Transforming Two-Way Alternating Finite Automata to One-Way Nondeterministic Automata

  • Viliam Geffert
  • Alexander Okhotin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8634)


It is proved that a two-way alternating finite automaton (2AFA) with n states can be transformed to an equivalent one-way nondeterministic finite automaton (1NFA) with f(n) = 2Θ(n logn) states, and that this number of states is necessary in the worst case already for the transformation of a two-way automaton with universal nondeterminism (2Π1FA) to a 1NFA. At the same time, an n-state 2AFA is transformed to a 1NFA with (2 n  − 1)2 + 1 states recognizing the complement of the original language, and this number of states is again necessary in the worst case. The difference between these two trade-offs is used to show that complementing a 2AFA requires at least Ω(n logn) states.


Transition Function Boolean Function Regular Language Finite Automaton Input String 
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.


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)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Birget, J.-C.: State-complexity of finite-state devices, state compressibility and incompressibility. Mathematical Systems Theory 26(3), 237–269 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Geffert, V.: An alternating hierarchy for finite automata. Theoretical Computer Science 445, 1–24 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Geffert, V., Mereghetti, C., Pighizzini, G.: Converting two-way nondeterministic unary automata into simpler automata. Theoretical Computer Science 295(1-3), 189–203 (2003)Google Scholar
  5. 5.
    Geffert, V., Mereghetti, C., Pighizzini, G.: Complementing two-way finite automata. Information and Computation 205(8), 1173–1187 (2007)Google Scholar
  6. 6.
    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
  7. 7.
    Kapoutsis, C.A.: Two-way automata versus logarithmic space. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 359–372. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Kunc, M., Okhotin, A.: Reversibility of computations in graph-walking automata. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 595–606. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    Ladner, R., Lipton, R., Stockmeyer, L.: Alternating pushdown and stack automata. SIAM Journal on Computing 13(1), 135–155 (1984)Google Scholar
  10. 10.
    Vardi, M.: A note on the reduction of two-way automata to one-way automata. Information Processing Letters 30(5), 261–264 (1989)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2014

Authors and Affiliations

  • Viliam Geffert
    • 1
  • Alexander Okhotin
    • 2
  1. 1.Department of Computer ScienceŠafárik UniversityKošiceSlovakia
  2. 2.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

Personalised recommendations