MaCE: General Mass Conserving Dynamics for Cellular Automata

TL;DR

MaCE is a versatile, simple method for enforcing mass conservation in cellular automata, leading to more stable and diverse behaviors, with applications demonstrated on Lenia, Neural-CAs, and discrete CAs.

Contribution

MaCE introduces a general, easy-to-implement mass conservation rule for cellular automata, enhancing their stability and behavioral richness.

Findings

MaCE is numerically stable and has a simple continuous limit.

Applying MaCE to Lenia yields diverse solitons and behaviors.

MaCE is versatile, applicable to Neural-CAs and discrete CAs.

Abstract

We present Mass-Conserving Evolution (MaCE), a general method for implementing mass conservation in Cellular Automata (CA). MaCE is a simple evolution rule that can be easily 'attached' to existing CAs to make them mass-conserving, which tends to produce interesting behaviours more often, as patterns can no longer explode or die out. We first show that MaCE is numerically stable and admits a simple continuous limit. We then test MaCE on Lenia, and through several experiments, we demonstrate that it produces a wide variety of interesting behaviours, starting from the variety and abundance of solitons up to hints of intrinsic evolution in resource-constrained environments. Finally, we showcase the versatility of MaCE by applying it to Neural-CAs and discrete CAs, and discuss promising research directions opened up by this scheme.