Abstract
We introduce the concepts of dependence and independence in a very general framework. We use a concept of rank to study dependence and independence. By means of the rank we identify (total) dependence with inability to create more diversity, and (total) 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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- 1.
Since set union is commutative, it also follows that if \(\Vert xyz\Vert = \Vert x\Vert + \Vert yz\Vert \) then \(\Vert xz\Vert = \Vert x\Vert + \Vert z\Vert \).
- 2.
Equivalently, it is possible to define matroids in terms of its independent sets (that is, the x such that \(r(x) = |x|\)), in terms of circuits (maximal independent sets), in terms of bases (minimal non-independent sets), or closure operations. All these definitions are equivalent. We refer the reader to [11] for more details.
- 3.
In general, in Dependence and Independence Logic teams do not necessarily have to be finite, but we will focus on the finite case in this example.
- 4.
In this work, \(\log \) will always represent the base-2 logarithm.
- 5.
Nothing in this example hinges on A being the same for all \(v \in M\), but we will assume so for simplicity.
- 6.
More precisely, this theorem shows that \(H(xy)-H(x) = -\sum _{m} P(x = m) \sum _{m'} P(y=m'|x=m)\log P(y=m'|x=m)\), and the right hand side is straightforwardly seen to be non-negative.
- 7.
Strictly speaking, this theorem states that \(H(x) - H(x|y) \ge 0\), but if we consider the above inequality with respect to distributions already conditioned on z the result follows at once.
- 8.
If one is uninterested in independence statements \(x \ \bot \ y\) in which x and y overlap, this axiom can be removed. Our proof of Theorem 2 then reduces essentially to the proof in [4].
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Acknowledgments
We thank the reviewers for a number of helpful comments and suggestions. The research of the second author was partially supported by grant 322795 of the Academy of Finland, and a grant of the Faculty of Science of the University of Helsinki.
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Galliani, P., Väänänen, J. (2020). Diversity, Dependence and Independence. In: Herzig, A., Kontinen, J. (eds) Foundations of Information and Knowledge Systems. FoIKS 2020. Lecture Notes in Computer Science(), vol 12012. Springer, Cham. https://doi.org/10.1007/978-3-030-39951-1_7
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