2019/12/17 by Nathan Grosshans, Grosshans, Nathan
Computer Science · #Computability, Logic, AI Algorithms #Computational Complexity (cs.CC) #FOS: Computer and information sciences #Formal Languages and Automata Theory (cs.FL) #Logic, programming, and type systems #Machine Learning and Algorithms #semigroups and automata theory
paper · doi:10.48550/arxiv.1912.07992
openalex publication_date 2019/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The model of programs over (finite) monoids, introduced by Barrington and Thérien, gives an interesting way to characterise the circuit complexity class NC1 and its subclasses and showcases deep connections with algebraic automata theory. In this article, we investigate the computational power of programs over monoids in 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 J, based on the length of programs but also some parametrisation of J. Second, and most importantly, we make progress in understanding what regular languages can be recognised by programs over monoids in 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 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 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.