Pushing Blocks via Checkable Gadgets: PSPACE-completeness of Push-1F and Block/Box Dude

TL;DR

The paper resolves longstanding questions about the complexity of pushing-block puzzles by proving PSPACE-completeness for Push-1F and all Push-k variants with $k \ge 2$, and for BlockDude, BloxDude, and BoxDude under gravity or lifting rules. It introduces a versatile checkable gadget framework that uses postselection to force end-stage checks, enabling robust nonlocal simulations that preserve planarity. The methodology builds on and extends the motion-planning-through-gadgets paradigm, employing gadgets such as diodes, self-closing doors, and locking 2-toggles to transfer PSPACE-hardness from known planar reachability problems. These results unify several puzzle families under a common gadget-based hardness approach and settle multiple open problems in the literature. The techniques open avenues for analyzing storage-variant puzzle versions and suggest further exploration of postselection-based gadget constructions.

Abstract

We prove PSPACE-completeness of the well-studied pushing-block puzzle Push-1F, a theoretical abstraction of many video games (introduced in 1999). The proof also extends to Push-$k$ for any $k \ge 2$. We also prove PSPACE-completeness of two versions of the recently studied block-moving puzzle game with gravity, Block Dude - a video game dating back to 1994 - featuring either liftable blocks or pushable blocks. Two of our reductions are built on a new framework for "checkable" gadgets, extending the motion-planning-through-gadgets framework to support gadgets that can be misused, provided those misuses can be detected later.

Figures (38)

• Figure 1: Sample Push-1F puzzle and solution sequence. In steps (c) and (e), for example, the agent cannot push right again. The agent is drawn as a robot head; the traversed path between steps is drawn as a gray line; pushable blocks are drawn as boxes; fixed blocks are drawn as brick walls; and the goal location is drawn as a flag. Robot and flag icons from Font Awesome under CC BY 4.0 License.
• Figure 2: A broken Push-1F diode gadget.
• Figure 3: The top three rows of the Push-1F diode gadget of Figure \ref{['fig:broken diode']}, as a checkable gadget. The checking traversals are "check 1 in $\rightarrow$ check 1 out" and "check 2 in $\rightarrow$ check 2 out", denoted by the hollow arrows.
• Figure 6: State diagram for the locking 2-toggle gadget. Each box represents the gadget in a different state, in this case labeled with the numbers $1,2,3$. Dots represent the four locations of the gadget. Arrows represent transitions in the gadget and are labeled with the states to which those transitions take the gadget. In state 2, the agent can traverse either tunnel going down, which blocks off both downward traversals until the agent reverses that traversal.
• Figure 7: State diagram for a nondeterministic locking 2-toggle. From state 1, the left tunnel can be traversed so as to leave the gadget in either state 2 or state 4. Formally, in the multigraph for state 1 there are two different edges, one labeled 2 and the other labeled 4.

Theorems & Definitions (14)

• Definition 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Definition 5.1
• Definition 5.2
• Theorem 5.3
• Lemma 5.4
• proof