Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible (1804.10193v4)

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 simple 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. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving "subset Hamiltonian path" for terminals on the boundary of an embedded planar graph in polynomial time.

• The paper establishes that distinct puzzle types in The Witness exhibit computational complexities ranging from P to Σ2-completeness through rigorous grid-graph analyses.
• The study adapts Slitherlink proofs to show NP-completeness for puzzles with squares, stars, and triangles under specific constraints.
• The introduction of antibodies and various polyomino configurations highlights novel complexity paradigms that inform both game design and theoretical computer science.

Computational Complexity of Puzzles in "The Witness"

The paper "Who witnesses The Witness?" provides a comprehensive analysis of the computational complexity underlying the puzzles in the 2016 video game "The Witness." The authors explore a variety of puzzle types, each defined by unique constraints on drawing paths through grid graphs, and they detail the complexity results for these puzzles both individually and in combination with others. Their findings indicate a range of complexity classes from P to $\Sigma_2$-completeness, revealing the depth and variety of logical reasoning embedded within the game's design.

Types of Puzzles and Complexity Results

1. Broken Edges and Hexagons: The paper demonstrates that puzzles consisting solely of broken edges can be solved in logarithmic space (L), while those with hexagons are shown to have varying complexities. Hexagons on the edges are NP-complete, yet they can be efficiently solved in specific boundary cases. Conversely, the complexity with hexagons on vertices, either exclusive or with broken edges, remains an open problem.
2. Squares and Stars: Building on known NP-completeness from Slitherlink for other shapes, the authors extend this to reveal that squares of two colors are NP-complete. Interestingly, the challenge extends into puzzles containing star clues, which are shown to be NP-hard for multiple colors, though the case for a single color remains open.
3. Triangles: The paper reaffirms known NP-completeness for Slitherlink with triangle clues, adapting proofs to cases unique to "The Witness" where 0-clue triangles aren’t available. Each type of triangle, ranging from 1- to 3-triangle configurations, has demonstrated NP-hardness.
4. Polyominoes and Antipolyominoes: For puzzles involving polyominoes, the landscape of computational complexity diversifies. While monominoes are solvable in polynomial time, the presence of antimonominoes or more complex polyomino shapes such as dominoes heralds NP-completeness.
5. Antibodies: The introduction of antibodies, which can cancel other constraints, adds a layer of complexity pushing problems into higher complexity classes. Even with multiple antibodies interacting with problems involving polyominoes, tasks become $\Sigma_2$-complete.

Implications and Future Directions

The results underscore that certain simple-looking puzzles can harbor inherently complex computational challenges. This has implications not only for game design but also offers insights into problem formulation in theoretical computer science. The paper frames how apparently straightforward pathfinding puzzles can encapsulate significant logical conundrums, a reflection upon the intricacies behind "The Witness."

Theoretical implications include foundational proofs and new problems in computational games, with suggested future inquiries into more complex puzzle architectures and universality challenges. For example, considerations around inter-puzzle interactions, recursion, and environmental aspects offer fertile ground for extending analysis to puzzles outside of "The Witness."

Moreover, open problems such as constraints enforced purely by symmetry or the universality of specific puzzle types invite further exploration—potentially integrating techniques from mixed problem instances or exploring fixed-parameter tractability under various constraints. These directions are poised to push forward our understanding of logical problem-solving within complex frameworks and their applications in entertainment, education, and beyond.