Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

2D Moore CA with new boundary conditions and its reversibility

B. A. Omirov, Sh. B. Redjepov, J. B. Usmonov

TL;DR

The paper addresses the reversibility and dynamics of two-dimensional linear Moore cellular automata over the finite field $\mathbb{Z}_p$ under mixed boundary conditions. It develops explicit, block-structured rule matrices $T_{\mathrm{R}}^{\varphi}$ for non-bijective boundary maps and their rotations, enabling a practical rank-based criterion to determine reversibility. Through analytical results on fixed points, nilpotency, and a concrete $m\!=\!4$, $n\!=\!3$ example over $\mathbb{Z}_3$, it demonstrates both reversible and irreversible Moore CA with Gardens of Eden. The approach provides a method to classify 2D CA reversibility under diverse boundary interactions, with potential implications for cryptography, image processing, and the study of CA dynamics on finite lattices.

Abstract

In this paper, under certain conditions we consider two-dimensional cellular automata with the Moore neighborhood. Namely, the characterization of 2D linear cellular automata defined by the Moore neighborhood with some mixed boundary conditions over the field $\mathbb{Z}_{p}$ is studied. Furthermore, we investigate the rule matrices of 2D Moore CA under some mixed boundary conditions by applying rotation. Finally, we give the conditions under which the obtained rule matrices for 2D finite CAs are reversible.

2D Moore CA with new boundary conditions and its reversibility

TL;DR

The paper addresses the reversibility and dynamics of two-dimensional linear Moore cellular automata over the finite field under mixed boundary conditions. It develops explicit, block-structured rule matrices for non-bijective boundary maps and their rotations, enabling a practical rank-based criterion to determine reversibility. Through analytical results on fixed points, nilpotency, and a concrete , example over , it demonstrates both reversible and irreversible Moore CA with Gardens of Eden. The approach provides a method to classify 2D CA reversibility under diverse boundary interactions, with potential implications for cryptography, image processing, and the study of CA dynamics on finite lattices.

Abstract

In this paper, under certain conditions we consider two-dimensional cellular automata with the Moore neighborhood. Namely, the characterization of 2D linear cellular automata defined by the Moore neighborhood with some mixed boundary conditions over the field is studied. Furthermore, we investigate the rule matrices of 2D Moore CA under some mixed boundary conditions by applying rotation. Finally, we give the conditions under which the obtained rule matrices for 2D finite CAs are reversible.
Paper Structure (8 sections, 11 theorems, 65 equations, 3 figures, 5 tables)

This paper contains 8 sections, 11 theorems, 65 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The von Neumann and Moore neighborhood on CA lattice.
  4. The rule matrix of Moore CA and mixed boundary condition associated with non-bijective map
  5. The rule matrices for boundary conditions by rotating $\phi$ on the lattice
  6. Dynamics of CAs
  7. Reversibility of the rule matrix $T^{\phi}_{\mathrm{R}}$.
  8. Conclusions

Key Result

Theorem 3.1

Let $T_{\mathrm{R}}^{\phi} \colon \mathbb{Z}_{p}^{mn} \to \mathbb{Z}_{p}^{mn}$ be the rule matrix which takes the finite Moore CA over the configuration $C(t)$ of order $m\times n$ to the configuration $C(t+1)$ under the boundary condition of ${\phi}$. Then $T_{\mathrm{R}}^{\phi}$ has the following where $A_1=A+d\epsilon_{n,n-1}, \ \ B_1=B+e \epsilon_{n,n-1}, \ \ C_1=C+c\epsilon_{n,n-1}, \ \ D_1

Figures (3)

  • Figure 1: von Neumann and Moore neighborhoods
  • Figure 2: Mixing boundary condition
  • Figure 3: The mixed boundary conditions by rotating $90^{\circ}$ the functions $\phi$.

Theorems & Definitions (21)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • Proposition 4.1
  • ...and 11 more