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Formalized Conceptual Spaces with a Geometric Representation of Correlations

  • Lucas BechbergerEmail author
  • Kai-Uwe Kühnberger
Chapter
Part of the Synthese Library book series (SYLI, volume 405)

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.

References

  1. Adams, B., & Raubal, M. (2009a). A metric conceptual space algebra. In 9th International Conference on Spatial Information Theory (pp. 51–68). Berlin/Heidelberg: Springer.Google Scholar
  2. Adams, B., & Raubal, M. (2009b). Conceptual space markup language (CSML): Towards the cognitive semantic web. In 2009 IEEE International Conference on Semantic Computing.Google Scholar
  3. 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.Google Scholar
  4. Aisbett, J., & Gibbon, G. (2001). A general formulation of conceptual spaces as a meso level representation. Artificial Intelligence, 133(1–2), 189–232.Google Scholar
  5. Attneave, F. (1950). Dimensions of similarity. The American Journal of Psychology, 63(4), 516–556.Google Scholar
  6. Bechberger, L. (2017). The size of a hyperball in a conceptual space. https://arxiv.org/abs/1708.05263.Google Scholar
  7. Bechberger, L. (2018). lbechberger/ConceptualSpaces: Version 1.1.0.  https://doi.org/10.5281/zenodo.1143978.
  8. Bechberger, L., & Kühnberger, K.-U. (2017a). A comprehensive implementation of conceptual spaces. In 5th International Workshop on Artificial Intelligence and Cognition.Google Scholar
  9. 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.Google Scholar
  10. 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.Google Scholar
  11. Bechberger, L., & Kühnberger, K.-U. (2017d). Towards grounding conceptual spaces in neural representations. In 12th International Workshop on Neural-Symbolic Learning and Reasoning.Google Scholar
  12. Bělohlávek, R., & Klir, G. J. (2011). Concepts and fuzzy logic. Cambridge: MIT Press.Google Scholar
  13. 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.Google Scholar
  14. 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.Google Scholar
  15. Bogart, K. P. (1989). Introductory combinatorics (2nd ed.). Philadelphia: Saunders College Publishing.Google Scholar
  16. Bouchon-Meunier, B., Rifqi, M., & Bothorel, S. (1996). Towards general measures of comparison of objects. Fuzzy Sets and Systems, 84(2), 143–153.Google Scholar
  17. Chella, A., Frixione, M., & Gaglio, S. (2001). Conceptual spaces for computer vision representations. Artificial Intelligence Review, 16(2), 137–152.Google Scholar
  18. 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.Google Scholar
  19. 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.Google Scholar
  20. 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 Google Scholar
  21. Derrac, J., & Schockaert, S. (2015). Inducing semantic relations from conceptual spaces: A data-driven approach to plausible reasoning. Artificial Intelligence, 228, 66–94.Google Scholar
  22. 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).Google Scholar
  23. Douven, I., Decock, L., Dietz, R., & Égré, P. (2011). Vagueness: A conceptual spaces approach. Journal of Philosophical Logic, 42(1), 137–160.Google Scholar
  24. Fiorini, S. R., Gärdenfors, P., & Abel, M. (2013). Representing part-whole relations in conceptual spaces. Cognitive Processing, 15(2), 127–142.Google Scholar
  25. Gärdenfors, P. (2000). Conceptual spaces: The geometry of thought. Cambridge: MIT press.Google Scholar
  26. Gärdenfors, P. (2014). The geometry of meaning: Semantics based on conceptual spaces. Cambridge: MIT Press.Google Scholar
  27. Gennari, J. H., Langley, P., & Fisher, D. (1989). Models of incremental concept formation. Artificial Intelligence, 40(1–3), 11–61.Google Scholar
  28. Harnad, S. (1990). The symbol grounding problem. Physica D: Nonlinear Phenomena, 42(1–3), 335–346.Google Scholar
  29. 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.Google Scholar
  30. 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.Google Scholar
  31. Kosko, B. (1992). Neural networks and fuzzy systems: A dynamical systems approach to machine intelligence. Englewood Cliffs: Prentice Hall.Google Scholar
  32. Lewis, M., & Lawry, J. (2016). Hierarchical conceptual spaces for concept combination. Artificial Intelligence, 237, 204–227.Google Scholar
  33. Lieto, A., Minieri, A., Piana, A., & Radicioni, D. P. (2015). A knowledge-based system for prototypical reasoning. Connection Science, 27(2), 137–152.Google Scholar
  34. 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.Google Scholar
  35. Love, B. C., Medin, D. L., & Gureckis, T. M. (2004). SUSTAIN: A network model of category learning. Psychological Review, 111(2), 309–332.Google Scholar
  36. Mas, M., Monserrat, M., Torrens, J., & Trillas, E. (2007). A survey on fuzzy implication functions. IEEE Transactions on Fuzzy Systems, 15(6):1107–1121.Google Scholar
  37. Medin, D. L., & Shoben, E. J. (1988). Context and structure in conceptual combination. Cognitive Psychology, 20(2):158–190.Google Scholar
  38. Murphy, G. (2002). The big book of concepts. Cambridge: MIT Press.Google Scholar
  39. Osherson, D. N., & Smith, E. E. (1982). Gradedness and conceptual combination. Cognition, 12(3), 299–318.Google Scholar
  40. Raubal, M. (2004). Formalizing conceptual spaces. In Third International Conference on Formal Ontology in Information Systems (pp. 153–164).Google Scholar
  41. Rickard, J. T. (2006). A concept geometry for conceptual spaces. Fuzzy Optimization and Decision Making, 5(4), 311–329.Google Scholar
  42. Rickard, J. T., Aisbett, J., & Gibbon, G. (2007). Knowledge representation and reasoning in conceptual spaces. In 2007 IEEE Symposium on Foundations of Computational Intelligence.Google Scholar
  43. Ruspini, E. H. (1991). On the semantics of fuzzy logic. International Journal of Approximate Reasoning, 5(1), 45–88.Google Scholar
  44. 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.Google Scholar
  45. Shepard, R. N. (1964). Attention and the metric structure of the stimulus space. Journal of Mathematical Psychology, 1(1), 54–87.Google Scholar
  46. Shepard, R. N. (1987). Toward a universal law of generalization for psychological science. Science, 237(4820), 1317–1323.Google Scholar
  47. Smith, C. R. (1968). A characterization of star-shaped sets. The American Mathematical Monthly, 75(4), 386.Google Scholar
  48. Wang, P. (2011). The assumptions on knowledge and resources in models of rationality. International Journal of Machine Consciousness, 3(1), 193–218.Google Scholar
  49. Warglien, M., Gärdenfors, P., & Westera, M. (2012). Event structure, conceptual spaces and the semantics of verbs. Theoretical Linguistics, 38(3–4), 159–193.Google Scholar
  50. Young, V. R. (1996). Fuzzy subsethood. Fuzzy Sets and Systems, 77(3), 371–384.Google Scholar
  51. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353.Google Scholar
  52. Zadeh, L. A. (1982). A note on prototype theory and fuzzy sets. Cognition, 12(3), 291–297.Google Scholar
  53. Zenker, F., & Gärdenfors, P. (Eds.). (2015). Applications of conceptual spaces. Cham: Springer Science + Business Media.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Cognitive ScienceOsnabrück UniversityOsnabrückGermany

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