Fractal Arrangement for 2D Cellular Automata and Its Implementation for Outer-Totalistic Rules

Abstract

Cellular automata (CAs) have played a significant role in studies of complex systems. Recently, a recursive estimation of neighbors algorithm that distinguishes the perception area of each cell from the CA rule neighborhood was introduced to extend CA. This framework makes it possible to construct non-uniform CA models composed of cells with different sizes of the perception area, which can be interpreted as an individual attribute of each cell. For example, focusing primarily on one-dimensional (1D) elementary CA, fractal CAs composed of self-similarly arranged cells have been proposed and their characteristics have been investigated. In this paper, 2D fractal CAs are defined and implemented for outer-totalistic CA rules. Fractal CAs derived from a linear rule inherit that rule’s features, including replicability and time reversibility, which indicate their applicability to various fields.