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Rodrigo A. Obando, Ph.D., TSYS School of Computer Science, Columbus State \
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If we arrange each \[OpenCurlyQuote]dimension\[CloseCurlyQuote] of our space \
using this lattice, we can partially-order the entire space where we can move \
from no-change rule 204 (identity) to total-change rule 51 (complement.) \
Huffman created a rule space crawler to navigate this space and visualize the \
neighboring rules. We found that adjacent rules exhibit similar global \
behavior. We decided to create an algorithm that would find the neighboring \
rules that would be similar in global behavior.\
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The clustering algorithm identifies the cluster by expanding \
\[OpenCurlyQuote]outward\[CloseCurlyQuote] from a seed rule. It \
systematically tries paths in the partially-ordered space and identifies \
which bits in the rule can change and still maintain the global behavior. \
There is a threshold that is used to measure percentage of similarity. The \
particular comparison algorithm can be independently defined.
The output of the algorithm is a Cluster Signature that encapsulates \
information about all the rules that are in the cluster. Each bit in the rule \
is identified as P, N or D. P indicates that the bit introduces change in the \
rule, i.e., a 1 in p0 or a 0 in p1. An N indicates that the bit should anchor \
the behavior or not introduce change, i.e. a 0 in p0 and a 1 in p1. A D \
indicates that the bit can be any value, either 0 or 1. Using the cluster \
signature we can extract all the rules that exist in the cluster. The size of \
the cluster can be found as follows.\
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There are some interesting clusters we found while testing the algorithm.\
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We tried finding the cluster signature around rule 90. We chose this rule \
because it had a recognizable global behavior for a single 1 as an input. We \
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Notice that we also complemented the input so that the output would be \
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These two signatures show how all these Sierpinski triangle rules are all \
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Cellular automata uses a single rule for each one of its cells. This is \
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cellular automata, or Non-Uniform, is a cellular automata where each cell \
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Changing the rule for each cell at every evaluation step is equivalent to \
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probabilistic cellular automaton.
Given these parameters there are a total of 6561 rules based on the ECA.\
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We believe that the global behavior of these cellular automata is \
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Triangles with tendrils: PCAs: PPPN, NDDD and complement DDDN, NPPP Input: \
DETERMINISTIC
Rips from top in triangles: PDDP, NPPP and complement PPPN, PDDP input: \
DETERMINISTIC
Rips on a veil: PDDD, NDDP and PDDN, DDDP (complements) Input: STOCHASTIC
Seashell patterns: DNNP, NPPP and complement PPPN, PNND Input: STOCHASTIC
Other seashell patterns: NPPN, DDDP and complement PDDD, NPPN Input: STOCHASTIC
Roots or lightning: DDDD, NDDD and complement DDDN, DDDD Input: STOCHASTIC
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