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Arimaa Is PSPACE-Hard

Authors Benjamin G. Rin , Atze Schipper

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Benjamin G. Rin
  • Utrecht University, The Netherlands
Atze Schipper
  • Utrecht University, The Netherlands

Acknowledgements

We would like to thank Rosalie Iemhoff for her useful feedback and suggestions, as well as Colin Caret, Dominik Klein, Johannes Korbmacher, Jaap van Oosten, and all other attendees at The Utrecht Logic in Progress Series for their comments on a presentation of an early version of this paper. We also would like to thank the anonymous referees for their feedback.

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Benjamin G. Rin and Atze Schipper. Arimaa Is PSPACE-Hard. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 27:1-27:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.FUN.2024.27

Abstract

Arimaa is a strategy board game developed in 2003 by Omar Syed, designed to be hard for AI to win because of its large branching factor. In this paper, its theoretical complexity is considered. We prove that Arimaa (suitably generalized to an n × n board) is PSPACE-hard. This result is found by reducing a known PSPACE-hard variant of Generalized Geography to a variant of Arimaa that we call Arimaa^′, which in turn is then reduced to (n × n) Arimaa. Since the game is easily seen to be solvable in exponential time, it follows that its complexity lies somewhere between being PSPACE-complete and EXPTIME-complete.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Arimaa
  • complexity theory
  • PSPACE-hardness
  • board games
  • Generalized Geography

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References

  1. Joshua Ani, Jeffrey Bosboom, Erik D. Demaine, Yevhenii Diomidov, Dylan H. Hendrickson, and Jayson Lynch. Walking through doors is hard, even without staircases: Proving PSPACE-hardness via planar assemblies of door gadgets. In Martin Farach-Colton, Giuseppe Prencipe, and Ryuhei Uehara, editors, 10th International Conference on Fun with Algorithms, FUN 2021, May 30 to June 1, 2021, Favignana Island, Sicily, Italy, volume 157 of LIPIcs, pages 3:1-3:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.FUN.2021.3.
  2. Arimaa.com. The 2015 Arimaa Challenge. http://arimaa.com/arimaa/challenge/2015/, 2015. [Online; accessed 20-02-2024].
  3. Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi. On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games. In Hiro Ito, Stefano Leonardi, Linda Pagli, and Giuseppe Prencipe, editors, 9th International Conference on Fun with Algorithms (FUN 2018), volume 100 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1-8:15, Dagstuhl, Germany, 2018. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.FUN.2018.8.
  4. Edouard Bonnet, Florian Jamain, and Abdallah Saffidine. Havannah and TwixT are PSPACE-complete. In H. Jaap van den Herik, Hiroyuki Iida, and Aske Plaat, editors, Computers and Games - 8th International Conference, CG 2013, Yokohama, Japan, August 13-15, 2013, Revised Selected Papers, volume 8427 of Lecture Notes in Computer Science, pages 175-186. Springer, 2013. URL: https://doi.org/10.1007/978-3-319-09165-5_15.
  5. Christ-Jan Cox. Analysis and implementation of the game Arimaa. MSc diss., Universiteit Maastricht, The Netherlands, 2006. Google Scholar
  6. David Fotland. Building a world-champion Arimaa program. In International Conference on Computers and Games, pages 175-186. Springer, 2004. Google Scholar
  7. David Fotland. Computer Arimaa: The beginning. International Computer Games Association Journal, 38(1):12-18, 2015. URL: https://doi.org/10.3233/icg-2015-38103.
  8. Aviezri S. Fraenkel and David Lichtenstein. Computing a perfect strategy for n × n chess requires time exponential in n. J. Comb. Theory, Ser. A, 31(2):199-214, 1981. URL: https://doi.org/10.1016/0097-3165(81)90016-9.
  9. Brian Haskin. A look at the Arimaa branching factor. http://arimaa.janzert.com/bf_study, 2006. [Online; accessed 20-02-2024].
  10. David Lichtenstein and Michael Sipser. Go is polynomial-space hard. Journal of the ACM, 27(2):393–401, apr 1980. URL: https://doi.org/10.1145/322186.322201.
  11. Stefan Reisch. Hex ist PSPACE-vollständig. Acta Informatica, 15:167-191, 1981. URL: https://doi.org/10.1007/BF00288964.
  12. J. M. Robson. The complexity of Go. In R. E. A. Mason, editor, Information Processing 83, Proceedings of the IFIP 9th World Computer Congress, Paris, France, September 19-23, 1983, pages 413-417. North-Holland/IFIP, 1983. Google Scholar
  13. J. M. Robson. N by N checkers is Exptime complete. SIAM Journal on Computing, 13(2):252-267, 1984. URL: https://doi.org/10.1137/0213018.
  14. Thomas J. Schaefer. Complexity of decision problems based on finite two-person perfect-information games. In Ashok K. Chandra, Detlef Wotschke, Emily P. Friedman, and Michael A. Harrison, editors, Proceedings of the 8th Annual ACM Symposium on Theory of Computing, May 3-5, 1976, Hershey, Pennsylvania, USA, pages 41-49. ACM, 1976. URL: https://doi.org/10.1145/800113.803629.
  15. Atze Schipper. n × n Arimaa is PSPACE-hard. Bachelor’s thesis, Utrecht University, 2021. Google Scholar
  16. James A. Storer. On the complexity of chess. Journal of Computer and System Sciences, 27(1):77-100, 1983. URL: https://doi.org/10.1016/0022-0000(83)90030-2.
  17. Omar Syed. The Arimaa Challenge: From inception to completion. International Computer Games Association Journal, 38(1):3-11, 2015. URL: https://doi.org/10.3233/ICG-2015-38102.
  18. Omar Syed and Aamir Syed. Arimaa - A new game designed to be difficult for computers. International Computer Games Association Journal, 26(2):138-139, 2003. URL: https://doi.org/10.3233/ICG-2003-26213.
  19. Gerhard Trippen. Plans, patterns, and move categories guiding a highly selective search. In H. Jaap van den Herik and Pieter Spronck, editors, Advances in Computer Games, 12th International Conference, ACG 2009, Pamplona, Spain, May 11-13, 2009. Revised Papers, volume 6048 of Lecture Notes in Computer Science, pages 111-122. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-12993-3_11.
  20. Jaap van den Herik. Table of contents / editorial. International Computer Games Association Journal, 38(1):1-2, March 2015. URL: https://doi.org/10.3233/icg-2015-38101.
  21. Wikibooks. Arimaa/Playing the game. https://en.wikibooks.org/wiki/Arimaa/Playing_The_Game, 2019. [Online; accessed 20-02-2024].
  22. David J. Wu. Move ranking and evaluation in the game of Arimaa. Bachelor’s thesis, Harvard College, 2011. URL: https://api.semanticscholar.org/CorpusID:504661.
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