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You Can't Solve These Super Mario Bros. Levels: Undecidable Mario Games

Authors MIT Hardness Group, Hayashi Ani, Erik D. Demaine , Holden Hall, Ricardo Ruiz, Naveen Venkat

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MIT Hardness Group
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Hayashi Ani
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Erik D. Demaine
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Holden Hall
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Ricardo Ruiz
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Naveen Venkat
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA

Acknowledgements

This paper was initiated during open problem solving in the MIT class on Algorithmic Lower Bounds: Fun with Hardness Proofs (6.5440) taught by Erik Demaine in Fall 2023. We thank the other participants of that class for helpful discussions and providing an inspiring atmosphere. Portions of this paper originally appeared in Ani’s master’s thesis [Hayashi Ani, 2023]. Several community level editors and emulators were very helpful in building and testing counters: - Reggie! by the NSMBW Community - https://github.com/NSMBW-Community/Reggie-Updated - CoinKiller by Arisotura - https://github.com/Arisotura/CoinKiller - Miyamoto by aboood40091 - https://github.com/aboood40091/Miyamoto - Dolphin by the Dolphin Emulator Project - https://dolphin-emu.org/ - Cemu by Team Cemu - https://cemu.info/ - Citra by Citra Emu - https://github.com/citra-emu/citra

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MIT Hardness Group, Hayashi Ani, Erik D. Demaine, Holden Hall, Ricardo Ruiz, and Naveen Venkat. You Can't Solve These Super Mario Bros. Levels: Undecidable Mario Games. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.FUN.2024.22

Abstract

We prove RE-completeness (and thus undecidability) of several 2D games in the Super Mario Bros. platform video game series: the New Super Mario Bros. series (original, Wii, U, and 2), and both Super Mario Maker games in all five game styles (Super Mario Bros. 1 and 3, Super Mario World, New Super Mario Bros. U, and Super Mario 3D World). These results hold even when we restrict to constant-size levels and screens, but they do require generalizing to allow arbitrarily many enemies at each location and onscreen, as well as allowing for exponentially large (or no) timer. In our Super Mario Maker reductions, we work within the standard screen size and use the property that the game engine remembers offscreen objects that are global because they are supported by "global ground". To prove these Mario results, we build a new theory of counter gadgets in the motion-planning-through-gadgets framework, and provide a suite of simple gadgets for which reachability is RE-complete.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • video games
  • computational complexity
  • undecidability

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References

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