Characterization of Uniform and Hybrid Cellular Automata with Reflecting Boundary

In this paper, we study the characterization of uniform cellular automata with the restricted vertical neighborhood and hybrid cellular automata with both restricted vertical neighborhood and Von Neumann Neighborhood using reflecting boundary condition over the eld ℤ2. The transition rule matrix for uniform and hybrid cellular automata with reflecting boundary condition is obtained and also reversibility of the uniform cellular automata is studied.


Introduction
Two dimensional cellular automata has a promising approach to solve some problems in human life, such as pecudo random generation, pattern classification, image en-compression and decompression. The notion of cellular automata (CA) was introduced by John Von Neumann and Stanislaw Ulam in 1950's. John Von Neumann [13] showed that a cellular automaton (CA) can be universal. Cellular Automata are also called Cellular Space, Tessellation Automata, Homogeneous structures, Cellular structure, Tessellation structures and Iterative arrays [14].
The study of CA has received remarkable attention in the last few years [1,6,15], because CA have been widely investigated in many disciplines (For Example Mathematics, physics, computer science, chemistry, etc.) with different purposes(For Example simulation of natural phenomena, pseudo-random number generation, image processing, analysis of universal model of computations, cryptography). Most of the work for CA s is done for one-dimensional (1-D) case. Recently, two dimensional (2-D) CA s have attracted much of the interest. Some basic and precise mathematical models using matrix algebra built on field with two elements Z 2 were reported for characterizing the behavior of two dimensional nearest neighborhood linear cellular automata with null or periodic boundary conditions [5,6,15].
The 2D finite cellular automata consists of (m × n) cells arranged. Where each cell takes one of the values of the field Z 2 . The relative positions of the cells is called neighbor of the cell given center cell. The state of these neighbors are used to compute the new state of the center cell. The paper is organized as follows. In the 2nd section, the concept used in the paper are formally defined. In 3rd section, the algebraic structure of restricted vertical neighborhood and Von Neumann neighborhood of 2D cellular automata is obtained. In 4th section, the rule of the uniform cellular automata with restricted vertical neighborhood is studied. In 5th section, using both Von Neumann and Restricted Vertical Neighborhood the rule matrix of the hybrid cellular automata is obtained. In 6th section, we compute the rank of rule matrix related to 2D uniform cellular automata. Also study the reversibility of the restricted vertical neighborhood of uniform CA.

Preliminaries
Definition 2.1 [9] A periodic Boundary CA is the one in which the extreme cells are adjacent to each other. Definition 2.2 [9] A null boundary CA is the one in which the extreme cells are connected to logic zero state. Definition 2.3 [2] Reflecting Boundary: In a reflecting boundary, the state of the opposite neighbor is replicated by the virtual cell. Definition 2.4 [15] Each state of a CA is called a configuration. In particular, each configuration of a (2-D) CA m×n is a binary information matrix of dimension (m × n). Definition 2.5[10] Uniform Cellular Automata: The uniform cellular automata have been presented by Nandi et al. If the same rule is applied to all the cells in a CA, then the CA is said to be uniform or regular CA.  A configuration is a mapping C : Z d → S which assigns each cell a state. Make C t denote the configuration at time t, then the state of cell ⃗ n at time is C t ⃗ (n) and its state at time (t+1) goes like this.
.., C t ( ⃗ n m )) now we consider the case in which the local rule f is a linear function ..,m. In this paper, we deal with CA defined by Von neumann rules and Restricted Vertical Neighborhood under reflecting Boundary condition (RB) of Uniform and Hybrid cellular automata. For convenience of analysis, the state of each cell is an element of a finite or an infinite state set. Moreover, the state of the cell (i, j) at time t is denoted by In [7], they consider the information matrix The matrix C (t) is called the configuration of the 2D finite CA at time t. We transition the row vectors of Hence, we consider the transition matrix T R that changes set of states of cellular automata from (t) to (t+1) such that

2D cellular automata over the field Z 2
In [12], the 2D finite CA consists of (m × n) cells arranged in m rows and n columns, where each cell takes one of the values of the field Z 2 . From now on, we denote 2D finite CA order (m × n) by 2D CA (m×n) . A configuration of the system is an assignment of the states to all cells. Every configuration determines a next configuration via a linear transition rule that is local in the sense that the state of a cell at time (t+1) depends only on the states of some of its neighbor at the time 't' using modulo 2 algebra. For 2D CA nearest neighbors, there are (mn) cells arranged in a (m × n) matrix centering that particular cell.

Restricted Vertical Neighborhood -162 Rule
In 2D CA theory, there are some classic types of neighborhood but in this paper we only restrict ourselves to the special neighbors which is called RVN. The RVN comprises the 3 cells surrounding the central cell on two-dimensional.  In Figure 2, we show the RVN which comprises 3 cells which surround the center cell X (p,q) .
The state X

Von Neumann Neighborhood -170 Rule
The Von Neumann neighborhood comprises the four cells surrounding the center cell on a 2D square lattice.

Case(ii). m is odd, the rule matrix T R is given in the following theorem
Rule 170RB

Rule 170RB
Rule 162RB Rule 162RB X 11 X 12 X 13 X 14 X 1n  . Hybrid rule means that it is applied 170RB and 162RB respectively for each rows when m is odd on (m × n) CA.

Reversibility of Uniform Cellular Automata
In this section our aim is to study whether the rule matrix is invertible or not. It is known that the 2D finite cellular automata is reversible if and only if its rule matrix is non singular. If the rule matrix has full rank, then it is invertible, so the 2D finite cellular automata is reversible, otherwise it is irreversible.
To determine the invertibility of the rule matrix associated to the Uniform Cellular Automata, we first study the submatrix A for all case. Lemma 6.1 For all n ≥ 3, rank(A) ̸ = n. Then A is not invertible.

Theorem 6.1
Prove that 2D uniform cellular automata with restricted vertical neighborhood is not reversible. Proof. The proof of above lemma can be obtained by following the theorem.

Conclusions
In this paper, 2D cellular automata of Restricted Vertical Neighborhood and Von Neumann Neighborhood are studied over the field Z 2 . The rule matrix of the transition T R matrix of 2D uniform cellular automata is computed. We prove that the RVN for RB rule matrix is not reversible.