Computational Complexity of Two-Dimensional Platform Games

  • Michal Forišek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6099)

Abstract

We analyze the computational complexity of various two-dimensional platform games. We state and prove several meta-theorems that identify a class of these games for which the set of solvable levels is NP-hard, and another class for which the set is even PSPACE-hard. Notably CommanderKeen is shown to be NP-hard, and PrinceOfPersia is shown to be PSPACE-complete.

We then analyze the related game Lemmings, where we construct a set of instances which only have exponentially long solutions. This shows that an assumption by Cormode in [3] is false and invalidates the proof that the general version of the Lemmings decision problem is in NP. We then augment our construction to only include one entrance, which makes our instances perfectly natural within the context of the original game.

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References

  1. 1.
    Bouton, C.L.: Nim, a game with a complete mathematical theory. Annals of Mathematics 3, 35–39 (1901/2002)Google Scholar
  2. 2.
    Conway, J.H.: On Numbers and Games. Academic Press, London (1976)MATHGoogle Scholar
  3. 3.
    Cormode, G.: The Hardness of the Lemmings Game, or Oh no, more NP-Completeness Proofs. In: Proceedings of Third International Conference on Fun with Algorithms, pp. 65–76 (2004)Google Scholar
  4. 4.
    Culbertson, J.: Sokoban is PSPACE-complete. In: Proceedings of the International Conference on Fun with Algorithms, pp. 65–76 (1998)Google Scholar
  5. 5.
    Demaine, E.D., Hearn, R.A.: Constraint logic: A uniform framework for modeling computation as games. In: Proceedings of the 23rd Annual IEEE Conference on Computational Complexity (2008)Google Scholar
  6. 6.
    Demaine, E.D., Hearn, R.A.: Playing games with algorithms: Algorithmic combinatorial game theory. In: Albert, M.H., Nowakowski, R.J. (eds.) Games of No Chance 3. Mathematical Sciences Research Institute Publications, vol. 56, pp. 3–56. Cambridge University Press, Cambridge (2009)Google Scholar
  7. 7.
    Eppstein, D.: Computational Complexity of Games and Puzzles (2009), http://www.ics.uci.edu/~eppstein/cgt/hard.html
  8. 8.
    Grundy, P.M.: Mathematics and games. Eureka 2, 6–8 (1939)Google Scholar
  9. 9.
    Itai, A., Papadimitriou, C.H., Szwarcfiter, J.L.: Hamilton Paths in Grid Graphs. SIAM Journal on Computing 11(4), 676–686 (1982)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kaye, R.: Minesweeper is NP-complete. Mathematical Intelligencer 22(2), 9–15 (2000)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kendall, G., Parkes, A., Spoerer, K.: A Survey of NP-Complete Puzzles. International Computer Games Association Journal 31(1), 13–34 (2008)Google Scholar
  12. 12.
    Lichtenstein, D.: Planar Formulae and Their Uses. SIAM Journal on Computing 11(2), 329–343 (1982)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    McCarthy, J.: Partial formalizations and the Lemmings game, Technical report, Stanford University, Formal Reasoning Group (1998)Google Scholar
  14. 14.
    Robin, G.: Estimation de la fonction de Tchebychef θ sur le k-ième nombre premier et grandes valeurs de la fonction ω(n) nombre de diviseurs premiers de n. Acta Arith. 42(4), 367–389 (1983)MATHMathSciNetGoogle Scholar
  15. 15.
    Robson, J.M.: The complexity of Go. In: Proceedings of the IFIP 9th World Computer Congress on Information Processing, pp. 413–417 (1983)Google Scholar
  16. 16.
    Robson, J.M.: N by N Checkers is EXPTIME complete. SIAM Journal on Computing 13(2), 252–267 (1984)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Spoerer, K.: The Lemmings Puzzle: Computational Complexity of an Approach and Identification of Difficult Instances. PhD thesis (2007)Google Scholar
  18. 18.
    Sprague, R.P.: Ueber mathematische Kampfspiele. Tohoku Mathematical Journal 41, 438–444 (1935/1936)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Michal Forišek
    • 1
  1. 1.Comenius UniversityBratislavaSlovakia

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