2-Dimensional Reversible Hexagonal Cellular Automata with Periodic Boundary

Abstract

The hexagonal nite cellular automata (shortly HFCA) are 2-dimensional (2D) cellular automata whose cells are of the form of hexagonal. Morita et al. [1] introduced this type of cellular automaton (CA) and they called it hexagonal partitioned CA (HPCA). A remarkable application of the family of these CAs is presented by Trun o [2] where a model is presented to simulate the evolution of forest res and Hernandez Encinas et al. [3] where they introduce a new mathematical model for predicting the spread of a re front in homogeneous and inhomogeneous environments. Also, in [4], debris ows are simulated and modeled by two-dimensional hexagonal cellular automata. These families of cellular automata are also applied to design discrete models of chemical reaction-di usion systems [5]. Also, 2D CAs have found applications in tra c modeling. For instance multi-value (including ternary) CA models for tra c ow are proposed in [6]. Recently, cellular automata have found applications in cryptography [7, 8], especially 2D CA have been proposed for multi-secret sharing scheme for colored images [9]. Due to the applications and modeling on hexagonal cellular automata, the algebraic structure of cellular automata has been of much interest to the researchers [10 13]. Algebraic representation of 2D CA helps in determining the characterization of CA. An important characterization is the determination of the reversibility of CA [13]. In [14], we have characterized a 2D nite CA by using matrix algebra built on Z3. Also, we have analyzed some results about the rule numbers 2460N and 2460P. In [15], we have obtained necessary and su cient conditions for the existence of Garden of Eden con gurations for 2D ternary CAs. In this paper, we deal with CA de ned by hexagonal rules under periodic boundary condition (PBC) and the ternary eld Z3. We obtain the rule matrix of the hexagonal nite periodic cellular automaton (HFPCA). We compute the rank of rule matrices related to HFPCA via an algorithm. Hence, we determine the reversibility of this type 2D CA which is one of the di cult problems in higher dimension as explained in the previous paragraph. Further, by using the matrix algebra it is shown that the HFPCA are reversible, if the number of columns is even and the HFPCA are not reversible, if the number of the columns is odd. A periodic boundary CA is the one where the extreme cells in the boundaries are adjacent to each other periodically [16]. A null boundary CA is the one where the extreme cells in the boundaries are connected to the zero states. The surrounding cells are all in zero state. For convenience of analysis, the state of each cell is an element of a nite or in nite state set. Moreover, the state of the cell (i, j) at time t is denoted by x (t) (i,j). The state of the cell (i, j) at time t+1 is denoted by x (t+1) (i,j) = y (t) (i,j). Let us consider the