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

TL;DR

This paper explores the closure properties of various classes of multi-tape automaton-recognized languages, focusing on their behavior under first-order logic and implications for automatic group theory.

Contribution

It provides new insights into the closure properties of regular, quasi-regular, and weakly regular languages under logical operations, with applications to decidability in group theory.

Findings

Regular languages are closed under certain logical operations.

Quasi-regular and weakly regular classes have distinct closure behaviors.

Results impact the decidability of problems in automatic group theory.

Abstract

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.