State complexity of cyclic shift

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 n2+n log n-o(n) for alphabets with at least 4 letters. For 2- and 3-letter alphabets, we prove 2 ⊖(n2) state complexity. We also establish a tight 2n 2 + 1 lower bound for the nondeterministic state complexity of this operation using a binary alphabet.

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