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

Authors
Type
Preprint
Publication Date
Mar 16, 2021
Submission Date
Dec 17, 2019
Source
arXiv
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.