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Ruliological resilience: Pattern restoration and robustness in Wolfram patterns. A basis for regeneration, not just
Syed Hussain Ather1, Richard Gordon2
1Master's Student in Analytics, Georgia Institute of Technology, Atlanta, GA, USA.
Bio Systems
|July 1, 2026
Summary
Elementary cellular automata (ECA) rules show varying recovery from disruptions. Simple rules quickly restore patterns, while complex rules diverge, offering insights into pattern maintenance.
Area of Science:
- Computational science
- Complex systems
- Theoretical computer science
Background:
- Elementary cellular automata (ECA) are simple computational models exhibiting complex emergent behavior.
- Understanding the robustness and recovery of ECA dynamics is crucial for complex systems research.
Purpose of the Study:
- To systematically investigate the recovery dynamics of all 256 elementary cellular automata (ECA) rules after localized perturbations.
- To quantify the resilience and restoration capabilities of different ECA rules.
Main Methods:
- Perturbing ECA simulations at specific pattern growth stages across all 256 Wolfram rules.
- Quantifying trajectory divergence using Boolean XOR difference mapping.
- Defining and calculating a 'restoration coefficient' based on normalized Hamming distance.
Main Results:
- ECA rules were classified by recovery behavior: Class I-II rules showed rapid restoration, Class III rules exhibited persistent divergence, and Class IV rules displayed intermediate, phase-dependent recovery.
- Rule 110 demonstrated enhanced resilience due to localized structures constraining divergence.
- Recovery properties were found to cluster by dynamical regime and perturbation conditions, not uniformly distributed across rule space.
Conclusions:
- The restoration coefficient provides a quantitative measure for comparing robustness in cellular automata.
- Findings offer an abstract computational perspective on pattern maintenance and regeneration in biological systems.
- Direct mechanistic interpretation of biological regeneration from these abstract models requires caution.
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