Advertisement

Dominance Constraints: Algorithms and Complexity

  • Alexander Koller
  • Joachim Niehren
  • Ralf Treinen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2014)

Abstract

Dominance constraints for finite tree structures are widely used in several areas of computational linguistics including syntax, semantics, and discourse. In this paper, we investigate algorithmic and complexity questions for dominance constraints and their first-order theory. The main result of this paper is that the satisfiability problem of dominance constraints is NP-complete. We present two NP algorithms for solving dominance constraints, which have been implemented in the concurrent constraint programming language Oz. Despite the intractability result, the more sophisticated of our algorithms performs well in an application to scope underspecification. We also show that the positive existential fragment of the first-order theory of dominance constraints is NP-complete and that the full first-order theory has non-elementary complexity.

Keywords

Dominance constraints complexity computational linguistics underspecification constraint programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Backofen, R., J. Rogers, and K. Vijay-Shanker. A first-order axiomatization of the theory of finite trees. Journal of Logic, Language, and Information, 4:5–39, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bos, J. Predicate logic unplugged. In Proceedings of the 10th Amsterdam Colloquium, pages 133–143, 1996.Google Scholar
  3. 3.
    Duchier, D., and C. Gardent. A constraint-based treatment of descriptions. In Proceedings of IWCS-3, Tilburg, 1999.Google Scholar
  4. 4.
    Duchier, D., and J. Niehren. Solving dominance constraints with finite set constraint programming. Technical report, Universität des Saarlandes, Programming Systems Lab, 1999. http://www.ps.uni-sb.de/Papers/abstracts/DomCP99.html.Google Scholar
  5. 5.
    Egg, M., J. Niehren, P. Ruhrberg, and F. Xu. Constraints over Lambda-Structures in Semantic Underspecification. In Proceedings COLING/ACL’98, Montreal, 1998.Google Scholar
  6. 6.
    Gardent, C., and B. Webber. Describing discourse semantics. In Proceedings of the 4th TAG+ Workshop, Philadelphia, 1998. University of Pennsylvania.Google Scholar
  7. 7.
    Johnson, D. S. A catalog of complexity classes. In Leeuwen, J. van, editor, Handbook of Theoretical Computer Science, vol A: Algorithms and Complexity, chapter 2, pages 67–161. Elsevier, 1990.Google Scholar
  8. 8.
    Koller, A. Constraint languages for semantic underspecification. Diplom thesis, Universität des Saarlandes, Saarbrücken, Germany, 1999. http://www.coli.uni-sb.de/~koller/papers/da.html.Google Scholar
  9. 9.
    Koller, A., and J. Niehren. Constraint programming in computational linguistics. Submitted. http://www.coli.uni-sb.de/~koller/cpcl.html, 1999.
  10. 10.
    Koller, A., and J. Niehren. Scope underspecification and processing. Lecture Notes, ESSLLI’ 99, Utrecht, 1999. http://www.coli.uni-sb.de/~koller/papers/esslli99.html
  11. 11.
    Marcus, M. P., D. Hindle, and M. M. Fleck. D-theory: Talking about talking about trees. In Proceedings of the 21st ACL, pages 129–136, 1983.Google Scholar
  12. 12.
    Meyer-Viol, W., and R. Kempson. Sequential construction of logical forms. In Proceedings of the Third Conference on Logical Aspects of Computational Linguistics, Grenoble, France, 1998.Google Scholar
  13. 13.
    Muskens, R. Underspecified semantics. Technical Report 95, Universität des Saarlandes, Saarbrücken, 1998. To appear.Google Scholar
  14. 14.
    Rabin, M. Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society, 141:1–35, 1969.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Reyle, U. Dealing with ambiguities by underspecification: construction, representation, and deduction. Journal of Semantics, 10:123–179, 1993.CrossRefGoogle Scholar
  16. 16.
    Rogers, J. Studies in the Logic of Trees with Applications to Grammar Formalisms. PhD thesis, University of Delaware, 1994.Google Scholar
  17. 17.
    Rogers, J., and K. Vijay-Shanker. Reasoning with descriptions of trees. In Proceedings of the 30th ACL, pages 72–80, University of Delaware, 1992.Google Scholar
  18. 18.
    Stockmeyer, L. J., and A. R. Meyer. Word problems requiring exponential time. In 5th Annual ACM Symposium on the Theory of Computing, pages 1–9, 1973.Google Scholar
  19. 19.
    Thatcher, J. W., and J. B. Wright. Generalized finite automata with an application to a decision problem of second-order logic. Mathematical Systems Theory, 2:57–68, 1968.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Vijay-Shanker, K. Using descriptions of trees in a tree adjoining grammar. Computational Linguistics, 18:481–518, 1992.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Alexander Koller
    • 1
  • Joachim Niehren
    • 2
  • Ralf Treinen
    • 3
  1. 1.Department of Computational LinguisticsUniversität des SaarlandesSaarbrückenGermany
  2. 2.Programming Systems LabUniversität des SaarlandesSaarbrückenGermany
  3. 3.Laboratoire de Recherche en InformatiqueUniversité Paris-SudOrsayFrance

Personalised recommendations