A New Kind of Science: The NKS Forum > Pure NKS > Generation new color with totalistic rules
Author
M.Abdeldjalil

Registered: Nov 2008
Posts: 39

Generation new color with totalistic rules

Hello,

In the paper of 'Two dimensional CA (1986)' there is:

'Totalistic cellular automata rules take the value of center site to depond only on the sum of the values of the sites in the neighborhood. With outer totalistic rules, sites are updated according to their previous values and the sum of the values of the other sites in the neighborhood'.

but in 'New kind of science ' in the page170 there is :
'Usually i consider so-called totalistic rules in which the new color of the center cell deponds only on the average of the previous color of its four neighbors,...'
and in the page 60 there is also:
'The idea of a totalistic rules is to take the new color of each cell to depond only on the average color of neighboring cells, and not on their individual color.'

So my question is: the new color deponds on the sum of the five colors of cells or the average (totalistic rules)?
And the avarege means division by 5?
And what is the role of codes or rules in this case where any color cell deponds on the sum or average of its neighbors?

For example with the rule 454(111000110) , and if the center cell is black with its right cell, the other cells are white, how will be the new color of the center cell?
** if with the sum: we have 0+0+1+1+0=2
what means this '2' because we have just 0(white) or 1(black).

** if with the average :we have 2/5(what that means).

So i don't understand how we can genere the new color of the cell according to the codes?

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03-08-2009 11:54 AM
Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

Neither totalistic nor outer totalistic depend on a literal average in the sense of mapping the color to just the mean of the previous colors. There will be individual rules that effectively do that in either space, but in each rule space there are others. Totalistic or outer totalistic do depend on an average, effectively, but they don't simply get the next cell by averaging. Their rule tells them what to do with any neighborhood total (which necessarily corresponds to a different value of an average, hence Wolfram's language, but that is just causing you confusion).

Totalistic means there are 6 possible distinct configurations on the previous step -

there are 5 black cells in the neighborhood
there are 4 black cells in the neighborhood
there are 3 black cells in the neighborhood
there are 2 black cells in the neighborhood
there is 1 black cell in the neighborhood
there are 0 black cells in the neighborhood.

There are therefore 2^6 = 64 possible totalistic rules with 2 colors, and using the 5-cell neighborhood, corners excluded.

Outer totalistic means the center cell varies independently, and only the outer 4 are lumped into the outer total. There are 10 possible distinct configurations on the previous step -

there are 4 black cells around the center and the center is black
there are 4 black cells around the center and the center is white
there are 3 black cells around the center and the center is black
there are 3 black cells around the center and the center is white
there are 2 black cells around the center and the center is black
there are 2 black cells around the center and the center is white
there are 1 black cell around the center and the center is black
there are 1 black cell around the center and the center is white
there are 0 black cells around the center and the center is black
there are 0 black cells around the center and the center is white

There are thus 2^10 = 1024 possible outer totalistic rules with 2 colors and corners excluded.

OK?

Now rule 454 is greater than 64, so it is an outer totalistic rule number. It has 10 digits, not 9. Those digits are {0,1,1,1,0,0,0,1,1,0}. That means its rule table is -

4 black cells outside, center is black -> white (0)
4 black cells outside, center is white -> black (1)
3 black cells outside, center is black -> black (1)
3 black cells outside, center is white -> black (1)
2 black cells outside, center is black -> white (0)
2 black cells outside, center is white -> white (0)
1 black cell outside, center is black -> white (0)
1 black cell outside, center is white -> black (1)
0 black cells outside, center is black -> black (1)
0 black cells outside, center is white -> white (0)

So, the right cell is black and the center cell is white, there is only 1 black cell outside. The third line from the bottom covers that case - 1 black cell outside, center is white - and it says "so the new cell is black". I've put in in bold above, in the entire rule table. The 3rd-to-last binary digit in the rule number says what to do for that combination of outer total and center cell.

If the one black cell were on the left instead of the right - no difference, same outcome. If the one black cell were above, not right or left - no difference, same outcome. Same for the bottom. An outer totalistic rule doesn't care which exterior cell is black, only how many exterior cells are black.

I hope this helps.

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03-08-2009 05:19 PM

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