Solving Proportional Analogies by E–Generalization

  • Stephan Weller
  • Ute Schmid
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4314)


We present an approach for solving proportional analogies of the form A : B :: C : D where a plausible outcome for D is computed. The core of the approach is E–Generalization. The generalization method is based on the extraction of the greatest common structure of the terms A, B and C and yields a mapping to compute every possible value for D with respect to some equational theory. This approach to analogical reasoning is formally sound and powerful and at the same time models crucial aspects of human reasoning, that is the guidance of mapping by shared roles and the use of re-representations based on a background theory. The focus of the paper is on the presentation of the approach. It is illustrated by an application for the letter string domain.


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  1. 1.
    Mitchell, M.: Analogy-Making as Perception: A Computer Model. MIT Press, Cambridge (1993)Google Scholar
  2. 2.
    French, R.M.: The computational modeling of analogy-making. Trends in Cognitive Sciences 6(5), 200–205 (2002)CrossRefGoogle Scholar
  3. 3.
    Evans, T.G.: A Program for the Solution of a Class of Gemetric-Analogy Intelligence-Test Questions. In: Minsky, M. (ed.) Semantic Information Processing, pp. 271–353. MIT Press, Cambridge (1968)Google Scholar
  4. 4.
    O’Hara, S.: A model of the redescription process in the context of geometric proportional analogy problems. In: Jantke, K.P. (ed.) AII 1992. LNCS, vol. 642, pp. 268–293. Springer, Heidelberg (1992)Google Scholar
  5. 5.
    Hofstadter, D.: The Fluid Analogies Research Gr.: Fluid Concepts and Creative Analogies. Basic Books, New York (1995)Google Scholar
  6. 6.
    Heinz, B.: Anti-Unifikation modulo Gleichungstheorie und deren Anwendung zur Lemmagenerierung. Technical report, GMD - Forschungszentrum Informationstechnick GmbH (1996)Google Scholar
  7. 7.
    Burghardt, J.: E-generalization using grammars. Artificial Intelligence Journal 165(1), 1–35 (2005)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Plotkin, G.D.: A note on inductive generalization. In: Machine Intelligence, vol. 5, pp. 153–163. Edinburgh University Press, Edinburgh (1970)Google Scholar
  9. 9.
    Reynolds, J.: Transformational Systems and the Algebraic Structure of Atomic Formulas. In: Machine Intelligence, vol. 5, Edinburgh University Press, Edinburgh (1970)Google Scholar
  10. 10.
    Falkenheimer, B., Forbus, K.D., Gentner, D.: The structure-mapping engine: Algorithm and examples. Artificial Intelligence 41, 1–63 (1989)CrossRefGoogle Scholar
  11. 11.
    Schmid, U., Gust, H.: An algebraic framework for solving proportional and predictive analogies. In: Schmalhofer, F., Young, R., Katz, G. (eds.) Proceedings of the First European Conference on Cognitive Science (EuroCogSci03), pp. 295–300. Lawrence Erlbaum, Mahwah (2003)Google Scholar
  12. 12.
    Brainerd, W.S.: The minimalization of tree automata. Information and Control 13, 484–491 (1968)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Thatcher, J.W., Wright, J.B.: Generalized finite automata theory with an application to a decision problem of second–order logic. Mathematical Systems Theory 2(1) (1968)Google Scholar
  14. 14.
    Comon, H., et al.: Tree automata techniques and applications (1997), Available on, release October, 1rst 2002
  15. 15.
    von Thaden, M., Weller, S.: Lösen von Intelligenztestaufgaben mit E-Generalisierung (Solving intelligence tasks by E-Generalization). In: Tagungsband der Informatiktage 2003, pp. 84–87. Gesellschaft für Informatik e.V. (2003)Google Scholar
  16. 16.
    Emmelmann, H.: Code Selection by Regularly Controlled Term Rewriting. In: Proc. of Int. Workshop on Code Generation (1991)Google Scholar
  17. 17.
    Weller, S.: Solving Proportional Analogies by Application of Anti-Unification modulo Equational Theory. Bachelor’s Thesis, unpublished (2005), Available on
  18. 18.
    Yan, J., Gentner, D.: A theory of rerepresentation in analogical matching. In: Proc. of the 25th Annual Conference of the Cognitive Science Society, Lawrence Erlbaum, Mahwah (2003)Google Scholar
  19. 19.
    Burns, B.D.: Meta-analogical transfer: Transfer between episodes of analogical reasoning. Journal of Experimental Psychology: Learning, Memory, and Cognition 22(4), 1032–1048 (1996)CrossRefGoogle Scholar
  20. 20.
    Cornuejlos, A.: Analogy as minimization of description length. In: Nakhaeizadeh, N., Taylor, C.C. (eds.) Machine Learning and Statistics. The Interface, pp. 321–335. Wiley, New York (1997)Google Scholar
  21. 21.
    Dastani, M., Indurkhya, B., Scha, R.: An Algebraic Approach to Modeling Analogical Projection in Pattern Perception. In: Proceedings of Mind II (1997)Google Scholar
  22. 22.
    Leeuwenberg, E.: A perceptual coding language for visual and auditory patterns. American Journal of Psychology 84, 307–349 (1971)CrossRefGoogle Scholar
  23. 23.
    Goldstein, E.B.: Sensation and Perception. Wadsworth Publishing Co., Belmont (1980)Google Scholar
  24. 24.
    Hasker, R.W.: The Replay of Program Derivations. PhD thesis, Univ. of Illinois at Urbana-Champaign (1995)Google Scholar
  25. 25.
    Schmid, U., Sinha, U., Wysotzki, F.: Program reuse and abstraction by anti-unification. In: Professionelles Wissensmanagement – Erfahrungen und Visionen, pp. 183–185. Shaker, Aachen (2001), Long Version: Google Scholar

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© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Stephan Weller
    • 1
  • Ute Schmid
    • 1
  1. 1.Department of Information Systems and Applied Computer Science, Otto-Friedrich-University, Bamberg 

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