Hyper-minimization for Deterministic Tree Automata

  • Artur Jeż
  • Andreas Maletti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7381)

Abstract

Hyper-minimization aims to reduce the size of the representation of a language beyond the limits imposed by classical minimization. To this end, the hyper-minimal representation can represent a language that has a finite difference to the original language. The first hyper-minimization algorithm is presented for (bottom-up) deterministic tree automata, which represent the recognizable tree languages. It runs in time \({\cal O}(\ell m n)\), where ℓ is the maximal rank of the input symbols, m is the number of transitions, and n is the number of states of the input tree automaton.

Keywords

Equivalent State Successor State Minimization Algorithm Reachable State Input Symbol 
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.

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References

  1. 1.
    Badr, A.: Hyper-minimization in O(n 2). Int. J. Found. Comput. Sci. 20(4), 735–746 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Badr, A., Geffert, V., Shipman, I.: Hyper-minimizing minimized deterministic finite state automata. RAIRO Theor. Inf. Appl. 43(1), 69–94 (2009)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Comon, H., Dauchet, M., Gilleron, R., Löding, C., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree automata: Techniques and applications (2007), http://tata.gforge.inria.fr/
  4. 4.
    Gawrychowski, P., Jeż, A.: Hyper-minimisation Made Efficient. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 356–368. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Gécseg, F., Steinby, M.: Tree Automata. Akadémiai Kiadó, Budapest (1984)Google Scholar
  6. 6.
    Gécseg, F., Steinby, M.: Tree languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, ch. 1, pp. 1–68. Springer (1997)Google Scholar
  7. 7.
    Högberg, J., Maletti, A., May, J.: Backward and forward bisimulation minimization of tree automata. Theoret. Comput. Sci. 410(37), 3539–3552 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Holzer, M., Maletti, A.: An n logn algorithm for hyper-minimizing a (minimized) deterministic automaton. Theoret. Comput. Sci. 411(38-39), 3404–3413 (2010)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Hopcroft, J.E.: An n logn algorithm for minimizing states in a finite automaton. In: Kohavi, Z., Paz, A. (eds.) Theory of Machines and Computations, pp. 189–196. Academic Press (1971)Google Scholar
  10. 10.
    Hosoya, H.: Foundations of XML Processing: The Tree-Automata Approach. Cambridge University Press (2011)Google Scholar
  11. 11.
    Knight, K.: Capturing practical natural language transformations. Machine Translation 21(2), 121–133 (2007)CrossRefGoogle Scholar
  12. 12.
    Maletti, A.: Notes on hyper-minimization. In: Proc. 13th Int. Conf. Automata and Formal Languages, pp. 34–49. Nyíregyháza College (2011)Google Scholar
  13. 13.
    Maletti, A., Quernheim, D.: Hyper-minimisation of deterministic weighted finite automata over semifields. In: Proc. 13th Int. Conf. Automata and Formal Languages, pp. 285–299. Nyíregyháza College (2011)Google Scholar
  14. 14.
    Maletti, A., Quernheim, D.: Optimal hyper-minimization. Int. J. Found. Comput. Sci. 22(8), 1877–1891 (2011)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Schewe, S.: Beyond hyper-minimisation — minimising DBAs and DPAs is NP-complete. In: Proc. 30th Int. Conf. Foundations of Software Technology and Theoretical Computer Science. LIPIcs, vol. 8, pp. 400–411. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)Google Scholar
  16. 16.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, ch. 2, pp. 41–110. Springer (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Artur Jeż
    • 1
  • Andreas Maletti
    • 2
  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland
  2. 2.Institute for Natural Language ProcessingUniversität StuttgartStuttgartGermany

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