Chapter

Foundations of Software Technology and Theoretical Computer Science

Volume 206 of the series Lecture Notes in Computer Science pp 303-328

Date:

The implication problem for functional and multivalued dependencies : An algebraic approach

  • V. S. LakshmananAffiliated withSchool of Automation, Indian Institute of Science
  • , C. E. Veni MadhavanAffiliated withSchool of Automation, Indian Institute of Science

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Abstract

Computation of the dependency basis is the fundamental step in solving the implication problem for MVDs in relational database theory. We examine this problem from an algebraic perspective. We introduce the notion of the inference basis of a set M of MVDs and show that it contains the maximum information about the logical consequences of M. We propose the notion of an MVD-lattice and develop an algebraic characterization of the inference basis using simple notions from lattice theory. We also establish several properties of MVD-lattices related to the implication problem. Founded on our characterization, we synthesize efficient algorithms for (a) computing the inference basis of a given set M of MVDs; (b) computing the dependency basis of a given attribute set w.r.t. M; and (c) solving the implication problem for MVDs. Finally, we show that our results naturally extend to incorporate FDs also in a way that enables the solution of the implication problem for both FDs and MVDs put together.