Selecting Continuous Life-Like Cellular Automata for Halting Unpredictability: Evolving for Abiogenesis (2204.07541v2)

Abstract: Substantial efforts have been applied to engineer CA with desired emergent properties, such as supporting gliders. Recent work in continuous CA has generated a wide variety of compelling bioreminiscent patterns, and the expansion of CA research into continuously-valued domains, multiple channels, and higher dimensions complicates their study. In this work we devise a strategy for evolving CA and CA patterns in two steps, based on the simple idea that CA are likely to be complex and computationally capable if they support patterns that grow indefinitely as well as patterns that vanish completely, and are difficult to predict the difference in advance. The second part of our strategy evolves patterns by selecting for mobility and conservation of mean cell value. We validate our pattern evolution method by re-discovering gliders in 17 of 17 Lenia CA, and also report 4 new evolved CA and 1 randomly evolved CA that support novel evolved glider patterns. The CA reported here share neighborhood kernels with previously described Lenia CA, but exhibit a wider range of typical dynamics than their Lenia counterparts. Code for evolving continuous CA is made available under an MIT License (https://github.com/rivesunder/yuca).

• The paper introduces evolutionary methods, including CMA-ES, to evolve continuous CA rules exhibiting unpredictable halting behavior.
• It employs a two-stage approach that first selects complex CA rule sets and then evolves glider patterns using compositional pattern-producing networks.
• Results reveal diverse dynamics, rediscovering known gliders and uncovering novel patterns that advance our understanding of artificial life systems.

This paper explores methods for automatically discovering interesting and complex continuous cellular automata (CA) rules, analogous to finding rules like Conway's Game of Life. The authors propose evolutionary strategies to select CA rules that are difficult to predict whether a given configuration will halt (reach a state where all cells are zero) or persist/grow indefinitely. This unpredictability is hypothesized to correlate with computational complexity and the potential to support complex, life-like patterns such as gliders.

The work utilizes the Glaberish framework, an extension of the continuous CA system called Lenia. While Lenia uses a single growth function based on the neighborhood state, Glaberish splits this into two functions: a genesis function $G_{gen}$ for cells with low values (analogous to 'birth') and a persistence function $P$ for cells with higher values (analogous to 'survival'). The update rule is given by $$A_{t + dt} = \rho(A_t + dt \cdot [(1 - A_t) \cdot G_{gen}(n) + A_t \cdot P(n)])$$, where $A_t$ is the grid state, $n$ is the neighborhood convolution result, $dt$ is the time step, and $\rho$ is a squashing function keeping values between 0 and 1.

Two methods for evolving CA rules are introduced:

1. Halting Prediction Evolution: This method uses an outer evolutionary algorithm (Covariance Matrix Adaptation Evolution Strategy - CMA-ES) to search for CA rule parameters (specifically, parameters for the genesis and persistence functions, often modeled as Gaussian peaks and widths). The fitness of a rule set is the negative average accuracy of three separately trained convolutional neural networks (CNNs) attempting to predict whether random initial patterns in that CA will eventually halt or persist. The goal is to find rules where prediction is difficult.
2. Simple Halting Evolution: A less computationally intensive approach where the fitness is based on minimizing the mean-squared error between the proportion of initial conditions that halt and a target proportion (set to 0.5). This directly encourages rules that support both halting and persistent behaviors.

After finding potentially complex CA rules, the paper uses a second evolutionary stage to search for mobile patterns, specifically gliders, within a fixed CA rule set. This is done by evolving compositional pattern-producing networks (CPPNs) which generate initial grid configurations. The fitness of a pattern is designed to reward:

• Motility: Measured by the displacement of the pattern's center of mass over time.
• Homeostasis: Penalizing changes in the pattern's average cell value.
• Survival: Severely penalizing patterns that vanish entirely.

The authors validated their pattern evolution method by successfully rediscovering gliders in 17 known Lenia CA rules. They then applied the pattern evolution to CA rules evolved using their halting unpredictability and simple halting evolution methods. They report finding several new evolved CA rules (derived from both evolution strategies and one random search) that support novel evolved glider patterns.

The evolved CA rules were characterized using metrics like fertility ratio (proportion of initial patterns that grow indefinitely, escaping a bounding box) and mortality ratio (proportion of initial patterns that halt, with all cell values becoming zero), inspired by Eppstein's heuristics (0911.2890). They also assigned putative Wolfram classes subjectively based on observed dynamics. The results showed that the evolved CA rules exhibit a wider range of dynamics (including Class I, III, and IV-like behaviors) compared to the tested Lenia rules, which predominantly showed Class II (static or oscillating pattern equilibria) characteristics. Despite the diverse dynamics, all the highlighted evolved CA rules were capable of supporting the evolved mobile patterns.

In summary, the paper presents novel evolutionary techniques for discovering continuous CA rules that are likely to be complex and support life-like dynamics. By selecting for unpredictable halting behavior or a balance between halting and persistence, and then evolving patterns within those rules, the authors demonstrated the ability to find new CA rule sets and mobile patterns. The work highlights the potential of evolutionary algorithms to explore the vast space of continuous CA dynamics and contributes to the understanding of what makes artificial life systems interesting and potentially computationally capable. Code for evolving continuous CA is available in the yuca library.