Extension of cellular automata by introducing an algorithm of recursive estimation of neighbors

This study focuses on an extended model of a standard cellular automaton (CA) that includes an extra index consisting of a radius that defines a perception area for each cell in addition to the radius defined by the CA rule. Extended standard CA rules form a sequence ordered by this index, which includes the CA rule as its first term. This extension aims at constructing a model that can be used within the CA framework to study the relationship between information processing and pattern formation in collective systems. Although the extension presented here is merely an extrapolation to a CA with a larger rule neighborhood, the extra radius can be interpreted as an individual difference of each cell, which provides a new perspective to CA. Some pattern formations in extended one-dimensional elementary CAs and two-dimensional Life-like CAs are presented. It is expected that the extended CA can be applied to various simulations of complex systems and other fields.

💡 Research Summary

The paper proposes an extension of the classic cellular automaton (CA) framework by introducing, for each cell, an additional “perception radius” (R) that defines a broader area from which the cell can gather information, in addition to the usual rule radius (r) that determines the immediate neighborhood used by the CA rule. The core idea is a recursive estimation algorithm: a cell first obtains the actual next‑state values of its direct neighbors (those within radius r) using the standard rule, then those neighbors, in turn, estimate the states of cells within their own perception areas, and this process propagates outward until the whole region of radius R reaches a consistent estimate. In practice the algorithm proceeds as follows: (1) initialise an “expected state” for every cell inside the perception region; (2) apply the original CA rule to the r‑neighborhood to obtain the true next state for the immediate neighbors; (3) compare each neighbor’s expected state with its true state, compute an error, and propagate a correction to the expectations of cells farther out; (4) repeat steps 2‑3 until convergence or a preset iteration limit.

This mechanism creates a dynamic, multi‑scale information flow that mimics predictive‑correction processes observed in biological and social collectives. By allowing R to differ from cell to cell, the model can encode individual differences: cells with large R act as “leaders” that integrate long‑range information, while cells with small R remain sensitive only to local fluctuations. Consequently, the extended CA generates a hierarchy of interaction scales that is absent from traditional CAs, where the interaction range is fixed and uniform.

The authors test the concept on one‑dimensional elementary CAs (e.g., rule 30 and rule 110) and on two‑dimensional Life‑like rules (Conway’s Game of Life B3/S23, B6/S5, etc.). For 1‑D rules, increasing R from 1 (the standard case) to 5 gradually suppresses the chaotic randomness of rule 30 and accentuates the coherent travelling structures of rule 110. In the 2‑D experiments, classic Life patterns such as still‑lifes, oscillators, and spaceships become “diffusive waves” or multi‑scale oscillations as R grows; rules that normally produce explosive growth (e.g., B6/S5) develop a mixture of rapid expansion and emergent stable islands when a larger perception radius is allowed. These observations demonstrate that the extended CA explores a richer pattern space than the original rule set.

From a theoretical standpoint, the extension maps the conventional rule space into a higher‑dimensional “perception space” defined by the pair (r, R). This mapping enables quantitative studies of the relationship between information‑processing capacity (average perception radius) and pattern complexity (entropy, spectral measures, etc.). Moreover, the recursive estimation process parallels predictive coding models in neuroscience and anticipatory signaling in cellular colonies, suggesting potential cross‑disciplinary applications in biology, sociology, and engineered swarm systems.

Computationally, the algorithm’s cost scales as O(R^d) where d is the lattice dimension, but because each cell’s update is independent, the method is highly amenable to parallelisation. The authors provide a Python prototype and discuss future C++/CUDA implementations that would allow real‑time simulation of large‑scale systems.

In conclusion, by adding a perception radius and a recursive neighbor‑estimation routine, the paper furnishes a versatile extension of cellular automata that captures heterogeneous information processing and long‑range coupling within the familiar CA paradigm. This framework opens new avenues for studying how local rules, extended perception, and recursive inference jointly shape emergent structures in complex systems across physics, biology, social science, and engineering.