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

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