On the State Complexity of the Shuffle of Regular Languages - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Communication Dans Un Congrès Année : 2016

On the State Complexity of the Shuffle of Regular Languages

Janusz Brzozowski
  • Fonction : Auteur
  • PersonId : 1022710
Galina Jirásková
  • Fonction : Auteur
  • PersonId : 1011781
Bo Liu
  • Fonction : Auteur
  • PersonId : 1022711
Aayush Rajasekaran
  • Fonction : Auteur
  • PersonId : 1022712
Marek Szykuła
  • Fonction : Auteur
  • PersonId : 1022713

Résumé

We investigate the shuffle operation on regular languages represented by complete deterministic finite automata. We prove that $$f(m,n)=2^{mn-1} + 2^{(m-1)(n-1)}(2^{m-1}-1)(2^{n-1}-1)$$ is an upper bound on the state complexity of the shuffle of two regular languages having state complexities m and n, respectively. We also state partial results about the tightness of this bound. We show that there exist witness languages meeting the bound if $$2\leqslant m\leqslant 5$$ and $$n\geqslant 2$$, and also if $$m=n=6$$. Moreover, we prove that in the subset automaton of the NFA accepting the shuffle, all $$2^{mn}$$ states can be distinguishable, and an alphabet of size three suffices for that. It follows that the bound can be met if all f(m, n) states are reachable. We know that an alphabet of size at least mn is required provided that $$m,n \geqslant 2$$. The question of reachability, and hence also of the tightness of the bound f(m, n) in general, remains open.
Fichier principal
Vignette du fichier
416473_1_En_6_Chapter.pdf (191.33 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01633943 , version 1 (13-11-2017)

Licence

Paternité

Identifiants

Citer

Janusz Brzozowski, Galina Jirásková, Bo Liu, Aayush Rajasekaran, Marek Szykuła. On the State Complexity of the Shuffle of Regular Languages. 18th International Workshop on Descriptional Complexity of Formal Systems (DCFS), Jul 2016, Bucharest, Romania. pp.73-86, ⟨10.1007/978-3-319-41114-9_6⟩. ⟨hal-01633943⟩
66 Consultations
51 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More