The Power of Programs over Monoids in J and Threshold Dot-depth One Languages

The model of programs over (finite) monoids, introduced by Barrington and Th\'erien, gives an interesting way to characterise the circuit complexity class $\mathsf{NC^1}$ and its subclasses and showcases deep connections with algebraic automata theory. In this article, we investigate the computational power of programs over monoids in $\mathbf{J}$, a small variety of finite aperiodic monoids. First, we give a fine hierarchy within the class of languages recognised by programs over monoids from $\mathbf{J}$, based on the length of programs but also some parametrisation of $\mathbf{J}$. Second, and most importantly, we make progress in understanding what regular languages can be recognised by programs over monoids in $\mathbf{J}$. To this end, we introduce a new class of restricted dot-depth one languages, threshold dot-depth one languages. We show that programs over monoids in $\mathbf{J}$ actually can recognise all languages from this class, using a non-trivial trick, and conjecture that threshold dot-depth one languages with additional positional modular counting suffice to characterise the regular languages recognised by programs over monoids in $\mathbf{J}$. Finally, using a result by J. C. Costa, we give an algebraic characterisation of threshold dot-depth one languages that supports that conjecture and is of independent interest.