First-order definability over constraint databases

Extended abstract
  • Stéphane Grumbach
  • Jianwen Su
Part of the Lecture Notes in Computer Science book series (LNCS, volume 976)


In this paper, we study the expressive power of first-order logic as a query language over constraint databases. We consider constraints over various domains (ℕ,2e, ℝ), and with various operations (⩽, +, ×, xy). We first tackle the problem of the definability of parity and connectivity, which are the most classical examples of queries not expressible in first-order logic over finite structures. We prove that these two queries are first-order expressible in presence of (enough) arithmetic. This is in sharp contrast with classical relational databases. Nevertheless, we show that they are not definable with constraints of interest for constraint databases such as linear constraints. We then develop reductions techniques for queries over constraint databases, that allow us to draw conclusions with respect to their undefinability in various constraint query languages.


Query Language Relation Symbol Connectivity Query Graph Query Real Closed Field 
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. [ACGK94]
    F. Afrati, S. Cosmadakis, S. Grumbach, and G. Kuper. Expressiveness of linear vs. polynomial constraints in database query languages. In Second Workshop on the Principles and Practice of Constraint Programming, 1994.Google Scholar
  2. [AHV94]
    S. Abiteboul, R. Hull, and V. Vianu. Foundations of Databases. Addison-Wesley, 1994.Google Scholar
  3. [CH80]
    A. Chandra and D. Harel. Computable Queries for Relational Data Bases. Journal of Computer and System Sciences, 21(2):156–178, Oct. 1980.CrossRefGoogle Scholar
  4. [CK95]
    J. Chomicki and G. Kuper. Measuring infinite relations. In Proc. 14th ACM Symp. on Principles of Database Systems, San Jose, May 1995.Google Scholar
  5. [Cod70]
    E.F. Codd. A relational model of data for large shared data banks. Communications of ACM, 13:6:377–387, 1970.CrossRefGoogle Scholar
  6. [CSV84]
    A. K. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM J. on Computing, 13(2):423–439, May 1984.CrossRefGoogle Scholar
  7. [Ehr61]
    A. Ehrenfeucht. An application of games to the completeness problem for formalized theories. Fund. Math, 49, 1961.Google Scholar
  8. [End72]
    H. Enderton. A Mathematical Introduction to Logic. Academic Press, 1972.Google Scholar
  9. [Fag74]
    R. Fagin. Generalized first-order spectra and polynomial-time recognizable sets. Complexity of Computations, SIAM-AMS Proceedings 7, pages 43–73, 1974.Google Scholar
  10. [Fag93]
    R. Fagin. Finite model theory — a personal perspective. Theoretical Computer Science, 116:3–31, 1993.CrossRefGoogle Scholar
  11. [Fra54]
    R. Fraïssé. Sur les classifications des systèmes de relations. Publ. Sci. Univ Alger, I:1, 1954.Google Scholar
  12. [FSS84]
    M. Furst, J. B. Saxe, and M. Sipser. Parity, circuits, and the polynomialtime hierarchy. Mathematical System Theory, 17:13–27, 1984.CrossRefGoogle Scholar
  13. [Gai81]
    H. Gaifman. On local and non local properties. In J. Stern, editor, Proc. Herbrand Symposium Logic Colloquium, pages 105–135. North Holland, 1981.Google Scholar
  14. [GS94]
    S. Grumbach and J. Su. Finitely representable databases. In 13th ACM Symp. on Principles of Database Systems, pages 289–300, Minneapolis, May 1994.Google Scholar
  15. [GS95]
    S. Grumbach and J. Su. Dense order constraint databases. In Proc. 14th ACM Symp. on Principles of Database Systems, San Jose, May 1995.Google Scholar
  16. [GST94]
    S. Grumbach, J. Su, and C. Tollu. Linear constraint databases. In D. Leivant, editor, Logic and Computational Complexity Workshop, Indianapolis, 1994. Springer Verlag. to appear in LNCS.Google Scholar
  17. [Gur88]
    Y. Gurevich. Current Trends in Theoretical Computer Science, E. Borger Ed., chapter Logic and the Challenge of Computer Science, pages 1–57. Computer Science Press, 1988.Google Scholar
  18. [Imm87]
    N. Immerman. Languages that capture complexity classes. SIAM J. of Computing, 16(4):760–778, Aug 1987.CrossRefGoogle Scholar
  19. [KG94]
    P. Kanellakis and D. Goldin. Constraint programming and database query languages. In Manuscript, 1994.Google Scholar
  20. [KKR90]
    P. Kanellakis, G Kuper, and P. Revesz. Constraint query languages. In Proc. 9th ACM Symp. on Principles of Database Systems, pages 299–313, Nashville, 1990.Google Scholar
  21. [Kup93]
    G.M. Kuper. Aggregation in constraint databases. In Proc. First Workshop on Principles and Practice of Constraint Programming, 1993.Google Scholar
  22. [PVV94]
    J. Paredaens, J. Van den Bussche, and D. Van Gucht. Towards a theory of spatial database queries. In Proc. 13th ACM Symp. on Principles of Database Systems, pages 279–288, 1994.Google Scholar
  23. [PVV95]
    J. Paredaens, J. Van den Bussche, and D. Van Gucht. First-order queries on finite structures over the reals. In Proc. IEEE Symposium on Logic In Computer Science, 1995.Google Scholar
  24. [Rev93]
    P. Revesz. A closed form for datalog queries with integer (gap)-order constraints. Theoretical Computer Science, 116(1):117–149, 1993.CrossRefGoogle Scholar
  25. [Rev95]
    P. Revesz. Datalog queries of set constraint databases. In Proc. Int. Conf. on Database Theory, 1995.Google Scholar
  26. [Rob49]
    J. Robinson. Decidability and decision problems in arithmetic. Journal of Symbolic Logic, 14:98–114, 1949.Google Scholar
  27. [Tar51]
    A. Tarski. A Decision method for elementary algebra and geometry. Univ. of California Press, Berkeley, California, 1951.Google Scholar
  28. [Yao90]
    F.F. Yao. Handbook of Theorical Computer Science, volume A, chapter 7 Computational Geometry, pages 343–389. J. Van Leeuwen, North Holland, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Stéphane Grumbach
    • 1
  • Jianwen Su
    • 2
  1. 1.I.N.R.I.A. RocquencourtI.N.R.I.A. and University of TorontoLe ChesnayFrance
  2. 2.Dept. of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA

Personalised recommendations