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Presenting Proofs with Adapted Granularity

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KI 2009: Advances in Artificial Intelligence (KI 2009)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5803))

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Abstract

When mathematicians present proofs they usually adapt their explanations to their didactic goals and to the (assumed) knowledge of their addressees. Modern automated theorem provers, in contrast, present proofs usually at a fixed level of detail (also called granularity). Often these presentations are neither intended nor suitable for human use. A challenge therefore is to develop user- and goal-adaptive proof presentation techniques that obey common mathematical practice. We present a flexible and adaptive approach to proof presentation based on classification. Expert knowledge for the classification task can be hand-authored or extracted from annotated proof examples via machine learning techniques. The obtained models are employed for the automated generation of further proofs at an adapted level of granularity.

This work was supported by a grant from Studienstiftung des Deutschen Volkes e.V.

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References

  1. Bartle, R.G., Sherbert, D.: Introduction to Real Analysis, 2nd edn. Wiley, Chichester (1982)

    MATH  Google Scholar 

  2. Benzmüller, C., Horacek, H., Kruijff-Korbayová, I., Pinkal, M., Siekmann, J.H., Wolska, M.: Natural language dialog with a tutor system for mathematical proofs. In: Lu, R., Siekmann, J.H., Ullrich, C. (eds.) Cognitive Systems: Joint Chinese-German Workshop. LNCS (LNAI), vol. 4429, pp. 1–14. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  3. Benzmüller, C., Horacek, H., Lesourd, H., Kruijff-Korbayová, I., Schiller, M., Wolska, M.: A corpus of tutorial dialogs on theorem proving; the influence of the presentation of the study-material. In: Proc. Intl. Conference on Language Resources and Evaluation (LREC 2006), Genoa, Italy, ELDA (2006)

    Google Scholar 

  4. Autexier, S., Benzmüller, C., Dietrich, D., Meier, A., Wirth, C.P.: A generic modular data structure for proof attempts alternating on ideas and granularity. In: [14], pp. 126–142

    Google Scholar 

  5. Fiedler, A.: P.rex: An interactive proof explainer. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 416–420. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  6. Denney, E., Power, J., Tourlas, K.: Hiproofs: A hierarchical notion of proof tree. In: Proc. of the 21st Annual Conf. on Mathematical Foundations of Progamming Semantics (MFPS XXI). ENTCS, vol. 155, pp. 341–359. Elsevier, Amsterdam (2006)

    Google Scholar 

  7. Autexier, S., Fiedler, A.: Textbook proofs meet formal logic - the problem of underspecification and granularity. In: [14], pp. 96–110

    Google Scholar 

  8. Quinlan, J.R.: C4.5: Programs for Machine Learning. Morgan Kaufmann, San Francisco (1993)

    Google Scholar 

  9. Schiller, M., Benzmüller, C.: Granularity-adaptive proof presentation. Technical report, SEKI Working-Paper (2009), http://arxiv.org/pdf/0903.0314v4

  10. Frank, E., Witten, I.H.: Generating accurate rule sets without global optimization. In: Proc. 15th Intl. Conf. on Machine Learning, pp. 144–151. Morgan Kaufmann, San Francisco (1998)

    Google Scholar 

  11. Platt, J.C.: Fast training of support vector machines using sequential minimal optimization. In: Schoelkopf, B., Burges, C., Smola, A. (eds.) Advances in Kernel Methods - Support Vector Learning, pp. 185–208. MIT Press, Cambridge (1998)

    Google Scholar 

  12. Hobbs, J.R.: Granularity. In: Joshi, A.K. (ed.) Proc. of the 9th Int. Joint Conf. on Artificial Intelligence (IJCAI), pp. 432–435. Morgan Kaufmann, San Francisco (1985)

    Google Scholar 

  13. McCalla, G., Greer, J., Barrie, B., Pospisil, P.: Granularity hierarchies. In: Computers & Mathematics with Applications, vol. 23(2-5), pp. 363–375. Elsevier, Amsterdam (1992)

    Google Scholar 

  14. Kohlhase, M. (ed.): MKM 2005. LNCS (LNAI), vol. 3863. Springer, Heidelberg (2006)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. German Research Center for Artificial Intelligence (DFKI), Bremen, Germany

    Marvin Schiller

  2. International University in Germany, Bruchsal, Germany

    Christoph Benzmüller

Authors
  1. Marvin Schiller
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  2. Christoph Benzmüller
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Editor information

Editors and Affiliations

  1. GET Lab, University of Paderborn, Pohlweg 47-49, 33098, Paderborn, Germany

    Bärbel Mertsching , Marcus Hund  & Zaheer Aziz ,  & 

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Schiller, M., Benzmüller, C. (2009). Presenting Proofs with Adapted Granularity. In: Mertsching, B., Hund, M., Aziz, Z. (eds) KI 2009: Advances in Artificial Intelligence. KI 2009. Lecture Notes in Computer Science(), vol 5803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04617-9_37

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  • DOI: https://doi.org/10.1007/978-3-642-04617-9_37

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04616-2

  • Online ISBN: 978-3-642-04617-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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