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

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LIPIcs.FUN.2021.3.pdf
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• 23 pages

## 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.

## Cite As

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
• 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.
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.
3. Erik D. Demaine, Martin L. Demaine, Michael Hoffmann, and Joseph O'Rourke. Pushing blocks is hard. Computational Geometry: Theory and Applications, 26(1):21-36, August 2003.
4. Erik D. Demaine, Isaac Grosof, Jayson Lynch, and Mikhail Rudoy. Computational complexity of motion planning of a robot through simple gadgets. In Proceedings of the 9th International Conference on Fun with Algorithms (FUN 2018), pages 18:1-18:21, La Maddalena, Italy, June 2018.
5. Erik D. Demaine, Dylan H. Hendrickson, and Jayson Lynch. Toward a general complexity theory of motion planning: Characterizing which gadgets make games hard. In Proceedings of the 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), pages 62:1-62:42, Seattle, January 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.62.
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.
7. Michal Forišek. Computational complexity of two-dimensional platform games. In Proceedings of the 5th International Conference on Fun with Algorithms (FUN 2010), volume 6099 of Lecture Notes in Computer Science, 2010. URL: https://doi.org/10.1007/978-3-642-13122-6_22.
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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