Remarks on recognizable subsets and local rank

TL;DR

This paper provides a model-theoretic characterization of recognizable subsets and regular languages in monoids, linking automata theory concepts with first-order logic ranks and multiplicities.

Contribution

It establishes a novel connection between recognizability in automata theory and first-order theory ranks, offering a new perspective on regular languages and their complexities.

Findings

Recognizable subsets correspond to zero $ ext{rank}$ in the first-order theory.

For finitely generated free monoids, this characterizes regular languages model-theoretically.

The $ ext{multiplicity}$ relates to state complexity and the size of the syntactic monoid.

Abstract

Given a monoid $(M,\varepsilon,\cdot )$ it is shown that a subset $A\subseteq M$ is recognizable in the sense of automata theory if and only if the $\varphi $-rank of $x=x$ is zero in the first-order theory $\operatorname{Th}(M,\varepsilon ,\cdot ,A)$, where $\varphi (x;u)$ is the formula $xu\in A$. In the case where $M$ is a finitely generated free monoid on a finite alphabet $\Sigma $, this gives a model-theoretic characterization of the regular languages over $\Sigma $. If $A$ is a regular language over $\Sigma $ then the $\varphi $-multiplicity of $x=x$ is the state complexity of $A$. Similar results holds for $\varphi' (x;u,v)$ given by $uxv\in A$, with the $\varphi' $-multiplicity now equal to the size of the syntactic monoid of $A$.