Walking through Doors is Hard, even without Staircases: Universality and PSPACE-hardness of Planar Door Gadgets
MIT Gadgets Group, Jeffrey Bosboom, Erik D. Demaine, Jenny Diomidova, Dylan Hendrickson, Hayashi Layers, Jayson Lynch

TL;DR
This paper proves that planning through planar door gadgets is PSPACE-hard and demonstrates the universality of door gadgets, simplifying previous complexity proofs and applying results to various video games.
Contribution
It establishes PSPACE-hardness and universality of planar door gadgets, simplifying previous proofs and extending results to multiple game scenarios.
Findings
Planar door gadgets are PSPACE-complete to analyze.
The open-close door gadget is a universal gadget for motion planning.
PSPACE-hardness results apply to several 3D Mario games and Sokobond.
Abstract
An open-close door gadget has two states and three tunnels that can be traversed by an agent (player, robot, etc.): the "opening" and "closing" tunnels set 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 system of any such door gadget, 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. Even stronger, we show that any gadget in the motion-planning-through-gadgets framework can be simulated by a planar system of door gadgets: the open-close door gadget is a universal gadget. We prove that these results hold for a variety of door…
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