A Thorough Formalization of Conceptual Spaces

  • Lucas BechbergerEmail author
  • Kai-Uwe Kühnberger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10505)


The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a high-dimensional 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 computationally efficient operations on concepts (intersection, union, and projection onto a subspace) and show that these operations can support both learning and reasoning processes.


Conceptual spaces Star-shaped sets Fuzzy sets 


  1. 1.
    Adams, B., Raubal, M.: A metric conceptual space algebra. In: Hornsby, K.S., Claramunt, C., Denis, M., Ligozat, G. (eds.) COSIT 2009. LNCS, vol. 5756, pp. 51–68. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-03832-7_4 CrossRefGoogle Scholar
  2. 2.
    Adams, B., Raubal, M.: Conceptual Space Markup Language (CSML): towards the cognitive semantic web. In: IEEE International Conference on Semantic Computing, September 2009Google Scholar
  3. 3.
    Aggarwal, C.C., Hinneburg, A., Keim, D.A.: On the surprising behavior of distance metrics in high dimensional space. In: Van den Bussche, J., Vianu, V. (eds.) ICDT 2001. LNCS, vol. 1973, pp. 420–434. Springer, Heidelberg (2001). doi: 10.1007/3-540-44503-X_27 CrossRefGoogle Scholar
  4. 4.
    Aisbett, J., Gibbon, G.: A general formulation of conceptual spaces as a meso level representation. Artif. Intell. 133(1–2), 189–232 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Attneave, F.: Dimensions of similarity. Am. J. Psychol. 63(4), 516 (1950)CrossRefGoogle Scholar
  6. 6.
    Bělohlávek, R., Klir, G.J.: Concepts and Fuzzy Logic. MIT Press, Cambridge (2011)zbMATHGoogle Scholar
  7. 7.
    Billman, D., Knutson, J.: Unsupervised concept learning and value systematicitiy: a complex whole aids learning the parts. J. Exp. Psychol. Learn. Mem. Cogn. 22(2), 458–475 (1996)CrossRefGoogle Scholar
  8. 8.
    Chella, A., Dindo, H., Infantino, I.: Anchoring by imitation learning in conceptual spaces. In: Bandini, S., Manzoni, S. (eds.) AI*IA 2005. LNCS, vol. 3673, pp. 495–506. Springer, Heidelberg (2005). doi: 10.1007/11558590_50 CrossRefGoogle Scholar
  9. 9.
    Chella, A., Frixione, M., Gaglio, S.: Conceptual spaces for computer vision representations. Artif. Intell. Rev. 16(2), 137–152 (2001)CrossRefzbMATHGoogle Scholar
  10. 10.
    Chella, A., Frixione, M., Gaglio, S.: Anchoring symbols to conceptual spaces: the case of dynamic scenarios. Robot. Auton. Syst. 43(2–3), 175–188 (2003)CrossRefGoogle Scholar
  11. 11.
    Derrac, J., Schockaert, S.: Inducing semantic relations from conceptual spaces: a data-driven approach to plausible reasoning. Artif. Intell. 228, 66–94 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dietze, S., Domingue, J.: Exploiting conceptual spaces for ontology integration. In: Data Integration Through Semantic Technology (DIST 2008) Workshop at 3rd Asian Semantic Web Conference (ASWC 2008) (2008)Google Scholar
  13. 13.
    Douven, I., Decock, L., Dietz, R., Égré, P.: Vagueness: a conceptual spaces approach. J. Philos. Logic 42(1), 137–160 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fiorini, S.R., Gärdenfors, P., Abel, M.: Representing part-whole relations in conceptual spaces. Cogn. Process. 15(2), 127–142 (2013)CrossRefGoogle Scholar
  15. 15.
    Gärdenfors, P.: Conceptual Spaces: The Geometry of Thought. MIT Press, Cambridge (2000)Google Scholar
  16. 16.
    Gärdenfors, P.: The Geometry of Meaning: Semantics Based on Conceptual Spaces. MIT Press (2014)Google Scholar
  17. 17.
    Harnad, S.: The symbol grounding problem. Phys. D Nonlinear Phenom. 42(1–3), 335–346 (1990)CrossRefGoogle Scholar
  18. 18.
    Hernández-Conde, J.V.: A case against convexity in conceptual spaces. Synthese 193, 1–27 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lewis, M., Lawry, J.: Hierarchical conceptual spaces for concept combination. Artif. Intell. 237, 204–227 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Medin, D.L., Shoben, E.J.: Context and structure in conceptual combination. Cogn. Psychol. 20(2), 158–190 (1988)CrossRefGoogle Scholar
  21. 21.
    Murphy, G.: The Big Book of Concepts. MIT Press, Cambridge (2002)Google Scholar
  22. 22.
    Osherson, D.N., Smith, E.E.: Gradedness and conceptual combination. Cognition 12(3), 299–318 (1982)CrossRefGoogle Scholar
  23. 23.
    Raubal, M.: Formalizing conceptual spaces. In: Third International Conference on Formal Ontology in Information Systems, vol. 114, pp. 153–164 (2004)Google Scholar
  24. 24.
    Rickard, J.T.: A concept geometry for conceptual spaces. Fuzzy Optim. Decis. Making 5(4), 311–329 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rickard, J.T., Aisbett, J., Gibbon G.: Knowledge representation and reasoning in conceptual spaces. In: 2007 IEEE Symposium on Foundations of Computational Intelligence, April 2007Google Scholar
  26. 26.
    Ruspini, E.H.: On the semantics of fuzzy logic. Int. J. Approx. Reason. 5(1), 45–88 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schockaert, S., Prade, H.: Interpolation and extrapolation in conceptual spaces: a case study in the music domain. In: Rudolph, S., Gutierrez, C. (eds.) RR 2011. LNCS, vol. 6902, pp. 217–231. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-23580-1_16 CrossRefGoogle Scholar
  28. 28.
    Shepard, R.N.: Attention and the metric structure of the stimulus space. J. Math. Psychol. 1(1), 54–87 (1964)CrossRefGoogle Scholar
  29. 29.
    Shepard, R.N.: Toward a universal law of generalization for psychological science. Science 237(4820), 1317–1323 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Smith, C.R.: A characterization of star-shaped sets. Am. Math. Mon. 75(4), 386 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Warglien, M., Gärdenfors, P., Westera, M.: Event structure, conceptual spaces and the semantics of verbs. Theor. Linguist. 38(3–4), 159–193 (2012)Google Scholar
  32. 32.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)CrossRefzbMATHGoogle Scholar
  33. 33.
    Zadeh, L.A.: A note on prototype theory and fuzzy sets. Cognition 12(3), 291–297 (1982)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Cognitive ScienceOsnabrück UniversityOsnabrückGermany

Personalised recommendations