You Can't Solve These Super Mario Bros. Levels: Undecidable Mario Games (2405.10546v1)

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. Our New Super Mario Bros. constructions fit within one standard screen size. 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.

• The paper demonstrates RE-completeness in Mario levels by reducing game mechanics to counter machines.
• It employs natural game elements like enemy behaviors, level timers, and obstacle positioning to construct counter gadgets simulating finite-state systems.
• The findings bridge video game design and theoretical computer science, offering new insights for AI applications and complex system analysis.

Undecidability in Super Mario Bros. Video Games

This paper explores the undecidability of several levels within the Super Mario Bros. video game series by drawing parallels with the computational concept of RE-completeness. This concept, related to the halting problem, indicates scenarios where it's impossible to determine through computational means whether a given procedure will eventually complete or continue indefinitely. Here, the authors extend previous work on Braid to newer Mario games, leveraging video game mechanics to illustrate complex computational concepts.

The authors demonstrate RE-completeness for games in the New Super Mario Bros. series (including variations across multiple platforms and generations) and both Super Mario Maker titles, addressing all available game styles. They establish the undecidability for these games by constructing a relationship between Mario game mechanics and counter machines using elements such as enemies, event mechanics, and memory persistence. A counter machine is a model of computation that employs a set number of instructions and counters to simulate more extensive computational processes, akin to a Turing machine's function.

The authors achieve their results even when constraining the games to constant-sized levels and screens, modifying the gameplay to allow unlimited enemies per screen and extended level timers. They use the concept of "global ground" and devise new counter gadget theories to simplify the reachability problem within these games. By correctly balancing in-game elements, such as Goombas, the paper reinterprets each counter gadget into RE-complete processes, thereby creating an equivalency between gameplay mechanics and non-trivial computational problems.

To support these claims, the paper discusses methods for constructing gadgets based on natural game mechanics and accounting for positioning constraints, obstacles, and traversal paths, which parallels the operation of finite-state machines. These constructions enable Mario games to simulate counter machine operations, further extending their undecidability. The construct approach leverages existing game physics and mathematical arrangements to systematically control enemy behavior, providing a tangible manifestation of computationally challenging problems within simple game scenarios.

Significant contributions include the breakdown of each Mario title's potential undecidability by utilizing existing game elements to simulate specific computational processes. For instance, the paper discusses using spawning and state-remembrance mechanics from Super Mario Maker to effectively build computational pathways, achieving RE-completeness. Through counter machines, the paper shows that Mario games can essentially reach undecidable scenarios, solidifying the link between video game design and computational complexity.

These findings not only reveal the inherent computational depth within such video games but also offer potential advancements in AI and video game design. The results may also inspire future work to explore computational complexity within similar digital environments, potentially broadening the application scope of game-derived paper methodologies. The documentation of practical reductions from computational problems to in-game environments presents a noteworthy intersection of entertainment and computer science, with implications for both theoretical research and engineering practices.

Future developments in AI could draw from these insights, using game environments as a testing ground for understanding complex decision-making algorithms and exploring undecidability in dynamic systems. There's an engaging potential in further exploring older or less canonical Mario series titles or similar games, where altering enemy behavior or environmental interaction could yield similarly fascinating computational insights.