Geodesic
State complexity of cyclic shift
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 2, pp. 335-360

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The cyclic shift of a language L, defined as shift(L)={vu|uv∈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)! · 2 (n-1)(n-2) , which shows that the state complexity of cyclic shift is 2 n 2 +nlogn-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.

DOI : 10.1051/ita:2007038
Classification : 68Q45, 68Q19
Keywords: finite automata, descriptional complexity, cyclic shift 
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     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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     zbl = {1144.68033},
     language = {en},
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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

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