Fractal Arrangement for 2D Cellular Automata and its Implementation in Outer-totalistic Rules

Cellular automata (CAs) are discrete computational structures that play a significant role in the study of complex systems. Recursive estimation of neighbors (REN), which distinguishes the perception area of each cell from the CA rule neighborhood, has recently been used in order to develop CA. This framework allows the construction of non-uniform CAs that are composed of cells with perception area sizes, which can be interpreted as an individual attribute of each cell. Focusing primarily on a 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 use in outer-to-talistic rules. Moreover, fractal CAs derived from multi-state linear rules are also presented. These CAs inherit the features of the linear rules, including replicability and time reversibility, which indicate their applicability to a wide variety of fields.