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Walking Through Doors Is Hard, Even Without Staircases: Proving PSPACE-Hardness via Planar Assemblies of Door Gadgets

Authors Joshua Ani, Jeffrey Bosboom, Erik D. Demaine, Yenhenii Diomidov, Dylan Hendrickson, Jayson Lynch



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Author Details

Joshua Ani
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Jeffrey Bosboom
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Erik D. Demaine
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Yenhenii Diomidov
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Dylan Hendrickson
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Jayson Lynch
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Acknowledgements

This work was initiated during open problem solving in the MIT class on Algorithmic Lower Bounds: Fun with Hardness Proofs (6.892) taught by Erik Demaine in Spring 2019. We thank the other participants of that class for related discussions and providing an inspiring atmosphere.

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Joshua Ani, Jeffrey Bosboom, Erik D. Demaine, Yenhenii Diomidov, Dylan Hendrickson, and Jayson Lynch. Walking Through Doors Is Hard, Even Without Staircases: Proving PSPACE-Hardness via Planar Assemblies of Door Gadgets. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 3:1-3:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.FUN.2021.3

Abstract

A door gadget has two states and three tunnels that can be traversed by an agent (player, robot, etc.): the "open" and "close" tunnel sets the gadget’s state to open and closed, respectively, while the "traverse" tunnel can be traversed if and only if the door is in the open state. We prove that it is PSPACE-complete to decide whether an agent can move from one location to another through a planar assembly of such door gadgets, removing the traditional need for crossover gadgets and thereby simplifying past PSPACE-hardness proofs of Lemmings and Nintendo games Super Mario Bros., Legend of Zelda, and Donkey Kong Country. Our result holds in all but one of the possible local planar embedding of the open, close, and traverse tunnels within a door gadget; in the one remaining case, we prove NP-hardness. We also introduce and analyze a simpler type of door gadget, called the self-closing door. This gadget has two states and only two tunnels, similar to the "open" and "traverse" tunnels of doors, except that traversing the traverse tunnel also closes the door. In a variant called the symmetric self-closing door, the "open" tunnel can be traversed if and only if the door is closed. We prove that it is PSPACE-complete to decide whether an agent can move from one location to another through a planar assembly of either type of self-closing door. Then we apply this framework to prove new PSPACE-hardness results for several 3D Mario games and Sokobond.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • gadgets
  • motion planning
  • hardness of games

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References

  1. Greg Aloupis, Erik D. Demaine, Alan Guo, and Giovanni Viglietta. Classic Nintendo games are (computationally) hard. Theoretical Computer Science, 586:135-160, 2015. Originally appeared at FUN 2014. Google Scholar
  2. Joshua Ani, Sualeh Asif, Erik D. Demaine, Yevhenii Diomidov, Dylan Hendrickson, Jayson Lynch, Sarah Scheffler, and Adam Suhl. PSPACE-completeness of pulling blocks to reach a goal. In Abstracts from the 22nd Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCGGG 2019), pages 31-32, Tokyo, Japan, September 2019. Google Scholar
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  6. Erik D. Demaine, Giovanni Viglietta, and Aaron Williams. Super Mario Bros. is harder/easier than we thought. In Proceedings of the 8th International Conference on Fun with Algorithms (FUN 2016), pages 13:1-13:14, La Maddalena, Italy, June 2016. Google Scholar
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  8. Alan Hazelden, Lee Shang Lun, and Allison Walker. Sokobond. https://www.sokobond.com/, 2014.
  9. Tom C. van der Zanden and Hand L. Bodlaender. PSPACE-completeness of Bloxorz and of games with 2-buttons. arXiv:1411.5951, 2014. URL: https://arXiv.org/abs/1411.5951.
  10. Giovanni Viglietta. Gaming is a hard job, but someone has to do it! Theory of Computing Systems, 54(4):595-621, 2014. Originally appeared at FUN 2012. URL: https://doi.org/10.1007/s00224-013-9497-5.
  11. Giovanni Viglietta. Lemmings is PSPACE-complete. Theoretical Computer Science, 586:120-134, 2015. Originally appeared at FUN 2014. URL: https://doi.org/10.1016/j.tcs.2015.01.055.
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