Pushing for Weighted Tree Automata

  • Andreas Maletti
  • Daniel Quernheim
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6907)

Abstract

Explicit pushing for weighted tree automata over semifields is introduced. A careful selection of the pushing weights allows a normalization of bottom-up deterministic weighted tree automata. Automata in the obtained normal form can be minimized by a simple transformation into an unweighted automaton followed by unweighted minimization. This generalizes results of Mohri and Eisner for deterministic weighted string automata to the tree case. Moreover, the new strategy can also be used to test equivalence of two bottom-up deterministic weighted tree automata M1 and M2 in time O(|M |log|Q|), where |M | = |M1| + |M2| and |Q| is the sum of the number of states of M1 and M2. This improves the previously best running time O(|M1|·|M2|).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berstel, J., Reutenauer, C.: Recognizable formal power series on trees. Theoret. Comput. Sci. 18(2), 115–148 (1982)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Borchardt, B.: The Myhill-Nerode theorem for recognizable tree series. In: Ésik, Z., Fülöp, Z. (eds.) DLT 2003. LNCS, vol. 2710, pp. 146–158. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Borchardt, B.: The Theory of Recognizable Tree Series. Ph.D. thesis, TU Dresden (2005)Google Scholar
  4. 4.
    Borchardt, B., Vogler, H.: Determinization of finite state weighted tree automata. J. Autom. Lang. Comb. 8(3), 417–463 (2003)MathSciNetMATHGoogle Scholar
  5. 5.
    Bozapalidis, S.: Equational elements in additive algebras. Theory Comput. Syst. 32(1), 1–33 (1999)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bozapalidis, S., Louscou-Bozapalidou, O.: The rank of a formal tree power series. Theoret. Comput. Sci. 27(1–2), 211–215 (1983)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Büchse, M., May, J., Vogler, H.: Determinization of weighted tree automata using factorizations. J. Autom. Lang. Comb. 15(3–4) (2010)Google Scholar
  8. 8.
    Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Löding, C., Lugiez, D., Tison, S., Tommasi, M.: Tree Automata Techniques and Applications (2007), http://www.grappa.univ-lille3.fr/tata
  9. 9.
    Drewes, F., Högberg, J., Maletti, A.: MAT learners for tree series — an abstract data type and two realizations. Acta Inform. 48(3), 165–189 (2011)CrossRefGoogle Scholar
  10. 10.
    Eisner, J.: Simpler and more general minimization for weighted finite-state automata. In: Hearst, M., Ostendorf, M. (eds.) HLT-NAACL 2003, pp. 64–71. ACL (2003)Google Scholar
  11. 11.
    Fülöp, Z., Vogler, H.: Weighted tree automata and tree transducers. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata. EATCS Monographs in Theoretical Computer Science, ch. 9, pp. 313–403. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Gécseg, F., Steinby, M.: Tree Automata. Akadémiai Kiadó, Budapest (1984)Google Scholar
  13. 13.
    Gécseg, F., Steinby, M.: Tree languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, ch. 1, vol. 3, pp. 1–68. Springer, Heidelberg (1997)Google Scholar
  14. 14.
    Golan, J.S.: Semirings and their Applications. Kluwer Academic Publishers, Dordrecht (1999)MATHGoogle Scholar
  15. 15.
    Hebisch, U., Weinert, H.J.: Semirings – Algebraic Theory and Applications in Computer Science. World Scientific, Singapore (1998)MATHGoogle Scholar
  16. 16.
    Högberg, J., Maletti, A., May, J.: Bisimulation minimisation for weighted tree automata. In: Harju, T., Karhumäki, J., Lepistö, A. (eds.) DLT 2007. LNCS, vol. 4588, pp. 229–241. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Högberg, J., Maletti, A., May, J.: Backward and forward bisimulation minimization of tree automata. Theoret. Comput. Sci. 410(37), 3539–3552 (2009)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kuich, W.: Formal power series over trees. In: Bozapalidis, S. (ed.) DLT 1997, pp. 61–101. Aristotle University of Thessaloniki (1998)Google Scholar
  19. 19.
    Maletti, A.: Minimizing deterministic weighted tree automata. Inform. Comput. 207(11), 1284–1299 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    May, J., Knight, K., Vogler, H.: Efficient inference through cascades of weighted tree transducers. In: Hajič, J., Carberry, S., Clark, S., Nivre, J. (eds.) ACL 2010, pp. 1058–1066. ACL (2010)Google Scholar
  21. 21.
    Mohri, M.: Finite-state transducers in language and speech processing. Comput. Linguist. 23(2), 269–311 (1997)MathSciNetGoogle Scholar
  22. 22.
    Post, M., Gildea, D.: Weight pushing and binarization for fixed-grammar parsing. In: de la Clergerie, E.V., Bunt, H. (eds.) IWPT 2009, pp. 89–98. ACL (2009)Google Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2011

Authors and Affiliations

  • Andreas Maletti
    • 1
  • Daniel Quernheim
    • 1
  1. 1.Institut für Maschinelle SprachverarbeitungUniversität StuttgartStuttgartGermany

Personalised recommendations