PSPACE-Hard 2D Super Mario Games: Thirteen Doors

TL;DR

This work extends the door-gadget framework to 15 two-dimensional Super Mario games, establishing $PSPACE$-hardness for thirteen titles and $NP$-hardness for two (Super Mario Land and Super Mario Run) under generalized game models. By formalizing generalized Mario with unbounded level size and time limits while constraining powerups and using planar door gadgets, the authors construct detailed open-close, self-closing, and symmetric self-closing gadgets across the titles, often adapting mechanics unique to each game. The results substantially map the computational hardness landscape of 2D Mario platforms, showing that a wide range of classic titles admit PSPACE-hardness reductions, with two exceptions remaining in the NP-hard regime. The paper outlines open problems and suggests that further work could resolve remaining PSPACE vs NP questions and explore undecidability in related settings, reinforcing the broader applicability of the door-gadget methodology.

Abstract

We prove PSPACE-hardness for fifteen games in the Super Mario Bros. 2D platforming video game series. Previously, only the original Super Mario Bros. was known to be PSPACE-hard (FUN 2016), though several of the games we study were known to be NP-hard (FUN 2014). Our reductions build door gadgets with open, close, and traverse traversals, in each case using mechanics unique to the game. While some of our door constructions are similar to those from FUN 2016, those for Super Mario Bros. 2, Super Mario Land 2, Super Mario World 2, and the New Super Mario Bros. series are quite different; notably, the Super Mario Bros. 2 door is extremely difficult. Doors remain elusive for just two 2D Mario games (Super Mario Land and Super Mario Run); we prove that these games are at least NP-hard.

Figures (15)

• Figure 1: State diagrams for the PSPACE-complete gadgets used in this paper. Each box denotes a state (labeled Open or Closed), and each arrow denotes a possible transition in that state, labeled with the new state that the gadget enters upon such traversal.
• Figure 2: An open-close door in Super Mario Bros.: The Lost Levels
• Figure 3: An open-close door in Super Mario Bros. 3
• Figure 4: An open-close door in New Super Mario Bros. Wii
• Figure 5: An open-close door in Super Mario World

Theorems & Definitions (28)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5
• proof