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

The cyclic shift of a language $L$, defined as $shift\left(L\right)$=$\left\{vu\phantom{\rule{0.222222em}{0ex}}|\phantom{\rule{0.222222em}{0ex}}uv\in L\right\}$, 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${}^{\left(n-1\right)\left(n-2\right)}$, which shows that the state complexity of cyclic shift is ${2}^{{n}^{2}+nlogn-O\left(n\right)}$ for alphabets with at least 4 letters. For 2- and 3-letter alphabets, we prove ${2}^{\Theta \left({n}^{2}\right)}$ 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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Jirásková, Galina; Okhotin, Alexander. State complexity of cyclic shift. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 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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