Query Languages for Constraint Databases: First-Order Logic, Fixed-Points, and Convex Hulls

  • Stephan Kreutzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1973)


We define various extensions of first-order logic on linear as well as polynomial constraint databases. First, we extend first-order logic by a convex closure operator and show this logic, FO(conv), to be closed and to have Ptime data-complexity. We also show that a weak form of multiplication is definable in this language and prove the equivalence between this language and the multiplication part of PFOL. We then extend FO(conv) by fixed-point operators to get a query languages expressive enough to capture Ptime. In the last part of the paper we lift the results to polynomial constraint databases.


Polynomial Time Convex Hull Query Language Region Extension Expressive Power 
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. [BCSS98]
    L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation. Springer, 1998.Google Scholar
  2. [BSS89]
    L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the American Mathmatical Society, 21:1–46, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [CJ98]
    B.F. Caviness and J.R. Johnson, editors. Quantifier Elimination and Cylindric Algebraic Decomposition. Springer, 1998.Google Scholar
  4. [Col75]
    George E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Automata Theory and Formal Languages, number 33 in LNCS, pages 134–183, Berlin, 1975. Springer-Verlag.Google Scholar
  5. [Ede87]
    H. Edelsbrunner. Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science. Springer, 1987.Google Scholar
  6. [EF95]
    H.-D. Ebbinghaus and J. Flum. Finite Model Theory. Springer, 1995.Google Scholar
  7. [GK97]
    S. Grumbach and G. M. Kuper. Tractable recursion over geometric data. In Principles and Practice of Constraint Programming, number 1330 in LNCS, pages 450–462. Springer, 1997.CrossRefGoogle Scholar
  8. [GK99]
    E. Grädel and S. Kreutzer. Descriptive complexity theory for constraint databases. In Computer Science Logic, number 1683 in as LNCS, pages 67–82. Springer, 1999.Google Scholar
  9. [GK00]
    F. Geerts and B. Kuijpers. Linear approximation of planar spatial databases using transitive-closure logic. In PODS 2000, pages 126–135. ACM Press, 2000.Google Scholar
  10. [GO97]
    Jacob E. Goodman and Joseph O’Rourke, editors. Handbook of Discrete and Computational Geometry. CRC Press, 1997.Google Scholar
  11. [GS97]
    S. Grumbach and J. Su. Finitely representable databases. Journal of Computer and System Sciences, 55:273–298, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [KKR90]
    P. C. Kanellakis, G. M. Kuper, and P. Z. Revesz. Constraint query languages. In PODS 1990, pages 299–313, 1990.Google Scholar
  13. [KLP00]
    G. Kuper, L. Libkin, and J. Paredaens, editors. Constraint Databases. Springer, 2000.Google Scholar
  14. [KPSV96]
    B. Kuijpers, J. Paredaens, M. Smits, and J. Van den Bussche. Termination properties of spatial datalog programs. In Logic in Databases, number 1154 in LNCS, pages 101–116, 1996.CrossRefGoogle Scholar
  15. [Kre00]
    S. Kreutzer. Fixed-point query languages for linear constraint databases. In PODS 2000, pages 116–125. ACM press, 2000.Google Scholar
  16. [Van99]
    L. Vandeurzen. Logic-Based Query Languages for the Linear Constraint Database Model. PhD thesis, Limburgs Universitair Centrum, 1999.Google Scholar
  17. [VGG98]
    L. Vandeurzen, M. Gyssens, and D. Van Gucht. An expressive language for linear spatial database queries. In PODS 1998, pages 109–118, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Stephan Kreutzer
    • 1
  1. 1.Lehrgebiet Mathematische Grundlagen der InformatikAachen

Personalised recommendations