Pushing Blocks Without Fixed Walls Via Checkable Gizmos: Push-1 Is PSPACE-Complete

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Abstract
We prove PSPACE -completeness of Push-1: given a rectangular grid of 1 × 1 cells, each possibly occupied by a movable block, can a robot move from one specified location to another, given the ability to push up to one block at a time? In particular, we remove the need for fixed (immovable) walls from a 2022 result. This fundamental model of block pushing, introduced in 1999, abstracts the mechanics of many video games. It was shown NP -hard in 2000, but its final complexity remained open for 25 years. Our result uses a new framework for checkable gadgets/gizmos, extending a prior framework for checkable gadgets to handle reconfiguration problems, at the cost of requiring a stronger auxiliary gadget. We also introduce a new connection between the motion-planning-through-gadgets framework (with an agent) and the Graph Orientation Reconfiguration Problem (with no agent), including Nondeterministic Constraint Logic.