PROOF GRANULARITY AS AN EMPIRICAL PROBLEM?

Marvin Schiller, Christoph Benzmüller

2009

Abstract

Even in introductory textbooks on mathematical proof, intermediate proof steps are generally skipped when this seems appropriate. This gives rise to different granularities of proofs, depending on the intended audience and the context in which the proof is presented. We have developed a mechanism to classify whether proof steps of different sizes are appropriate in a tutoring context. The necessary knowledge is learnt from expert tutors via standard machine learning techniques from annotated examples. We discuss the ongoing evaluation of our approach via empirical studies.

References

  1. Autexier, S. (2005). The core calculus. In Nieuwenhuis, R., editor, Automated Deduction - CADE-20, 20th International Conference on Automated Deduction, Tallinn, Estonia, July 22-27, 2005, Proceedings, volume 3632 of LNCS, pages 84-98. Springer.
  2. Benzmüller, C., Horacek, H., Kruijff-Korbayová, I., Pinkal, M., Siekmann, J. H., and Wolska, M. (2007). Natural language dialog with a tutor system for mathematical proofs. In Lu, R., Siekmann, J. H., and Ullrich, C., editors, Cognitive Systems, Joint Chinese-German Workshop, Shanghai, China, March 7-11, 2005, Revised Selected Papers, volume 4429 of LNCS, pages 1-14. Springer.
  3. Benzmüller, C., Horacek, H., Lesourd, H., KruijffKorbajová, I., Schiller, M., and Wolska, M. (2006). A corpus of tutorial dialogs on theorem proving; the influence of the presentation of the study-material. In Proceedings of International Conference on Language Resources and Evaluation (LREC), Genoa, Italy. ELDA.
  4. Bundy, A., van Harmelen, F., Hesketh, J., and Smaill, A. (1991). Experiments with proof plans for induction. J. Autom. Reasoning, 7(3):303-324.
  5. Gordon, M. J. C., Milner, R., and Wadsworth, C. P. (1979). Edinburgh LCF, volume 78 of LNCS. Springer.
  6. Hobbs, J. R. (1985). Granularity. In Proc. of the 9th Int. Joint Conf. on Artificial Intelligence (IJCAI), pages 432-435.
  7. Huang, X. (1994). Reconstruction proofs at the assertion level. In Bundy, A., editor, Automated Deduction - CADE-12, 12th International Conference on Automated Deduction, Nancy, France, June 26 - July 1, 1994, Proceedings, volume 814 of LNCS, pages 738- 752. Springer.
  8. Keet, M. C. (2008). A Formal Theory of Granularity. PhD thesis, Free University of Bozen-Bolzano, Italy.
  9. Mccalla, G., Greer, J., Barrie, B., and Pospisil, P. (1992). Granularity hierarchies. In Computers and Mathematics with Applications: Special Issue on Semantic Networks, pages 363-375.
  10. McCune, W. (2003). Otter 3.3 reference manual. www.mcs.anl.gov/AR/otter/otter33.pdf.
  11. McMath, D., Rozenfeld, M., and Sommer, R. (2001). A computer environment for writing ordinary mathematical proofs. In Proc. of LPAR 2001, volume 2250 of LNCS, pages 507-516. Springer.
  12. Melis, E. (1999). Knowledge-based proof planning. Artificial Intelligence, 115:494-498.
  13. Schiller, M. and Benzmüller, C. (2009). Granularityadaptive proof presentation. In 14th Int. Conference on Artificial Intelligence in Education (AIED). Submitted to The 14th International Conference on Artificial Intelligence in Education (AIED 2009).
  14. Schiller, M., Dietrich, D., and Benzmüller, C. (2008). Proof step analysis for proof tutoring - a learning approach to granularity. Teaching Mathematics and Computer Science. In print.
  15. Sieg, W. (2007). The AProS project: Strategic thinking & computational logic. Logic Journal of the IGPL, 15(4):359-368.
  16. Siekmann, J., Benzmüller, C., and Autexier, S. (2006). Computer supported mathematics with Omega. Journal of Applied Logic, 4(4):533-559.
  17. Wack, B. (2005). Typage et deduction dans le calcul de reecriture. PhD thesis, Univ. Henri Poincare Nancy 1.
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Paper Citation


in Harvard Style

Schiller M. and Benzmüller C. (2009). PROOF GRANULARITY AS AN EMPIRICAL PROBLEM? . In Proceedings of the First International Conference on Computer Supported Education - Volume 1: CSEDU, ISBN 978-989-8111-82-1, pages 350-354. DOI: 10.5220/0002159603500354


in Bibtex Style

@conference{csedu09,
author={Marvin Schiller and Christoph Benzmüller},
title={PROOF GRANULARITY AS AN EMPIRICAL PROBLEM?},
booktitle={Proceedings of the First International Conference on Computer Supported Education - Volume 1: CSEDU,},
year={2009},
pages={350-354},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0002159603500354},
isbn={978-989-8111-82-1},
}


in EndNote Style

TY - CONF
JO - Proceedings of the First International Conference on Computer Supported Education - Volume 1: CSEDU,
TI - PROOF GRANULARITY AS AN EMPIRICAL PROBLEM?
SN - 978-989-8111-82-1
AU - Schiller M.
AU - Benzmüller C.
PY - 2009
SP - 350
EP - 354
DO - 10.5220/0002159603500354