Journal of Logic, Language and Information

, Volume 22, Issue 3, pp 297–314 | Cite as

An Analytic Tableaux Model for Deductive Mastermind Empirically Tested with a Massively Used Online Learning System

  • Nina Gierasimczuk
  • Han L. J. van der Maas
  • Maartje E. J. Raijmakers
Article

Abstract

The paper is concerned with the psychological relevance of a logical model for deductive reasoning. We propose a new way to analyze logical reasoning in a deductive version of the Mastermind game implemented within a popular Dutch online educational learning system (Math Garden). Our main goal is to derive predictions about the difficulty of Deductive Mastermind tasks. By means of a logical analysis we derive the number of steps needed for solving these tasks (a proxy for working memory load). Our model is based on the analytic tableaux method, known from proof theory. We associate the difficulty of Deductive Mastermind game-items with the size of the corresponding logical trees obtained by the tableaux method. We derive empirical hypotheses from this model. A large group of students (over 37 thousand children, 5–12 years of age) played the Deductive Mastermind game, which gave empirical difficulty ratings of all 321 game-items. The results show that our logical approach predicts these item ratings well, which supports the psychological relevance of our model.

Keywords

Deductive Mastermind Mastermind game Deductive reasoning Analytic tableaux Math Garden (Rekentuin) Educational tools 

References

  1. Barton, E., Berwick, R., & Ristad, E. (1987). Computational complexity and natural language. Cambridge, MA: The MIT Press.Google Scholar
  2. Berwick, R., & Weinberg, A. (1984). The grammatical basis of linguistic performance. Cambridge, MA: The MIT Press.Google Scholar
  3. Beth, E. W. (1955). Semantic entailment and formal derivability. Mededelingen van de Koninklijke Nederlandse Akademie van Wetenschappen Afdeling Letterkunde, 18(13), 309–342.Google Scholar
  4. Cherniak, C. (1986). Minimal rationality. Cambridge, MA: The MIT Press.Google Scholar
  5. Chvatal, V. (1983). Mastermind. Combinatorica, 3, 325–329.Google Scholar
  6. Elo, A. (1978). The rating of chessplayers, past and present. Arco.Google Scholar
  7. Ghosh, S., & Meijering, B. (2011). On combining cognitive and formal modeling: A case study involving strategic reasoning. In J. Van Eijck, & R. Verbrugge (Eds.) Proceedings of the workshop on reasoning about other minds: Logical and cognitive perspectives (RAOM-2011), Groningen, The Netherlands, July 11th, 2011, CEUR-WS.org. CEUR Workshop Proceedings, vol 751 (pp. 79–92).Google Scholar
  8. Ghosh, S., Meijering, B., & Verbrugge, R. (2010). Logic meets cognition: Empirical reasoning in games. In O. Boissier, A. E. Fallah-Seghrouchni, S. Hassas & N. Maudet (Eds.) Proceedings of the multi-agent logics, languages, and organisations federated workshops (MALLOW 2010), Lyon, France, August 30–September 2, 2010, CEUR-WS.org, CEUR Workshop Proceedings, vol, 627 (pp. 15–34).Google Scholar
  9. Gierasimczuk, N., & Szymanik, J. (2009). Branching quantification v. two-way quantification. Journal of Semantics, 26(4), 367–392.CrossRefGoogle Scholar
  10. Greenwell, D. L. (1999–2000). Mastermind. Journal of Recreational Mathematics, 30, 191–192.Google Scholar
  11. Irving, R. W. (1978–79). Towards an optimum Mastermind strategy. Journal of Recreational Mathematics, 11, 81–87.Google Scholar
  12. Jansen, B., & Van der Maas, H. (1997). A statistical test of the rule assessment methodology by latent class analysis. Developmental Review, 17, 321–357.CrossRefGoogle Scholar
  13. Klinkenberg, S., Straatemeier, M., & Van der Maas, H. L. J. (2011). Computer adaptive practice of maths ability using a new item response model for on the fly ability and difficulty estimation. Computers in Education, 57, 1813–1824.CrossRefGoogle Scholar
  14. Knuth, D. E. (1977). The computer as master mind. Journal of Recreational Mathematics, 9(1), 1–6.Google Scholar
  15. Kooi, B. (2005). Yet another Mastermind strategy. ICGA Journal, 28(1), 13–20.Google Scholar
  16. Koyama, M., & Lai, T. (1993). An optimal Mastermind strategy. Journal of Recreational Mathematics, 25, 251–256.Google Scholar
  17. Maris, G., & Van der Maas, H. L. J. (2012). Speed-accuracy response models: Scoring rules based on response time and accuracy. Psychometrika, 77(4), 615–633.CrossRefGoogle Scholar
  18. Meijering, B., van Rijn, H., Taatgen, N. A., & Verbrugge, R. (2012). What eye movements can tell about theory of mind in a strategic game. PLoS One, 7(9), e45961Google Scholar
  19. Pelánek, R. (2011). Difficulty rating of Sudoku puzzles by a computational model. In R. C. Murray & P. M. McCarthy (Eds.), Proceedings of the twenty-fourth international Florida artificial intelligence research society conference, May 18–20, 2011. Palm Beach, Florida, USA: AAAI Press.Google Scholar
  20. Ristad, E. (1993). The language complexity game. Cambridge, MA: The MIT Press.Google Scholar
  21. Schmittmann, V. D., Van der Maas, H. L. J., & Raijmakers, M. E. J. (2012). Distinct discrimination learning strategies and their relation with spatial memory and attentional control in 4- to 14-year-olds. Journal of Experimental Child Psychology, 111(4), 644–62.CrossRefGoogle Scholar
  22. Smullyan, R. (1968). First-order logic. Berlin: Springer.CrossRefGoogle Scholar
  23. Stuckman, J., & Zhang, G. (2006). Mastermind is NP-complete. INFOCOMP Journal of Computer Science, 5, 25–28.Google Scholar
  24. Szymanik, J. (2009). Quantifiers in time and space. Computational complexity of generalized quantifiers in natural language. PhD thesis, Universiteit van Amsterdam.Google Scholar
  25. Szymanik, J. (2010). Computational complexity of polyadic lifts of generalized quantifiers in natural language. Linguistics and Philosophy, 33(3), 215–250.CrossRefGoogle Scholar
  26. Szymanik, J., & Zajenkowski, M. (2010). Comprehension of simple quantifiers. Empirical evaluation of a computational model. Cognitive Science, 34(3), 521–532.CrossRefGoogle Scholar
  27. Van Benthem, J. (1974). Semantic tableaus. Nieuw Archief voor Wiskunde, 22, 44–59.Google Scholar
  28. Van Bers, B. M. C. W., Visser, I., Van Schijndel, T. J. P., Mandell, D. J., & Raijmakers, M. E. J. (2011). The dynamics of development on the dimensional change card sorting task. Developmental Science, 14(5), 960–71.CrossRefGoogle Scholar
  29. Van der Maas, H., & Molenaar, P. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99(3), 395–417.CrossRefGoogle Scholar
  30. Van der Maas, H., Klinkenberg, S., & Straatemeier, M. (2010). Rekentuin.nl: Combinatie van oefenen en toetsen. Examens, 4, 10–14.Google Scholar
  31. Van Rooij, I. (2008). The tractable cognition thesis. Cognitive Science, 32, 939–984.CrossRefGoogle Scholar
  32. Van Rooij, I., Kwisthout, J., Blokpoel, M., Szymanik, J., Wareham, T., & Toni, I. (2011). Intentional communication: Computationally easy or difficult? Frontiers in Human Neuroscience, 5, 1–18.Google Scholar
  33. Verbrugge, R., & Mol, L. (2008). Learning to apply theory of mind. Journal of Logic, Language and Information, 17(4), 489–511.CrossRefGoogle Scholar
  34. Zajenkowski, M., Styła, R., & Szymanik, J. (2011). A computational approach to quantifiers as an explanation for some language impairments in schizophrenia. Journal of Communication Disorders, 44(6), 595–600.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Nina Gierasimczuk
    • 1
  • Han L. J. van der Maas
    • 2
  • Maartje E. J. Raijmakers
    • 2
  1. 1.Institute for Logic, Language and ComputationUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Department of PsychologyUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations