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)


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.


State Complexity Regular Language Input String Letter Alphabet Power Operation 
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  1. 1.
    Chrobak, M.: Finite automata and unary languages. Theoretical Computer Science 47, 149–158 (1986); Errata 302, 497–498 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Domaratzki, M., Okhotin, A.: State complexity of power. Theoretical Computer Science 410(24-25), 2377–2392 (2009)CrossRefzbMATHMathSciNetGoogle 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)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Geffert, V., Mereghetti, C., Pighizzini, G.: Complementing two-way finite automata. Information and Computation 205(8), 1173–1187 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Holzer, M., Kutrib, M.: Nondeterministic descriptional complexity of regular languages. International Journal of Foundations of Computer Science 14, 1087–1102 (2003)CrossRefzbMATHMathSciNetGoogle 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)zbMATHGoogle Scholar
  11. 11.
    Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Mathematics Doklady 11, 1373–1375 (1970)zbMATHGoogle Scholar
  12. 12.
    Miller, W.: The maximum order of an element of a finite symmetric group. American Mathematical Monthly 94(6), 497–506 (1987)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Rampersad, N.: The state complexity of L 2 and L k. Information Processing Letters 98, 231–234 (2006)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle Scholar

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© 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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