Myhill-Nerode Theorem for Recognizable Tree Series Revisited

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


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.


Finite Index Congruence Relation Tree Series Tree Automaton Tree Transducer 
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  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

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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

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

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