Skip to main content
SpringerLink
Log in

Theory blending: extended algorithmic aspects and examples

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

In Cognitive Science, conceptual blending has been proposed as an important cognitive mechanism that facilitates the creation of new concepts and ideas by constrained combination of available knowledge. It thereby provides a possible theoretical foundation for modeling high-level cognitive faculties such as the ability to understand, learn, and create new concepts and theories. Quite often the development of new mathematical theories and results is based on the combination of previously independent concepts, potentially even originating from distinct subareas of mathematics. Conceptual blending promises to offer a framework for modeling and re-creating this form of mathematical concept invention with computational means. This paper describes a logic-based framework which allows a formal treatment of theory blending (a subform of the general notion of conceptual blending with high relevance for applications in mathematics), discusses an interactive algorithm for blending within the framework, and provides several illustrating worked examples from mathematics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdel-Fattah, A.M.H., Krumnack, U., Kühnberger, K.U.: The importance of two cognitive mechanisms in analyzing counterfactuals: an implementation-oriented explication. In: Advances in cognitive systems, Baltimore (2013)

  2. Alexander, J.: Blending in Mathematics. Semiotica 2011(187), 1–48 (2011)

    Article  Google Scholar 

  3. Besold, T.R., Kühnberger, K.U., Plaza, E.: Analogy, amalgams, and concept blending. In: Proceedings of the third annual conference on advances in cognitive systems (ACS 2015), Poster Collection. CogSys.org (2015)

  4. Besold, T.R., Plaza, E.: Generalize and blend: concept blending based on generalization, analogy, and amalgams. In: Proceedings of the sixth international conference on computational creativity (ICCC 2015). Brigham Young University (2015)

  5. Bou, F., Schorlemmer, M., Corneli, J., Gómez-Ramírez, D., Maclean, E., Smaill, A., Pease, A.: The role of blending in mathematical invention. In: Proceedings of the 6th international conference on computational creativity (ICCC), Lecture Notes in Computer Science, vol. 1342 (2015)

  6. Cicerone, S., Parisi-Presicce, F.: On the complexity of specification morphisms. Theor. Comput. Sci. 189(1–2), 239–248 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Colton, S., Ramezani, R., Llano, M.T.: The HR3 discovery system: design decisions and implementation details. In: Kibble, R. (ed.) Proceedings of the 50th anniversary convention of the AISB (2014)

  8. Confalonieri, R., Corneli, J., Pease, A., Plaza, E., Schorlemmer, M.: Using argumentation to evaluate concept blends in combinatorial creativity. In: Proceedings of the sixth international conference on computational creativity (ICCC 2015). Brigham Young University (2015)

  9. Coulson, S.: Semantic leaps: frame-shifting and conceptual blending in meaning construction. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  10. Diaconescu, R.: Extra theory morphisms for institutions: logical semantics for multi-paradigm languages. Appl. Categ. Struct. 6(4), 427–453 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eppe, M., Maclean, E., Confalonieri, R., Schorlemmer, M., Kutz, O., Plaza, E.: ASP, amalgamation and the conceptual blending workflow. In: Calimeri, F., Ianni, G., Truszczynski, M. (eds.) Logic programming and nonmonotonic reasoning: 13th international conference, LPNMR 2015, Lecture Notes in Artificial Intelligence, vol. 9345. Springer (2015)

  12. Fauconnier, G., Turner, M.: The way we think: conceptual blending and the mind’s hidden complexities. Basic Books, New York (2002)

    Google Scholar 

  13. Gentner, D.: Structure-mapping: a theoretical framework for analogy. Cogn. Sci. 7(2), 155–170 (1983)

    Article  Google Scholar 

  14. Goguen, J., Andler, D., Ogawa, Y., Okada, M., Watanabe S.: Mathematical models of cognitive space and time. In: Reasoning and cognition: proceedings of the interdisciplinary conference on reasoning and cognition, pp 125–128. Keio University Press (2006)

  15. Gómez-Ramírez, D.: Conceptual blending as a creative meta-generator of mathematical concepts: prime ideals and dedekind domains as a blend. In: Besold, T.R., Kühnberger, K.U., Schorlemmer, M., Smaill, A. (eds.) Proceedings of the 4th international workshop on computational creativity, concept invention, and general intelligence (C3GI), Publications of the Institute of Cognitive Science, vol. 02-2015, pp 1–11. Institute of Cognitive Science (2015)

  16. Kutz, O., Neuhaus, F., Mossakowski, T., Codescu, M.: Blending in the hub: Towards a computational concept invention platform. In: Proceedings of the fifth international conference on computational creativity (ICCC 2015). Jozef Stefan Institute (2014)

  17. Lakoff, G., Nuñeź, R.: Where mathematics comes from: how the embodied mind brings mathematics into being. Basic Books, New York (2000)

    MATH  Google Scholar 

  18. Lee, M., Barnden, J.: A computational approach to conceptual blending within counterfactuals. cognitive science research papers CSRP-01-10, School of Computer Science. University of Birmingham (2001)

  19. Li, B., Zook, A., Davis, N., O’ Riedl, M.: Goal-driven conceptual blending: A computational approach for creativity. In: Proceedings of the third international conference on computational creativity (ICCC 2015). University College Dublin (2012)

  20. MacNish, C.K., Antoniou, G.: Specification morphisms for nonmonotonic knowledge systems. In: Sattar, A. (ed.) Knowledge representation and reasoning, lecture notes in computer science, vol. 1342, pp 246–254. Springer (2005)

  21. Manna, Z., Zarba, C.: Combining decision procedures. In: Formal methods at the crossroads: from panacea to foundational support, LNCS 2787, pp 381–422. Springer (2003)

  22. Martinez, M., Krumnack, U., Smaill, A., Besold, T.R., Abdel-Fattah, A., Schmidt, M., Gust, H., Kühnberger, K.U., Guhe, M., Pease, A.: Algorithmic aspects of theory blending. In: Aranda-Corral, G., Martın-Mateos, F., Calmet, J. (eds.) Proceedings of the 12th international conference on artificial intelligence and symbolic computation (AISC). LNAI, Springer (2014)

  23. Martins, P., Urbanci, T., Pollak, S., Lavrac, N., Cardoso, A.: The good, the bad, and the aha! blends. In: Proceedings of the sixth international conference on computational creativity (ICCC 2015). Brigham Young University (2015)

  24. Mossakowski, T., Maeder, C., Codescu, M.: Hets user guide version 0.99 (2014). http://www.informatik.uni-bremen.de/agbkb/forschung/formal_methods/CoFI/hets/UserGuide.pdf

  25. Pereira, F.C.: Creativity and AI: a conceptual blending approach. applications of cognitive linguistics (ACL). Mouton de Gruyter, Berlin (2007)

    Google Scholar 

  26. Plotkin, G.D.: A note on inductive generalization. Mach. Intell. 5, 153–163 (1970)

    MathSciNet  MATH  Google Scholar 

  27. Schiralli, M., Sinclair, N.: A constructive response to “where mathematics comes from”. Educ. Stud. Math. 52, 79–91 (2003)

    Article  Google Scholar 

  28. Schmidt, M., Krumnack, U., Gust, H., Kühnberger, K.U.: Heuristic-driven theory projection: an overview. In: Computational approaches to analogical reasoning: current trends, studies in computational intelligence, vol. 548, pp 163–194. Springer (2014)

  29. Veale, T.: Computability as a test on linguistic theories. In: Kristiansen, G., Achard, M., Dirven, R., Ruiz de Mendoza Ibanez, F. (eds.) Cognitive linguistics: current applications and future perspectives. Mouton DeGruyter, The Hague (2006)

  30. Veale, T.: A robust computational model of conceptual blending. In: Proceedings of the third international conference on computational creativity (ICCC 2015). University College Dublin (2012)

  31. Veale, T., O’Donoghue, D.: Computation and blending. Comput. Linguis. 11 (3–4), 253–282 (2000). special issue on Conceptual Blending

    Google Scholar 

  32. Xiao, P., Linkola, S.: Vismantic: meaning-making with images. In: Proceedings of the sixth international conference on computational creativity (ICCC 2015). Brigham Young University (2015)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Universidad de los Andes, Bogotá, Colombia

    M. Martinez

  2. Faculty of Science, Ain Shams University, Cairo, Egypt

    A. M. H. Abdel-Fattah

  3. Institute of Cognitive Science, University of Osnabrück, Osnabrück, Germany

    U. Krumnack, D. Gómez-Ramírez, M. Schmidt & K.-U. Kühnberger

  4. School of Informatics, University of Edinburgh, Edinburgh, UK

    A. Smaill & M. Guhe

  5. The KRDB Research Centre, Free University of Bozen-Bolzano, Bolzano, Italy

    T. R. Besold

  6. School of Science and Engineering, University of Dundee, Dundee, UK

    A. Pease

Authors
  1. M. Martinez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. M. H. Abdel-Fattah
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. U. Krumnack
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. Gómez-Ramírez
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. Smaill
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. T. R. Besold
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. A. Pease
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. M. Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. M. Guhe
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. K.-U. Kühnberger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. M. H. Abdel-Fattah.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Martinez, M., Abdel-Fattah, A.M.H., Krumnack, U. et al. Theory blending: extended algorithmic aspects and examples. Ann Math Artif Intell 80, 65–89 (2017). https://doi.org/10.1007/s10472-016-9505-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-016-9505-y

Keywords

Search

Navigation