# A Markovian Framing of WaveFunctionCollapse for Procedurally Generating Aesthetically Complex Environments

arXiv: https://arxiv.org/abs/2509.09919 · HTML: https://mlsanity.com/paper/2509.09919/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 30 references that are in the map.
- First posted on arXiv 12 Sep 2025.

## Builds on (most cited first, 12 of 12)

- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Leveraging Procedural Generation to Benchmark Reinforcement Learning](https://mlsanity.com/paper/1912.01588/) (2019)
- [Procedural Content Generation via Machine Learning (PCGML)](https://mlsanity.com/paper/1702.00539/) (2017)
- [PCGRL: Procedural Content Generation via Reinforcement Learning](https://mlsanity.com/paper/2001.09212/) (2020)
- [Level Generation Through Large Language Models](https://mlsanity.com/paper/2302.05817/) (2023)
- [Learning Controllable Content Generators](https://mlsanity.com/paper/2105.02993/) (2021)
- [Adversarial Reinforcement Learning for Procedural Content Generation](https://mlsanity.com/paper/2103.04847/) (2021)
- [Experience-Driven PCG via Reinforcement Learning: A Super Mario Bros Study](https://mlsanity.com/paper/2106.15877/) (2021)
- [DreamCraft: Text-Guided Generation of Functional 3D Environments in Minecraft](https://mlsanity.com/paper/2404.15538/) (2024)
- [Mutation Models: Learning to Generate Levels by Imitating Evolution](https://mlsanity.com/paper/2206.05497/) (2022)
- [Moonshine: Distilling Game Content Generators into Steerable Generative Models](https://mlsanity.com/paper/2408.09594/) (2024)
- [Extend Wave Function Collapse to Large-Scale Content Generation](https://mlsanity.com/paper/2308.07307/) (2023)

## Abstract (arXiv)

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.