State complexity of cyclic shift

Jirásková, Galina; Okhotin, Alexander

doi:10.1051/ita:2007038

Jirásková, Galina ; Okhotin, Alexander

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 2, pp. 335-360.

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.

EuDML MR Zbl

DOI : 10.1051/ita:2007038

Classification : 68Q45, 68Q19
Mots clés : finite automata, descriptional complexity, cyclic shift

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     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},
     pages = {335--360},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {2},
     year = {2008},
     doi = {10.1051/ita:2007038},
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Jirásková, Galina; Okhotin, Alexander. State complexity of cyclic shift. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 2, pp. 335-360. doi : 10.1051/ita:2007038. http://www.numdam.org/articles/10.1051/ita:2007038/

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