Bisimulation Minimization of Tree Automata

  • Parosh Aziz Abdulla
  • Lisa Kaati
  • Johanna Högberg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4094)


We extend an algorithm by Paige and Tarjan that solves the coarsest stable refinement problem to the domain of trees. The algorithm is used to minimize non-deterministic tree automata (NTA) with respect to bisimulation. We show that our algorithm has an overall complexity of \(O(\hat{r}m \log n)\), where \(\hat{r}\) is the maximum rank of the input alphabet, m is the total size of the transition table, and n is the number of states.


Transition Rule Maximum Rank Minimization Algorithm Tree Automaton Tree Language 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdulla, P.A., Legay, A., d’Orso, J., Rezine, A.: Tree regular model checking: A simulation-based approach. Jour. of Logic and Alg. Programming (to appear, 2006)Google Scholar
  2. 2.
    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
  3. 3.
    Abdulla, P.A., Jonsson, B., Nilsson, M., Saksena, M.: A survey of regular model checking. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 35–48. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Abdulla, P.A., Kaati, L., Högberg, J.: Minimization of tree automata. UMINF 06.25, Department of Computer Science, Umeå University (2006)Google Scholar
  5. 5.
    Biehl, M., Klarlund, N., Rauhe, T.: Algorithms for guided tree automata. In: Raymond, D.R., Yu, S., Wood, D. (eds.) WIA 1996. LNCS, vol. 1260. Springer, Heidelberg (1997)Google Scholar
  6. 6.
    Brainerd, W.S.: The minimalization of tree automata. Information and Computation 13, 484–491 (1968)MATHMathSciNetGoogle Scholar
  7. 7.
    Cristau, J., Löding, C., Thomas, W.: Deterministic automata on unranked trees. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Gramlich, G., Schnitger, G.: Minimizing nfa’s and regular expressions. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Hopcroft, J.E.: An \(n \ log \ n\) algorithm for minimizing states in a finite automaton. In: Kohavi, Z. (ed.) Theory of Machines and Computations. Academic Press, London (1971)Google Scholar
  10. 10.
    Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions. In: Proc. 13th Ann. IEEE Symp. on Switching and Automata Theory, pp. 125–129 (1972)Google Scholar
  11. 11.
    Murata, M.: Hedge Automata: a Formal Model for XML Schemata. Web page (2000)Google Scholar
  12. 12.
    Nivat, M., Podelski, A.: Minimal ascending and descending tree automata. SIAM Journal on Computing 26, 39–58 (1997)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Paige, R., Tarjan, R.: Three partition refinement algorithms. SIAM Journal on Computing 16, 973–989 (1987)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Parosh Aziz Abdulla
    • 1
  • Lisa Kaati
    • 1
  • Johanna Högberg
    • 2
  1. 1.Dept. of Information TechnologyUppsala UniversitySweden
  2. 2.Dept. of Computing ScienceUmeå UniversitySweden

Personalised recommendations