The genus of regular languages

TL;DR

This paper introduces the concept of genus for regular languages and automata, establishing bounds, exploring its implications on automata transformations, and demonstrating a hierarchy of languages based on genus.

Contribution

It defines the genus of regular languages, analyzes bounds related to alphabet size, and constructs languages with arbitrarily large genus, revealing a new hierarchy.

Findings

Genus grows at least linearly with automata size under certain conditions.

Powerset determinization incurs exponential topological cost.

Minimization is orthogonal to the genus of regular languages.

Abstract

The article defines and studies the genus of finite state deterministic automata (FSA) and regular languages. Indeed, a FSA can be seen as a graph for which the notion of genus arises. At the same time, a 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.