Lower Bound Methods for the Size of Nondeterministic Finite Automata Revisited

  • Hellis Tamm
  • Brink van der Merwe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10168)


We revisit the following lower bound methods for the size of a nondeterministic finite automaton: the fooling set technique, the extended fooling set technique, and the biclique edge cover technique, presenting these methods in terms of quotients and atoms of regular languages. Although the lower bounds obtained by these methods are not necessarily tight, some classes of languages for which tight bounds can be achieved, are known. We show that languages with maximal reversal complexity belong to the class of languages for which the fooling set technique provides a tight bound. We also show that the extended fooling set technique is tight for a subclass of unary cyclic languages.


  1. 1.
    Birget, J.C.: Intersection and union of regular languages and state complexity. Inf. Process. Lett. 43, 185–190 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brzozowski, J., Davies, S.: Quotient complexities of atoms in regular ideal languages. Acta Cybernetica 22(2), 293–311 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brzozowski, J., Tamm, H.: Theory of átomata. Theoret. Comput. Sci. 539, 13–27 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Denis, F., Lemay, A., Terlutte, A.: Residual finite state automata. Fundamenta Informaticae 51, 339–368 (2002)MathSciNetMATHGoogle Scholar
  5. 5.
    Glaister, I., Shallit, J.: A lower bound technique for the size of nondeterministic finite automata. Inf. Process. Lett. 59, 75–77 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gruber, H.: On the descriptional and algorithmic complexity of regular languages. Ph.D. thesis, Gießener dissertation, Fachbereich Mathematik und Informatik, Physik, Geographie, Justus-Liebig-Universität Gießen (D26) (2009)Google Scholar
  7. 7.
    Gruber, H., Holzer, M.: Finding lower bounds for nondeterministic state complexity is hard. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 363–374. Springer, Heidelberg (2006). doi: 10.1007/11779148_33 CrossRefGoogle Scholar
  8. 8.
    Iván, S.: Complexity of atoms, combinatorially. Inf. Process. Lett. 116, 356–360 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jiang, T., McDowell, E., Ravikumar, B.: The structure and complexity of minimal NFA’s over a unary alphabet. Int. J. Found. Comput. Sci. 2(2), 163–182 (1991)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kameda, T., Weiner, P.: On the state minimization of nondeterministic finite automata. IEEE Trans. Comput. C–19(7), 617–627 (1970)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Latteux, M., Roos, Y., Terlutte, A.: Minimal NFA and biRFSA languages. RAIRO - Theoret. Inform. Appl. 43(2), 221–237 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Nerode, A.: Linear automaton transformations. Proc. Am. Math. Soc. 9, 541–544 (1958)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Salomaa, A., Wood, D., Yu, S.: On the state complexity of reversals of regular languages. Theoret. Comput. Sci. 320, 315–329 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Salomaa, A.: Mirror images and schemes for the maximal complexity of nondeterminism. Fundamenta Informaticae 116, 237–249 (2012)MathSciNetMATHGoogle Scholar
  15. 15.
    Tamm, H.: Some minimality results on biresidual and biseparable automata. In: Dediu, A.-H., Fernau, H., Martín-Vide, C. (eds.) LATA 2010. LNCS, vol. 6031, pp. 573–584. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-13089-2_48 CrossRefGoogle Scholar
  16. 16.
    Tamm, H.: Generalization of the double-reversal method of finding a canonical residual finite state automaton. In: Shallit, J., Okhotin, A. (eds.) DCFS 2015. LNCS, vol. 9118, pp. 268–279. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-19225-3_23 CrossRefGoogle Scholar
  17. 17.
    Tamm, H.: New interpretation and generalization of the Kameda-Weiner method. In: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), vol. 55, pp. 116:1–116:12. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of CyberneticsTallinn University of TechnologyTallinnEstonia
  2. 2.Department of Computer ScienceStellenbosch UniversityMatielandSouth Africa

Personalised recommendations