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State complexity of cyclic shift
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- Journal:
- RAIRO - Theoretical Informatics and Applications / Volume 42 / Issue 2 / April 2008
- Published online by Cambridge University Press:
- 13 December 2007, pp. 335-360
- Print publication:
- April 2008
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- Article
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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'spioneering 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 is2n2+ nlogn - O(n) for alphabets with at least 4 letters.For 2- and 3-letter alphabets, we prove $2^{\Theta(n^2)}$
state complexity.We also establish a tight 2n2+1 lower bound forthe nondeterministic state complexity of this operation using a binary alphabet.
