When Almost Is Not Even Close: Remarks on the Approximability of HDTP

  • Tarek Richard Besold
  • Robert Robere
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7999)


A growing number of researchers in Cognitive Science advocate the thesis that human cognitive capacities are constrained by computational tractability. If right, this thesis also can be expected to have far-reaching consequences for work in Artificial General Intelligence: Models and systems considered as basis for the development of general cognitive architectures with human-like performance would also have to comply with tractability constraints, making in-depth complexity theoretic analysis a necessary and important part of the standard research and development cycle already from a rather early stage. In this paper we present an application case study for such an analysis based on results from a parametrized complexity and approximation theoretic analysis of the Heuristic Driven Theory Projection (HDTP) analogy-making framework.


Function Symbol General Intelligence Theory Projection Admissible Sequence Term Algebra 
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  1. 1.
    Wang, P., Goertzel, B.: Introduction: Aspects of Artificial General Intelligence. In: Goertzel, B., Wang, P. (eds.) Advances in Artificial General Intelligence - Proc. of the AGI Workshop 2006. Frontiers in Artificial Intelligence and Applications, vol. 157, pp. 1–16. IOS Press (2007)Google Scholar
  2. 2.
    van Rooij, I.: The tractable cognition thesis. Cognitive Science 32, 939–984 (2008)CrossRefGoogle Scholar
  3. 3.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)Google Scholar
  4. 4.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)Google Scholar
  5. 5.
    Robere, R., Besold, T.R.: Complex Analogies: Remarks on the Complexity of HDTP. In: Thielscher, M., Zhang, D. (eds.) AI 2012. LNCS, vol. 7691, pp. 530–542. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Schwering, A., Kühnberger, K.U., Kokinov, B.: Analogies: Integrating multiple cognitive abilities - guest editorial. Journal of Cognitive Systems Research 10(3) (2009)Google Scholar
  7. 7.
    Hofstadter, D.: Epilogue: Analogy as the core of cognition. In: Gentner, D., Holyoak, K., Kokinov, B. (eds.) The Analogical Mind: Perspectives from Cognitive Science, pp. 499–538. MIT Press, Cambridge (2001)Google Scholar
  8. 8.
    Hofstadter, D., Mitchell, M.: The copycat project: a model of mental fluidity and analogy-making. In: Holyoak, K., Barnden, J. (eds.) Advances in Connectionist and Neural Computation Theory. Analogical Connections, vol. 2, pp. 31–112. Ablex, New York (1994)Google Scholar
  9. 9.
    Falkenhainer, B., Forbus, K., Gentner, D.: The structure-mapping engine: Algorithm and examples. Artificial Intelligence 41(1), 1–63 (1989)zbMATHCrossRefGoogle Scholar
  10. 10.
    Gentner, D., Forbus, K.: MAC/FAC: A Model of Similarity-based Retrieval. Cognitive Science 19, 141–205 (1991)Google Scholar
  11. 11.
    Gentner, D.: Structure-mapping: A theoretical framework for analogy. Cognitive Science 7(2), 155–170 (1983)CrossRefGoogle Scholar
  12. 12.
    Gust, H., Kühnberger, K.U., Schmid, U.: Metaphors and Heuristic–Driven Theory Projection (HDTP). Theoretical Computer Science 354, 98–117 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Schwering, A., Krumnack, U., Kühnberger, K.U., Gust, H.: Syntactic principles of Heuristic-Driven Theory Projection. Journal of Cognitive Systems Research 10(3), 251–269 (2009)CrossRefGoogle Scholar
  14. 14.
    Guhe, M., Pease, A., Smaill, A., Martinez, M., Schmidt, M., Gust, H., Kühnberger, K.U., Krumnack, U.: A computational account of conceptual blending in basic mathematics. Journal of Cognitive Systems Research 12(3), 249–265 (2011)CrossRefGoogle Scholar
  15. 15.
    Plotkin, G.D.: A note on inductive generalization. Machine Intelligence 5, 153–163 (1970)MathSciNetGoogle Scholar
  16. 16.
    Krumnack, U., Schwering, A., Gust, H., Kühnberger, K.-U.: Restricted higher-order anti-unification for analogy making. In: Orgun, M.A., Thornton, J. (eds.) AI 2007. LNCS (LNAI), vol. 4830, pp. 273–282. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Schmidt, M., Gust, H., Kühnberger, K.-U., Krumnack, U.: Refinements of restricted higher-order anti-unification for heuristic-driven theory projection. In: Bach, J., Edelkamp, S. (eds.) KI 2011. LNCS, vol. 7006, pp. 289–300. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Guhe, M., Pease, A., Smaill, A., Schmidt, M., Gust, H., Kühnberger, K.U., Krumnack, U.: Mathematical reasoning with higher-order anti-unifcation. In: Proc. of the 32nd Annual Conference of the Cognitive Science Society, pp. 1992–1997 (2010)Google Scholar
  19. 19.
    Besold, T.R., Gust, H., Krumnack, U., Abdel-Fattah, A., Schmidt, M., Kühnberger, K.U.: An argument for an analogical perspective on rationality & decision-making. In: van Eijck, J., Verbrugge, R. (eds.) Proc. of the Workshop on Reasoning About Other Minds (RAOM 2011). CEUR Workshop Proceedings, vol. 751, pp. 20–31. (July 2011)Google Scholar
  20. 20.
    Martinez, M., Besold, T.R., Abdel-Fattah, A., Kühnberger, K.U., Gust, H., Schmidt, M., Krumnack, U.: Towards a Domain-Independent Computational Framework for Theory Blending. AAAI Technical Report of the AAAI Fall 2011 Symposium on Advances in Cognitive Systems, pp. 210–217 (2011)Google Scholar
  21. 21.
    Martinez, M., Besold, T.R., Abdel-Fattah, A., Gust, H., Schmidt, M., Krumnack, U., Kühnberger, K.U.: Theory Blending as a Framework for Creativity in Systems for General Intelligence. In: Wang, P., Goertzel, B. (eds.) Theoretical Foundations of AGI. Atlantis Press (2012)Google Scholar
  22. 22.
    Vazirani, V.: Approximation Algorithms. Springer (2001)Google Scholar
  23. 23.
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic numbers. In: Proc. of the 38th ACM Symposium on Theory of Computing, pp. 681–690 (2006)Google Scholar
  24. 24.
    Tohill, J., Holyoak, K.: The impact of anxiety on analogical reasoning. Thinking & Reasoning 6(1), 27–40 (2000)CrossRefGoogle Scholar
  25. 25.
    van Rooij, I., Wareham, T.: Intractability and approximation of optimization theories of cognition. Journal of Mathematical Psychology 56(4), 232–247 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tarek Richard Besold
    • 1
  • Robert Robere
    • 2
  1. 1.Institute of Cognitive ScienceUniversity of OsnabrückGermany
  2. 2.Department of Computer ScienceUniversity of TorontoCanada

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