Propositional Tree Automata

  • Joe Hendrix
  • Hitoshi Ohsaki
  • Mahesh Viswanathan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4098)


In the paper, we introduce a new tree automata framework, called propositional tree automata, capturing the class of tree languages that are closed under an equational theory and Boolean operations. This framework originates in work on developing a sufficient completeness checker for specifications with rewriting modulo an equational theory. Propositional tree automata recognize regular equational tree languages. However, unlike regular equational tree automata, the class of propositional tree automata is closed under Boolean operations. This extra expressiveness does not affect the decidability of the membership problem. This paper also analyzes in detail the emptiness problem for propositional tree automata with associative theories. Though undecidable in general, we present a semi-algorithm for checking emptiness based on machine learning that we have found useful in practice.


Boolean Operation Equational Theory Regular Language 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.
    Angluin, D.: Learning Regular Sets from Queries and Counterexamples. Information and Computation 75, 87–106 (1987)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Armando, A., Basin, D., Boichut, Y., Chevalier, Y., Compagna, L., Cuellar, J., Hankes Drielsma, P., Heám, P.-C., Kouchnarenko, O., Mantovani, J., Mödersheim, S., von Oheimb, D., Rusinowitch, M., Santiago, J., Turuani, M., Viganò, L., Vigneron, L.: The AVISPA Tool for the Automated Validation of Internet Security Protocols and Applications. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 281–285. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Autebert, J., Berstel, J., Boasson, L.: Context-Free Languages and Push-Down Automata, Handbook of Formal Languages, vol. 1, pp. 111–174. Springer, Heidelberg (1997)Google Scholar
  4. 4.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Google Scholar
  5. 5.
    Boichut, Y., Heám, P.-C., Kouchnarenko, O.: Automatic Verification of Security Protocols Using Approximations, technical report RR-5727, INRIA (October 2005)Google Scholar
  6. 6.
    Bouhoula, A., Jouannaud, J.P., Meseguer, J.: Specification and Proof in Membership Equational Logic. TCS, vol. 236, pp. 35–132. Elsevier, Amsterdam (2000)Google Scholar
  7. 7.
    Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree Automata Techniques and Applications, incomplete draft (2005), available at:
  8. 8.
    Devienne, P., Talbot, J.-M., Tison, S.: Set-Based Analysis for Logic Programming and Tree Automata. In: Van Hentenryck, P. (ed.) SAS 1997. LNCS, vol. 1302, pp. 127–140. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  9. 9.
    Du, D.-Z., Ko, K.: Theory of Computational Complexity. John Wiley and Sons, Chichester (2000)MATHGoogle Scholar
  10. 10.
    Gallagher, J.P., Puebla, G.: Abstract Interpretation over Non-deterministic Finite Tree Automata for Set-Based Analysis of Logic Programs. In: Krishnamurthi, S., Ramakrishnan, C.R. (eds.) PADL 2002. LNCS, vol. 2257, pp. 243–261. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Genet, T., Klay, F.: Rewriting for Cryptographic Protocol Verification. In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, pp. 271–290. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  12. 12.
    Ginsburg, S.: The Mathematical Theory of Context-Free Languages. McGraw-Hill, New York (1966)MATHGoogle Scholar
  13. 13.
    Hendrix, J., Ohsaki, H., Meseguer, J.: Sufficient Completeness Checking with Propositional Tree Automata, technical report UIUCDCS-R-2005-2635, Department of Computer Science, University of Illinois at Urbana-Champaign (2005), available at:
  14. 14.
    Hendrix, J., Ohsaki, H., Viswanathan, M.: Propositional Tree Automat, technical report UIUCDCS-R-2006-2695, University of Illinios at Urbana-Champaign (2005), available at:
  15. 15.
    Hendrix, J.: CETA: A Library for Equational Tree Automata, Department of Computer Science, University of Illinois at Urbana-Champaign (2006), Software available under GPL license at:
  16. 16.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley Publishing Company, Reading (1979)MATHGoogle Scholar
  17. 17.
    Hosoya, H., Vouillon, J., Pierce, B.C.: Regular Expression Types for XML. In: Proc. of 5th ICFP, SIGPLAN, Montreal (Canada), vol. 35(9), pp. 11–22. ACM, New York (2000)Google Scholar
  18. 18.
    Kearns, M., Vazirani, U.: An Introduction to Computational Learning Theory. MIT Press, Cambridge (1994)Google Scholar
  19. 19.
    Klarlund, N., Møller, A.: MONA Version 1.4 User Manual, BRICS Notes Series NS-01-1, Department of Computer Science, University of Aarhus (2001)Google Scholar
  20. 20.
    Lugiez, D.: Multitree Automata That Count. TCS, vol. 333. Elsevier, Amsterdam (2005)Google Scholar
  21. 21.
    Nederhof, M.: Practical Experiments with Regular Approximation of Context-Free Languages. Computational Linguistics 26(1), 17–44 (2000)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Ohsaki, H., Talbot, J.-M., Tison, S., Roos, Y.: Monotone AC-Tree Automata. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 337–351. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  23. 23.
    Ohsaki, H., Takai, T.: ACTAS: A System Design for Associative and Commutative Tree Automata Theory. In: Proc. of 5th RULE, Aachen (Germany). ENTCS, vol. 124, pp. 97–111. Elsevier, Amsterdam (2005)Google Scholar
  24. 24.
    Ohsaki, H., Takai, T.: Decidability and Closure Properties of Equational Tree Languages. In: Tison, S. (ed.) RTA 2002. LNCS, vol. 2378, pp. 114–128. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  25. 25.
    Ohsaki, H.: Beyond Regularity: Equational Tree Automata for Associative and Commutative Theories. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 539–553. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  26. 26.
    Parikh, R.: On Context-Free Languages. JACM 13(4), 570–581 (1966)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Seidl, H., Schwentick, T., Muscholl, A.: Numerical Document Queries. In: Proc. of 22nd PODS, SanDiego (USA), pp. 155–166. ACM, New York (2003)Google Scholar
  28. 28.
    Yagi, I., Takata, Y., Seki, H.: A Static Analysis Using Tree Automata for XML Access Control. In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 234–247. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  29. 29.
    Verma, K.N.: Two-Way Equational Tree Automata for AC-Like Theories: Decidability and Closure Properties. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 180–197. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Joe Hendrix
    • 1
  • Hitoshi Ohsaki
    • 2
  • Mahesh Viswanathan
    • 1
  1. 1.University of Illinois at Urbana-Champaign 
  2. 2.National Institute of Advanced Industrial Science and Technology 

Personalised recommendations