An automata-theoretic approach to the study of the intersection of two submonoids of a free monoid

Laura Giambruno; Antonio Restivo

RAIRO - Theoretical Informatics and Applications (2008)

  • Volume: 42, Issue: 3, page 503-524
  • ISSN: 0988-3754

Abstract

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We investigate the intersection of two finitely generated submonoids of the free monoid on a finite alphabet. To this purpose, we consider automata that recognize such submonoids and we study the product automata recognizing their intersection. By using automata methods we obtain a new proof of a result of Karhumäki on the characterization of the intersection of two submonoids of rank two, in the case of prefix (or suffix) generators. In a more general setting, for an arbitrary number of generators, we prove that if H and K are two finitely generated submonoids generated by prefix sets such that the product automaton associated to H K has a given special property then r k ˜ ( H K ) r k ˜ ( H ) r k ˜ ( K ) where r k ˜ ( L ) = max ( 0 , r k ( L ) - 1 ) for any submonoid L.

How to cite

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Giambruno, Laura, and Restivo, Antonio. "An automata-theoretic approach to the study of the intersection of two submonoids of a free monoid." RAIRO - Theoretical Informatics and Applications 42.3 (2008): 503-524. <http://eudml.org/doc/250329>.

@article{Giambruno2008,
abstract = { We investigate the intersection of two finitely generated submonoids of the free monoid on a finite alphabet. To this purpose, we consider automata that recognize such submonoids and we study the product automata recognizing their intersection. By using automata methods we obtain a new proof of a result of Karhumäki on the characterization of the intersection of two submonoids of rank two, in the case of prefix (or suffix) generators. In a more general setting, for an arbitrary number of generators, we prove that if H and K are two finitely generated submonoids generated by prefix sets such that the product automaton associated to $H \cap K$ has a given special property then $\widetilde\{rk\}(H \cap K) \leq \widetilde\{rk\}(H) \widetilde\{rk\}(K)$ where $\widetilde\{rk\}(L)=\max(0,rk(L)-1)$ for any submonoid L. },
author = {Giambruno, Laura, Restivo, Antonio},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Automata; free monoids; rank; intersection of submonoids; product automata},
language = {eng},
month = {6},
number = {3},
pages = {503-524},
publisher = {EDP Sciences},
title = {An automata-theoretic approach to the study of the intersection of two submonoids of a free monoid},
url = {http://eudml.org/doc/250329},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Giambruno, Laura
AU - Restivo, Antonio
TI - An automata-theoretic approach to the study of the intersection of two submonoids of a free monoid
JO - RAIRO - Theoretical Informatics and Applications
DA - 2008/6//
PB - EDP Sciences
VL - 42
IS - 3
SP - 503
EP - 524
AB - We investigate the intersection of two finitely generated submonoids of the free monoid on a finite alphabet. To this purpose, we consider automata that recognize such submonoids and we study the product automata recognizing their intersection. By using automata methods we obtain a new proof of a result of Karhumäki on the characterization of the intersection of two submonoids of rank two, in the case of prefix (or suffix) generators. In a more general setting, for an arbitrary number of generators, we prove that if H and K are two finitely generated submonoids generated by prefix sets such that the product automaton associated to $H \cap K$ has a given special property then $\widetilde{rk}(H \cap K) \leq \widetilde{rk}(H) \widetilde{rk}(K)$ where $\widetilde{rk}(L)=\max(0,rk(L)-1)$ for any submonoid L.
LA - eng
KW - Automata; free monoids; rank; intersection of submonoids; product automata
UR - http://eudml.org/doc/250329
ER -

References

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