Defying Gravity: The Complexity of the Hanano Puzzle

TL;DR

This paper introduces a new property of Nondeterministic Constraint Logic instances using visibility representations, reducing gadget complexity, and establishes the PSPACE-completeness of the Hanano Puzzle through a novel reduction.

Contribution

It presents a simplified gadget construction for NCL-based hardness proofs and demonstrates PSPACE-completeness of the Hanano Puzzle via an innovative reduction.

Findings

Reduced the number of gadgets from 32 to 3 in general cases

Established the PSPACE-completeness of the Hanano Puzzle

Proposed a new framework for studying games with irreversible gravity

Abstract

Using the notion of visibility representations, our paper establishes a new property of instances of the Nondeterministic Constraint Logic (NCL) problem (a PSPACE-complete problem that is very convenient to prove the PSPACE-hardness of reversible games with pushing blocks). Direct use of this property introduces an explosion in the number of gadgets needed to show PSPACE-hardness, but we show how to bring that number from 32 down to only three in general, and down to two in a specific case! We propose it as a step towards a broader and more general framework for studying games with irreversible gravity, and use this connection to guide an indirect polynomial-time many-one reduction from the NCL problem to the Hanano Puzzle -- which is NP-hard -- to prove it is in fact PSPACE-complete.