Advertisement

Myhill-Nerode Theorem for Recognizable Tree Series Revisited

  • Andreas Maletti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4957)

Abstract

In this contribution the Myhill-Nerode congruence relation on tree series is reviewed and a more detailed analysis of its properties is presented. It is shown that, if a tree series is deterministically recognizable over a zero-divisor free and commutative semiring, then the Myhill-Nerode congruence relation has finite index. By [Borchardt: Myhill-Nerode Theorem for Recognizable Tree Series. LNCS 2710. Springer 2003] the converse holds for commutative semifields, but not in general. In the second part, a slightly adapted version of the Myhill-Nerode congruence relation is defined and a characterization is obtained for all-accepting weighted tree automata over multiplicatively cancellative and commutative semirings.

Keywords

Finite Index Congruence Relation Tree Series Tree Automaton Tree Transducer 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kozen, D.: On the Myhill-Nerode theorem for trees. Bulletin of the EATCS 47, 170–173 (1992)zbMATHGoogle Scholar
  2. 2.
    Eisner, J.: Simpler and more general minimization for weighted finite-state automata. In: HLT-NAACL (2003)Google Scholar
  3. 3.
    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
  4. 4.
    Hebisch, U., Weinert, H.J.: Semirings—Algebraic Theory and Applications in Computer Science. World Scientific, Singapore (1998)zbMATHGoogle Scholar
  5. 5.
    Golan, J.S.: Semirings and their Applications. Kluwer Academic, Dordrecht (1999)zbMATHGoogle Scholar
  6. 6.
    Knight, K., Graehl, J.: An overview of probabilistic tree transducers for natural language processing. In: Gelbukh, A. (ed.) CICLing 2005. LNCS, vol. 3406, Springer, Heidelberg (2005)Google Scholar
  7. 7.
    Borchardt, B.: The Theory of Recognizable Tree Series. PhD thesis, Technische Universität Dresden (2005)Google Scholar
  8. 8.
    Drewes, F., Vogler, H.: Learning deterministically recognizable tree series. J. Autom. Lang. Combin (to appear, 2007)Google Scholar
  9. 9.
    Berstel, J., Reutenauer, C.: Rational Series and Their Languages. In: EATCS Monographs on Theoret. Comput. Sci., vol. 12, Springer, Heidelberg (1988)Google Scholar
  10. 10.
    Bozapalidis, S.: Equational elements in additive algebras. Theory Comput. Systems 32(1), 1–33 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Borchardt, B., Vogler, H.: Determinization of finite state weighted tree automata. J. Autom. Lang. Combin. 8(3), 417–463 (2003)zbMATHMathSciNetGoogle 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, vol. 3, pp. 1–68. Springer, Heidelberg (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Andreas Maletti
    • 1
  1. 1.International Computer Science InstituteBerkeleyUSA

Personalised recommendations