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The genus of regular languages

Published online by Cambridge University Press:  20 May 2016

GUILLAUME BONFANTE  and
FLORIAN DELOUP
GUILLAUME BONFANTE
Affiliation:
LORIA, Université de Lorraine, Campus scientifique, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France Email: guillaume.bonfante@loria.fr
FLORIAN DELOUP
Affiliation:
Institut de Mathématiques, Université Paul Sabatier, Toulouse III, 118 Route de Narbonne, 31069 Toulouse cedex 9, France Email: florian.deloup@math.univ-toulouse.fr
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Abstract

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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.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Issue 1 , January 2018 , pp. 14 - 44
Copyright
Copyright © Cambridge University Press 2016 

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