engleski

On dimensionality of cellular automata

The oldest and very much applied topological index, the Wiener index W [1-4], is still a subject of intensive researches in mathematics and mathematical chemistry. W it is defined as (a half of) the sum of all distances in a graph G. it is known that W can be expressed as a polynomial in N [5], where N is the number of vertices in G, and where the leading exponent is a number simply related to an integer dimensionality [6-9], we call Wiener dimensionality, of the graph under study. Recently, graph of fractal objects like Sierpinski gasket and carpet have been studied [10] and it was shown that the leading exponent is then a number related to fractal dimensionality. Here we generalize the idea of Wiener dimensionality to cellular automata [11] with finite number of cells in each their row. In mathematics, the dimension is the property of topological space. Hence, in order to define dimension, we need to have topology, However, cellular automata do not have topology that can be easily defined since they are union of cells (of course trivial topology in which every cell is open set is not plausible). Hence, there is not a straightforward way to see how to define the dimension of the cellular automata. On the other hand, cellular automata result in structures that strongly resemble plane, line, Sierpinski gasket and so on. Hence, one is tempted to assign the dimension to it. Here, we provide exact mathematical method to define the dimension of the cellular automata that is in accordance with our intuition.

cellular automata ; dimensionality

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