Minimal DFA for Symmetric Difference NFA

  • Brink van der Merwe
  • Hellis Tamm
  • Lynette van Zijl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7386)


Recently, a characterization of the class of nondeterministic finite automata (NFAs) for which determinization results in a minimal deterministic finite automaton (DFA), was given in [2]. We present a similar result for the case of symmetric difference NFAs. Also, we show that determinization of any minimal symmetric difference NFA produces a minimal DFA.


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  1. 1.
    Brzozowski, J.: Canonical regular expressions and minimal state graphs for definite events. In: Proceedings of the Symposium on the Mathematical Theory of Automata. MRI Symposia Series, pp. 529–561. Polytechnic Press of Polytechnic Institute of Brooklyn (1963)Google Scholar
  2. 2.
    Brzozowski, J., Tamm, H.: Theory of Átomata. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 105–116. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Brzozowski, J., Tamm, H.: Quotient complexities of atoms of regular languages. In: Proceedings of the 16th International Conference on Developments in Language Theory (DLT). Springer (August 2012)Google Scholar
  4. 4.
    Denis, F., Lemay, A., Terlutte, A.: Residual Finite State Automata. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 144–157. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Dornhoff, L., Hohn, F.: Applied Modern Algebra. Macmillan Publishing Company (1978)Google Scholar
  6. 6.
    Droste, M., Kuich, W., Vogler, H.: Handbook of Weighted Automata, 1st edn. Springer Publishing Company, Incorporated (2009)Google Scholar
  7. 7.
    Droste, M., Rahonis, G.: Weighted Automata and Weighted Logics on Infinite Words. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 49–58. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Hopcroft, J., Ullman, J.: Introduction to Automata Theory, Languages and Computation. Addison Wesley (1979)Google Scholar
  9. 9.
    Ilie, L., Navarro, G., Yu, S.: On NFA Reductions. In: Karhumäki, J., Maurer, H., Păun, G., Rozenberg, G. (eds.) Salomaa Festschrift. LNCS, vol. 3113, pp. 112–124. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Jiang, T., Ravikumar, B.: Minimal NFA problems are hard. SIAM Journal on Computing 22(6), 1117–1141 (1993)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Kirsten, D., Mäurer, I.: On the determinization of weighted automata. Journal of Automata, Languages and Combinatorics 10(2/3), 287–312 (2005)MathSciNetMATHGoogle Scholar
  12. 12.
    Mohri, M.: Finite-state transducers in language and speech processing. Computational Linguistics 23(2), 269–311 (1997)MathSciNetGoogle Scholar
  13. 13.
    Stone, H.: Discrete Mathematical Structures. Science Research Associates (1973)Google Scholar
  14. 14.
    van der Merwe, B., van Zijl, L., Geldenhuys, J.: Ambiguity of Unary Symmetric Difference NFAs. In: Cerone, A., Pihlajasaari, P. (eds.) ICTAC 2011. LNCS, vol. 6916, pp. 256–266. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Van Zijl, L.: Generalized Nondeterminism and the Succinct Representation of Regular Languages. Ph.D. thesis, Stellenbosch University (November 1997)Google Scholar
  16. 16.
    Van Zijl, L.: On binary symmetric difference NFAs and succinct representations of regular languages. Theoretical Computer Science 328(1), 161–170 (2004)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Vuillemin, J., Gama, N.: Compact Normal Form for Regular Languages as Xor Automata. In: Maneth, S. (ed.) CIAA 2009. LNCS, vol. 5642, pp. 24–33. Springer, Heidelberg (2009)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Brink van der Merwe
    • 1
  • Hellis Tamm
    • 2
  • Lynette van Zijl
    • 1
  1. 1.Department of Computer ScienceStellenbosch UniversityMatielandSouth Africa
  2. 2.Institute of CyberneticsTallinn University of TechnologyTallinnEstonia

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