Commutative Cellular Automata (original) (raw)

Abstract

Commutative cellular automata are a class of cellular automata that portray certain characteristics of commutative behavior. We develop the notion of neighborhood partitions and neighborhood equivalence classes to analyze and enumerate these automata.

Figures (17)

tic cellular automata over the color set s = {0, 1, 2}. The notation g(y) has For the partition shown in Figure 1, there are 3” = 2187 rules, as there are k = 3 colors and seven equivalence classes. The number of rules for a k-color cellular automaton with n neighborhood equiva- lence classes in its partition is k”. If 2 < k, then the cellular automa- ton effectively becomes an -color cellular automaton, as there are only 7 maximum mappings in the neighborhood function f:T > s to the color set s. To remove this degenerate case, it is required that

tic cellular automata over the color set s = {0, 1, 2}. The notation g(y) has For the partition shown in Figure 1, there are 3” = 2187 rules, as there are k = 3 colors and seven equivalence classes. The number of rules for a k-color cellular automaton with n neighborhood equiva- lence classes in its partition is k”. If 2 < k, then the cellular automa- ton effectively becomes an -color cellular automaton, as there are only 7 maximum mappings in the neighborhood function f:T > s to the color set s. To remove this degenerate case, it is required that

Figure 6. The neighborhood partition for radius r = 1 totalistic cellular au- tomata over the color set s = {0, 1, 4}. Equivalent partition to multiplistic cel- lular automata over color set s = {1, 2, 3} (Figure 5), and therefore will be able to produce the exact same cellular automata. Other possible color sets that produce this same partition for totalistic cellular automata are {log(1), log(2), log(3)}, {1, 4, 13}.

Figure 6. The neighborhood partition for radius r = 1 totalistic cellular au- tomata over the color set s = {0, 1, 4}. Equivalent partition to multiplistic cel- lular automata over color set s = {1, 2, 3} (Figure 5), and therefore will be able to produce the exact same cellular automata. Other possible color sets that produce this same partition for totalistic cellular automata are {log(1), log(2), log(3)}, {1, 4, 13}.

Figure 7. The neighborhood partition for radius r = 1 commutative cellular au- tomata over the color set s = {0, 1, 2}. This partition is the same as the parti- tions seen in multiplistic (Figure 5) and totalistic (Figure 6) cellular automata.

Figure 7. The neighborhood partition for radius r = 1 commutative cellular au- tomata over the color set s = {0, 1, 2}. This partition is the same as the parti- tions seen in multiplistic (Figure 5) and totalistic (Figure 6) cellular automata.

where g’ is a commutative function.

where g’ is a commutative function.

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References (6)

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