Complexity in the Universe of Cellular Automata
The colored toy cubes contain all the information about the complex dynamics of cellular automata. An important advantage of the Boolean cube representation is that it allows us to define an index of complexity (Chua et al. 2002). Each one of the 256 cubes is obviously characterized by different clusters of red or blues vertices which can be separated by parallel planes. On the other hand, the separating planes can be analytically defined in the coordinate system of the Boolean cubes. Therefore, the complexity index of a cellular automaton with local rule N is defined by the minimum number of parallel planes needed to separate the red vertices of the corresponding Boolean cube N from the blue vertices. Figure 1 shows three examples of Boolean cubes for the three possible complexity indices κ = 1, 2, 3 with one, two and three separating parallel planes. There are 104 local rules with complexity index κ = 1. Similarly, there are 126 local rules with complexity index κ = 2, and only 26 local rules with complexity index κ = 3. This analytically defined complexity index is to be distinguished from Wolfram’s complexity index based on phenomenological estimations of pattern formation.
KeywordsCellular Automaton Turing Machine Local Rule Complexity Index Colored Vertex
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