State Complexity of Operations on Two-Way Deterministic Finite Automata over a Unary Alphabet

  • Michal Kunc
  • Alexander Okhotin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6808)

Abstract

The paper determines the number of states in a two-way deterministic finite automaton (2DFA) over a one-letter alphabet sufficient and in the worst case necessary to represent the results of the following operations: (i) intersection of an m-state 2DFA and an n-state 2DFA requires between m + n and m + n + 1 states; (ii) union of an m-state 2DFA and an n-state 2DFA, between m + n and 2m + n + 4 states; (iii) Kleene star of an n-state 2DFA, (g(n) + O(n))2 states, where \(g(n)=e^{\sqrt{n \ln n}(1+o(1))}\) is the maximum value of lcm(p 1, …, p k ) for \(\sum p_i \leqslant n\), known as Landau’s function; (iv) k-th power of an n-state 2DFA, between (k − 1)g(n) − k and k(g(n) + n) states; (v) concatenation of an m-state and an n-state 2DFAs, \(e^{(1+o(1)) \sqrt{(m+n)\ln(m+n)}}\) states.

Keywords

State Complexity Regular Language Input String Letter Alphabet Power Operation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Chrobak, M.: Finite automata and unary languages. Theoretical Computer Science 47, 149–158 (1986); Errata 302, 497–498 (2003)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Domaratzki, M., Okhotin, A.: State complexity of power. Theoretical Computer Science 410(24-25), 2377–2392 (2009)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Dressler, R.E.: A stronger Bertrand’s postulate with an application to partitions. Proceedings of the AMS 33(2), 226–228 (1972)MATHMathSciNetGoogle Scholar
  4. 4.
    Geffert, V., Mereghetti, C., Pighizzini, G.: Complementing two-way finite automata. Information and Computation 205(8), 1173–1187 (2007)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Holzer, M., Kutrib, M.: Nondeterministic descriptional complexity of regular languages. International Journal of Foundations of Computer Science 14, 1087–1102 (2003)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Jirásková, G., Okhotin, A.: On the state complexity of operations on two-way finite automata. In: Ito, M., Toyama, M. (eds.) DLT 2008. LNCS, vol. 5257, pp. 443–454. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: 38th Annual Symposium on Foundations of Computer Science (FOCS 1997), Miami Beach, Florida, USA, October 19-22, pp. 66–75. IEEE, Los Alamitos (199719-22)CrossRefGoogle Scholar
  8. 8.
    Kunc, M., Okhotin, A.: Describing periodicity in two-way deterministic finite automata using transformation semigroups. In: Leporati, A. (ed.) DLT 2011. LNCS, vol. 6795, pp. 324–336. Springer, Heidelberg (2011)Google Scholar
  9. 9.
    Kunc, M., Okhotin, A.: Reversible two-way finite automata over a unary alphabet (manuscript in preparation)Google Scholar
  10. 10.
    Landau, E.: Über die Maximalordnung der Permutationen gegebenen Grades (On the maximal order of permutations of a given degree). Archiv der Mathematik und Physik, Ser. 3(5), 92–103 (1903)MATHGoogle Scholar
  11. 11.
    Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Mathematics Doklady 11, 1373–1375 (1970)MATHGoogle Scholar
  12. 12.
    Miller, W.: The maximum order of an element of a finite symmetric group. American Mathematical Monthly 94(6), 497–506 (1987)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Okhotin, A.: Unambiguous finite automata over a unary alphabet. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 556–567. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Pighizzini, G., Shallit, J.: Unary language operations, state complexity and Jacobsthal’s function. International Journal of Foundations of Computer Science 13(1), 145–159 (2002)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Rampersad, N.: The state complexity of L 2 and L k. Information Processing Letters 98, 231–234 (2006)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexity of some basic operations on regular languages. Theoretical Computer Science 125, 315–328 (1994)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Michal Kunc
    • 1
  • Alexander Okhotin
    • 2
    • 3
  1. 1.Masaryk UniversityCzech Republic
  2. 2.Department of MathematicsUniversity of TurkuFinland
  3. 3.Academy of FinlandFinland

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