It is well known that context-free languages (CFLs) can be recognized by pushdown automata. Such machines maintain a stack such that the top of the stack affects the state transitions. During the state transitions, the stack may get modified as well. In the classical model, after a single pass over the input, the automaton will be in an accepting state for valid strings.

This paper considers automata that make multiple passes over the input. Motivation here seems to be theoretical, with a goal to investigate the classes of languages that can be accepted by such machines. It is not yet clear if these enhanced machines can exhibit superiority over the single pass versions in practical applications.

The authors have delivered a well-written work that is full of mathematical rigor. The first section provides some introduction and the second defines the automata, but the details quickly become intense. Some background in the algebra of groups will be helpful to follow the content. Essentially, deterministic multipass automata can recognize the Boolean closure of the class of deterministic CFLs. The nondeterministic ones can accept languages, which are the intersection of finitely many CFLs.

The focus here is on group word problems. Specifically, the authors consider the class of finitely generated groups whose word problem is in the Boolean closure of deterministic CFLs. Sections 3 and 4 provide proofs of theorems related to this. Section 5 reviews theory related to Abelianization maps, with further results covered in sections 6 and 7.

Overall, an impressive academic contribution that will be best enjoyed by researchers pursuing formal language theory.