2.4. Phenomenology of cellular automata—Wolfram's taxonomy

Probably the most important contribution in understanding cellular automata capabilities as a computer model is Wolfram's experimental approach [8]. Wolfram explored the behavior of linear cellular automata for all rules (with a neighborhood of dimension 3), starting from different "seeds" or initial states. Wolfram established a classification based on the statistical

Figure 3. Neighborhoods in two-dimensional cellular automata: von Neumann (left) and Moore (right) neighborhoods.

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

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

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 serve as "filters."

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 pattern generation [8, 10].

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 cycle of configurations.

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 specific local laws [1].
