On the State Complexity of Operations on Two-Way Finite Automata

  • Galina Jirásková
  • Alexander Okhotin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5257)


The number of states in two-way deterministic finite automata (2DFAs) is considered. It is shown that the state complexity of basic operations is: at least m + n − o(m + n) and at most 4m + n + 1 for union; at least m + n − o(m + n) and at most m + n + 1 for intersection; at least n and at most 4n for complementation; at least \(\Omega(\frac{m}{n}) + \frac{2^{\Omega(n)}}{\log m}\) and at most \(2m^{m+1}\cdot 2^{n^{n+1}}\) for concatenation; at least \(\frac{1}{n} 2^{\frac{n}{2}-1}\) and at most \(2^{O(n^{n+1})}\) for both star and square; between n and n + 2 for reversal; exactly 2n for inverse homomorphism. In each case m and n denote the number of states in 2DFAs for the arguments.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Birget, J.C.: Partial orders on words, minimal elements of regular languages, and state complexity. Theoretical Computer Science 119, 267–291 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Domaratzki, M., Okhotin, A.: State complexity of power, TUCS Technical Report No 845, Turku Centre for Computer Science, Turku, Finland (January 2007)Google Scholar
  3. 3.
    Geffert, V.: Nondeterministic computations in sublogarithmic space and space constructibility. SIAM Journal on Computing 20(3), 484–498 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Geffert, V.: Personal communication (March 2008)Google Scholar
  5. 5.
    Geffert, V., Mereghetti, C., Pighizzini, G.: Complementing two-way finite automata. Information and Computation 205(8), 1173–1187 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Holzer, M., Kutrib, M.: Nondeterministic descriptional complexity of regular languages. International Journal of Foundations of Computer Science 14, 1087–1102 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kapoutsis, C.A.: Removing bidirectionality from nondeterministic finite automata. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 544–555. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Leiss, E.L.: Succinct representation of regular languages by Boolean automata. Theoretical Computer Science 13, 323–330 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Mathematics Doklady 11, 1373–1375 (1970)zbMATHGoogle Scholar
  10. 10.
    Moore, F.R.: On the bounds for state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way finite automata. IEEE Transactions on Computers 20, 1211–1214 (1971)zbMATHCrossRefGoogle Scholar
  11. 11.
    Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM Journal of Research and Development 3, 114–125 (1959)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Rampersad, N.: The state complexity of L 2 and L k. Information Processing Letters 98, 231–234 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Sakoda, W.J., Sipser, M.: Nondeterminism and the size of two way finite automata. In: 10th ACM Symposium on Theory of Computing (STOC 1978), pp. 275–286 (1978)Google Scholar
  14. 14.
    Sipser, M.: Halting space-bounded computations. Theoretical Computer Science 10(3), 335–338 (1980)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Galina Jirásková
    • 1
  • Alexander Okhotin
    • 2
    • 3
  1. 1.Mathematical InstituteSlovak Academy of SciencesKošiceSlovakia
  2. 2.Academy of Finland 
  3. 3.Department of MathematicsUniversity of TurkuFinland

Personalised recommendations