2.3. Taxonomy of cellular automata

The classification of cellular automata is based either on structure (topology) or behavior (function and/or phenomenology).

properties of the linear cellular automata evolution, which is particularly important from the perspective of applications of cellular automata in generation of pseudorandom sequences.

Cellular Automata and Randomization: A Structural Overview

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Class 1. Cellular automata that evolve to a homogenous finite state. Automata belonging to this first class evolve from almost all initial states, after a finite number of steps, to a unique homogenous configuration, in which all cells have the same state or value. The evolution of

Class 2. Cellular automata that have a periodical behavior. The configurations are divided in basins of attraction, the periodical evolution of the global configuration depending on the initial state. The states' space being classified in attractor cycles; such cellular automata can

Class 3. Cellular automata exhibiting chaotic or pseudorandom behavior. This class is particularly significant for ideal (infinite) cellular automata. Cellular automata belonging to the third class evolve, from almost all possible initial states, not periodically, leading to "chaotic" patterns. After an enough long evolution (number of steps), the statistical properties of these patterns are typically the same for all initial configurations, according to Wolfram. Although in practice cellular automata will always be limited to a finite number of cells, this class can be extrapolated maintaining the local rules. Such automata can be used for pseudorandom pat-

Class 4. Cellular automata having complicated localized and propagating structures. They can be viewed as computing systems in which data represented by the initial configuration are processed by time evolution, the result being contained in the final configuration of attractor

In conclusion, cellular automata are systems consisting of a regular network of identical simple cells that evolve synchronously according to local rules that depend on local conditions. Although the structure is very simple and regular, it produces a vast phenomenology, and therefore, cellular automata are often described as an "artificial universe" that has its own

This section discusses specific issues of cellular automata that one should be aware before

Massive parallelism is one of the definitory features of cellular automata. However, simulations are often used instead of actual implementations, and for very good reasons: there are no commercially available hardware versions, hardware accelerators, or co-processors for the cellular automata computational model. The full performance of the massive parallel architecture will never be reached by simulation; in fact, the high granularity will make the simulations slow, even

class 1 cellular automata completely "destroys" any information on the initial state.

serve as "filters."

tern generation [8, 10].

cycle of configurations.

specific local laws [1].

for small dimensions.

3. Theoretical and practical issues

deciding to use or investigate this model.

3.1. Parallelism: hardware vs. simulation

The topology refers mainly to the type of network and local connections (neighborhoods and boundary conditions). In linear cellular automata, the cells are connected in a row (vector) to their nearest neighbors. Further subdivision of linear cellular automata is based on the neighborhood dimension, which is one of the main factors that affect the complexity of the cell. In two-dimensional cellular automata models, the interconnection network is two dimensional, typically rectangular, but also hexagonal networks have been explored for specific applications. However, topologically any two-dimensional network can be transformed in a rectangular one, by choosing an appropriate neighborhood [5]. In typical rectangular connections, there are two most used neighborhoods: von Neumann neighborhood which contains the four adjacent cells on the vertical and horizontal lines, and the central cell itself; and Moore neighborhood that contains the lateral neighbors, the central cell, and the cells adjacent at corners (Figure 3).

The theoretical analysis of two-dimensional cellular automata is an open field of research, and most often, the results are extensions of the better-known case of linear automata. Twodimensional cellular automata are very important in applications, as for instance in image processing, where the image corresponds directly to the configuration of the system. Most of modeling applications also involve two-dimensional extensions, therefore, use two-dimensional cellular automata [11].
