On the Logical Implication of Multivalued Dependencies with Null Values

The implication of multivalued dependencies (MVDs) in relational databases has originally been defined in the context of some fixed finite universe (Fagin 1977, Zaniolo 1976). While axiomatisability, implication problem and many design problems have been intensely studied with respect to this notion, almost no research has been devoted towards the alternative notion of implication in which the underlying universe of attributes is left undetermined (Biskup 1980). A milestone in the advancement of database systems was the permission of null values in databases. In particular, many achievements on MVDs have been extended to encompass incomplete information. Multivalued dependencies with null values (NMVDs) were defined and axiomatised in (Lien 1982). The definition of NMVDs is again based on a fixed underlying universe of attributes, and any complete set of inference rules requires therefore some version of the complementation rule. In this paper we show that the axiomatisation in (Lien 1982) does not reflect the fact that the complementation rule is merely a means to achieve database normalisation. Moreover, we provide an alternative axiomatisation for NMVDs that does reflect this property. We also suggest an alternative notion for the implication of NMVDs in which the underlying universe is left undetermined, and propose several sound and complete sets of inference rules for this notion. Moreover, a correspondence between (minimal) axiomatisations in fixed universes that do reflect the property of complementation and (minimal) axiomatisations in undetermined universes is shown.

[1]  A BernsteinPhilip,et al.  Computational problems related to the design of normal form relational schemas , 1979 .

[2]  John Grant,et al.  Null Values in a Relational Data Base , 1977, Inf. Process. Lett..

[3]  Joachim Biskup.  Inferences of Multivalued Dependencies in Fixed and Undetermined Universes , 1980, Theor. Comput. Sci..

[4]  Paolo Atzeni,et al.  Functional Dependencies and Constraints on Null Values in Database Relations , 1986, Inf. Control..

[5]  Mark Levene,et al.  Semantics for null extended nested relations , 1993, TODS.

[6]  Yatsuka Nakamura,et al.  Armstrong's Axioms , 2007 .

[7]  E. F. CODD,et al.  A relational model of data for large shared data banks , 1970, CACM.

[8]  Joachim Biskup.  On the complementation rule for multivalued dependencies in database relations , 2004, Acta Informatica.

[9]  Y. Edmund Lien,et al.  On the Equivalence of Database Models , 1982, JACM.

[10]  Chengfei Liu,et al.  A Redundancy Free 4NF for XML , 2003, Xsym.

[11]  Catriel Beeri,et al.  A complete axiomatization for functional and multivalued dependencies in database relations , 1977, SIGMOD '77.

[12]  Sebastian Link.  On Multivalued Dependencies in Fixed and Undetermined Universes , 2006, FoIKS.

[13]  Carmem S. Hara,et al.  Reasoning about nested functional dependencies , 1999, PODS '99.

[14]  Bernhard Thalheim,et al.  Dependencies in relational databases , 1991, Teubner-Texte zur Mathematik.

[15]  Bernhard Thalheim.  Conceptual Treatment of Multivalued Dependencies , 2003, ER.

[16]  E. F. Codd,et al.  Extending the database relational model to capture more meaning , 1979, ACM Trans. Database Syst..

[17]  Claude Delobel,et al.  Normalization and hierarchical dependencies in the relational data model , 1978, TODS.

[18]  Carlo Zaniolo,et al.  Database relations with null values , 1982, J. Comput. Syst. Sci..

[19]  Mark Levene,et al.  Axiomatisation of Functional Dependencies in Incomplete Relations , 1998, Theor. Comput. Sci..

[20]  Sven Hartmann,et al.  Multi-valued dependencies in the presence of lists , 2004, PODS '04.

[21]  Carlo Zaniolo,et al.  Analysis and design of relational schemata for database systems. , 1976 .

[22]  Gösta Grahne,et al.  Dependency Satisfaction in Databases with Incomplete Information , 1984, VLDB.

[23]  Alberto O. Mendelzon.  On Axiomatizing Multivalued Dependencies in Relational Databases , 1979, JACM.

[24]  Jixue Liu,et al.  Multivalued Dependencies in XML , 2003, BNCOD.

[25]  Zahir Tari,et al.  Object normal forms and dependency constraints for object-oriented schemata , 1997, TODS.

[26]  Sven Hartmann,et al.  On a problem of Fagin concerning multivalued dependencies in relational databases , 2006, Theor. Comput. Sci..

[27]  Akifumi Makinouchi,et al.  A Consideration on Normal Form of Not-Necessarily-Normalized Relation in the Relational Data Model , 1977, VLDB.

[28]  Ronald Fagin,et al.  Multivalued dependencies and a new normal form for relational databases , 1977, TODS.

[29]  Marcelo Arenas,et al.  A normal form for XML documents , 2002, PODS '02.

[30]  Philip A. Bernstein,et al.  Computational problems related to the design of normal form relational schemas , 1979, TODS.

[31]  Georg Gottlob,et al.  Closed World Databases Opened Through Null Values , 1988, VLDB.

[32]  Philip A. Bernstein,et al.  What does Boyce-Codd Normal Form Do? , 1980, VLDB.

[33]  Jef Wijsen,et al.  Temporal FDs on complex objects , 1999, TODS.

[34]  Ronald Fagin,et al.  The theory of data dependencies - a survey , 1984 .

[35]  E. F. Codd,et al.  Understanding relations , 1973, SGMD.

[36]  Grant E. Weddell,et al.  Reasoning about functional dependencies generalized for semantic data models , 1992, TODS.

[37]  Michael Johnson,et al.  Three approaches to Partiality in the Sketch Data Model , 2003, CATS.

[38]  Philip A. Bernstein,et al.  Synthesizing third normal form relations from functional dependencies , 1976, TODS.

[39]  Patrick C. Fischer,et al.  Interactions between Dependencies and Nested Relational Structures , 1985, J. Comput. Syst. Sci..

[40]  Zvi Galil,et al.  An Almost Linear-Time Algorithm for Computing a Dependency Basis in a Relational Database , 1982, JACM.