Abstract
The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a similarity space and concepts are represented by convex regions in this space. After pointing out a problem with the convexity requirement, we propose a formalization of conceptual spaces based on fuzzy star-shaped sets. Our formalization uses a parametric definition of concepts and extends the original framework by adding means to represent correlations between different domains in a geometric way. Moreover, we define various operations for our formalization, both for creating new concepts from old ones and for measuring relations between concepts. We present an illustrative toy-example and sketch a research project on concept formation that is based on both our formalization and its implementation.
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- 1.
- 2.
Please note that this is a very simplified artificial example to illustrate our main point.
- 3.
For instance, there is an obvious correlation between a banana’s color and its taste. If you replace the “age” dimension with “hue” and the “height” dimension with “sweetness” in Fig. 3.1, you will observe similar encoding problems for the “banana” concept as for the “child” concept.
- 4.
Please note that although the sketched sets are still convex under the Euclidean metric, they are star-shaped but not convex under the Manhattan metric.
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All of the following definitions and propositions hold for any number of dimensions.
- 6.
We will drop the modifier “axis-parallel” from now on.
- 7.
Note that if the two cores are defined on completely different domains (i.e., \(\Delta _{S_1} \cap \Delta _{S_2} = \emptyset \)), then their central regions intersect (i.e., P 1 ∩ P 2 ≠ ∅), because we can find at least one point in the overall space that belongs to both P 1 and P 2.
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In some cases, the normalization constraint of the resulting domain weights might be violated. We can enforce this constraint by manually normalizing the weights afterwards.
- 9.
A strict inequality in the definition of \(C^{\left (+\right )}\) or \(C^{\left (-\right )}\) would not yield a cuboid.
- 10.
Note that the intersection of two overlapping fuzzified cuboids is again a fuzzified cuboid.
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Note that ellipses under d M have the form of stretched diamonds.
- 12.
One could also say that the fuzzified cuboids \(\widetilde {C}_i\) are sub-concepts of \(\widetilde {S}\), because \(\widetilde {C}_i \subseteq \widetilde {S}\).
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- 14.
- 15.
The source code of an earlier and more limited version of their system can be found here: http://www.di.unito.it/~lieto/cc_classifier.html.
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Bechberger, L., Kühnberger, KU. (2019). Formalized Conceptual Spaces with a Geometric Representation of Correlations. In: Kaipainen, M., Zenker, F., Hautamäki, A., Gärdenfors, P. (eds) Conceptual Spaces: Elaborations and Applications. Synthese Library, vol 405. Springer, Cham. https://doi.org/10.1007/978-3-030-12800-5_3
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