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

TL;DR

This paper explores the computational capabilities of programs over monoids in the variety J, introducing threshold dot-depth one languages and conjecturing their role in characterizing regular languages recognized by such programs.

Contribution

It establishes a hierarchy within languages recognized by monoid programs in J and introduces threshold dot-depth one languages, linking algebraic automata theory with circuit complexity.

Findings

Programs over monoids in J recognize all threshold dot-depth one languages.

A new algebraic characterization of threshold dot-depth one languages is provided.

Conjecture: Additional modular counting extends recognition to all regular languages.

Abstract

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.