Structure and Reversibility of 2D von Neumann Cellular Automata Over Triangular Lattice

Abstract

Even though the fundamental main structure of cellular automata (CA) is a discrete special model, the global behaviors at many iterative times and on big scales could be a close, nearly a continuous, model system. CA theory is a very rich and useful phenomena of dynamical model that focuses on the local information being relayed to the neighboring cells to produce CA global behaviors. 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. A possible case is when CA is to be a reversible one. In this paper, we investigate the structure and the reversibility of two-dimensional (2D) finite, linear, triangular von Neumann CA with null boundary case. It is considered on ternary field [Formula: see text] (i.e. 3-state). We obtain their transition rule matrices for each special case. For given special triangular information (transition) rule matrices, we prove which triangular linear 2D von Neumann CAs are reversible or not. It is known that the reversibility cases of 2D CA are generally a much challenged problem. In the present study, the reversibility problem of 2D triangular, linear von Neumann CA with null boundary is resolved completely over ternary field. As far as we know, there is no structure and reversibility study of von Neumann 2D linear CA on triangular lattice in the literature. Due to the main CA structures being sufficiently simple to investigate in mathematical ways, and also very complex to obtain in chaotic systems, it is believed that the present construction can be applied to many areas related to these CA using any other transition rules.

Reversibility Algorithm for 2D Cellular Automata with Reflective Condition

“…New other results is another goal of the future study. These CA results can be applied successfully in especially image processing area [9][10][11][12][13] and the other science branches in near future [14][15][16][17][18][19][20][21].…”

Real Linear Automata with a Continuum of Periodic Solutions for Every Period

“…Martinez summarised several classifications of elementary CA in [Martinez, 2013]. Uguz and collaborators investigated the effect the underlying lattice grid has on the CA dynamics in [Uguz et al, 2013] and [Uguz et al, 2017], and linear two-dimensional (2D) CA over ternary field [Sahin et al, 2015]. The de Bruijn graph, an alternative representation of the local rule table, can be used to determine fixed points and the reversibility of 1D CA ([Sutner, 1991], [Bhattacharjee et al, 2018], [Martínez et al, 2018], [McIntosh, 1991] and [McIntosh, 2009]).…”