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

TL;DR

The paper investigates the decidability of solving levels in several 2D Mario games by proving $RE$-completeness (undecidability) through a novel counter-gadget framework that encodes Minsky-style counter machines inside game levels. It introduces a compact set of infinite-state gadgets (e.g., Inc-Dec-JZ, Inc-JZDec, Inc-DecNZ-PZ, and Inc$[a,b]$-DecNZ$[c,d]$-PZ) and uses planar, single-player reductions to establish $RE$-hardness, including constant-size level results via Korec’s universal counter machines. The results extend across the New Super Mario Bros. series and all Super Mario Maker styles (with both normal and 3D World variants), often leveraging global memory effects and event-driven mechanics to store and manipulate counter values. These findings reveal substantial computational universality in mainstream video games and highlight how careful gadget design can encode complex Decision/Computability problems within seemingly simple platformers.

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.

Figures (25)

• Figure 1: An Inc-JZDec counter gadget in Braid, built from the Level Pull, Counter, and Branch gadgets of braid.
• Figure 2: A 1-toggle, along with its state diagram on the right. When the player goes from $a$ to $b$, the arrow flips.
• Figure 3: Counter components that are allowed in counter gadgets, along with their sets $T$ of allowed traversals.
• Figure 4: The Inc-Dec-JZ gadget, shown in states 0 and 1.
• Figure 5: The Inc-DecNZ-DecNZ gadget, along with its simulation by Inc-Dec-JZ gadgets.

Theorems & Definitions (17)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Theorem 3.1