Visibly Pushdown Languages in Groups
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866915961856065536 |
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| author | Ciobanu, Laura Turaev, Daniel |
| author_facet | Ciobanu, Laura Turaev, Daniel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22375 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Visibly Pushdown Languages in Groups Ciobanu, Laura Turaev, Daniel Group Theory Formal Languages and Automata Theory 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. |
| title | Visibly Pushdown Languages in Groups |
| topic | Group Theory Formal Languages and Automata Theory |
| url | https://arxiv.org/abs/2604.22375 |