Multipass automata and group word problems
Tullio Ceccherini-Silberstein, Michel Coornaert, Francesca Fiorenzi, Paul E. Schupp, Nicholas W. M. Touikan
TL;DR
This work introduces multipass automata, a natural extension of pushdown automata that read input over multiple passes, and establishes sharp language-class characterizations: $\mathcal{DM}=\mathcal{BDC}$ (Boolean closure of deterministic CFLs) and $\mathcal{NM}=\mathcal{PCF}$ (poly-context-free languages). It then applies these models to finitely generated groups by defining $\mathcal{BDG}$ and $\mathcal{PG}$ as classes with word problems in $\mathcal{BDC}$ and $\mathcal{PCF}$, respectively, proving presentation-independence and closure under direct products, finite extensions, and finitely generated subgroups. The paper further shows that certain HNN extensions preserve membership in $\mathcal{BDG}$ and $\mathcal{PG}$, and that the broader class $\mathcal{D}$ is closed under a general family of HNN extensions, offering strong evidence for Brough's conjecture. Parikh’s theorem is employed to identify groups not in $\mathcal{PG}$ (e.g., certain Baumslag–Solitar groups), while the authors analyze decision problems—membership, emptiness, and the order problem—within these classes and relate them to matrix-group representations, illustrating the boundaries of these language-theoretic group properties.
Abstract
We introduce the notion of multipass automata as a generalization of pushdown automata and study the classes of languages accepted by such machines. The class of languages accepted by deterministic multipass automata is exactly the Boolean closure of the class of deterministic context-free languages while the class of languages accepted by nondeterministic multipass automata is exactly the class of poly-context-free languages, that is, languages which are the intersection of finitely many context-free languages. We illustrate the use of these automata by studying groups whose word problems are in the above classes.