[en] We study the structure of automata accepting the greedy representations of N in a
wide class of numeration systems. We describe the conditions under which such automata can
have more than one strongly connected component and the form of any such additional components. Our characterization applies, in particular, to any automaton arising from a Bertrand
numeration system. Furthermore, we show that for any automaton A arising from a system
with a dominant root > 1, there is a morphism mapping A onto the automaton arising from
the Bertrand system associated with the number . Under some mild assumptions, we also
study the state complexity of the trim minimal automaton accepting the greedy representations
of the multiples of m>=2 for a wide class of linear numeration systems. As an example, the
number of states of the trim minimal automaton accepting the greedy representations of mN in
the Fibonacci system is exactly 2m^2.
Disciplines :
Mathematics Computer science
Author, co-author :
Charlier, Emilie ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Rampersad, Narad ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Rigo, Michel ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Waxweiler, Laurent ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Language :
English
Title :
Structure of the minimal automaton of a numeration language and applications to state complexity
Publication date :
2010
Event name :
Journées montoises d'informatique théorique 2010
Event organizer :
Fabien Durand, Richard Groult, Julien Leroy, Florence Levé, Samuel Petite, Gwénaël Richomme, Michel Rigo
Event place :
Amiens, France
Event date :
du 6 septembre 2010 au 10 septembre 2010
Audience :
International
Main work title :
Actes des Journées Montoises d'Informatique Théorique
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