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Lepton
1:41+ Arse Scratcher

Posted: Fri May 29, 2009 11:07 am    Post subject: 1

I was stuck in a meeting today with a new notebook, so I decided to see if I could create the cellular automata that are introduced in Wolfram's "A New Kind of Science". I used functions in each cell to display either a 1 or a 0 (if you actually do this, set up your spreadsheet so that 0s aren't displayed) depending on the values of the three cells directly and diagonally above. The simplest example would be:
 Code: =mod((abs(A1+B1+C1),2)

Using the modular algebra available -- addition and subtraction, mod 2 -- can you create Wolfram's interesting Rule 30 or Rule 110 automatons?

Quick overview of cellular automata:
Imagine a big grid of squares that can be either black or white. In the middle of the first row is a single black square. Below that row, each square is colored according to a particular set of rules that take as input the values of the cell directly above, the cell diagonally above left, and the cell diagonally above right.
ralphmerridew
Daedalian Member

 Posted: Fri May 29, 2009 3:59 pm    Post subject: 2 With just addition and subtraction, mod 2, there are only 8 rules you can create. I'm not sure whether adding abs improves that any. It's fairly easy to create any binary rule with addition and multiplication, mod 2, though.
ChickenMarengo
Daedalian Member

 Posted: Sat May 30, 2009 10:52 am    Post subject: 3 Can't you create 16 rules, as you can also optionally add 1 to invert a rule? Allowing multiplication, I get: Rule 30: =mod(A1+B1+C1+B1*C1,2) Rule 110: =mod(A1*B1*C1+B1*C1+B1+C1,2) To get these, start with rules that will only match one pattern from the previous generation. e.g. A*(B+1)*(C+1) mod 2 will only give you a 1 when the previous generation is 100. Add up these rules for all the patterns that you want to give a 1, then multiply out and simplify by removing pairs of duplicate terms.
lostdummy
Daedalian Member

Posted: Mon Jun 15, 2009 2:31 pm    Post subject: 4

Well, it seems like it is possible to do it even without multiplication, if what is given in example is allowed.

In fact, if ABS could be used, then I can make it even without "mod 2" .

I will show example with defining "functions" to shorten up writing, but in excel cells those functions can be expanded to be entire formula instead:

sgn(n)= abs(n+0.5)-abs(n-0.5)
as(n)= abs( sgn(n) )
is0(n)= 1-as(n)

sumN= A1+A1+A1+A1+B1+B1+C1

cell= is0(sumN-1) + is0(sumN-2) + is0(sumN-3) + is0(sumN-4)

This will put in "cell" value needed for Rule 30. One expanded cell in real excel sheet at position I2 would look like:

 Code: =(1-ABS(ABS((H1+H1+H1+H1+I1+I1+J1-1)+0.5)-ABS((H1+H1+H1+H1+I1+I1+J1-1)-0.5)))+ (1-ABS(ABS((H1+H1+H1+H1+I1+I1+J1-2)+0.5)-ABS((H1+H1+H1+H1+I1+I1+J1-2)-0.5)))+ (1-ABS(ABS((H1+H1+H1+H1+I1+I1+J1-3)+0.5)-ABS((H1+H1+H1+H1+I1+I1+J1-3)-0.5)))+ (1-ABS(ABS((H1+H1+H1+H1+I1+I1+J1-4)+0.5)-ABS((H1+H1+H1+H1+I1+I1+J1-4)-0.5)))

where H1=0, I1=1, J1=0, and I2 is first cell under initial cell. All other cells can be just copy from this one.
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