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

TL;DR

This paper establishes the PSPACE-completeness of Push-1F and Block Dude puzzles, using a novel framework of checkable gadgets to analyze the computational complexity of these popular video game-inspired puzzles.

Contribution

It introduces a new framework for checkable gadgets that extends existing motion-planning frameworks, proving PSPACE-completeness for Push-1F, Push-k, and Block Dude puzzles.

Findings

Push-1F is PSPACE-complete.

Block Dude with liftable or pushable blocks is PSPACE-complete.

New checkable gadgets framework supports complex reductions.

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.