The model of programs over (finite) monoids, introduced by Barrington and Th{e}rien, 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 $\mathbf{J}$. We show that those programs actually can recognise all languages from a class of restricted dot-depth one languages, using a non-trivial trick, and conjecture that this class suffices to characterise the regular languages recognised by programs over monoids in $\mathbf{J}$.

The Power of Programs over Monoids in J

The PLP conjecture for monoids states that for every monoid M, either M is universal (that is, for every language $L \subseteq \varSigma ^*$ there is a program over M which accepts the language L) or it has the polynomial length property (that is, every program over the monoid M has an equivalent program of length ${\mathsf {poly}}(n)$). The conjecture has been confirmed (Tesson-Therien (2001)) for the case of groups and several subclasses of aperiodic monoids such as the variety DA and the monoids divided by the monoid U. However, the case of the set of monoids divided by the monoid $\mathsf {BA}_2$ is still open, which if resolved, confirms the conjecture for all aperiodic monoids.

Janani Sundaresan

Papers

On the Computational Power of Programs over \(\mathsf {BA}_2\) Monoid