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.