NeurIPS Workshop on Agent Learning in Open-Endedness (ALOE) 2023
Keith Y. Patarroyo · Abhishek Sharma · Sara I. Walker · Leroy Cronin
Complex Chemistry Labs, University of Glasgow | School of Earth and Space Exploration, Arizona State University
AssemblyCA introduces a framework for measuring open-endedness in discrete dynamical systems by combining Cellular Automata (CA) with Assembly Theory — a formalism originally developed to quantify molecular complexity and the origins of life.
The central idea is that truly open-ended systems indefinitely generate novel, diverse, and complex objects. We use an assembly theory derived metric as a rigorous complexity measure that captures historical contingency — something traditional entropy or diversity metrics cannot. This allows us to distinguish between:
| Exploration Type | Description | Complexity Outcome |
|---|---|---|
| Undirected (random soup) | Random initial conditions, no selection | Low complexity, high copy number |
| Open-ended (LifeWiki community) | Human-curated discovery over 50 years | High complexity, systematically increasing |
| Algorithmic agents | Programmed modification of patterns | Moderate complexity, plateaus without selection |
The benchmark uses data collected up to 2023, drawing from the Game of Life Catagolue census, the LifeWiki pattern database, and agent-based searches.
- Hash Assembly algorithm — an efficient method to compute the assembly index of CA configurations using a hierarchical hashing approach inspired by Hashlife
- Benchmark across three exploration regimes — random soups, community-discovered patterns, and algorithmic agents
- Assembly Theory metric as an open-endedness metric — captures selection and historical contingency beyond entropy
- Analysis of Game of Life Patterns, iconic Game of Life patterns (gliders, guns, breeders, oscillators), soup searches from Catagolue and culturally significant structures from LifeWiki (up to 2023)
- Agent-based search for finding high-assembly CA objects by agents querying a library
Requirements: Python 3.10+
- Clone the repository and its submodules:
git clone https://github.com/assemblyca/assemblyca.git
cd assemblyca
git submodule update --init --recursive- Install Python dependencies:
pip install -r requirements.txtAll analyses from the paper are reproduced in a single self-contained Jupyter notebook:
jupyter notebook HashAssembly.ipynbThe notebook walks through:
- Hash Assembly algorithm — definition and implementation
- CA analysis — assembly index across different CA rules, 1D,2D,3D with different rules and neighborhoods
- Benchmarking — comparison of hash assembly against the exact assembly index and entropy
- Game of Life patterns — assembly of gliders, guns, oscillators, and breeders
- Soup Search Experiments — assembly complexity of soup Experiments (up to 2023)
- Cultural patterns — assembly complexity of Wikipedia structures over time (up to 2023)
- Agent search — searching CA high-assembly objects by agents querying a library
assemblyca/
├── HashAssembly.ipynb # Main notebook (reproduces all paper results)
├── assemblyca_tools.py # Core library: hash assembly, CA tools, plotting
├── hashlife/ # Hashlife submodule (fast CA simulation)
├── agent_search/ # Agent-based search for high-assembly CA objects
├── apgmera/ # apgmera submodule (finding patterns in long simulations)
├── benchmark_data/ # Benchmarking results vs. exact assembly index
├── rle_files/ # Game of Life patterns (RLE format)
├── rule_files/ # CA rule tables
├── soup_search/ # Random soup search experiments
├── wiki_patterns/ # Cultural patterns from Wikipedia (up to 2023)
├── pathway_images/ # Assembly pathway visualizations
└── requirements.txt # Python dependencies
| Name | Affiliation |
|---|---|
| Keith Y. Patarroyo | Complex Chemistry Labs, University of Glasgow |
| Abhishek Sharma | Complex Chemistry Labs, University of Glasgow |
| Sara I. Walker | School of Earth and Space Exploration, Arizona State University |
| Leroy Cronin | Complex Chemistry Labs, University of Glasgow |
If you use this code or benchmark, please cite:
@inproceedings{patarroyo2023assemblyca,
title = {AssemblyCA: A Benchmark of Open-Endedness for Discrete Cellular Automata},
author = {Patarroyo, Keith Y. and Sharma, Abhishek and Walker, Sara I. and Cronin, Leroy},
booktitle = {NeurIPS Workshop on Agent Learning in Open-Endedness (ALOE)},
year = {2023},
url = {https://openreview.net/pdf?id=5cEQ4ZOsIN}
}This project is released for academic use. See the paper for details.