The decidability of the genus of regular languages and directed emulators
Guillaume Bonfante, Florian Deloup
TL;DR
This work investigates the genus of regular languages, introducing directed emulators and automatic relations as graph-theoretic tools to connect minimal deterministic automata with topological genus. The authors establish that the genus $g(L)$ of a language $L$ equals the minimum genus among directed emulators of the underlying graph $G(L)$ of the minimal automaton, and they show that directed covers suffice to realize this genus; they also relate these directed notions to undirected counterparts. A key contribution is the automatic-description framework, linking directed emulators to automatic and MN-recursive relations, and proving a lattice structure that yields canonical decompositions $ ext{φ}= ext{ι}\circ[-]_{ ilde{ o}}$. The main theoretical payoff is the equivalence between the language-genus problem and a graph-genus problem for directed covers (hence planarity), enabling potential reductions via directed-minor-type techniques and providing a principled path toward decidability questions in the genus of regular languages. These results illuminate deep connections between automata theory, graph topology, and category-theoretic formalisms, with implications for planarity tests and possibly for related formalisms such as linear logic.
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.