Walking through Doors is Hard, even without Staircases: Universality and PSPACE-hardness of Planar Door Gadgets

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 gadgets. In particular, the opening, closing, and traverse tunnel locations can have an arbitrary cyclic order around the door; each tunnel can be directed or undirected; and the opening tunnel can instead be an optional button (with identical entrance and exit locations). Furthermore, we show the same hardness and universality results for two simpler types of door gadgets: self-closing door gadgets and symmetric self-closing door gadgets. Again we show that any self-closing door gadget planarly simulates any gadget, and thus the reachability motion planning problem is PSPACE-complete. Then we apply this framework to prove new PSPACE-hardness results for eight different 3D Mario video games and Sokobond.

Figures (35)

• Figure 1: State diagram of a 2-state 6-location 5-transition gadget (the directed open-tunneled open--close door from Figure \ref{['fig:different-doors-directed']}). The left and right copies represent the two states ("closed" and "open"). In each state, we draw the allowed transitions as arrows, labeled with the resulting state change, if any. Dotted transitions are forbidden and are just drawn for intuition (and to clarify locations).
• Figure 2: State diagrams for a few different open--close doors. The opening button/tunnel is green, the traverse tunnel is blue, and the closing tunnel is red.
• Figure 3: State diagrams for a few different self-closing doors. The opening button/tunnel is green, and self-opening/self-closing tunnel(s) are purple with an X (to indicate that they close themselves when traversed). Dotted traversals are forbidden.
• Figure 4: Implementation of an open--close door (a--b), self-closing door (c--d), and symmetric self-closing door (e--f) in any pushing-block game that allows pushing a polyomino piece (drawn here as a molecule, in the style of Sokobond Sokobond). The symmetric self-closing door gadget is itself $180^\circ$ rotationally symmetric.
• Figure 5: Any self-closing door or symmetric self-closing door planarly simulates an open-buttoned self-closing door.

Theorems & Definitions (47)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Lemma 2.5
• proof