Closure properties of predicates recognized by deterministic and non-deterministic asynchronous automata

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.