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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Notes
- 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.
- 5.
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.
- 8.
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.
- 11.
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 \(\widetilde {C}_i \subseteq \widetilde {S}\).
- 13.
- 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.
References
Adams, B., & Raubal, M. (2009a). A metric conceptual space algebra. In 9th International Conference on Spatial Information Theory (pp. 51–68). Berlin/Heidelberg: Springer.
Adams, B., & Raubal, M. (2009b). Conceptual space markup language (CSML): Towards the cognitive semantic web. In 2009 IEEE International Conference on Semantic Computing.
Aggarwal, C. C., Hinneburg, A., & Keim, D. A. (2001). On the surprising behavior of distance metrics in high dimensional space. In 8th International Conference on Database Theory (pp. 420–434). Berlin/Heidelberg: Springer.
Aisbett, J., & Gibbon, G. (2001). A general formulation of conceptual spaces as a meso level representation. Artificial Intelligence, 133(1–2), 189–232.
Attneave, F. (1950). Dimensions of similarity. The American Journal of Psychology, 63(4), 516–556.
Bechberger, L. (2017). The size of a hyperball in a conceptual space. https://arxiv.org/abs/1708.05263.
Bechberger, L. (2018). lbechberger/ConceptualSpaces: Version 1.1.0. https://doi.org/10.5281/zenodo.1143978.
Bechberger, L., & Kühnberger, K.-U. (2017a). A comprehensive implementation of conceptual spaces. In 5th International Workshop on Artificial Intelligence and Cognition.
Bechberger, L., & Kühnberger, K.-U. (2017b). A thorough formalization of conceptual spaces. In G. Kern-Isberner, J. Fürnkranz, & M. Thimm (Eds.), KI 2017: Advances in Artificial Intelligence: 40th Annual German Conference on AI, Proceedings, Dortmund, 25–29 Sept 2017 (pp. 58–71). Springer International Publishing.
Bechberger, L., & Kühnberger, K.-U. (2017c). Measuring relations between concepts in conceptual spaces. In M. Bramer & M. Petridis (Eds.), Artificial intelligence XXXIV: 37th SGAI International Conference on Artificial Intelligence, AI 2017, Proceedings, Cambridge, 12–14 Dec 2017 (pp. 87–100). Springer International Publishing.
Bechberger, L., & Kühnberger, K.-U. (2017d). Towards grounding conceptual spaces in neural representations. In 12th International Workshop on Neural-Symbolic Learning and Reasoning.
Bělohlávek, R., & Klir, G. J. (2011). Concepts and fuzzy logic. Cambridge: MIT Press.
Bengio, Y., Courville, A., & Vincent, P. (2013). Representation learning: A review and new perspectives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(8), 1798–1828.
Billman, D., & Knutson, J. (1996). Unsupervised concept learning and value systematicitiy: A complex whole aids learning the parts. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22(2), 458–475.
Bogart, K. P. (1989). Introductory combinatorics (2nd ed.). Philadelphia: Saunders College Publishing.
Bouchon-Meunier, B., Rifqi, M., & Bothorel, S. (1996). Towards general measures of comparison of objects. Fuzzy Sets and Systems, 84(2), 143–153.
Chella, A., Frixione, M., & Gaglio, S. (2001). Conceptual spaces for computer vision representations. Artificial Intelligence Review, 16(2), 137–152.
Chella, A., Frixione, M., & Gaglio, S. (2003). Anchoring symbols to conceptual spaces: The case of dynamic scenarios. Robotics and Autonomous Systems, 43(2–3), 175–188.
Chella, A., Dindo, H., & Infantino, I. (2005). Anchoring by imitation learning in conceptual spaces. In S. Bandini, & S. Manzoni (Eds.), AI*IA 2005: Advances in Artificial Intelligence (pp. 495–506). Berlin/New York: Springer.
Chen, X., Duan, Y., Houthooft, R., Schulman, J., Sutskever, I., & Abbeel, P. (2016). InfoGAN: Interpretable representation learning by information maximizing generative adversarial nets. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, & R. Garnett (Eds.), Advances in neural information processing systems (Vol. 29, pp. 2172–2180). Curran Associates, Inc. http://papers.nips.cc/paper/6399-infogan-interpretable-representation-learning-by-information-maximizing-generative-adversarial-nets.pdf
Derrac, J., & Schockaert, S. (2015). Inducing semantic relations from conceptual spaces: A data-driven approach to plausible reasoning. Artificial Intelligence, 228, 66–94.
Dietze, S., & Domingue, J. (2008). Exploiting conceptual spaces for ontology integration. In Data Integration Through Semantic Technology (DIST2008) Workshop at 3rd Asian Semantic Web Conference (ASWC 2008).
Douven, I., Decock, L., Dietz, R., & Égré, P. (2011). Vagueness: A conceptual spaces approach. Journal of Philosophical Logic, 42(1), 137–160.
Fiorini, S. R., Gärdenfors, P., & Abel, M. (2013). Representing part-whole relations in conceptual spaces. Cognitive Processing, 15(2), 127–142.
Gärdenfors, P. (2000). Conceptual spaces: The geometry of thought. Cambridge: MIT press.
Gärdenfors, P. (2014). The geometry of meaning: Semantics based on conceptual spaces. Cambridge: MIT Press.
Gennari, J. H., Langley, P., & Fisher, D. (1989). Models of incremental concept formation. Artificial Intelligence, 40(1–3), 11–61.
Harnad, S. (1990). The symbol grounding problem. Physica D: Nonlinear Phenomena, 42(1–3), 335–346.
Higgins, I., Matthey, L., Pal, A., Burgess, C., Glorot, X., Botvinick, M., Mohamed, S., & Lerchner, A. (2017). β-VAE: Learning basic visual concepts with a constrained variational framework. In 5th International Conference on Learning Representations.
Johannesson, M. (2001). The problem of combining integral and separable dimensions. Technical Report HS-IDA-TR-01-002, University of Skövde, School of Humanities and Informatics.
Kosko, B. (1992). Neural networks and fuzzy systems: A dynamical systems approach to machine intelligence. Englewood Cliffs: Prentice Hall.
Lewis, M., & Lawry, J. (2016). Hierarchical conceptual spaces for concept combination. Artificial Intelligence, 237, 204–227.
Lieto, A., Minieri, A., Piana, A., & Radicioni, D. P. (2015). A knowledge-based system for prototypical reasoning. Connection Science, 27(2), 137–152.
Lieto, A., Radicioni, D. P., & Rho, V. (2017). Dual PECCS: A cognitive system for conceptual representation and categorization. Journal of Experimental & Theoretical Artificial Intelligence, 29(2), 433–452.
Love, B. C., Medin, D. L., & Gureckis, T. M. (2004). SUSTAIN: A network model of category learning. Psychological Review, 111(2), 309–332.
Mas, M., Monserrat, M., Torrens, J., & Trillas, E. (2007). A survey on fuzzy implication functions. IEEE Transactions on Fuzzy Systems, 15(6):1107–1121.
Medin, D. L., & Shoben, E. J. (1988). Context and structure in conceptual combination. Cognitive Psychology, 20(2):158–190.
Murphy, G. (2002). The big book of concepts. Cambridge: MIT Press.
Osherson, D. N., & Smith, E. E. (1982). Gradedness and conceptual combination. Cognition, 12(3), 299–318.
Raubal, M. (2004). Formalizing conceptual spaces. In Third International Conference on Formal Ontology in Information Systems (pp. 153–164).
Rickard, J. T. (2006). A concept geometry for conceptual spaces. Fuzzy Optimization and Decision Making, 5(4), 311–329.
Rickard, J. T., Aisbett, J., & Gibbon, G. (2007). Knowledge representation and reasoning in conceptual spaces. In 2007 IEEE Symposium on Foundations of Computational Intelligence.
Ruspini, E. H. (1991). On the semantics of fuzzy logic. International Journal of Approximate Reasoning, 5(1), 45–88.
Schockaert, S., & Prade, H. (2011). Interpolation and extrapolation in conceptual spaces: A case study in the music domain. In 5th International Conference on Web Reasoning and Rule Systems (pp. 217–231). Springer Nature.
Shepard, R. N. (1964). Attention and the metric structure of the stimulus space. Journal of Mathematical Psychology, 1(1), 54–87.
Shepard, R. N. (1987). Toward a universal law of generalization for psychological science. Science, 237(4820), 1317–1323.
Smith, C. R. (1968). A characterization of star-shaped sets. The American Mathematical Monthly, 75(4), 386.
Wang, P. (2011). The assumptions on knowledge and resources in models of rationality. International Journal of Machine Consciousness, 3(1), 193–218.
Warglien, M., Gärdenfors, P., & Westera, M. (2012). Event structure, conceptual spaces and the semantics of verbs. Theoretical Linguistics, 38(3–4), 159–193.
Young, V. R. (1996). Fuzzy subsethood. Fuzzy Sets and Systems, 77(3), 371–384.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353.
Zadeh, L. A. (1982). A note on prototype theory and fuzzy sets. Cognition, 12(3), 291–297.
Zenker, F., & Gärdenfors, P. (Eds.). (2015). Applications of conceptual spaces. Cham: Springer Science + Business Media.
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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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