examples

John Conway's "Game of Life"

Every empty cell with exactly three neighbors is a birth cell.
Each cell with four or more neighbors dies from overpopulation.
Each cell with one or less neighbors dies from isolation.

Its neighborhood consists of the eight cells surrounding a cell -> all the rules will start with an '8'.

Every empty cell with exactly three neighbors is a birth cell.

This rule is applied to any empty cell:

'V'

To have an action, there must be three cells (color 1) in the neighborhood:

'A=3'

The action is the birth of a new cell (color 1):

'A'

The first rule is: 8VA=3A

Each cell with four or more neighbors dies from overpopulation.

This rule is applied to cells of color 1:

'A'

To have an action, there must be more than 3 cells (color 1) in the neighborhood:

'A>3'

The action is the death of the cell:

'V'

The second rule is: 8AA>3V

Each cell with one or less neighbors dies from isolation.

The third rule is: 8AA<2V

"Game of life":

8VA=3A
8AA>3V
8AA<2V

Edward Fredkin's rules

Each cell with an even number of live neighbors becomes or remains empty
Each cell with an odd number of live neighbors becomes or remains live

Its neighborhood consists of the four orthogonally adjacent cells -> all the rules will start with a '4'.

Each cell with an even number of live neighbors becomes or remains empty

This rule is applied to any cell (empty or not):

'T'

To have an action, there must be an even number of neighbors:

'AM2'

The action is the death of the cell:

'V'

The first rule is: 4TAM2V

Each cell with an odd number of live neighbors becomes or remain live

This rule is also applied to any cell:

'T'

To have an action, there must be an odd number of neighbors:

'AN2'

The action is the birth of the cell:

'A'

The second rule is: 4TAN2A

Edward Fredkin's rules:

4TAM2V
4TAN2A

These rules can easily be simplified:

4AAM2V
4VAN2A

2 colors version of John Conway's "Game of Life"

8VN=3P
8PN<2V
8PN>3V

These rules work if you have only 2 colors. A third color could interact and create mistakes in patterns

The rules can be modified as following:

Every empty cell with exactly three neighbors of colors 1 or 2 and at least 2 cells of color 1 will become a cell of color 1
Every empty cell with exactly three neighbors of colors 1 or 2 and at least 2 cells of color 2 will become a cell of color 2
Each color 1 or 2 cell with four or more neighbors of colors 1 or 2 dies from overpopulation.
Each color 1 or 2 cell with one or less neighbors of colors 1 or 2 dies from isolation.

Colors 1 & 2 cells can't interact with other cells now

Every empty cell with exactly three neighbors of colors 1 or 2 and at least 2 cells of color 1 will become a cell of color 1

There are 2 condition for a unique cell. The solution is to 'mark' this cell:

8VN=3W

This marks (mark 1) all cells with exactly three neighbors of colors 1 or 2

to complete the first rule:

8WA>1A

The other rules are easy to write:

2 (or more) colors version of John Conway's "Game of Life":

8VN=3W
8WA>1A
8WB>1B (Each mark may be used by several rules)
8AN<2V
8BN<2V
8AN>3V
8BN>3V

Stanislaw Ulam's rules

Births occur on cells that have only one neighbor
All live cells of generation n vanish when generation n+2 is born

We can represent each generation by a specific color -state-:

color 1 for generation n
color 2 for generation n+1

The first rule is:

4VN=1A

We now just have to change the color -state- of the cell:

0A1=1B

0B1=1V

There is no condition on the neighborhood

-> 0 neighbors

-> 1=1 is always true

Stanislaw Ulam's rules:

4VN=1A
0A1=1B
0B1=1V

As only cells whose neighborhood has changed during the last turn are tested, some rules may not work: 0V1=1A wouldn't do anything on an empty map (Try with only one cell) - I promise to correct that on the next version -