A Semigroup Approach to Automaticity

Abstract.

We present an abstract, yet rather natural concept of automaticity which is based on semigroup actions. The standard notion of k-automaticity as well as many, if not all of its generalizations are recovered by specifying appropriate semigroup actions on $ \mathbb{N}_0 $ or $ \mathbb{N}^N_0 $ or other similar objects—e.g., the standard notion of k-automaticity is recovered by considering a certain "natural" action of the free semigroup with k generators on $ \mathbb{N}_0 $. We show that the main results characterizing automatic sequences (Cobham, 1972, Math. Systems Theory 6) hold almost verbatim in the general context, too.