  Let A be a finite alphabet and let L contained in (A*)^n be an n-variable
language over A. We say that L is regular if it is the language accepted by a
synchronous n-tape finite state automaton, it is quasi-regular if it is
accepted by an asynchronous n-tape automaton, and it is weakly regular if it is
accepted by a non-deterministic asynchronous n-tape automaton. We investigate
the closure properties of the classes of regular, quasi-regular, and weakly
regular languages under first-order logic, and apply these observations to an
open decidability problem in automatic group theory.
