Journal of Automated Reasoning

, Volume 51, Issue 4, pp 371–400 | Cite as

Emptiness and Finiteness for Tree Automata with Global Reflexive Disequality Constraints

  • Carles Creus
  • Adrià Gascón
  • Guillem Godoy


In recent years, several extensions of tree automata have been considered. Most of them are related with the capability of testing equality or disequality of certain subterms of the term evaluated by the automaton. In particular, tree automata with global constraints are able to test equality and disequality of subterms depending on the state to which they are evaluated. The emptiness problem is known decidable for this kind of automata, but with a non-elementary time complexity, and the finiteness problem remains unknown. In this paper, we consider the particular case of tree automata with global constraints when the constraint is a conjunction of disequalities between states, and the disequality predicate is forced to be reflexive. This restriction is significant in the context of XML definitions with monadic key constraints. We prove that emptiness and finiteness are decidable in triple exponential time for this kind of automata.


Tree automata Global constraints Disequality constraints Decision problems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, New York (1998)Google Scholar
  2. 2.
    Barguñó, L., Creus, C., Godoy, G., Jacquemard, F., Vacher, C.: The emptiness problem for tree automata with global constraints. In: Logic in Computer Science (LICS), pp. 263–272 (2010)Google Scholar
  3. 3.
    Bogaert, B., Tison, S.: Equality and disequality constraints on direct subterms in tree automata. In: Symposium on Theoretical Aspects of Computer Science (STACS), pp. 161–171 (1992)Google Scholar
  4. 4.
    Bojańczyk, M., Muscholl, A., Schwentick, T., Segoufin, L.: Two-variable logic on data trees and XML reasoning. J. ACM 56(3), 13:1–13:48 (2009)Google Scholar
  5. 5.
    Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Löding, C., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications (2007).
  6. 6.
    Comon, H., Jacquemard, F.: Ground reducibility and automata with disequality constraints. In: Symposium on Theoretical Aspects of Computer Science (STACS), pp. 151–162 (1994)Google Scholar
  7. 7.
    Comon, H., Jacquemard, F.: Ground reducibility is EXPTIME-complete. In: Logic in Computer Science (LICS), pp. 26–34 (1997)Google Scholar
  8. 8.
    Dauchet, M., Caron, A.C., Coquidé, J.L.: Automata for reduction properties solving. J. Symbol. Comput. 20(2), 215–233 (1995)CrossRefzbMATHGoogle Scholar
  9. 9.
    David, C., Libkin, L., Tan, T.: Efficient reasoning about data trees via integer linear programming. In: ICDT, pp. 18–29 (2011)Google Scholar
  10. 10.
    Filiot, E., Talbot, J., Tison, S.: Tree automata with global constraints. Int. J. Found. Comput. Sci. 21(4), 571–596 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jacquemard, F., Klay, F., Vacher, C.: Rigid tree automata and applications. Inf. Comput. 209(3), 486–512 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mongy, J.: Transformation de noyaux reconnaissables d’arbres. forêts rateg. Ph.D. thesis, Laboratoire d’Informatique Fondamentale de Lille, Université des Sciences et Technologies de Lille, Villeneuve d’Ascq, France (1981)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations