State complexity of cyclic shift

Jirásková, Galina; Okhotin, Alexander

doi:10.1051/ita:2007038

Jirásková, Galina ; Okhotin, Alexander

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 2, pp. 335-360.

Résumé

The cyclic shift of a language $L$ , defined as $s h i f t (L)$ = ${v u | u v \in L}$ , 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)! $\cdot$ 2 $^{(n - 1) (n - 2)}$ , which shows that the state complexity of cyclic shift is $2^{n^{2} + n log n - O (n)}$ 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.

EuDML MR Zbl

DOI : 10.1051/ita:2007038

Classification : 68Q45, 68Q19
Mots clés : finite automata, descriptional complexity, cyclic shift

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     author = {Jir\'askov\'a, Galina and Okhotin, Alexander},
     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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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. http://archive.numdam.org/articles/10.1051/ita:2007038/

Bibliographie
Cité par

[1] A.V. Aho, J.D. Ullman and M. Yannakakis, On notions of information transfer in VLSI circuits, in Proceedings of 15th ACM STOC, ACM (1983) 133-139.

[2] J.-C. Birget, Intersection and union of regular languages and state complexity. Inform. Process. Lett. 43 (1992) 185-190. | MR | Zbl

[3] J.-C. Birget, Partial orders on words, minimal elements of regular languages, and state complexity. Theor. Comput. Sci. 119 (1993) 267-291. | MR | Zbl

[4] J.-C. Birget, The state complexity of $\bar{Σ^{*} \bar{L}}$ and its connection with temporal logic. Inform. Process. Lett. 58 (1996) 185-188. | MR

[5] C. Câmpeanu, K. Salomaa and S. Yu, Tight lower bound for the state complexity of shuffle of regular languages. J. Autom. Lang. Comb. 7 (2002) 303-310. | MR | Zbl

[6] M. Domaratzki, State complexity and proportional removals. J. Autom. Lang. Comb. 7 (2002) 455-468. | MR | Zbl

[7] M. Domaratzki, D. Kisman and J. Shallit, On the number of distinct languages accepted by finite automata with $n$ states. J. Autom. Lang. Comb. 7 (2002) 469-486. | MR | Zbl

[8] V. Geffert, Magic numbers in the state hierarchy of finite automata, Mathematical Foundations of Computer Science, MFCS 2006, Stará Lesná, Slovakia, August 28-September 1, 2006, Springer, Berlin. Lect. Notes Comput. Sci. 4162 (2006) 312-423. | MR | Zbl

[9] I. Glaister and J. Shallit, A lower bound technique for the size of nondeterministic finite automata. Inform. Process. Lett. 59 (1996) 75-77. | MR | Zbl

[10] M. Holzer and M. Kutrib, Nondeterministic descriptional complexity of regular languages. Int. J. Found. Comput. Sci. 14 (2003) 1087-1102. | MR | Zbl

[11] J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979). | MR | Zbl

[12] M. Hricko, G. Jirásková and A. Szabari, Union and intersection of regular languages and descriptional complexity, in Proceedings of DCFS 2005, Como, Italy, June 30-July 2, 2005, 170-181.

[13] J. Hromkovič, Communication Complexity and Parallel Computing. Springer-Verlag, Berlin, Heidelberg (1997). | MR | Zbl

[14] J. Hromkovič, Descriptional complexity of finite automata: concepts and open problems. J. Autom. Lang. Comb. 7 (2002) 519-531. | MR | Zbl

[15] K. Iwama, A. Matsuura and M. Paterson, A family of NFAs which need $2^{n} - α$ deterministic states. Theor. Comput. Sci. 301 (2003), 451-462. | MR | Zbl

[16] G. Jirásková, State complexity of some operations on binary regular languages. Theor. Comput. Sci. 330 (2005) 287-298. | MR | Zbl

[17] A.N. Maslov, Estimates of the number of states of finite automata. Soviet Mathematics Doklady 11 (1970) 1373-1375. | MR | Zbl

[18] A.N. Maslov, Cyclic shift operation for languages. Probl. Inf. Transm. 9 (1973) 333-338. | MR

[19] T. Oshiba, Closure property of the family of context-free languages under the cyclic shift operation. T. IECE 55D (1972) 119-122. | MR

[20] A. Salomaa, D. Wood and S. Yu, On the state complexity of reversals of regular languages. Theor. Comput. Sci. 320 (2004) 315-329. | MR | Zbl

[21] A. Salomaa, K. Salomaa and S. Yu, State complexity of combined operations. Theor. Comput. Sci. 383 (2007) 140-152. | MR | Zbl

[22] S. Yu, State complexity: recent results and open problems. Fund. Inform. 64 (2005) 471-480. | MR | Zbl

[23] S. Yu, Q. Zhuang and K. Salomaa, The state complexity of some basic operations on regular languages. Theor. Comput. Sci. 125 (1994) 315-328. | MR | Zbl

[24] L. Van Zijl, Magic numbers for symmetric difference NFAs. Int. J. Found. Comput. Sci. 16 (2005) 1027-1038. | MR | Zbl

Cité par Sources :