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.

💡 Research Summary

The paper investigates three increasingly expressive classes of multi‑tape languages over a finite alphabet A: (i) regular languages, accepted by synchronous n‑tape finite‑state automata; (ii) quasi‑regular languages, accepted by deterministic asynchronous n‑tape automata; and (iii) weakly regular languages, accepted by nondeterministic asynchronous n‑tape automata. After formalising these notions, the authors study how each class behaves under the standard first‑order logical operations: union, intersection, complement, existential quantification, and universal quantification (i.e., projection and its complement).

For union and intersection the paper shows that all three classes are closed. The construction is straightforward for synchronous automata, while for deterministic asynchronous automata the authors introduce a “synchronised product” that allows the two independent head movements to be simulated in parallel without forcing a global clock. The same technique, extended with nondeterministic branching, yields closure for weakly regular languages.

Complementation reveals a strict separation. Regular languages are closed under complement, as expected. Deterministic asynchronous automata, however, fail to be closed: the authors present a language L = {(w₁,w₂) | |w₁| = |w₂|} whose complement cannot be recognised by any deterministic asynchronous automaton because such a device cannot globally compare the lengths of two independently moving heads. In contrast, the nondeterministic asynchronous model regains closure under complement, since a nondeterministic choice can guess a witness to the violation of the original property and reject accordingly.

Existential quantification (∃) corresponds to projecting away one component of an n‑tuple. The paper proves that deterministic asynchronous automata are not closed under projection. A concrete counterexample is the language L₁ = {(x,y) | x = y·a}, whose projection onto the first component yields {x | x ends with a}, a language that cannot be recognised by any deterministic asynchronous automaton without losing the ability to “guess” the missing y. By contrast, weakly regular languages are closed under ∃‑quantification: the nondeterministic automaton can guess a suitable y on the fly, thereby simulating the projection. This result highlights the extra expressive power supplied by nondeterminism in the asynchronous setting.

Universal quantification (∀) is the complement of an existential projection, and consequently none of the three classes is closed under ∀. The authors give a simple example, ∀y (x ≠ y), whose language is not weakly regular, demonstrating that the lack of closure under complement propagates to universal quantification.

Having established these logical closure properties, the authors turn to an application in automatic group theory. A group is called automatic if its word problem and the graph of its multiplication can be recognised by a synchronous automaton; a “asynchronous automatic group” relaxes the synchrony requirement. The paper observes that the open decidability problem—whether a given weakly regular language can serve as the multiplication table of a group—can be reframed in terms of the closure results above. Since weakly regular languages are closed under existential projection, one can eliminate one coordinate of the multiplication table while preserving recognisability, but the lack of closure under complement and universal quantification prevents a straightforward decision procedure. The authors suggest that further study of the structural properties of nondeterministic asynchronous automata might eventually yield an algorithmic characterisation of asynchronous automatic groups, thereby solving the longstanding open problem.

In summary, the work delineates a clear hierarchy of language classes defined by deterministic versus nondeterministic asynchronous automata, maps their exact closure behaviour under first‑order operations, and connects these theoretical insights to a concrete open question in the theory of automatic groups. The results both clarify the limits of asynchronous computation models and point toward promising directions for future research.