We have already seen with S states per cell and N cells per nbhd, there are
SN neighborhood configurations. |
A rule
specifies exactly which of the S possible states results from each of the SN
neighborhood configurations. |
Suppose we number the neighborhood configurations 1 through
SN. |
Then there are S outcomes for configuration 1 and S outcomes for
configuration 2, so S2 combinations of outcomes for configurations 1 and 2. |
For example, take S = 2, and write the states as 0 and 1. Then the combinations
for the first two neighborhood configurations are |
00, 10, 01, 11 |
Continuing, there are S3 combinations of outcomes
for configurations 1, 2, and 3. |
For S = 2, here are the combinations for the first
three neighborhood configurations |
000, 100, 010, 110, 001, 101, 011, 111 |
Because there are SN neighborhood configurations, |
|
How large is this number? |