Resource-Bounded Modelling and Analysis of Human-Level Interactive Proofs

  • Christoph BenzmüllerEmail author
  • Marvin Schiller
  • Jörg Siekmann
Part of the Cognitive Technologies book series (COGTECH)


Mathematics is the lingua franca of modern science, not least because of its conciseness and abstractive power. The ability to prove mathematical theorems is a key prerequisite in many fields of modern science, and the training of how to do proofs therefore plays a major part in the education of students in these subjects. Computer-supported learning is an increasingly important form of study since it allows for independent learning and individualised instruction.


Natural Deduction Tutor System Student Model Proof Step Dialogue State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We are grateful to the reviewers of this paper who provided many useful comments and suggestions. The second author gratefully acknowledges the support from the German National Merit Foundation (Studienstiftung des deutschen Volkes e.V.).


  1. 1.
    Abel A, Chang BYE, Pfenning F (2001) Human-readable machine-verifiable proofs for teaching constructive logic. In: Egly U, Fiedler A, Horacek H, Schmitt S (eds) Proceedings of the Workshop on Proof Transformations, Proof Presentations and Complexity of Proofs (PTP’01), Universitá degli studi di SienaGoogle Scholar
  2. 2.
    Andrews P, Bishop M, Brown C, Issar S, Pfenning F, Xi H (2004) Etps: A system to help students write formal proofs. Journal of Automated Reasoning 32: 75–92CrossRefGoogle Scholar
  3. 3.
    Autexier S, Benzmüller CE, Fiedler A, Horacek H, Vo BQ (2004) Assertion-level proof representation with under-specification. Electronic Notes in Theoretical Computer Science 93:5–23CrossRefGoogle Scholar
  4. 4.
    Autexier S, Benzmüller C, Dietrich D, Meier A, Wirth CP (2006) A generic modular data structure for proof attempts alternating on ideas and granularity. In: Kohlhase M (ed) Proceedings of the 5th International Conference on Mathematical Knowledge Management (MKM’05), Springer, LNAI, 1vol. 3863,pp. 126–142Google Scholar
  5. 5.
    Benzmüller C, Vo Q (2005) Mathematical domain reasoning tasks in natural language tutorial dialog on proofs. In: Veloso M, Kambhampati S (eds) Proceedings of the Twentieth National Conference on Artificial Intelligence (AAAI-05), AAAI Press / The MIT Press, Pittsburgh, Pennsylvania, USA, pp. 516–522Google Scholar
  6. 6.
    Benzmüller C, Fiedler A, Gabsdil M, Horacek H, Kruijff-Korbayová I, Pinkal M, Siekmann J, Tsovaltzi D, Vo BQ, Wolska M Tutorial dialogs on mathematical proofs. In: Proceedings of IJCAI-03 Workshop on Knowledge Representation and Automated Reasoning for E-Learning Systems, Acapulco, Mexico, pp. 12–22Google Scholar
  7. 7.
    Benzmüller C, Fiedler A, Gabsdil M, Horacek H, Kruijff-Korbayová I, Pinkal M, Siekmann J, Tsovaltzi D, Vo BQ, Wolska M A wizard of oz experiment for tutorial dialogues in mathematics. In: Proceedings of AI in Education (AIED 2003) Workshop on Advanced Technologies for Mathematics Education, Sydney, AustraliaGoogle Scholar
  8. 8.
    Benzmüller C, Fiedler A, Gabsdil M, Horacek H, Kruijff-Korbayová I, Tsovaltzi D, Vo BQ, Wolska M Towards a principled approach to tutoring mathematical proofs. In: Proceedings of the Workshop on Expressive Media and Intelligent Tools for Learning, German Conference on AI (KI 2003), Hamburg, GermanyGoogle Scholar
  9. 9.
    Benzmüller C, Horacek H, Lesourd H, Kruijff-Korbajova I, Schiller M, 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 2006), ELDA, Genoa, ItalyGoogle Scholar
  10. 10.
    Benzmüller C, Horacek H, Lesourd H, Kruijff-Korbayová I, Schiller M, Wolska M (2006) Diawoz-II - a tool for wizard-of-oz experiments in mathematics. In: Freksa C, Kohlhase M, Schill K (eds) KI 2006: Advances in Artificial Intelligence. 29th Annual German Conference on AI, Springer, LNAI, vol. 4314Google Scholar
  11. 11.
    Benzmüller C, Dietrich D, Schiller M, Autexier S Deep inference for automated proof tutoring? In: Hertzberg J, Beetz M, Englert R (eds) KI, Springer, Lecture Notes in Computer Science, Springer, New York, vol. 4667, pp. 435–439Google Scholar
  12. 12.
    Benzmüller C, Horacek H, Kruijff-Korbayová I, Pinkal M, Siekmann J, Wolska M (2007) Natural language dialog with a tutor system for mathematical proofs. In: Lu R, Siekmann J, Ullrich C (eds) Cognitive Systems, Springer, LNAI, New York, vol. 4429Google Scholar
  13. 13.
    Billingsley W, Robinson P (2007) Student proof exercises using mathtiles and isabelle/hol in an intelligent book. Journal of Automated Reasoning 39:181–218zbMATHCrossRefGoogle Scholar
  14. 14.
    D’Agostino M, Endriss U (1998) Winke: A proof assistant for teaching logic. In: Proceedings of the First International Workshop on Labelled DeductionGoogle Scholar
  15. 15.
    Dietrich D, Buckley M (2007) Verification of proof steps for tutoring mathematical proofs. In: Luckin R, Koedinger KR, Greer J (eds) Proceedings of the 13th International Conference on Artificial Intelligence in Education, IOS Press, Los Angeles, USA, vol. 158, pp. 560–562Google Scholar
  16. 16.
    Melis E, Siekmann J (2004) Activemath: An intelligent tutoring system for mathematics. In: Rutkowski L, Siekmann J, Tadeusiewicz R, Zadeh L (eds) Seventh International Conference ‘Artificial Intelligence and Soft Computing’ (ICAISC), Springer-Verlag, LNAI, vol. 3070, pp. 91–101Google Scholar
  17. 17.
    Fiedler A, Tsovaltzi D (2005) Domain-knowledge manipulation for dialogue-adaptive hinting. In: Looi CK, McCalla G, Bredeweg B, Breuker J (eds) Artificial Intelligence in Education — Supporting Learning through Intelligent and Socially Informed Technology, IOS Press, no. 125 in Frontiers in Artificial Intelligence and Applications, Amsterdam, pp. 801–803Google Scholar
  18. 18.
    Fiedler A, Gabsdil M, Horacek H (2004) A tool for supporting progressive refinement of wizard-of-oz experiments in natural language. In: Lester JC, Vicari RM, Paraguaçu F (eds) Intelligent Tutoring Systems — 7th International Conference (ITS 2004), Springer, no. 3220 in LNCS, New York, pp. 325–335Google Scholar
  19. 19.
    Gentzen G (1934) Untersuchungen über das logische schließen. Math Zeitschrift 39:176–210, 405–431MathSciNetCrossRefGoogle Scholar
  20. 20.
    Goguadze G, Ullrich C, Melis E, Siekmann J, Gross C, Morales R (2004) Leactivemath structure and metadata model. Tech. Rep., Saarland UniversityGoogle Scholar
  21. 21.
    Van der Hoeven (2001) J.GNU texmacs: A free, structured, wysiwyg and technical text editor. In: Flipo D (ed) Le document au XXI-ième siècle, Metz, vol 39-40, pp. 39–50, actes du congrès GUTenbergGoogle Scholar
  22. 22.
    Huang X (1994) Human oriented proof presentation: A reconstructive approach. Phd thesis, Universität des Saarlandes, Saarbrücken, Germany, published by infix, St. Augustin, Germany, Dissertationen zur Künstlichen Intelligenz, Vol. 112, 1996Google Scholar
  23. 23.
    Kakas A, Kowalski R, Toni F (1995) The role of abduction in logic programming. In: Handbook of Logic in Artificial Intelligence and Logic Programming, Oxford University, OxfordGoogle Scholar
  24. 24.
    Kelley JF (1984) An iterative design methodology for user-friendly natural language office information applications. ACM Transactions on Information and system 2(1):26–41MathSciNetCrossRefGoogle Scholar
  25. 25.
    Liu CH, Matthews R (2005) Vygotsky’s philosophy: Constructivism and its criticisms examined. Int Education J 6(3):{386–399}Google Scholar
  26. 26.
    Makatchev M, Jordan P, VanLehn K (2004) Abductive theorem proving for analyzing student explanations to guide feedback in intelligent tutoring systems. Journal of Automated Reasoning 32:187–226Google Scholar
  27. 27.
    Normand-Assadi S, Coulange L, Delozanne É, Grugeon B (2004) In: Linguistic markers to improve the assessment of students in mathematics: An exploratory study. Intelligent Tutoring Systems, Springer, New York, pp. 380–389Google Scholar
  28. 28.
    OECD (2004) Learning for tomorrow’s world – first results from PISA 2003. OECD PublishingGoogle Scholar
  29. 29.
    Platt J (1998) 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, MIT Press, Cambridge, MA, pp. 185–208Google Scholar
  30. 30.
    Quinlan JR (1993) C4.5: Programs for Machine Learning. Morgan Kaufmann, San Francisco, CAGoogle Scholar
  31. 31.
    Rips LJ (1994) The Psychology of Proof : Deductive Reasoning in Human Thinking. MIT Press, Cambridge, MAzbMATHGoogle Scholar
  32. 32.
    Rosé CP, Moore JD, VanLehn K, Allbritton D (2001) A comparative evaluation of socratic versus didactic tutoring. In: Proceedings of Cognitive Sciences SocietyGoogle Scholar
  33. 33.
    RuleQuest Research (2007) Data mining tools see5 and c5.0.
  34. 34.
    Schiller M (2005) Mechanizing proof step evaluation for mathematics tutoring - the case of granularity. Master’s thesis, Universität des Saarlandes, Saarbrücken, GermanyGoogle Scholar
  35. 35.
    Schiller M, Benzmüller C (2006) Granularity judgments in proof tutoring. In: Poster papers at KI 2006: Advances in Artificial Intelligence: 29th Annual German Conference on AI, Bremen, GermanyGoogle Scholar
  36. 36.
    Schiller M, Benzmüller C, de Veire AV (2006) Judging granularity for automated mathematics teaching. In: LPAR 2006 Short Papers Proceedings, Phnom Pehn, Cambodia,
  37. 37.
    Schiller M, Dietrich D, Benzmüller C (2008) Proof step analysis for proof tutoring – a learning approach to granularity. Teaching Mathematics and Computer Science 6(2):325–343Google Scholar
  38. 38.
    Sieg W (2007) The apros project: Strategic thinking & computational logic. Logic Journal of IGPL 15(4):359–368zbMATHCrossRefGoogle Scholar
  39. 39.
    Sommer R, Nickols G (2004) A proof environment for teaching mathematics. Journal of Automated Reasoning 32:227–258CrossRefGoogle Scholar
  40. 40.
    Tsovaltzi D, Fiedler A (2005) Human-adaptive determination of natural language hints. In: Reed C (ed) Proceedings of IJCAI-05 Workshop on Computational Models of Natural Argument (CMNA), Edinburgh, UK, pp. 84–88Google Scholar
  41. 41.
    Tsovaltzi D, Fiedler A, Horacek H (2004) A multi-dimensional taxonomy for automating hinting. In: Lester JC, Vicari RM, Paraguaçu F (eds) Intelligent Tutoring Systems — 7th International Conference (ITS 2004), Springer, no. 3220 in LNCS, pp. 772–781Google Scholar
  42. 42.
    Wolska M, Vo BQ, Tsovaltzi D, Kruijff-Korbayová I, Karagjosova E, Horacek H, Gabsdil M, Fiedler A, Benzmüller C (2004) An annotated corpus of tutorial dialogs on mathematical theorem proving. In: Proceedings of International Conference on Language Resources and Evaluation (LREC 2004), ELDA, Lisbon, PortugalGoogle Scholar
  43. 43.
    Zinn C (2006) Supporting the formal verification of mathematical texts. Journal of Applied Logic 4(4):592–621MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Christoph Benzmüller
    • 1
    Email author
  • Marvin Schiller
    • 1
  • Jörg Siekmann
    • 2
  1. 1.Department of Computer ScienceSaarland UniversitySaarbrückenGermany
  2. 2.DFKI GmbH and Saarland UniversitySaarbrückenGermany

Personalised recommendations