Advertisement

Inspection and Selection of Representations

  • Daniel RaggiEmail author
  • Aaron Stockdill
  • Mateja Jamnik
  • Grecia Garcia Garcia
  • Holly E. A. Sutherland
  • Peter C.-H. Cheng
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11617)

Abstract

We present a novel framework for inspecting representations and encoding their formal properties. This enables us to assess and compare the informational and cognitive value of different representations for reasoning. The purpose of our framework is to automate the process of representation selection, taking into account the candidate representation’s match to the problem at hand and to the user’s specific cognitive profile. This requires a language for talking about representations, and methods for analysing their relative advantages. This foundational work is first to devise a computational end-to-end framework where problems, representations, and user’s profiles can be described and analysed. As AI systems become ubiquitous, it is important for them to be more compatible with human reasoning, and our framework enables just that.

Keywords

Representation in reasoning Heterogeneous reasoning Representation selection Representational system 

Notes

Acknowledgements

We thank the 3 anonymous reviewers for their comments, which helped to improve the presentation of this paper.

References

  1. 1.
    Ainsworth, S.: The functions of multiple representations. Comput. Educ. 33(2–3), 131–152 (1999)CrossRefGoogle Scholar
  2. 2.
    Barker-Plummer, D., Etchemendy, J., Liu, A., Murray, M., Swoboda, N.: Openproof - a flexible framework for heterogeneous reasoning. In: Stapleton, G., Howse, J., Lee, J. (eds.) Diagrams 2008. LNCS (LNAI), vol. 5223, pp. 347–349. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-87730-1_32CrossRefGoogle Scholar
  3. 3.
    Bengio, Y., Courville, A., Vincent, P.: Representation learning: a review and new perspectives. IEEE Trans. Pattern Anal. Mach. Intell. 35(8), 1798–1828 (2013)CrossRefGoogle Scholar
  4. 4.
    Blackwell, A., Green, T.: Notational systems-the cognitive dimensions of notations framework. In: HCI Models, Theories, and Frameworks: Toward an Interdisciplinary Science. Morgan Kaufmann (2003)Google Scholar
  5. 5.
    Cheng, P.C.-H.: Unlocking conceptual learning in mathematics and science with effective representational systems. Comput. Educ. 33(2–3), 109–130 (1999)CrossRefGoogle Scholar
  6. 6.
    Cheng, P.C.-H.: Probably good diagrams for learning: representational epistemic recodification of probability theory. Top. Cogn. Sci. 3(3), 475–498 (2011)CrossRefGoogle Scholar
  7. 7.
    Cheng, P.C.-H.: What constitutes an effective representation? In: Jamnik, M., Uesaka, Y., Elzer Schwartz, S. (eds.) Diagrams 2016. LNCS (LNAI), vol. 9781, pp. 17–31. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-42333-3_2CrossRefGoogle Scholar
  8. 8.
    Chomsky, N.: Three models for the description of language. IRE Trans. Inf. Theory 2(3), 113–124 (1956)CrossRefGoogle Scholar
  9. 9.
    Chomsky, N.: On certain formal properties of grammars. Inf. Control 2(2), 137–167 (1959)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Coquand, T.: Type theory. In: Stanford Encyclopedia of Philosophy (2006)Google Scholar
  11. 11.
    Harrison, J.: HOL light: an overview. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 60–66. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-03359-9_4CrossRefGoogle Scholar
  12. 12.
    Huffman, B., Kunčar, O.: Lifting and transfer: a modular design for quotients in Isabelle/HOL. In: Gonthier, G., Norrish, M. (eds.) CPP 2013. LNCS, vol. 8307, pp. 131–146. Springer, Cham (2013).  https://doi.org/10.1007/978-3-319-03545-1_9CrossRefzbMATHGoogle Scholar
  13. 13.
    Jamnik, M., Bundy, A., Green, I.: On automating diagrammatic proofs of arithmetic arguments. J. Log. Lang. Inf. 8(3), 297–321 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jupyter. jupyter.org
  15. 15.
    Kaufmann, M., Moore, J.S.: ACL2: an industrial strength version of Nqthm. In: Proceedings of 11th Annual Conference on Computer Assurance, COMPASS 1996, pp. 23–34. IEEE (1996)Google Scholar
  16. 16.
    Kovács, L., Voronkov, A.: First-order theorem proving and Vampire. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 1–35. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39799-8_1CrossRefGoogle Scholar
  17. 17.
    Kühlwein, D., Blanchette, J.C., Kaliszyk, C., Urban, J.: MaSh: machine learning for sledgehammer. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 35–50. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39634-2_6CrossRefGoogle Scholar
  18. 18.
  19. 19.
    Mossakowski, T., Maeder, C., Lüttich, K.: The heterogeneous tool set, Hets. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 519–522. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-71209-1_40CrossRefGoogle Scholar
  20. 20.
    Newell, A.: Human Problem Solving. Prentice-Hall Inc., Upper Saddle River (1972)Google Scholar
  21. 21.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic, vol. 2283. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-45949-9CrossRefzbMATHGoogle Scholar
  22. 22.
    GNU Octave. octave.org
  23. 23.
    Raggi, D., Bundy, A., Grov, G., Pease, A.: Automating change of representation for proofs in discrete mathematics (extended version). Math. Comput. Sci. 10(4), 429–457 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    SageMath. sagemath.org
  25. 25.
    Stapleton, G., Jamnik, M., Shimojima, A.: What makes an effective representation of information: a formal account of observational advantages. J. Log. Lang. Inf. 26(2), 143–177 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Urbas, M., Jamnik, M.: A framework for heterogeneous reasoning in formal and informal domains. In: Dwyer, T., Purchase, H., Delaney, A. (eds.) Diagrams 2014. LNCS (LNAI), vol. 8578, pp. 277–292. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-44043-8_28CrossRefGoogle Scholar
  27. 27.
    Urbas, M., Jamnik, M., Stapleton, G., Flower, J.: Speedith: a diagrammatic reasoner for spider diagrams. In: Cox, P., Plimmer, B., Rodgers, P. (eds.) Diagrams 2012. LNCS (LNAI), vol. 7352, pp. 163–177. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-31223-6_19CrossRefGoogle Scholar
  28. 28.
    Winterstein, D., Bundy, A., Gurr, C.: Dr.Doodle: a diagrammatic theorem prover. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 331–335. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-25984-8_24CrossRefGoogle Scholar
  29. 29.
    WordNet (2010). wordnet.princeton.edu

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Daniel Raggi
    • 1
    Email author
  • Aaron Stockdill
    • 1
  • Mateja Jamnik
    • 1
  • Grecia Garcia Garcia
    • 2
  • Holly E. A. Sutherland
    • 2
  • Peter C.-H. Cheng
    • 2
  1. 1.University of CambridgeCambridgeUK
  2. 2.University of SussexBrightonUK

Personalised recommendations