Simulations of Weighted Tree Automata

  • Zoltán Ésik
  • Andreas Maletti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6482)


Simulations of weighted tree automata (wta) are considered. It is shown how such simulations can be decomposed into simpler functional and dual functional simulations also called forward and backward simulations. In addition, it is shown in several cases (fields, commutative rings, Noetherian semirings, semiring of natural numbers) that all equivalent wta M and N can be joined by a finite chain of simulations. More precisely, in all mentioned cases there is a single wta that simulates both M and N. Those results immediately yield decidability of equivalence provided that the semiring is finitely (and effectively) presented.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdulla, P.A., Jonsson, B., Mahata, P., d’Orso, J.: Regular tree model checking. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 555–568. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Alexandrakis, A., Bozapalidis, S.: Représentations matricielles des séries d’arbre reconnaissables. Informatique Théorique et Applications 23(4), 449–459 (1989)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Béal, M.P., Lombardy, S., Sakarovitch, J.: On the equivalence of ℤ-automata. In: Caires, L., et al. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 397–409. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Béal, M.P., Lombardy, S., Sakarovitch, J.: Conjugacy and equivalence of weighted automata and functional transducers. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 58–69. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Berstel, J., Reutenauer, C.: Rational Series and Their Languages. EATCS Monographs on Theoret. Comput. Sci., vol. 12. Springer, Heidelberg (1984)zbMATHGoogle Scholar
  6. 6.
    Bloom, S.L., Ésik, Z.: Iteration theories: The Equational Logic of Iterative Processes. Springer, Heidelberg (1993)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bloom, S.L., Ésik, Z.: An extension theorem with an application to formal tree series. J. Autom. Lang. Combin. 8(2), 145–185 (2003)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bozapalidis, S.: Effective construction of the syntactic algebra of a recognizable series on trees. Acta Inform. 28(4), 351–363 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buchholz, P.: Bisimulation relations for weighted automata. Theoret. Comput. Sci. 393(1-3), 109–123 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cleophas, L.: Forest FIRE and FIRE wood: Tools for tree automata and tree algorithms. In: FSMNLP, pp. 191–198 (2008)Google Scholar
  11. 11.
    Eilenberg, S.: Automata, Languages, and Machines. Academic Press, London (1974)zbMATHGoogle Scholar
  12. 12.
    Ésik, Z.: Axiomatizing the equational theory of regular tree languages. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 455–465. Springer, Heidelberg (1998)Google Scholar
  13. 13.
    Ésik, Z.: Axiomatizing the equational theory of regular tree languages. J. Log. Algebr. Program. 79(2), 189–213 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ésik, Z.: Fixed point theory. In: Handbook of Weighted Automata. EATCS Monographs on Theoret. Comput. Sci., pp. 29–66. Springer, Heidelberg (2010)Google Scholar
  15. 15.
    Ésik, Z., Kuich, W.: A generation of Kozen’s axiomatization of the equational theory of the regular sets. In: Words, Semigroups, and Transductions, pp. 99–114. World Scientific, Singapore (2001)CrossRefGoogle Scholar
  16. 16.
    Ésik, Z., Maletti, A.: Simulation vs. equivalence. In: FCS, pp. 119–122. CSREA Press (2010) (preprint),
  17. 17.
    Hebisch, U., Weinert, H.J.: Semirings—Algebraic Theory and Applications in Computer Science. World Scientific, Singapore (1998)zbMATHGoogle Scholar
  18. 18.
    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
  19. 19.
    Karner, G.: Continuous monoids and semirings. Theoret. Comput. Sci. 318(3), 355–372 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Klarlund, N., Møller, A.: MONA Version 1.4 User Manual (2001)Google Scholar
  21. 21.
    Knight, K., Graehl, J.: An overview of probabilistic tree transducers for natural language processing. In: Gelbukh, A. (ed.) CICLing 2005. LNCS, vol. 3406, pp. 1–24. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  22. 22.
    Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Inform. and Comput. 110(2), 366–390 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lang, S.: Algebra, 2nd edn. Addison Wesley, Reading (1984)zbMATHGoogle Scholar
  24. 24.
    May, J., Knight, K.: TIBURON: A weighted tree automata toolkit. In: Ibarra, O.H., Yen, H.-C. (eds.) CIAA 2006. LNCS, vol. 4094, pp. 102–113. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  25. 25.
    Milner, R.: A Calculus of Communicating Systems. Springer, Heidelberg (1980)CrossRefzbMATHGoogle Scholar
  26. 26.
    Park, D.M.R.: Concurrency and automata on infinite sequences. In: Deussen, P. (ed.) GI-TCS 1981. LNCS, vol. 104, pp. 167–183. Springer, Heidelberg (1981)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Zoltán Ésik
    • 1
  • Andreas Maletti
    • 2
  1. 1.Department of Computer ScienceUniversity of SzegedSzegedHungary
  2. 2.Departament de Filologies RomàniquesUniversitat Rovira i VirgiliTarragonaSpain

Personalised recommendations