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Diversity, dependence and independence

Annals of Mathematics and Artificial Intelligence (2021)Cite this article

Abstract

We propose a very general, unifying framework for the concepts of dependence and independence. For this purpose, we introduce the notion of diversity rank. By means of this diversity rank we identify total determination with the inability to create more diversity, and independence with the presence of maximum diversity. We show that our theory of dependence and independence covers a variety of dependence concepts, for example the seemingly unrelated concepts of linear dependence in algebra and dependence of variables in logic.

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Acknowledgements

We thank the reviewers for a number of helpful comments and suggestions. We are particularly grateful to Reviewer 2 for their careful reading of the manuscript and for providing a clearer proof for Theorem 3. The second author would like to thank the Academy of Finland, grant no: 322795. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 101020762).

Funding

Open access funding provided by Libera Università di Bolzano within the CRUI-CARE Agreement.

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Author notes

    Affiliations

    1. Faculty of Computer Science, Free University of Bozen-Bolzano, piazza Domenicani, 3, Bozen-Bolzano, 39100, Italy

      Pietro Galliani

    2. Department of Mathematics and Statistics, University of Helsinki, Pietari Kalmin katu 5, Helsinki, PL 68 FIN-00014, Finland

      Jouko Väänänen

    3. FNWI, ILLC, Universiteit van Amsterdam, P.O. Box 94242, Amsterdam, 1090 GE, The Netherlands

      Jouko Väänänen

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    1. Pietro Galliani
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    2. Jouko Väänänen
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    Correspondence to Pietro Galliani.

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    Cite this article

    Galliani, P., Väänänen, J. Diversity, dependence and independence. Ann Math Artif Intell (2021). https://doi.org/10.1007/s10472-021-09778-8

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    • DOI: https://doi.org/10.1007/s10472-021-09778-8

    Keywords

    • Dependence logic
    • Matroids
    • Independence
    • Team semantics

    Mathematics Subject Classification (2010)

    • 03B60