Tameness and the power of programs over monoids in DA

The program-over-monoid model of computation originates with Barrington's proof that the model captures the complexity class $\mathsf{NC^1}$. Here we make progress in understanding the subtleties of the model. First, we identify a new tameness condition on a class of monoids that entails a natural characterization of the regular languages recognizable by programs over monoids from the class. Second, we prove that the class known as $\mathbf{DA}$ satisfies tameness and hence that the regular languages recognized by programs over monoids in $\mathbf{DA}$ are precisely those recognizable in the classical sense by morphisms from $\mathbf{QDA}$. Third, we show by contrast that the well studied class of monoids called $\mathbf{J}$ is not tame. Finally, we exhibit a program-length-based hierarchy within the class of languages recognized by programs over monoids from $\mathbf{DA}$.