Skip to main content
SpringerLink
Log in
Book cover

International Conference on Artificial General Intelligence

AGI 2018: Artificial General Intelligence pp 62–76Cite as

Solving Tree Problems with Category Theory

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10999))

Abstract

Artificial Intelligence (AI) has long pursued models, theories, and techniques to imbue machines with human-like general intelligence. Yet even the currently predominant data-driven approasches in AI seem to be lacking humans’ unique ability to solve wide ranges of problems. This situation begs the question of the existence of principles that underlie general problem-solving capabilities. We approach this question through the mathematical formulation of analogies across different problems and solutions. We focus in particular on problems that could be represented as tree-like structures. Most importantly, we adopt a category-theoretic approach in formalising tree problems as categories, and in proving the existence of equivalences across apparently unrelated problem domains. We prove the existence of a functor between the category of tree problems and the category of solutions. We also provide a weaker version of the functor by quantifying equivalences of problem categories using a metric on tree problems.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Arjonilla, F.J., Ogata, T.: General problem solving with category theory. arXiv preprint arXiv:1709.04825 (2017)

  2. Arjonilla García, F.: A three-component cognitive theory. Master’s thesis (2015)

    Google Scholar 

  3. Arzi-Gonczarowski, Z.: Perceive this as that-analogies, artificial perception, and category theory. Ann. Math. Artif. Intell. 26(1–4), 215–252 (1999)

    Article  Google Scholar 

  4. Bonet, B., Geffner, H.: Solving POMDPs: RTDP-Bel vs. point-based algorithms, pp. 641–1646 (2009)

    Google Scholar 

  5. Bonet, B., Geffner, H.: Policies that generalize: solving many planning problems with the same policy, vol. 15 (2015)

    Google Scholar 

  6. Cardona, G., Mir, A., Rosselló, F., Rotger, L., Sánchez, D.: Cophenetic metrics for phylogenetic trees, after sokal and rohlf. BMC Bioinform. 14(1), 3 (2013)

    Article  Google Scholar 

  7. Csikvári, P., Lin, Z.: Graph homomorphisms between trees. arXiv preprint arXiv:1307.6721 (2013)

  8. Diuk, C., Schapiro, A., Córdova, N., Ribas-Fernandes, J., Niv, Y., Botvinick, M.: Divide and conquer: hierarchical reinforcement learning and task decomposition in humans. In: Baldassarre, G., Mirolli, M. (eds.) Computational and Robotic Models of the Hierarchical Organization of Behavior, pp. 271–291. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39875-9_12

    Chapter  Google Scholar 

  9. Eilenberg, S., MacLane, S.: General theory of natural equivalences. Trans. Am. Math. Soc. 58, 231–294 (1945)

    Article  MathSciNet  Google Scholar 

  10. Fodor, J.A., Pylyshyn, Z.W.: Connectionism and cognitive architecture: a critical analysis. Cognition 28(1–2), 3–71 (1988)

    Article  Google Scholar 

  11. Gordon, G., McMahon, E.: A greedoid polynomial which distinguishes rooted arborescences. Proc. Am. Math. Soc. 107(2), 287–298 (1989)

    Article  MathSciNet  Google Scholar 

  12. Greville, T.N.: Generalized inverses: theory and applications Adi Ben-Israel

    Google Scholar 

  13. Healy, M.J.: Category theory applied to neural modeling and graphical representations. In: Proceedings of the IEEE-INNS-ENNS International Joint Conference on Neural Networks, IJCNN 2000, vol. 3, pp. 35–40. IEEE (2000)

    Google Scholar 

  14. Hutter, M.: Universal Artificial Intelligence: Sequential Decisions Based on Algorithmic Probability. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  15. Izbicki, M.: Algebraic classifiers: a generic approach to fast cross-validation, online training, and parallel training. In: International Conference on Machine Learning, pp. 648–656 (2013)

    Google Scholar 

  16. Izbicki, M.: Two monoids for approximating NP-complete problems (2013)

    Google Scholar 

  17. Keisler, H.J., Chang, C.C.: Model Theory, North-Holland, Amsterdam (1990)

    Google Scholar 

  18. Kendall, M., Colijn, C.: A tree metric using structure and length to capture distinct phylogenetic signals. arXiv preprint arXiv:1507.05211 (2015)

  19. Laird, J.E.: The Soar cognitive architecture (2012)

    Google Scholar 

  20. Laird, J.E., Wray III, R.E.: Cognitive architecture requirements for achieving AGI. In: Proceedings of the Third Conference on Artificial General Intelligence, pp. 79–84 (2010)

    Google Scholar 

  21. Liao, J., Yao, Y., Yuan, L., Hua, G., Kang, S.B.: Visual attribute transfer through deep image analogy. arXiv preprint arXiv:1705.01088 (2017)

  22. Mac Lane, S.: Categories for the Working Mathematician, vol. 5. Springer, New York (2013)

    MATH  Google Scholar 

  23. Magnan, F., Reyes, G.E.: Category theory as a conceptual tool in the study of cognition. In: The Logical Foundations of Cognition, pp. 57–90 (1994)

    Google Scholar 

  24. MAZEMASTERS: Maze to tree [online] (2007). https://www.youtube.com/watch?v=k1tSK5V1pds

  25. Newell, A.: Unified theories of cognition and the role of soar. In: Michon, J.A., Akyürek, A. (eds.) Soar: A Cognitive Architecture in Perspective. Studies in Cognitive Systems, vol. 10, pp. 25–79. Springer, Dordrecht (1992). https://doi.org/10.1007/978-94-011-2426-3_3

  26. Newell, A., Shaw, J.C., Simon, H.A.: Report on a general problem solving program. In: IFIP Congress, vol. 256, p. 64 (1959)

    Google Scholar 

  27. Phillips, S.: A general (category theory) principle for general intelligence: duality (adjointness). In: Everitt, T., Goertzel, B., Potapov, A. (eds.) AGI 2017. LNCS (LNAI), vol. 10414, pp. 57–66. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63703-7_6

    Chapter  Google Scholar 

  28. Phillips, S., Wilson, W.H.: Categorial compositionality: a category theory explanation for the systematicity of human cognition. PLoS Comput. Biol. 6(7), e1000858 (2010)

    Article  Google Scholar 

  29. Phillips, S., Wilson, W.H.: Systematicity and a categorical theory of cognitive architecture: universal construction in context. Front. Psychol. 7, 1139 (2016)

    Google Scholar 

  30. Pierce, B.C.: Basic Category Theory for Computer Scientists. MIT Press, Cambridge (1991)

    MATH  Google Scholar 

  31. Ramırez, J.D.G.: A new foundation for representation in cognitive and brain science: category theory and the hippocampus. Departamento de Automatica, Ingenierıa Electronica e Informatica Industrial Escuela Tecnica Superior de Ingenieros Industriales (2010)

    Google Scholar 

  32. Rasmussen, D.: Hierarchical reinforcement learning in a biologically plausible neural architecture (2014)

    Google Scholar 

  33. Reed, S.E., Zhang, Y., Zhang, Y., Lee, H.: Deep visual analogy-making. In: Advances in Neural Information Processing Systems, pp. 1252–1260 (2015)

    Google Scholar 

  34. Rosa, M., Feyereisl, J., Collective, T.G.: A framework for searching for general artificial intelligence. arXiv preprint arXiv:1611.00685 (2016)

  35. Sharma, M., Holmes, M.P., Santamaría, J.C., Irani, A., Ram, A.: Transfer learning in real-time strategy games using hybrid CBR/RL

    Google Scholar 

  36. Taylor, M.E., Stone, P.: Transfer learning for reinforcement learning domains: a survey. J. Mach. Learn. Res. 10, 1633–1685 (2009)

    MathSciNet  MATH  Google Scholar 

  37. Tsuchiya, N., Taguchi, S., Saigo, H.: Using category theory to assess the relationship between consciousness and integrated information theory. Neurosci. Res. 107, 1–7 (2016)

    Article  Google Scholar 

  38. Veness, J., Ng, K.S., Hutter, M., Uther, W., Silver, D.: A monte-carlo AIXI approximation (2010)

    Google Scholar 

  39. Walters, R.F.C.: Categories and Computer Science. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Psychological Sciences, Faculty of Medicine Nursing and Health Sciences, Monash University, Melbourne, Australia

    Rafik Hadfi

Authors
  1. Rafik Hadfi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rafik Hadfi .

Editor information

Editors and Affiliations

  1. Adams State University, Alamosa, Colorado, USA

    Matthew Iklé

  2. Odessa Competence Center for Artificial Intelligence (OCCAM), Odessa, Ukraine

    Arthur Franz

  3. Hokkaido University, Sapporo, Japan

    Rafal Rzepka

  4. SingularityNET; OpenCog; Hanson Robotics, Hong Kong, China

    Ben Goertzel

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Hadfi, R. (2018). Solving Tree Problems with Category Theory. In: Iklé, M., Franz, A., Rzepka, R., Goertzel, B. (eds) Artificial General Intelligence. AGI 2018. Lecture Notes in Computer Science(), vol 10999. Springer, Cham. https://doi.org/10.1007/978-3-319-97676-1_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-97676-1_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-97675-4

  • Online ISBN: 978-3-319-97676-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation