Tameness and the power of programs over monoids in DA

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