2D Triangular von Neumann Cellular Automata with Periodic Boundary

Abstract

Cellular automata (CA) theory is a very rich and useful model of a discrete dynamical system that focuses on their local information relying on the neighboring cells to produce CA global behaviors. Although the main structure of CA is a discrete special model, the global behaviors at many iterative times and on big scales can be close to nearly a continuous system. The mathematical points of the basic model imply the computable values of the mathematical structure of CA. After modeling the CA structure, an important problem is to be able to move forwards and backwards on CA to understand their behaviors in more elegant ways. This happens in the possible case if CA is a reversible one. In this paper, we investigate the structure and the reversibility cases of two-dimensional (2D) finite, linear, and triangular von Neumann CA with periodic boundary case. It is considered on ternary field [Formula: see text] (i.e. 3-state). We obtain the transition rule matrices for each special case. It is known that the reversibility cases of 2D CA is generally a very challenging problem. For given special triangular information (transition) rule matrices, we prove which triangular linear 2D von Neumann CA is reversible or not. In other words, the reversibility problem of 2D triangular, linear von Neumann CA with periodic boundary is resolved completely over ternary field. However, the general transition rule matrices are also presented to establish the reversibility cases of these special 3-states CA. Since the main CA structures are sufficiently simple to investigate in mathematical ways and also very complex for obtaining chaotic models, we believe that these new types of CA can be found in many different real life applications in special cases e.g. mathematical modeling, theoretical biology and chemistry, DNA research, image science, textile design, etc. in the near future.