On the implication problem for cardinality constraints and functional dependencies

  • Sven Hartmann


In database design, integrity constraints are used to express database semantics. They specify the way by that the elements of a database are associated to each other. The implication problem asks whether a given set of constraints entails further constraints. In this paper, we study the finite implication problem for cardinality constraints. Our main result is a complete characterization of closed sets of cardinality constraints. Similar results are obtained for constraint sets containing cardinality constraints, but also key and functional dependencies. Moreover, we construct Armstrong databases for these constraint sets, which are of special interest for example-based deduction in database design.

cardinality constraint key functional dependency implication problem Armstrong database transversal design clique graph 


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  1. [1]
    B. Alspach, K. Heinrich and G. Liu, Orthogonal factorizations of graphs, in: Contemporary Design Theory, eds. Dinitz and Stinson (Wiley, New York, 1992) chapter 2.Google Scholar
  2. [2]
    W.W. Armstrong, Dependency structures of database relationship, Inform. Process. 74 (1974) 580-583.Google Scholar
  3. [3]
    P. Atzeni and V. De Antonelles, Relational Database Theory (Benjamin/Cummings, Redwood, 1993).Google Scholar
  4. [4]
    C. Batini, S. Ceri and S. Navathe, eds., Conceptual Database Design (Benjamin/Cummings, Redwood, 1992).Google Scholar
  5. [5]
    C. Beeri and B. Thalheim, Identification as a primitive of database models, in: Fundamentals of Information Systems, eds. T. Polle et al. (Kluwer, Dordrecht, 1999) pp. 19-36.Google Scholar
  6. [6]
    F.E. Bennett and L. Wu, On minimum matrix representation of closure operations, Discrete Appl. Math. 26 (1990) 25-40.Google Scholar
  7. [7]
    F.E. Bennett and L. Wu, On minimum matrix representation of Sperner systems, Discrete Appl.Math. 81 (1998) 9-17.Google Scholar
  8. [8]
    T. Beth, D. Jungnickel and H. Lenz, Design Theory (BI, Mannheim, 1985).Google Scholar
  9. [9]
    A. Binemann-Zdanowicz, Systematization of approaches to equality-generating constraints, in: Current Issues in Databases and Information Systems, eds. J. Stuller et al., Lecture Notes in Computer Science, Vol. 1884 (Springer, Berlin, 2000) pp. 307-314.Google Scholar
  10. [10]
    J. Biskup, R. Menzel, T. Polle and Y. Sagiv, Decomposition of relationships through pivoting, in: Conceptual Modeling, ed. B. Thalheim, Lecture Notes in Computer Science, Vol. 823 (Springer, Berlin, 1996) pp. 28-41.Google Scholar
  11. [11]
    B. Bollobás, Extremal Graph Theory (Academic Press, London, 1978).Google Scholar
  12. [12]
    G. Burosch, J. Demetrovics and G.O.H. Katona, The poset of closures as a model of changing databases, Order 4 (1987) 127-142.Google Scholar
  13. [13]
    D. Calvanese and M. Lenzerini, On the interaction between ISA and cardinality constraints, in: Proc. of Tenth Int. Conf. on Data Engin. (1994) pp. 204-213.Google Scholar
  14. [14]
    P.P. Chen, The entity-relationship model: towards a unified view of data, ACM Trans. Database Systems 1 (1976) 9-36.Google Scholar
  15. [15]
    S. Chowla, P. Erdős and E.G. Straus, On the maximal number of pairwise orthogonal Latin squares of a given order, Canad. J. Math. 12 (1960) 204-208.Google Scholar
  16. [16]
    E.F. Codd, A relation model of data for large shared data banks, Comm. ACM 13 (1970) 377-387.Google Scholar
  17. [17]
    C.J. Colbourn and J.H. Dinitz, eds., The CRC Handbook of Combinatorial Designs (CRC Press, Boca Raton, 1996).Google Scholar
  18. [18]
    C. Delobel and R.G. Casey, Decompositions of a database and the theory of Boolean switching functions, IBM J. Res. Develop. 17 (1973) 374-386.Google Scholar
  19. [19]
    J. Demetrovics, On the equivalence of candidate keys and Sperner systems, Acta Cybernet. 4 (1979) 247-252.Google Scholar
  20. [20]
    J. Demetrovics, Z. Füredi and G.O.H. Katona, Minimum matrix representation of closure operations, Discrete Appl. Math. 11 (1985) 115-128.Google Scholar
  21. [21]
    J. Demetrovics and G.O.H. Katona, A survey on some combinatorial problems concerning functional dependencies in database relations, Ann. Math. Artificial Intelligence 7 (1993) 63-82.Google Scholar
  22. [22]
    J. Demetrovics, G.O.H. Katona and A. Sali, Design type problems motivated by database theory, J. Statist. Plann. Inference 72 (1998) 149-164.Google Scholar
  23. [23]
    J. Demetrovics and V.D. Thi, Relations and minimal keys, Acta Cybernet. 8 (1988) 279-285.Google Scholar
  24. [24]
    J. Demetrovics and V.D. Thi, Some results about functional dependencies, Acta Cybernet. 8 (1988) 273-278.Google Scholar
  25. [25]
    J.H. Dinitz and D.R. Stinson, eds., Contemporary Design Theory (Wiley, New York, 1992).Google Scholar
  26. [26]
    P. Erdős, Extremal problems in graph theory, in: A Seminar in Graph Theory, ed. F. Harary (Holt, Rinehart and Winston, 1967) pp. 54-64.Google Scholar
  27. [27]
    P.P. Chen et al., eds., Conceptual Modeling, Lecture Notes in Computer Science, Vol. 1565 (Springer, Berlin, 1996).Google Scholar
  28. [28]
    B. Ganter and H.-D.O.F. Gronau, On two conjectures of Demetrovics, Füredi and Katona on partitions, Discrete Math. 88 (1991) 149-155.Google Scholar
  29. [29]
    M. Gondran and M. Minoux, Graphs and Algorithms (Wiley, Chichecter, 1990).Google Scholar
  30. [30]
    H.-D.O.F. Gronau, M. Grüttmüller, S. Hartmann, U. Leck and V. Leck, On orthogonal double covers of graphs, Des. Codes Cryptogr. (2001) to appear.Google Scholar
  31. [31]
    H.-D.O.F. Gronau, R.C. Mullin and A. Rosa, Orthogonal double covers of complete graphs by trees, Graphs Combin. 13 (1997) 251-262.Google Scholar
  32. [32]
    H.-D.O.F. Gronau, R.C. Mullin and P.J. Schellenberg, On orthogonal double covers and a conjecture of Chung and West, J. Combin. Designs 3 (1995) 213-231.Google Scholar
  33. [33]
    A. Hajnal and E. Szemerédi, Proof of a conjecture of Erdős, in: Combinatorial Theory and its Applications, eds. A. Renyi, P. Erdős and V.T. Sós, Colloq. Math. Soc. János Bolyai, Vol. 4 (North-Holland, Amsterdam, 1970) pp. 601-623.Google Scholar
  34. [34]
    S. Hartmann, Graphtheoretic methods to construct entity-relationship databases, in: Graphtheoretic Concepts in Computer Science, ed. M. Nagl, Lecture Notes in Computer Science, Vol. 1017 (Springer, Berlin, 1995) pp. 131-145.Google Scholar
  35. [35]
    S. Hartmann, On the consistency of int-cardinality constraints, in: Conceptual Modeling, eds. T.W. Ling, S. Ram and M.L. Li, Lecture Notes in Computer Science, Vol. 1507 (Springer, Berlin, 1998) pp. 150-163.Google Scholar
  36. [36]
    S. Hartmann, Orthogonal decompositions of complete digraphs, Graphs Combin. (1998) to appear.Google Scholar
  37. [37]
    D. Jungnickel, Graphen, Netzwerke und Algorithmen (BI, Mannheim, 1994).Google Scholar
  38. [38]
    M. Lenzerini and P. Nobili, On the satisfiability of dependency constraints in entity-relationship schemata, Inform. Syst. 15 (1990) 453-461.Google Scholar
  39. [39]
    S.W. Liddle, D.W. Embley and S.N. Woodfield, Cardinality constraints in semantic data models, Data Knowledge Engrg. 11 (1993) 235-270.Google Scholar
  40. [40]
    H.F. MacNeish, Euler's squares, Ann. of Math. 23 (1922) 221-227.Google Scholar
  41. [41]
    H. Mannila and K. Räihä, Design by example: an application of Armstrong relations, J. Comput. Syst. Sci. 33 (1986) 126-141.Google Scholar
  42. [42]
    H. Mannila and K. Räihä, The Design of Relational Databases (Addison-Wesley, Reading, 1992).Google Scholar
  43. [43]
    A. McAllister, Complete rules for n-ary relationship cardinality constraints, Data Knowledge Engrg. 27 (1998) 255-288.Google Scholar
  44. [44]
    H. Noltemeier, Graphentheorie (de Gruyter, Berlin, 1976).Google Scholar
  45. [45]
    A. Schrijver, Theory of Linear and Integer Programming (Wiley, Chichecter, 1986).Google Scholar
  46. [46]
    T. Takaoka, Subcubic cost algorithms for the all pairs shortest path problem, Algorithmica 20 (1998) 309-318.Google Scholar
  47. [47]
    B. Thalheim, Dependencies in Relational Databases (Teubner, Stuttgart, 1991).Google Scholar
  48. [48]
    B. Thalheim, Foundations of entity-relationship modeling, Ann. Math. Artificial Intelligence 6 (1992) 197-256.Google Scholar
  49. [49]
    B. Thalheim, Fundamentals of cardinality constraints, in: Entity-Relationship Approach, eds. G. Pernul and A.M. Tjoa, Lecture Notes in Computer Science, Vol. 645 (Springer, Berlin, 1992) pp. 7-23.Google Scholar
  50. [50]
    B. Thalheim, Entity-Relationship Modeling (Springer, Berlin, 2000).Google Scholar
  51. [51]
    D. Theodorates, Deductive object oriented schemas, in: Conceptual Modeling, ed. B. Thalheim, Lecture Notes in Computer Science, Vol. 1157 (Springer, Berlin, 1996) pp. 58-72.Google Scholar
  52. [52]
    K. Tichler, Minimum matrix representation of some key system, in: Foundations of Information and Knowledge Systems, eds. K.-D. Schewe and B. Thalheim, Lecture Notes in Computer Science, Vol. 1762 (Springer, Berlin, 2000) pp. 275-287.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Sven Hartmann
    • 1
  1. 1.FB MathematikUniversität RostockRostockGermany

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