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International Conference on Developments in Language Theory

DLT 2007: Developments in Language Theory pp 254–265Cite as

Deterministic Blow-Ups of Minimal Nondeterministic Finite Automata over a Fixed Alphabet

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4588))

Abstract

We show that for all integers n and α such that \(n \leqslant \alpha \leqslant 2^n,\) there exists a minimal nondeterministic finite automaton of n states with a four-letter input alphabet whose equivalent minimal deterministic finite automaton has exactly α states. It follows that in the case of a four-letter alphabet, there are no “magic numbers”, i.e., the holes in the hierarchy. This improves a similar result obtained by Geffert for a growing alphabet of size n + 2 (Proc. 7th DCFS, Como, Italy, 23–37).

Research supported by the VEGA grants 1/3129/06 and 2/6089/26, and the VEGA grant “Combinatorial Structures and Complexity of Algorithms”.

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References

  1. Adorna, H.N.: 3-Party Message Complexity is Better than 2-Party Ones for Proving Lower Bounds on the Size of Minimal Nondeterministic Finite Automata. J. Autom. Lang. Comb. 7, 419–432 (2002)

    MATH  MathSciNet  Google Scholar 

  2. Aho, A.V., Ullman, J.D., Yannakakis, M.: On notions of informations transfer in VLSI circuits. In: Proc.15th ACM STOC, pp. 133–139. ACM, New York (1983)

    Google Scholar 

  3. Berman, P., Lingas, A.: On the complexity of regular languages in terms of finite automata. Technical Report 304, Polish Academy of Sciences (1977)

    Google Scholar 

  4. Birget, J.C.: Intersection and union of regular languages and state complexity. Inform. Process. Lett. 43, 185–190 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Birget, J.C.: Partial orders on words, minimal elements of regular languages, and state complexity. Theoret. Comput. Sci. 119, 267–291 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chrobak, M.: Finite automata and unary languages. Theoret. Comput. Sci. 47, 149–158 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  7. Geffert, V. (Non)determinism and the size of one-way finite automata. In: Proc. 7th DCFS, Como, Italy, pp. 23–37 ( 2006)

    Google Scholar 

  8. Geffert, V.: Magic numbers in the state hierarchy of finite automata. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 412–423. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  9. Glaister, I., Shallit, J.: A lower bound technique for the size of nondeterministic finite automata. Inform. Process. Lett. 59, 75–77 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hromkovič, J.: Communication Complexity and Parallel Computing. Springer-Verlag, Berlin, Heidelberg (1997)

    MATH  Google Scholar 

  11. Hromkovič, J.: Descriptional complexity of finite automata: concepts and open problems. J. Autom. Lang. Comb. 7, 519–531 (2002)

    MATH  MathSciNet  Google Scholar 

  12. Hromkovič, J., Seibert, S., Karhumäki, J., Klauck, H., Schnitger, G.: Communication complexity method for measuring nondeterminism in finite automata. Inform. and Comput. 172, 202–217 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Iwama, K., Kambayashi, Y., Takaki, K.: Tight bounds on the number of states of DFAs that are equivalent to n-state NFAs. Theoret. Comput. Sci. 237, 485–494 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Iwama, K., Matsuura, A., Paterson, M.: A family of NFAs which need 2n − α deterministic states. Theoret. Comput. Sci. 301, 451–462 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jirásková, G.: Note on minimal finite automata. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 421–431. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  16. Jirásková, G.: State complexity of some operations on binary regular languages. Theoret. Comput. Sci. 330, 287–298 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lupanov, O.B.: A comparison of two types of finite automata. Problemy Kibernetiki 9, 321–326 (in Russian) (1963)

    Google Scholar 

  18. Lyubich, Y.I.: Estimates for optimal determinization of nondeterministic autonomous automata. Sibirskii Matematichskii Zhurnal 5, 337–355 (in Russian) (1964)

    MATH  Google Scholar 

  19. Moore, F.R.: On the bounds for state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way finite automata. IEEE Trans. Comput. 20, 1211–1214 (1971)

    Article  MATH  Google Scholar 

  20. Rabin, M., Scott, D.: Finite automata and their decision problems. IBM Res. Develop. 3, 114–129 (1959)

    Article  MathSciNet  Google Scholar 

  21. Sakoda, W.J., Sipser, M.: Nondeterminism and the size of two-way finite automata. In: Proc. 10th Annual ACM Symp. on Theory of Computing, pp. 275–286 (1978)

    Google Scholar 

  22. Sipser, M.: Introduction to the theory of computation. PWS Publishing Company, Boston (1997)

    MATH  Google Scholar 

  23. Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. I, ch. 2, pp. 41–110. Springer-Verlag, Berlin Heidelberg (1997)

    Google Scholar 

  24. Yu, S.: A renaissance of automata theory? Bull. Eur. Assoc. Theor. Comput. Sci. EATCS 72, 270–272 (2000)

    Google Scholar 

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Authors and Affiliations

  1. Institute of Computer Science, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovakia

    Jozef Jirásek & Alexander Szabari

  2. Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01 Košice, Slovakia

    Galina Jirásková

Authors
  1. Jozef Jirásek
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  2. Galina Jirásková
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  3. Alexander Szabari
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Tero Harju Juhani Karhumäki Arto Lepistö

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Jirásek, J., Jirásková, G., Szabari, A. (2007). Deterministic Blow-Ups of Minimal Nondeterministic Finite Automata over a Fixed Alphabet. In: Harju, T., Karhumäki, J., Lepistö, A. (eds) Developments in Language Theory. DLT 2007. Lecture Notes in Computer Science, vol 4588. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73208-2_25

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  • DOI: https://doi.org/10.1007/978-3-540-73208-2_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73207-5

  • Online ISBN: 978-3-540-73208-2

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