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Combinatorial Games: From Theoretical Solving to AI Algorithms

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Scalable Uncertainty Management (SUM 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9858))

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Abstract

Combinatorial game solving is a research field that is frequently highlighted each time a program defeats the best human player: Deep Blue (IBM) vs Kasparov for Chess in 1997, and Alpha Go (Google) vs Lee Sedol for the game of Go in 2016. But what is hidden behind these success stories? First of all, I will consider combinatorial games from a theoretical point of view. We will see how to proceed to properly define and deal with the concepts of outcome, value, and winning strategy. Are there some games for which an exact winning strategy can be expected? Unfortunately, the answer is no in many cases (including some of the most famous ones like Go, Othello, Chess or Checkers), as exact game solving belongs to the problems of highest complexity. Therefore, finding out an effective approximate strategy has highly motivated the community of AI researchers. In the current survey, the basics of the best AI programs will be presented, and in particular the well-known Minimax and Monte-Carlo Tree Search approaches.

Supported by the ANR-14-CE25-0006 project of the French National Research Agency and the CNRS PICS-07315 project.

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References

  1. Allis, L.V.: Searching for solutions in games an artificial intelligence. Ph.D. Maastricht, Limburg University, Netherland (1994)

    Google Scholar 

  2. Berlekamp, E., Conway, J.H., Guy, R.K.: Winning ways for your mathematical plays, vol. 1, 2nd edn. A K Peters Ltd., Natick (2001)

    MATH  Google Scholar 

  3. Bernstein, A., Roberts, M.: Computer V. Chess player. Sci. Am. 198, 96–105 (1958)

    Article  Google Scholar 

  4. Bodlaender, H.L., Kratsch, D.: Kayles and numbers. J. Algorithms 43, 106–119 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonnet, E., Saffidine, A.: Complexit des Jeux (in french). Bulletin de la ROADEF 31, 9–12 (2014)

    Google Scholar 

  6. Bouton, C.L.: Nim, a game with a complete mathematical theory. Ann. Math. 3, 35–39 (1905)

    Article  MathSciNet  MATH  Google Scholar 

  7. Browne, C., Powley, E., Whitehouse, D., Lucas, S., Cowling, P.I., Rohlfshagen, P., Tavener, S., Perez, D., Samothrakis, S., Colton, S.: A survey of monte carlo tree search methods. IEEE Trans. Comput. Intell. AI Games 4(1), 1–43 (2012)

    Article  Google Scholar 

  8. Conway, J.H.: On Number and Games. Academic Press Inc., Cambridge (1976)

    MATH  Google Scholar 

  9. Coulom, R.: Efficient selectivity and backup operators in Monte-Carlo tree search. In: Proceedings of the 5th International Conference on Computers and Games, Turin, Italy, pp. 72–83 (2006)

    Google Scholar 

  10. Coulom, R.: Criticality: a Monte-Carlo Heuristic for Go Programs. Invited talk in University of Electro-Communication, Tokyo (2009)

    Google Scholar 

  11. Duchêne, E., Fraenkel, A.S., Gurvich, V., Ho, N.B., Kimberling, C., Larsson, U.: Wythoff Wisdom (preprint)

    Google Scholar 

  12. Edwards, D.J., Hart, T.P.: The \(\alpha - \beta \) heuristic. In: Artificial intelligence project RLE and MIT Computation Centre, Memo 30 (1963)

    Google Scholar 

  13. Enzenberger, M., Muller, M., Arneson, B., Segal, R.: Fuego an open-source framework for board games and Go engine based on Monte Carlo tree search. IEEE Trans. Comput. Intell. AI Games 2(4), 259–270 (2010)

    Article  Google Scholar 

  14. Fabbri, A., Armetta, F., Duchne, E., Hassas, S.: A self-acquiring knowledge process for MCTS. Int. J. Artif. Intell. Tools 25(1), 1660007 (2016)

    Article  Google Scholar 

  15. Gelly, S., Silver, D.: Combining online and offline knowledge in UCT. In: Ghahramani, Z. (ed.) Proceedings of the International Conference on Machine Learning (ICML), pp. 273–280. ACM, New York (2007)

    Google Scholar 

  16. Hearn, R.A., Demaine, E.D.: Games, Puzzles, and Computation. A K Peters (2009)

    Google Scholar 

  17. Fraenkel, A.S.: Nim is easy, chess is hard - but why? J. Int. Comput. Games Assoc. 29, 203–206 (2006)

    Google Scholar 

  18. Fraenkel, A.S.: Complexity, appeal and challenges of combinatorial games. Theor. Comput. Sci. 313, 393–415 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fraenkel, A.S., Simonson, S.: Geography. Theor. Comput. Sci. 110, 197–214 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gardner, M.: Mathematical games: cram, crosscram and quadraphage: new games having elusive winning strategies. Sci. Am. 230, 106–108 (1974)

    Article  Google Scholar 

  21. Junghanns, A., Schaeffer, J.: Sokoban: enhancing general single-agent search methods using domain knowledge. Artif. Intell. 129(1), 219–251 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kocsis, L., Szepesvári, C.: Bandit based Monte-Carlo planning. In: Fürnkranz, J., Scheffer, T., Spiliopoulou, M. (eds.) ECML 2006. LNCS (LNAI), vol. 4212, pp. 282–293. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  23. Larsson, U., Nowakowski, R.J., Santos, C.: Theory, Absolute Combinatorial Game. arXiv:1606.01975 (2016)

  24. Larsson, U., Nowakowski, R.J., Santos, C.: When waiting moves you in scoring combinatorial games. arXiv:1505.01907 (2015)

  25. Renault, G., Schmidt, S.: On the complexity of the misre version of three games played on graphs (preprint)

    Google Scholar 

  26. Rimmel, A., Teytaud, F., Teytaud, O.: Biasing Monte-Carlo simulations through RAVE values. In: van den Herik, H.J., Iida, H., Plaat, A. (eds.) CG 2010. LNCS, vol. 6515, pp. 59–68. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  27. Rougetet, L.: Combinatorial games and machines. In: Pisano, R. (ed.) A Bridge between Conceptual Frameworks, Sciences, Society and Technology Studies, Pisano, vol. 27, pp. 475–494. Springer, Dordrecht (2015)

    Chapter  Google Scholar 

  28. Schaeffer, T.J.: On the complexity of some two-person perfect-information games. J. Comput. Syst. Sci. 16, 185–225 (1978)

    Article  MathSciNet  Google Scholar 

  29. Schaeffer, J., Burch, N., Bjrnsson, Y., Kishimoto, A., Mller, M., Lake, R., Lu, P., Sutphen, S.: Checkers is solved. Science 317(5844), 1518–1522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shannon, C.: Programming a computer for playing chess. Philos. Mag. Ser. 7 41(314), 256–275 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  31. Siegel, A.N.: Combinatorial Game Theory. American Mathematical Society, San Francisco (2013)

    MATH  Google Scholar 

  32. Silver, D., Huang, A., Maddison, C.J., Guez, A., Sifre, L., Van Den Driessche, G., Schrittwieser, J., Antonoglou, I., Panneershelvam, V., Lanctot, M., et al.: Mastering the game of Go with deep neural networks and tree search. Nature 529(7587), 484–489 (2016)

    Article  Google Scholar 

  33. Tak, M.J.W., Winands, M.H.M., Bjornsson, Y.: N-grams and the last-good-reply policy applied in general game playing. IEEE Trans. Comput. Intell. AI Games 4(2), 73–83 (2012)

    Article  Google Scholar 

  34. Tsuruoka, Y., Yokoyama, D., Chikayama, T.: Game-tree search algorithm based on realization probability. ICGA J. 25(3), 132–144 (2002)

    Google Scholar 

  35. Wang, Y., Gelly, S.: Modifications of UCT and sequence-like simulations for Monte-Carlo Go. In: IEEE Symposium on Computational Intelligence and Games, Honolulu, Hawai, pp. 175–182 (2007)

    Google Scholar 

  36. Winands, M.: Monte-Carlo tree search in board games. In: Nakatsu, R., Rauterberg, M., Ciancarini, P. (eds.) Handbook of Digital Games and Entertainment Technologies, pp. 1–30. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Université de Lyon, CNRS Université Lyon 1, LIRIS, UMR5205, 69622, Lyon, France

    Eric Duchêne

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  1. Eric Duchêne
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Correspondence to Eric Duchêne .

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Editors and Affiliations

  1. Cardiff University , Cardiff, United Kingdom

    Steven Schockaert

  2. Télécom ParisTech , Paris, Paris, France

    Pierre Senellart

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Duchêne, E. (2016). Combinatorial Games: From Theoretical Solving to AI Algorithms. In: Schockaert, S., Senellart, P. (eds) Scalable Uncertainty Management. SUM 2016. Lecture Notes in Computer Science(), vol 9858. Springer, Cham. https://doi.org/10.1007/978-3-319-45856-4_1

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  • DOI: https://doi.org/10.1007/978-3-319-45856-4_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45855-7

  • Online ISBN: 978-3-319-45856-4

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