Extension of cellular automata via the introduction of an algorithm for the recursive estimation of neighbors

This study focuses on an extended model of standard cellular automaton (CA), which includes an extra index, comprising a radius that defines a perception area for each cell in addition to the radius defined by the CA rule. Such an extension can be realized by introducing a recursive algorithm called the "Recursive Estimation of Neighbors." The extended CA rules form a sequence ordered by this index, which includes the CA rule as its first term. This extension aims to construct a model that can be used within the CA framework to study the relation between information processing and pattern formation in collective systems. Even though the extension presented here is merely an extrapolation to a CA having a larger rule neighborhood identical to the perception area, the extra radius can be interpreted as an individual attribute of each cell. The novel perspective to CA provided here makes it possible to build heterogeneous CAs, which contain cells having different extra radii. Several pattern formations in the extension of one-dimensional elementary CAs and two-dimensional Life-like CAs are presented. It is expected that the extended model can be applied to various simulations of complex systems and in other fields.