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Constructing Armstrong tables for general cardinality constraints and not-null constraints

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Abstract

Integrity constraints capture relevant requirements of an application that should be satisfied by every state of the database. The theory of integrity constraints is largely a theory over relations. To make data processing more efficient, SQL permits database states to be partial bags that can accommodate incomplete and duplicate information. Integrity constraints, however, interact differently on partial bags than on the idealized special case of relations. In this current paper, we study the implication problem of the combined class of general cardinality constraints and not-null constraints on partial bags. We investigate structural properties of Armstrong tables for general cardinality constraints and not-null constraints, and prove exact conditions for their existence. For the fragment of general max-cardinality constraints, unary min-cardinality constraints and not-null constraints we show that the effort for constructing Armstrong tables is precisely exponential. For the same fragment we provide an axiomatic characterization of the implication problem. The major tool for establishing our results is the Hajnal and Szemerédi theorem on the equitable colorings of graphs.

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Authors and Affiliations

  1. Department of Informatics, Clausthal University of Technology, Clausthal-Zellerfeld, Germany

    Sven Hartmann

  2. School of Engineering and Advanced Technology, Massey University, Wellington, New Zealand

    Henning Köhler

  3. Department of Mathematics and Computer Science, University of Wisconsin, Superior, USA

    Uwe Leck

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

    Sebastian Link

  5. Department of Computer Science, University of Kiel, Kiel, Germany

    Bernhard Thalheim

  6. Institute of Environmental Science and Research, Upper Hutt, New Zealand

    Jing Wang

Authors
  1. Sven Hartmann
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  2. Henning Köhler
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  3. Uwe Leck
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  4. Bernhard Thalheim
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  5. Jing Wang
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Correspondence to Sven Hartmann.

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Hartmann, S., Köhler, H., Leck, U. et al. Constructing Armstrong tables for general cardinality constraints and not-null constraints. Ann Math Artif Intell 73, 139–165 (2015). https://doi.org/10.1007/s10472-014-9423-9

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  • DOI: https://doi.org/10.1007/s10472-014-9423-9

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