The decidability of the genus of regular languages and directed emulators

TL;DR

This paper investigates the genus of regular languages by introducing directed emulators and automatic relations, establishing their correspondence, and analyzing how these concepts determine the minimal genus of automata recognizing the language.

Contribution

It introduces directed emulators and automatic relations, proving their equivalence and their role in determining the minimal genus of regular languages' automata.

Findings

The genus of a regular language equals the minimum genus among all directed emulators of its minimal automaton.

The minimal genus is attained within the class of directed covers of the automaton's underlying graph.

Deciding the minimal genus for directed emulators relates to the undirected case, linking directed and undirected graph problems.

Abstract

The article continues our study of the genus of a regular language $L$, defined as the minimal genus among all genera of all finite deterministic automata recognizing $L$. Here we define and study two closely related tools on a directed graph: directed emulators and automatic relations. A directed emulator morphism essentially encapsulates at the graph-theoretic level an epimorphism onto the minimal deterministic automaton. An automatic relation is the graph-theoretic version of the Myhill-Nerode relation. We show that an automatic relation determines a directed emulator morphism and respectively, a directed emulator morphism determines an automatic relation up to isomorphism. Consider the set $S$ of all directed emulators of the underlying directed graph of the minimal deterministic automaton for $L$. We prove that the genus of $L$ is $\underset{G \in S}{\min}\ g(G)$. We also consider the more restrictive notion of directed cover and prove that the genus of $L$ is reached in the class of directed covers of the underlying directed graph of the minimal deterministic automaton for $L$. This stands in sharp contrast to undirected emulators and undirected covers which we also consider. Finally we prove that if the problem of determining the minimal genus of a directed emulator of a directed graph has a solution then the problem of determining the minimal genus of an undirected emulator of an undirected graph has a solution.