# State complexity of cyclic shift

Galina Jirásková; Alexander Okhotin

RAIRO - Theoretical Informatics and Applications (2007)

- Volume: 42, Issue: 2, page 335-360
- ISSN: 0988-3754

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topJirásková, Galina, and Okhotin, Alexander. "State complexity of cyclic shift." RAIRO - Theoretical Informatics and Applications 42.2 (2007): 335-360. <http://eudml.org/doc/92875>.

@article{Jirásková2007,

abstract = {
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
2n2+ 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 for
the nondeterministic state complexity of this operation using a binary alphabet.
},

author = {Jirásková, Galina, Okhotin, Alexander},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Finite automata; descriptional complexity; cyclic shift},

language = {eng},

month = {12},

number = {2},

pages = {335-360},

publisher = {EDP Sciences},

title = {State complexity of cyclic shift},

url = {http://eudml.org/doc/92875},

volume = {42},

year = {2007},

}

TY - JOUR

AU - Jirásková, Galina

AU - Okhotin, Alexander

TI - State complexity of cyclic shift

JO - RAIRO - Theoretical Informatics and Applications

DA - 2007/12//

PB - EDP Sciences

VL - 42

IS - 2

SP - 335

EP - 360

AB -
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
2n2+ 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 for
the nondeterministic state complexity of this operation using a binary alphabet.

LA - eng

KW - Finite automata; descriptional complexity; cyclic shift

UR - http://eudml.org/doc/92875

ER -

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