The genus of regular languages and directed graph emulators

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 minG∈Sg(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.