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Appropriate inferences of data dependencies in relational databases

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Abstract

We study inference systems for the combined class of functional and full hierarchical dependencies in relational databases. Two notions of implication are considered: the original notion in which a dependency is implied by a given set of dependencies and the underlying set of attributes, and the alternative notion in which a dependency is implied by a given set of dependencies alone. The first main result establishes a finite axiomatisation for the original notion of implication which clarifies the role of the complementation rule in the combined setting. In fact, we identify inference systems that are appropriate in the following sense: full hierarchical dependencies can be inferred without use of the complementation rule at all or with a single application of the complementation rule at the final step of the inference only; and functional dependencies can be inferred without any application of the complementation rule. The second main result establishes a finite axiomatisation for the alternative notion of implication. We further show how inferences of full hierarchical dependencies can be simulated by inferences of multivalued dependencies, and vice versa. This enables us to apply both of our main results to the combined class of functional and multivalued dependencies. Furthermore, we establish a novel axiomatisation for the class of non-trivial functional dependencies.

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References

  1. Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, Reading, MA (1995)

    MATH  Google Scholar 

  2. Arenas, M., Libkin, L.: A normal form for XML documents. ACM Trans. Database Syst. 29(1), 195–232 (2004)

    Article  Google Scholar 

  3. Armstrong, W.W.: Dependency structures of database relationships. Inf. Process. 74, 580–583 (1974)

    MathSciNet  Google Scholar 

  4. Armstrong, W.W., Nakamura, Y., Rudnicki, P.: Armstrong’s axioms. Formaliz. Math 11(1), 39–51 (2003)

    Google Scholar 

  5. Beeri, C.: On the membership problem for functional and multivalued dependencies in relational databases. ACM Trans. Database Syst. 5(3), 241–259 (1980)

    Article  MATH  Google Scholar 

  6. Beeri, C., Bernstein, P.A.: Computational problems related to the design of normal form relational schemata. ACM Trans. Database Syst. 4(1), 30–59 (1979)

    Article  Google Scholar 

  7. Beeri, C., Fagin, R., Howard, J.H.: A complete axiomatization for functional and multivalued dependencies in database relations. In: Proceedings of the SIGMOD International Conference on Management of Data, pp. 47–61. ACM (1977)

  8. Beeri, C., Vardi, M.Y.: Formal systems for join dependencies. Theor. Comput. Sci. 38, 99–116 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bernstein, P.: Synthesizing third normal form relations from functional dependencies. ACM Trans. Database Syst. 1(4), 277–298 (1976)

    Article  Google Scholar 

  10. Bernstein, P.A., Goodman, N.: What does Boyce-Codd normal form do? In: Proceedings of the 6th International Conference on Very Large Data Bases, pp. 245–259. IEEE Computer Society (1980)

  11. Biskup, J.: On the complementation rule for multivalued dependencies in database relations. Acta Inf. 10(3), 297–305 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  12. Biskup, J.: Inferences of multivalued dependencies in fixed and undetermined universes. Theor. Comput. Sci. 10(1), 93–106 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. Biskup, J.: Grundlagen von Informationssystemen. Vieweg (1995)

  14. Biskup, J., Dayal, U., Bernstein, P.: Synthesizing independent database schemas. In: Proceedings of the ACM SIGMOD International Conference on Management of Data, pp. 143–151 (1979)

  15. Biskup, J., Meyer, R.: Design of relational database schemes by deleting attributes in the canonical decomposition. J. Comput. Syst. Sci. 35(1), 1–22 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Buneman, P., Davidson, S.B., Fan, W., Hara, C.S., Tan, W.C.: Keys for XML. Comput. Netw. 39(5), 473–487 (2002)

    Article  Google Scholar 

  17. Chandra, A.K., Vardi, M.Y.: The implication problem for functional and inclusion dependencies is undecidable. SIAM J. Comput. 14(3), 671–677 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  18. Codd, E.F.: A relational model of data for large shared data banks. Commun. ACM 13(6), 377–387 (1970)

    Article  MATH  Google Scholar 

  19. Codd, E.F.: Further normalization of the database relational model. In: Courant Computer Science Symposia 6: Data Base Systems, pp. 33–64. Prentice-Hall (1972)

  20. Delobel, C.: Normalisation and hierarchical dependencies in the relational data model. ACM Trans. Database Syst. 3(3), 201–222 (1978)

    Article  Google Scholar 

  21. Demetrovics, J., Molnar, A., Thalheim, B.: Graphical reasoning for sets of functional dependencies. In: Proceedings of the 23rd International Conference on Conceptual Modeling (ER). Lecture Notes in Computer Science, vol. 3288, pp. 166–179. Springer (2004)

  22. Demetrovics, J., Molnar, A., Thalheim, B.: Relationship design using spreadsheet reasoning for sets of functional dependencies. In: Proceedings of the 10th East-European Conference on Advances in Databases and Information Systems (AdBIS). Lecture Notes in Computer Science, vol. 4152, pp. 108–123. Springer (2006)

  23. Fagin, R.: Multivalued dependencies and a new normal form for relational databases. ACM Trans. Database Syst. 2(3), 262–278 (1977)

    Article  MathSciNet  Google Scholar 

  24. Ferrarotti, F., Hartmann, S., Link, S.: On the role of the complementation rule for data dependencies over incomplete relations. In: Proceedings of the 17th International Workshop on Logic, Language, Information and Computation (WoLLIC). Lecture Notes in Computer Science, vol. 6188, pp. 136–147. Springer (2010)

  25. Fischer, P.C., Gucht, D.V.: Weak multivalued dependencies. In: Proceedings of the Third ACM SIGACT-SIGMOD Symposium on Principles of Database Systems (PoDS), pp. 266–274 (1984)

  26. Galil, Z.: An almost linear-time algorithm for computing a dependency basis in a relational database. J. ACM 29(1), 96–102 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  27. Graetzer, G.: General Lattice Theory. Birkhauser, Cambridge, MA (1998)

    MATH  Google Scholar 

  28. Hara, C., Davidson, S.: Reasoning about nested functional dependencies. In: Proceedings of the 18th SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pp. 91–100. ACM (1999)

  29. Hartmann, S., Köhler, H., Trinh, T.: On the existence of Armstrong data trees for XML functional dependencies. In: Proceedings of the 6th International Symposium on Foundations of Information and Knowledge Systems (FoIKS). Lecture Notes in Computer Science, vol. 5956, pp. 94–113. Springer (2010)

  30. Hartmann, S., Link, S.: On a problem of Fagin concerning multivalued dependencies in relational databases. Theor. Comput. Sci. 353(1–3), 53–62 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hartmann, S., Link, S.: Characterising nested database dependencies by fragments of propositional logic. Ann. Pure Appl. Logic 152(1–3), 84–106 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hartmann, S., Link, S.: Efficient reasoning about a robust XML key fragment. ACM Trans. Database Syst. 34(2), Article 10 (2009)

    Google Scholar 

  33. Hartmann, S., Link, S.: On inferences of weak multivalued dependencies. Fundam. Inform. 92 (1–2), 83–102 (2009)

    MathSciNet  MATH  Google Scholar 

  34. Hartmann, S., Link, S.: Numerical constraints on XML data. Inf. Comput. 208(5), 521–544 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hartmann, S., Link, S.: When data dependencies over SQL tables meet the logics of paradox and S-3. In: Proceedings of the Twenty-Ninth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PoDS), pp. 317–326 (2010)

  36. Hartmann, S., Link, S., Köhler, H.: Full hierarchical dependencies in fixed and undetermined universes. Ann. Math. Artif. Intell. 50(1–2), 195–226 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hartmann, S., Link, S., Schewe, K.-D.: Functional and multivalued dependencies in nested databases generated by record and list constructor. Ann. Math. Artif. Intell. 46(1–2), 114–164 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hartmann, S., Link, S., Trinh, T.: Solving the implication problem for XML functional dependencies with properties. In: Proceedings of the 17th International Workshop on Logic, Language, Information and Computation (WoLLIC), Lecture Notes in Computer Science, vol. 6188, pp. 161–175. Springer (2010)

  39. Herrmann, C.: On the undecidability of implications between embedded multivalued database dependencies. Inf. Comput. 122(2), 221–235 (1995)

    Article  MATH  Google Scholar 

  40. Herrmann, C.: Corrigendum to “On the undecidability of implications between embedded multivalued database dependencies”. Inf. Comput. 204(12), 1847–1851 (2006)

    Article  MATH  Google Scholar 

  41. Klug, A.C., Price, R.: Determining view dependencies using tableaux. ACM Trans. Database Syst. 7(3), 361–380 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  42. Levene, M., Loizou, G.: Axiomatisation of functional dependencies in incomplete relations. Theor. Comput. Sci. 206(1–2), 283–300 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  43. Levene, M., Vincent, M.: Justification for inclusion dependency normal form. IEEE Trans. Knowl. Data Eng. 12(2), 281–291 (2000)

    Article  Google Scholar 

  44. Lien, Y.E.: On the equivalence of data models. J. ACM 29(2), 333–363 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  45. Link, S.: On multivalued dependencies in fixed and undetermined universes. In: Proceedings of the Fourth International Symposium on Foundations of Information and Knowledge Systems (FoIKS). Lecture Notes in Computer Science, vol. 3861, pp. 257–276 (2006)

  46. Link, S.: Charting the completeness frontier of inference systems for multivalued dependencies. Acta Inf. 45(7–8), 565–591 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. Link, S.: On the implication of multivalued dependencies in partial database relations. Int. J. Found. Comput. Sci. 19(3), 691–715 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  48. Maier, D., Sagiv, Y., Yannakakis, M.: On the complexity of testing implications of functional and join dependencies. J. ACM 28(4), 680–695 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  49. Mendelzon, A.: On axiomatising multivalued dependencies in relational databases. J. ACM 26(1), 37–44 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  50. Niepert, M., Gucht, D.V., Gyssens, M.: Logical and algorithmic properties of stable conditional independence. Int. J. Approx. Reason. 51(5), 531–543 (2010)

    Article  MATH  Google Scholar 

  51. Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA (1988)

    Google Scholar 

  52. Petrov, S.V.: Finite axiomatisation of languages for representation of system properties. Inf. Sci. 47(3), 339–372 (1989)

    Article  MATH  Google Scholar 

  53. Saxton, L.V., Tang, X.: Tree multivalued dependencies for XML datasets. In: Proceedings of the 5th International Conference on Advances in Web-Age Information Management (WAIM). Lecture Notes in Computer Science, vol. 3129, pp. 357–367. Springer (2004)

  54. Tari, Z., Stokes, J., Spaccapietra, S.: Object normal forms and dependency constraints for object-oriented schemata. ACM Trans. Database Syst. 22, 513–569 (1997)

    Article  Google Scholar 

  55. Thalheim, B.: A compelte axiomatization for full join dependencies in relations. Bull. EATCS 24, 109–114 (1984)

    Google Scholar 

  56. Thalheim, B.: Dependencies in Relational Databases. Teubner-Verlag (1991)

  57. Thalheim, B.: Entity-Relationship Modeling: Foundations of Database Technology. Springer (2000)

  58. Thalheim, B.: Conceptual treatment of multivalued dependencies. In: Proceedings of the 22nd International Conference on Conceptual Modeling (ER). Lecture Notes in Computer Science, vol. 2813, pp. 363–375. Springer (2003)

  59. Trinh, T.: Using transversals for discovering XML functional dependencies. In: Proceedings of the 5th International Symposium on Foundations of Information and Knowledge Systems (FoIKS). Lecture Notes in Computer Science, vol. 4932 pp. 199–218. Springer (2008)

  60. Vincent, M., Liu, J.: Multivalued dependencies in XML. In: Proceedings of the 20th British National Conference on Databases. Lecture Notes in Computer Science, vol. 2712, pp. 4–18. Springer (2003)

  61. Vincent, M.W., Liu, J., Liu, C.: Strong functional dependencies and their application to normal forms in XML. ACM Trans. Database Syst. 29(3), 445–462 (2004)

    Article  Google Scholar 

  62. Weddell, G.: Reasoning about functional dependencies generalized for semantic data models. ACM Trans. Database Syst. 17(1), 32–64 (1992)

    Article  MathSciNet  Google Scholar 

  63. Wijsen, J.: Temporal FDs on complex objects. ACM Trans. Database Syst. 24(1), 127–176 (1999)

    Article  Google Scholar 

  64. Zaniolo, C., Melkanoff, M.A.: On the design of relational database schemata. ACM Trans. Database Syst. 6(1), 1–47 (1981)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Fakultät für Informatik, Technische Universität Dortmund, 44221, Dortmund, Germany

    Joachim Biskup

  2. Department of Computer Science, The University of Auckland, Auckland, New Zealand

    Sebastian Link

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  1. Joachim Biskup
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Correspondence to Sebastian Link.

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This research is supported by the Marsden fund council from Government funding, administered by the Royal Society of New Zealand.

A preliminary version of this paper was presented at FoIKS 2008, Pisa, Italy.

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Biskup, J., Link, S. Appropriate inferences of data dependencies in relational databases. Ann Math Artif Intell 63, 213–255 (2011). https://doi.org/10.1007/s10472-012-9275-0

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