Definability and Descriptive Complexity on Databases of Bounded Tree-Width

  • Martin Grohe
  • Julian Mariño
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1540)


We study the expressive power of various query languages on relational databases of bounded tree-width.

Our first theorem says that fixed-point logic with counting captures polynomial time on classes of databases of bounded tree-width. This result should be seen on the background of an important open question of Chandra and Harel [7] asking whether there is a query language capturing polynomial time on unordered databases. Our theorem is a further step in a larger project of extending the scope of databases on which polynomial time can be captured by reasonable query languages.

We then prove a general definability theorem stating that each query on a class of databases of bounded tree-width which is definable in monadic second-order logic is also definable in fixed-point logic (or datalog). Furthermore, for each k ≥ 1 the class of databases of tree-width at most k is definable in fixed-point logic. These results have some remarkable consequences concerning the definability of certain classes of graphs.

Finally, we show that each database of tree-width at most k can be characterized up to isomorphism in the language Ck+3, the (k + 3)-variable fragment of firstorder logic with counting.


Polynomial Time Planar Graph Query Language Database Schema Relation Symbol 
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  1. 1.
    S. Abiteboul and V. Vianu. Fixpoint extensions of first order logic and datalog-like languages. In Proceedings of the 4th IEEE Symposium on Logic in Computer Science, pages 71–79, 1989.Google Scholar
  2. 2.
    S. Arnborg, D. Corneil, and A. Proskurowski. Complexity of finding embeddings in a k-tree. SIAM Journal on Algebraic Discrete Methods, 8:277–284, 1987.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    H.L. Bodländer. NC-algorithms for graphs with small treewidth. In J. van Leeuwen, editor, Proceedings of the 14th International Workshop on Graph theoretic Concepts in Computer Science WG’88, volume 344 of Lecture Notes in Computer Science, pages 1–10. Springer-Verlag, 1988.Google Scholar
  4. 4.
    H.L. Bodländer. Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees. Journal of Algorithms, 11:631–643, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    H.L. Bodländer. Treewidth: Algorithmic techniques and results. In Proceedings 22nd International Symposium on Mathematical Foundations of Computer Science, MFCS’97, volume 1295 of Lecture Notes in Computer Science, pages 29–36. Springer-Verlag, 1997.Google Scholar
  6. 6.
    J. Cai, M. Fürer, and N. Immerman. An optimal lower bound on the number of variables for graph identification. Combinatorica, 12:389–410, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. Chandra and D. Harel. Structure and complexity of relational queries. Journal of Computer and System Sciences, 25:99–128, 1982.zbMATHCrossRefGoogle Scholar
  8. 8.
    B. Courcelle. Graph rewriting: An algebraic and logic approach. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume 2, pages 194–242. Elsevier Science Publishers, 1990.Google Scholar
  9. 9.
    R. Diestel. Graph Theory. Springer-Verlag, 1997.Google Scholar
  10. 10.
    T. Feder and M.Y Vardi. Monotone monadic SNP and constraint satisfaction. In Proceedings of the 25th ACM Symposium on Theory of Computing, pages 612–622, 1993.Google Scholar
  11. 11.
    E. Grädel. On the restraining power of guards, 1998.Google Scholar
  12. 12.
    E. Grädel and M. Otto. Inductive definability with counting on finite structures. In E. Börger, G. Jäger, H. Kleine Büning, S. Martini, and M.M. Richter, editors, Computer Science Logic, 6th Workshop, CSL '92, San Miniato 1992, Selected Papers, volume 702 of Lecture Notes in Computer Science, pages 231–247. Springer-Verlag, 1993.Google Scholar
  13. 13.
    M. Grohe. Finite-variable logics in descriptive complexity theory 1998.Google Scholar
  14. 14.
    M. Grohe. Fixed-point logics on planar graphs. In Proceedings of the 13th IEEE Symposium on Logic in Computer Science, pages 6–15, 1998.Google Scholar
  15. 15.
    R. Halin. S-Functions for graphs. Journal of Geometry, 8:171–186, 1976.CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    N. Immerman. Relational queries computable in polynomial time. Information and Control, 68:86–104, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    N. Immerman. Expressibility as a complexity measure: results and directions. In Proceedings of the 2nd IEEE Symposium on Structure in Complexity Theory, pages 194–202, 1987.Google Scholar
  18. 18.
    N. Immerman and E. Lander. Describing graphs: A first-order approach to graph canonization. In A. Selman editor, Complexity theory retrospective, pages 59–81. Springer-Verlag, 1990Google Scholar
  19. 19.
    B. Reed. Tree width and tangles: A new connectivity measure and some applications. In R.A. Bailey editor, Surveys in Combinatorics, volume 241 of LMS Lecture Note Series, pages 87–162. Cambridge University Press, 1997.Google Scholar
  20. 20.
    N. Robertson and P.D. Seymour. Graph minors II. Algorithmic aspects of tree-width. Journal of Algorithms, 7:309–322, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    N. Robertson and P.D. Seymour. Graph minors IV. Tree-width and well-quasi-ordering. Journal of Combinatorial Theory, Series B 48:227–254, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    N. Robertson and P.D. Seymour. Graph minors V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41:92–114, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    N. Robertson and P.D. Seymour. Graph minors XX. Wagner’s conjecture, 1988. unpublished manuscript.Google Scholar
  24. 24.
    M. Y. Vardi. The complexity of relational query languages. In Proceedings of the 14th ACM Symposium on Theory of Computing, pages 137–146, 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Martin Grohe
    • 1
  • Julian Mariño
    • 1
  1. 1.Institut für mathematische LogikFreiburgGermany

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