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.

Abstract This article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|gen to be the minimal size of all finite deterministic automata of genus g(L) computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus.

The paper defines and studies the genus of finite state deterministic automata (FSA) and regular languages. Indeed, an FSA can be seen as a graph for which the notion of genus arises. At the same time, an FSA has a semantics via its underlying language. It is then natural to make a connection between the languages and the notion of genus. After we introduce and justify the the notion of the genus for regular languages, the following questions are addressed. First, depending on the size of the alphabet, we provide upper and lower bounds on the genus of regular languages: we show that under a relatively generic condition on the alphabet and the geometry of the automata, the genus grows at least linearly in terms of the size of the automata. Second, we show that the topological cost of the powerset determinization procedure is exponential. Third, we prove that the notion of minimization is orthogonal to the notion of genus. Fourth, we build regular languages of arbitrary large genus: the notion of genus defines a proper hierarchy of regular languages.