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A Markovian Framing of WaveFunctionCollapse for Procedurally Generating Aesthetically Complex Environments

Franklin Yiu, Mohan Lu, Nina Li, Kevin Joseph, Tianxu Zhang, Julian Togelius, Timothy Merino, Sam Earle

TL;DR

This work tackles procedural content generation by reformulating WaveFunctionCollapse (WFC) as a Markov Decision Process (MDP), enabling objective-driven generation to run alongside WFC's constraint propagation. By decoupling constraint satisfaction from optimization, the authors demonstrate that evolving an action sequence within the WFC-MDP yields faster and more reliable convergence than traditional joint optimization approaches, across binary and biome-rich domains, with performance degrading as path-length targets grow. The study carefully analyzes representation choices (1D vs 2D action encodings) and demonstrates the framework's flexibility by extending to additional biomes, while acknowledging limitations in exploration for the hardest tasks and suggesting future RL or novelty-search approaches. Practically, this decoupled, constraint-aware formulation provides a scalable, ML-friendly blueprint for creating aesthetically complex environments that meet designer-specified objectives, with potential for interactive and scalable game-map generation on consumer hardware.

Abstract

Procedural content generation often requires satisfying both designer-specified objectives and adjacency constraints implicitly imposed by the underlying tile set. To address the challenges of jointly optimizing both constraints and objectives, we reformulate WaveFunctionCollapse (WFC) as a Markov Decision Process (MDP), enabling external optimization algorithms to focus exclusively on objective maximization while leveraging WFC's propagation mechanism to enforce constraint satisfaction. We empirically compare optimizing this MDP to traditional evolutionary approaches that jointly optimize global metrics and local tile placement. Across multiple domains with various difficulties, we find that joint optimization not only struggles as task complexity increases, but consistently underperforms relative to optimization over the WFC-MDP, underscoring the advantages of decoupling local constraint satisfaction from global objective optimization.

A Markovian Framing of WaveFunctionCollapse for Procedurally Generating Aesthetically Complex Environments

TL;DR

This work tackles procedural content generation by reformulating WaveFunctionCollapse (WFC) as a Markov Decision Process (MDP), enabling objective-driven generation to run alongside WFC's constraint propagation. By decoupling constraint satisfaction from optimization, the authors demonstrate that evolving an action sequence within the WFC-MDP yields faster and more reliable convergence than traditional joint optimization approaches, across binary and biome-rich domains, with performance degrading as path-length targets grow. The study carefully analyzes representation choices (1D vs 2D action encodings) and demonstrates the framework's flexibility by extending to additional biomes, while acknowledging limitations in exploration for the hardest tasks and suggesting future RL or novelty-search approaches. Practically, this decoupled, constraint-aware formulation provides a scalable, ML-friendly blueprint for creating aesthetically complex environments that meet designer-specified objectives, with potential for interactive and scalable game-map generation on consumer hardware.

Abstract

Procedural content generation often requires satisfying both designer-specified objectives and adjacency constraints implicitly imposed by the underlying tile set. To address the challenges of jointly optimizing both constraints and objectives, we reformulate WaveFunctionCollapse (WFC) as a Markov Decision Process (MDP), enabling external optimization algorithms to focus exclusively on objective maximization while leveraging WFC's propagation mechanism to enforce constraint satisfaction. We empirically compare optimizing this MDP to traditional evolutionary approaches that jointly optimize global metrics and local tile placement. Across multiple domains with various difficulties, we find that joint optimization not only struggles as task complexity increases, but consistently underperforms relative to optimization over the WFC-MDP, underscoring the advantages of decoupling local constraint satisfaction from global objective optimization.

Paper Structure

This paper contains 37 sections, 4 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. WaveFunctionCollapse Algorithm and Modifications
  4. WaveFunctionCollapse
  5. Enhancements to WaveFunctionCollapse
  6. Evolutionary Approaches to PCG
  7. Evolutionary Algorithms
  8. FI-2Pop for Constrained Optimization
  9. Procedural Content Generation via Reinforcement Learning (PCGRL)
  10. Problem Domain
  11. Binary
  12. Biome/Binary Hybrid
  13. Methods
  14. Direct Map Evolution
  15. Baseline Evolution.
  16. ...and 22 more sections

Figures (4)

  • Figure 1: Biome Tileset Pack B - Grassland, Savannah, and Scrubland. Unused tiles are darkened. Path tiles are marked in orange, grass tiles are marked in green, water tiles are marked in blue, and hill tiles are marked in brown. The water center tile is marked in light blue.
  • Figure 2: Optimizing for target path-lengths in the Binary domain. The red line shows the longest shortest path.
  • Figure 4: Outputs resulting from the optimization of other Biome objectives
  • Figure 5: Plot which serve as a visual representation of the convergence behavior expressed in a corresponding tables. The lines correlate with the fraction of converged training samples (left y axis — higher is better) and the bars correlate to the mean generations to successfully converge (right y axis — lower is better).