Advertisement

Some Remarks on Relational Database Schemes Having Few Minimal Keys

  • Joachim Biskup
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7260)

Abstract

Relational database schemes comprise semantic constraints to formally capture at least part of the “real-world” semantics of an application. Functional dependencies constitute a basic and widely studied class of such constraints. Accordingly, many properties of this class are known, including the insight that the number of minimal keys – as determined by a declared set of functional dependencies – might vary extensively, from just one to exponentially many (in the number of the underlying attributes). The case of just one minimal key is known to be characterized by the set of extremal attributes forming a minimal key. Starting from this result, the present work studies schemes having only a few minimal keys. In particular, we consider the cases of schemes having two and three minimal keys, and then suggest some research for dealing with the more general case of n minimal keys. Furthermore we study the impact of additionally requiring the schemes to be in Boyce-Codd normal form or Third normal form.

Keywords

Boyce-Codd normal form computational complexity extremal attribute functional dependency functional relationship implicational closure logical implication minimal key minimal-key equivalence NP-completeness object normal form prime attribute relational database relation scheme semantic modeling Sperner system superprime attribute Third normal form 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, Reading (1995)zbMATHGoogle Scholar
  2. 2.
    Atzeni, P., Antonellis, V.D.: Relational Database Theory. Benjamin/Cummings, Redwood City (1993)zbMATHGoogle Scholar
  3. 3.
    Biskup, J.: Boyce-Codd normal form and object normal forms. Inf. Process. Lett. 32(1), 29–33 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Biskup, J., Demetrovics, J., Libkin, L., Muchnik, I.B.: On relational database schemes having unique minimal key. Elektronische Informationsverarbeitung und Kybernetik 27(4), 217–225 (1991)zbMATHGoogle Scholar
  5. 5.
    Biskup, J., Embley, D.W., Lochner, J.-H.: Reducing inference control to access control for normalized database schemas. Inf. Process. Lett. 106(1), 8–12 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Buneman, P., Davidson, S.B., Fan, W., Hara, C.S., Tan, W.C.: Reasoning about keys for XML. Inf. Syst. 28(8), 1037–1063 (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, P.P.: The entity-relationship model – toward a unified view of data. ACM Trans. Database Syst. 1(1), 9–36 (1976)CrossRefGoogle Scholar
  8. 8.
    Demetrovics, J.: On the number of candidate keys. Inf. Process. Lett. 7(6), 266–269 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Demetrovics, J.: On the equivalence of candidate keys with Sperner systems. Acta Cybern. 4, 247–252 (1980)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Demetrovics, J., Katona, G.O.H., Miklós, D., Seleznjev, O., Thalheim, B.: Asymptotic properties of keys and functional dependencies in random databases. Theor. Comput. Sci. 190(2), 151–166 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Hartmann, S., Leck, U., Link, S.: On Codd families of keys over incomplete relations. Comput. J. 54(7), 1166–1180 (2011)CrossRefGoogle Scholar
  12. 12.
    Hartmann, S., Link, S.: Efficient reasoning about a robust XML key fragment. ACM Trans. Database Syst. 34(2) (2009)Google Scholar
  13. 13.
    Lausen, G., Meier, M., Schmidt, M.: SPARQLing constraints for RDF. In: Kemper, A., et al. (eds.) International Conference on Extending Database Technology, EDBT 2008. ACM International Conference Proceeding Series, vol. 261, pp. 499–509. ACM, New York (2008)Google Scholar
  14. 14.
    Lucchesi, C.L., Osborn, S.L.: Candidate keys for relations. J. Comput. Syst. Sci. 17(2), 270–279 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Mannila, H., Räihä, K.-J.: The Design of Relational Databases. Addison-Wesley, Wokingham (1992)zbMATHGoogle Scholar
  16. 16.
    Paredaens, J., Bra, P.D., Gyssens, M., Gucht, D.V.: The Structure of the Relational Database Model. Springer, Heidelberg (1989)CrossRefzbMATHGoogle Scholar
  17. 17.
    Thalheim, B.: On semantic issues connected with keys in relational databases permitting null values. Elektronische Informationsverarbeitung und Kybernetik 25(1/2), 11–20 (1989)MathSciNetGoogle Scholar
  18. 18.
    Thalheim, B.: Dependencies in Relational Databases. Teubner, Stuttgart (1991)CrossRefzbMATHGoogle Scholar
  19. 19.
    Thalheim, B.: The number of keys in relational and nested relational databases. Discrete Applied Mathematics 40(2), 265–282 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Thalheim, B.: Entity-Relationship Modeling – Foundations of Database Technology. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  21. 21.
    Thalheim, B.: Towards a Theory of Conceptual Modelling. In: Heuser, C.A., Pernul, G. (eds.) ER 2009. LNCS, vol. 5833, pp. 45–54. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  22. 22.
    Toman, D., Weddell, G.E.: On keys and functional dependencies as first-class citizens in description logics. J. Autom. Reasoning 40(2-3), 117–132 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Vincent, M.W., Srinivasan, B.: A note on relation schemes which are in 3NF but not in BCNF. Inf. Process. Lett. 48(6), 281–283 (1993)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Joachim Biskup
    • 1
  1. 1.Technische Universität DortmundDortmundGermany

Personalised recommendations