathbf{VPL}$, and that finding solutions to equations in a free group with $\mathbf{VPL}$ constraints (as reduced words) is undecidable. We explore the structure of sets whose full preimage is $\mathbf{VPL}$, showing these are often recognisable sets. We conjecture that, in any group, this class is precisely the recognisable sets.