On a problem of Fagin concerning multivalued dependencies in relational databases

Multivalued dependencies (MVDs) are an important class of relational constraints that is fundamental to relational database design. Reflexivity axiom, complementation rule, and pseudo-transitivity rule form a minimal set of inference rules for the implication of MVDs. The complementation rule plays a distinctive role as it takes into account the underlying relation schema R which the MVDs are defined on. The R-axiom 0 ↠--R is much weaker than the complementation rule, but is sufficient to form a minimal set of inference rules together with augmentation and pseudo-difference rule. Fagin has asked whether it is possible to reduce the power of the complementation rule and drop the augmentation rule at the same time and still obtain a complete set. It was argued that there is a trade-off between complementation rule and augmentation rule, and one can only dispense with one of these rules at the same time. It is shown in this paper that an affirmative answer to Fagin's problem can nevertheless be achieved. In fact, it is proven that R-axiom together with a weaker form of the reflexivity axiom, pseudo-transitivity rule and exactly one of union, intersection or difference rule form such desirable minimal sets. The positive solution to this problem gives further insight into the difference between the notions of functional and multivalued dependencies.

[1]  Klaus-Dieter Schewe,et al.  Erratum to "Axiomatisations of functional dependencies in the presence of records, lists, sets and multisets" , 2006, Theor. Comput. Sci..

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

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

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

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

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

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

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

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

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

[11]  Paris C. Kanellakis,et al.  Elements of Relational Database Theory , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

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

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

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

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

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

[17]  Klaus-Dieter Schewe,et al.  Axiomatisations of functional dependencies in the presence of records, lists, sets and multisets , 2006, Theor. Comput. Sci..

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

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

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

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

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

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

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

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

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

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

[29]  Catriel Beeri,et al.  On the menbership problem for functional and multivalued dependencies in relational databases , 1980, TODS.

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

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

[32]  Graeme C. Simsion Data modeling essentials - analysis, design, and innovation , 1993, VNR computer library.