[Paper Review] Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

Zachary Abel, Jeffrey Bosboom+6 more|arXiv (Cornell University)|Jan 1, 2018

Advanced Image and Video Retrieval Techniques5 citations

TL;DR

This paper analyzes the computational complexity of pencil-and-paper puzzles in The Witness video game, showing that most puzzle types—such as hexagons, triangles, squares, stars, polyominoes, and antipolyominoes—are NP-complete to solve, meaning witnesses (valid solutions) exist but are hard to find. The introduction of antibodies, which cancel other clues, elevates the problem to Σ₂-complete, indicating that witnesses may not exist at all, even with a single antibody.

ABSTRACT

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding Sigma_2-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies.

Motivation & Objective

• To systematically analyze the computational complexity of all single-panel puzzle types in The Witness.
• To determine whether valid solution paths (witnesses) exist and can be efficiently found for each puzzle type.
• To explore the impact of new clue types—especially antibodies—on solution existence and complexity.
• To identify cases where polynomial-time algorithms exist, such as for monomino clues or boundary hexagons.
• To classify puzzle types by their expressive power in representing paths and region decompositions.

Proposed method

• Reduction-based proofs to establish NP-completeness for most puzzle types, including hexagons, triangles, squares, stars, and polyominoes.
• Construction of puzzle instances that simulate known NP-complete problems like Exact Cover and 3-Partition.
• Use of planar graph embeddings and boundary terminal constraints to reduce subset Hamiltonian path problems to polynomial time.
• Introduction of 'antibody' clues that cancel the effect of other clues, leading to Σ₂-completeness via existential and universal quantifier alternation.
• Reduction of monomino and antimonomino puzzles to boundary hexagon problems, enabling polynomial-time solvability.
• Use of component puzzles with disjoint and connected constraints to enforce specific path and region decompositions.

Experimental results

Research questions

• RQ1 Is the path-finding problem for Witness puzzles with only vertex hexagons solvable in polynomial time?
• RQ2 Is the problem NP-hard when only a constant number of star colors are used?
• RQ3 What is the complexity of light-bridge metapuzzles in The Witness?
• RQ4 Can the puzzle design problem (constructing puzzles with specific solution structures) be solved efficiently?
• RQ5 Are polyomino clues 1-region-universal, meaning can they express any desired region decomposition?

Key findings

• Any puzzle type other than broken edges (which are trivial) is NP-complete, even on rectangular grids.
• The presence of antibodies makes the problem Σ₂-complete, indicating that solution existence is not guaranteed and is computationally harder than NP.
• Monomino clues can be solved in polynomial time by reducing them to boundary hexagon problems.
• Polyomino and antipolyomino clues are NP-complete, even when restricted to monominoes or dominoes.
• The problem remains Σ₂-complete even with a single antibody, and adding more antibodies does not increase complexity.
• The complexity of the puzzle design problem remains open, though it is suspected to be hard.