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Decidability of regular language genus computation

Published online by Cambridge University Press:  10 April 2019

Guillaume Bonfante  and
Florian L. Deloup
Guillaume Bonfante
Affiliation:
LORIA, Université de Lorraine, Campus scientifique, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex
Florian L. Deloup*
Affiliation:
IMT, Université Paul Sabatier – Toulouse III, 118 Route de Narbonne, 31069 Toulouse cedex 9
*
*Corresponding author. Email: florian.deloup@math.univ-toulouse.fr
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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.

Keywords

Deterministic finite automatonregular languagetopological genusgraph embeddingplanarity
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Issue 9 , October 2019 , pp. 1428 - 1443
Copyright
© Cambridge University Press 2019 

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