State complexity of cyclic shift

Jirásková, Galina; Okhotin, Alexander

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008), p. 335-360 / Harvested from Numdam

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Résumé

The cyclic shift of a language $L$ , defined as $s h i f t (L)$ = ${v u | u v \in L}$ , is an operation known to preserve both regularity and context-freeness. Its descriptional complexity has been addressed in Maslov’s pioneering paper on the state complexity of regular language operations [Soviet Math. Dokl. 11 (1970) 1373-1375], where a high lower bound for partial DFAs using a growing alphabet was given. We improve this result by using a fixed 4-letter alphabet, obtaining a lower bound (n-1)! $\cdot$ 2 $^{(n - 1) (n - 2)}$ , which shows that the state complexity of cyclic shift is $2^{n^{2} + n log n - O (n)}$ for alphabets with at least 4 letters. For 2- and 3-letter alphabets, we prove $2^{Θ (n^{2})}$ state complexity. We also establish a tight 2n $^{2} + 1$ lower bound for the nondeterministic state complexity of this operation using a binary alphabet.

Publié le : 2008-01-01

MR 2401266 | Zbl 1144.68033

DOI : https://doi.org/10.1051/ita:2007038
Classification: 68Q45, 68Q19

@article{ITA_2008__42_2_335_0,
     author = {Jir\'askov\'a, Galina and Okhotin, Alexander},
     title = {State complexity of cyclic shift},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     volume = {42},
     year = {2008},
     pages = {335-360},
     doi = {10.1051/ita:2007038},
     mrnumber = {2401266},
     zbl = {1144.68033},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ITA_2008__42_2_335_0}
}

Jirásková, Galina; Okhotin, Alexander. State complexity of cyclic shift. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) pp. 335-360. doi : 10.1051/ita:2007038. http://gdmltest.u-ga.fr/item/ITA_2008__42_2_335_0/

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