A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications

TL;DR

The surveyed work catalogs cellular automata (CAs) across types, analytical tools, global behaviors, and non-uniform extensions, linking simple local rules to rich collective dynamics. It formalizes CA definitions and catalogs neighborhood, boundary, and rule variations, then surveys characterization tools such as De Bruijn graphs, matrix algebra, and reachability trees, along with dynamical notions like universality, chaos, and conservation laws. A major focus is non-uniform CAs, especially non-uniform ECAs, detailing linear/additive versus nonlinear classes, group versus non-group structures, and their implications for reversibility and applications. The paper then surveys CA-based technology in electronics, computer vision, and medical science, highlighting practical hardware-era patterns, pattern classifiers, and bioinformatics modeling, before summarizing challenges and future directions. Together, the work provides a comprehensive blueprint for understanding CA diversity, analysis, and real-world deployment.

Abstract

Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation. Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modelling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been introduced in update pattern, lattice structure, neighborhood dependency and local rule. In this survey, we tour to the various types of CAs introduced till date, the different characterization tools, the global behaviors of CAs, like universality, reversibility, dynamics etc. Special attention is given to non-uniformity in CAs and especially to non-uniform elementary CAs, which have been very useful in solving several real-life problems.

Figures (8)

• Figure 1: Neighborhood dependencies: \ref{['1way']} and \ref{['2way']} are $1$-dimensional CAs, and \ref{['von']} and \ref{['moore']} are $2$-dimensional CAs. The black cell is the cell under consideration, and its neighbors are the shaded cells and the black cell itself
• Figure 2: Neighborhood dependencies: \ref{['Fig:Margolus']} is a block CA, \ref{['Fig:Hexagonal']} is hexagonal CA, and \ref{['Fig:PartitionedCA']} is a partitioned CA, where a square cell is partitioned into four triangular parts
• Figure 3: Boundary conditions of 1-D finite CAs. Arrows pointing to a cell indicate the dependencies of the cell. Here all the CAs use 3-neighborhood dependency
• Figure 4: Triangular partitioned CA and its rule (as given by morita2016universality). The rule is rotation symmetric
• Figure 5: The de Bruijn Graph of CA with rule $90$ ($2^{nd}$ row of Table \ref{['rt2']})

Theorems & Definitions (8)

• definition 1
• definition 2
• definition 3
• definition 4
• definition 5
• definition 6
• definition 7
• definition 8