Reversible Cellular Automata Part 1

by **hbyte** » Sun Feb 26, 2023 3:21 pm

Cellular Automata are exciting and have been discovered in the 60's

Each CA rule begins from a well ordered state and some evolve into a chaotic one.

Initial conditions are important.

They are deterministic they apply a simply rule to determine the value of each cell
using the values of neighbouring cells at time t-1

E.g Rule 214

111 110 101 100 011 010 001 000
1-----1-----0-----1-----0-----1-----1-----0

Left=1 Center=0 Right=1 ==> Cell=0

They are identified by the value of the each rules binary number e.g 11010110 = 214

I have written a short function that takes any rule and generates its boolean form

Code: Select all: int rule_gen(int rule,int left,int center,int right,int top){ //Generate rules from rule code int out; if(bitset<8>(rule)[0]==1){ if(left==1 & center==1 & right==1){out=1;} }else{ if(left==1 & center==1 & right==1){out=0;} } if(bitset<8>(rule)[1]==1){ if(left==1 & center==1 & right==0){out=1;} }else{ if(left==1 & center==1 & right==0){out=0;} } if(bitset<8>(rule)[2]==1){ if(left==1 & center==0 & right==1){out=1;} }else{ if(left==1 & center==0 & right==1){out=0;} } if(bitset<8>(rule)[3]==1){ if(left==1 & center==0 & right==0){out=1;} }else{ if(left==1 & center==0 & right==0){out=0;} } if(bitset<8>(rule)[4]==1){ if(left==0 & center==1 & right==1){out=1;} }else{ if(left==0 & center==1 & right==1){out=0;} } if(bitset<8>(rule)[5]==1){ if(left==0 & center==1 & right==0){out=1;} }else{ if(left==0 & center==1 & right==0){out=0;} } if(bitset<8>(rule)[6]==1){ if(left==0 & center==0 & right==1){out=1;} }else{ if(left==0 & center==0 & right==1){out=0;} } if(bitset<8>(rule)[7]==1){ if(left==0 & center==0 & right==0){out=1;} }else{ if(left==0 & center==0 & right==0){out=0;} } return out; }

So now you can generate any rule

Now we need to add a 4th value that being the cells value 2 time steps back at t-2

Using this and incorporating it into the rule of any CA will make it fully reversable:

Code: Select all: out = (out==1 ^ top==1)? 0:1;

top being the value of the cell at t-2

Here is the output of a program that applies a reverse rule 214 to recover a random seed:

Code: Select all: ---Seed Before 214--- ....................0.....................0....... ..00.....................................0........ ........0.....................................0... .......0.......................................... .......0.....0.0.................................. ...............0.........................0........ .0..................................0.....0....... .................................................. .0...........................0.................... ......................0................0.........0 .................................................0 .................................................. .........0.0.........................0............ .................................0...0............ ..........0...............0....................... .....................................0............ ...0........0....0.............0......0........... ............0.............................0....... ....................0.......0..................... ......................................0........... ..........................00......0............... ...................0.............................. 0..................0.........................0.... ........0..........................0.......0...... ......0........................................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0.........................

Now lets apply the rule using the seed:

Code: Select all: ---Output from Rule 214--- ........................0......................... ........................0......................... ........0..............0.0....................0... ........00............00000..................000.. ........000..0.0.....0..0.00................0..00. .......00.0000..0...00...0000............0.00..000 00....0..0000.0.00.0..0.....00......0...000..000.0 000..00......00.00.0..00.00.000....000.0..0...0.00 .0000..0.000....0...0000..0.0000..0..0.0...0...0.. .0.00..0..000000.....00.0...0..0000....00..00..0.. 0.00.00.0.0.0000.....0.00000...000.0.0000000000.00 00..0.000..00...0...0.0..000..00..00..0..00..000.0 .0..0000...0..0.00.0000..0..00......00...0...0.000 0.0000....0..00.00...0.00....0.0.00.....0...0.000. 00.0...0.0..0...0.00..0.0.0....0..0000.00..000...0 .00..000.0.00.00.0.00.00..00.00.0.000....00..0.0.. .0000.0.....0..00.000.00.000..0.0.0...00..0....000 .0.000...0..00.000....0.....0.......00.00.00.00000 ..00.0..000000.0.0.0.0...00..00.0.00.0000.00..0... ....0.00..00....0..0.0..0.00..0.0..000....0.0...00 0.0.00.0..0..0..0.0...0000..0....0000..000.00000.0 0.0.000.00..0000..00...0.0..000...0...0.000..00000 0...0.00.0.0..00.00.0...0.000000....00000.0.....0. .000.000...0..0....000..00.0...00.0..0.0.0...00..0 0.000.0.000.00...0...000000..0.00.00..0.00..0.00.0 000.00.0.000.0..0000.0000.0.00.0..000.00..000000.0 .0.0000000.00.00..00.0..00.......00.0.00...0.0...0 0.0...0.0.0000.0..0......0..00.00..0..0.0...0..00. 000.0..0.0...00.00...0..0.00.0..0..0.0.000..0.0.00 .0..00.000.0.000.0..0000.0.00.0..00..00..000..0000 0..000.00..0.0.00.00..00.0000000..0.000..000.000.. 0.0....0..0...0000.0..0.000..0.00......000........ ..0.000..00......00.00.00.....00000.00..0..0.000.0 .0...00.0..0..00.000.00.0.0......00..00...00..00.0 .0...0..0..000.0.0.000000.00..00.0.0.00000..0.0... 0.0.0..0.000000...000..00.0000.0..00.0.000..0....0 000.0.000.0...0........00.0.000.0.....00..00.00.0. 00......00..0..0..000.000..00.000000.......00.0.00 ....0.0.00.000.000.00...00...000.0000.00...000..00 .0.00.0.0....0.00000.00..000....00..0..00.00.0.00. 00.........0...0..0.0.000000000......0.00...0..... ..000.0.0.00000....00000....000000...0.0.00.0..000 ...00.0.0...000......0..000.0..0000.0...0.0..00.00 0..0.....00.0..0...0....000....0..0.0...00.0..000. 000...0...0....00.0000.0...00.0......0.000000.00.0 .00..000...00.000...00.000..0.0...0..0....0.0.0.0. .0.00..00...0....00.0..0.00....0.0000.000..0...00. 0.0.0..000...000..0...0..0000..0...000.000.0.....0 0000.000.00...000..0000..0.0000.00.0.000.0..0..0.. 00.00.0.0000..0.00..00.0..00.000.0..000.0.0.00000.

Ok now we want to recover the seed so we shall apply the same rule in reverse to do this I will
generate a mask - when applied to the output it will reveal the seed. The mask is made from the reverse application of the rule on the above output:

Code: Select all: ---Mask from Seed 214 Reverse--- .................................................. .................................................. 00000000000000000000000.0.000000000000000000000000 0000000...000000000000.....000000000000000000...00 0000000....0000000000.00.0..0000000000000000.00..0 0000000..0....0..000..000....00000000000000..00... .00000.00....0.0..0.00.00000..0000000000..000...0. ...00..000000..0..0.00..0..0...0000...0.00.000.0.. 00...00.0...0000.000....00.0.0..00.00.0.000.000.00 0.0..00.00......00000.00.000.00....00000.00..00.0. .0..0..0.0.0....00000.0.....000...0.0..........0.0 ..00.0...00..000.000.0.00...00..00..00.00..00...0. 0.00....0.0000.0..0....00.00..000000.0000.000.0... .0....0000.00..0..000.0..0000.0.00.00.00.000.0...0 ..0.000.0..0.000.0..00.0.000000.00....0..00...000. 0..00...0.0..0..0.0..0..00..0..0.0....000..00.0.00 0..0.0.00000000......0..0...00...0.0000.00.0000... 0.0...000.000.0...0000.00000.0000000..0..00.0..... 00..0.00......0.0.0...000..0...0.0..0....0..00.000 0000.0..00..0000.00.0.00.0..00.0.00....000.0.000.. .0.0..0.00.00.00.0.000.......0000.0..00...0.....0. .0.0...0..00....00.0000.0.00...000.000.0...00..... 0000.0..0.0.00..0....000.0......0000.....0.00.00.0 0...0....00.00.0000...00..0.000..0..0.0.0.0.0..00. .0...000...0..000.000......00.0..0..00.0..00.0..0. ...0..0.0...0.00....0...00.0..0.00...0..00......0. 0.0.......0..0..00..0.000.0000000..0.0..000.0.000. .0.000.0.0....0.00.00000..00..0..00.00.0.000.00..0 ...0.00.0.000..0..000.0000..0.00.00.0.0...00.0.0.. 0.00..0...0.0...0.00......0..0.00..00..00...00.... .00...0..00.0.0..0..00..........00.0...00...0...00 .0.0000.00.000....0.00.00..00.0..000000...00000000 00.0...00..000000..0..0.000000.....0..00.00.0...0. 0.000..0.00.00..0...0..000.000000..00..000..00..0. 0.000.00.00...0.0.0.....00..00..0.0.0.....00.0.000 .0.0.00.0......000...00.00....0.00..0.0...00.0000. ...0.0...0.000.00000000.00.0...0.00000..00..0..0.0 ..000000..00.00.00...0..000..0......0000000..0.0.. 0000.0.0..0...0...0..0000.000...0....0..000...00.. 0.0..0.0.0000.0.....0..0....0000..00.00..0..0.0..0 ..000000000.000.00.0.0..0......000000.0..000.00000 00...0.0.0.....0000......000......000.0.0..0.00... 000..0.0.000...000000.000..0.00....0.000.0.00..0.. .00.00000..0.00.000.00000..0000.00.0.000..0.00...0 ...000.000.0000..0....0..00..0.000000.0......0..0. 0..00...000..0...000..0.0.00.0.000.00.0000.0.0.0.0 0.0..00..000.0000..0.00....0000.0....0...00.000..0 .0.0.00...000...00.000.0.....00.000...0...0.00000. ....0...0..000...00....0..0....0..0.0...0.00.00.00 ..0..0.0....00.0..00..0..0..0...0.00...0.0.0.....0

Now we simply apply the mask to recover the hidden seed using the mask!!

Code: Select all: ---Mask applied to Output 214 Reverse--- ........0.....................................0... .......0.......................................... .......0.....0.0.................................. ...............0.........................0........ .0..................................0.....0....... .................................................. .0...........................0.................... ......................0................0.........0 .................................................0 .................................................. .........0.0.........................0............ .................................0...0............ ..........0...............0....................... .....................................0............ ...0........0....0.............0......0........... ............0.............................0....... ....................0.......0..................... ......................................0........... ..........................00......0............... ...................0.............................. 0..................0.........................0.... ........0..........................0.......0...... ......0........................................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0......................... ........................0.........................

As you can see by using the reverse rule you can exactly recover the seed after applying the rule and then reversing the rule.

When you apply the rule to the seed use the following:

Code: Select all: rule = 214 seed==1 ^ top==1 ^ rule==1

This will seed the rule 214 and make it reversable

To reverse the rule and recover the seed use this:

Code: Select all: NOT (top==1 ^ rule==1)

(NB. ^ = XOR)

This might be a useful way to encrypt your data so long as the rule
remains secret!

by **hbyte** » Tue Feb 28, 2023 1:30 pm

In part 1 we recovered a seed by performing a reverse rule, generating a mask and then

revealing the seed from the output

Now we will see that we can alter the output state (future state) and generate a new seed by

using the reverse rule.

Instead of applying random [0,1] to any cell as we did earlier to create a seed we will instead

create a logical rule where

random [0,1] ^ output==0 ? output=0

This will alter the future state in such a way that we can apply the reverse rule and

recover a brand new seed using

NOT (top==1 ^ rule==1)

where rule = 214

Here is a new seed recovered from the above altered future state using this method:

Code: Select all: ........................0......................... ........................0......................... .....................0..0......................... ....................000.0.0....................... .....................0..0000...................... ..........................0....................... .......................0.......................... .................................................. ................0.......0.....0................... ...............0.0......0....000.................. ................0.......0.....0................... ............0...........0......................... ...........000..........0.0....................... ............0..0........0000..........0........... ...............0........0.0............0.......... ...............0........0.............0........... ........................0.....0................... ........................0....000.0.....0.......... ........................0.....0.000...000......... ........................0........0.....0.......... ....0...........0.......0......................... ...0.0.........000......0.........0............... ....0...........00......0........00............... .................00.....0.........0............... .................0......0......................... ........................0......................... ........................0.........................

As the altered future state was made using an XOR the recovered seed is made of cross's.

You can verify this is the correct seed as applying the forward rule will regenerate the altered future state.

Now we can turn back time and recover the a state in the past and change the past by altering the future!!!

by **hbyte** » Sun Apr 16, 2023 2:07 pm

Ok so the next stage was for me to start building a Cellular Automata that can have non binary values
in order to use non binary we need to use fuzzy logic in the rules instead of binary logic.

So for Rule 90:

Code: Select all: Rule 90 = Left Cell Xor Right Cell (L ^ R)

Its a simple rule that only uses two cells.

Now lets make it reversible by adding the value of the cell 2 steps back (T):

Code: Select all: Rule 90 R = (T ^ L ^ R)

Ok so thats fully reversible. In order to make it fuzzy we initialize the first 2 lines of our cells with random values within the range [0:0.25] - I only got this to work with this range(so far).

To Fuzzify we are using a parametric function 1-X^2.
To Defuzzify we are using X^2.

To apply the rule to our Fuzzyfied values we are using multiplicative operators where:

Code: Select all: x AND y = x*y NOT x = 1 - x Hence, x OR y = NOT( AND( NOT(x), NOT(y) ) ) x OR y = NOT( AND(1-x, 1-y) ) x OR y = NOT( (1-x)*(1-y) ) x OR y = 1-(1-x)*(1-y) x OR y = x+y-xy

So if we know how to do AND, OR, and NOT we can definately do XOR:

Code: Select all: Fuzzy Rule 90 Cell0 XOR Cell2 IF CELL0 AND CELL2 OUT=0 IF CELL0 AND NOT CELL2 OUT=1 IF NOT CELL0 AND CELL2 OUT=1 IF NOT CELL0 AND NOT CELL2 OUT=0 IF (CELL0 AND NOT CELL2) OR (NOT CELL0 AND CELL2)

Ok so here is the rule applied using C++

Code: Select all: float applyrule(float left,float center, float right,float top,int rule,int rev){ if(rule==1){ //fuzzy conways rule 90 = left cell xor right cell float comp1 = (left * (1-right)); float comp2 = ((1-left) * right); /*Apply rule without top = out*/ float myout = comp1+comp2-(comp1*comp2); float myout1,myout2,myout3; comp1 = (myout * (1-top)); comp2 = ((1-myout) * top); /*Apply rule with top= outr*/ myout1 = comp1+comp2-(comp1*comp2); /*Create output*/ if(rev==1){ float comp3 = 1-top; /*Invert tophat = ~top*/ comp1 = (myout1 * (1-comp3)); comp2 = ((1-myout1) * comp3); /*XOR mask = ~top ^ outr*/ myout2 = comp1+comp2-(comp1*comp2); /*Create Mask*/ comp1 = (myout1 * (1-myout2)); comp2 = ((1-myout1) * myout2); /*XOR mask xor output = mask ^ outr*/ myout3 = comp1+comp2-(comp1*comp2); /*Reverse Rule using Mask*/ return myout3; }else{ return myout1; } }else{ return 0; } }

Step 1. Init first 2 layers of CA [0:0.25]
Step 2. For Rule 90 FuzzyFy Cell Left, Cell Right and Cell Center 2 Layers back using 1-X^2
Step 3. Apply Rule 90 using Multiplication (Cell Left ^ Cell Right ^ Cell Top)
Step 4. Defuzzyfy the Output for Cell using X^2

Step 5. Repeat the above Steps using Reverse Rule = NOT (Cell Top ^ Cell)

Here is an example of a Fuzzy CA using a seed to generate an output, alter the output and then applying a reverse rule and recovering an altered seed.

Alter the future state using the following Rule Cell = NOT(Rnd ^ Cell)

It works!!!!

Code: Select all: Seed: --Result-- 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0.361423,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0.481429,0,0,0,0,0,0,0,0, 0.499924,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0.437887,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0.420797,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0.32445,0,0,0,0,0,0,0,0, 0,0.393147,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0.42394,0,0,0,0,0.36436,0.274377,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, The End --Fuzzy CA-- Output: --Result-- 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0.361423,0,0,0,0,0,0,0,0,0, 0,1,0,0,0,0.684988,0,0,0,1, 1,0.481429,1,0,1,0,1,0,1,0, 0.500076,1,1,0,0,1,0,0,0,0.725028, 1,1,0,1,0,0,0,0.927732,0,1, 1,0,1,0,1,1,1,0,0,0, 1,1,0,1,1,0,1,0,0.437887,0, 0,1,1,1,0,1,1,0,0,0, 0.369152,0,0,0,0.176763,1,0,1,1,0, 0,0,1,1,1,1,1,1,1,0, 1,1,1,0,1.24974e-16,1,0,1,0,1, 0,0,0,0,0,1,1,1,1,1, 0,0.67555,1,0,1,0,0,1,0,0, 1,0.606853,1,0,0,0,0,1,0,1, 0,0.0814234,0,1,1,0,1,1,0,1, 0,1,0,1,1,0,1,0,0,1, 0,2.47773e-37,0,0,0,0,1,0.601306,1,1, 1,1,0,1,1,1,0,0,0,0, 0.57606,1,0,1,0,0.63564,0.274377,1,1,0, 0,0,0,1,1,5.06639e-06,0,5.06639e-06,1,0, 1,1,1,0,1,0,0.999995,0,0.292245,1, 0,0,1,1,1,0,0,0,0,0, 1,0,0,0.224128,0,1,1,0,1,1, 1,0.252083,0.999998,1,2.38419e-06,1,1,0,1,0, 0.260872,0,0.260872,0.999998,0,0,0,0,1,1, 0,0.733358,0,0.00433832,1,1,1,1,0,0.00433832, 0.00433832,0,0.00433832,0,1,0,0,1,0,1, 1,1,0,1,1,0,0,1,0,0.814692, 0,1,0,1,0,1,1,1,0,0, 0,1,0,1,1,1,0,0,1,1, 0,1,0,0,0,0,0,0,1,1, 0,1,1,1,1,1,0,1,0,0, 1,0,0,0,0,1,0,0,0,1, 1,0,1,1,0,1,1,0.96997,1,1, 0,0,1,1,0,0,0,0,0,1, 0,1,0,0,1,1,1,1,0,1, 0,0,0,0.946532,1,0,0,1,0,1, 1,1,1,1,0,0,0,1,0,1, 0,0,0,0,0,0,1,1,0,0, 1,1,1,1,0,1,1,0,1,1, 0,0,0,1,0,1,0,0.901547,1,0.151189, 1,1,0,1,0,1,1,1,0,0, 1,1,0,1,0,0,0,0,0,1, 0.647633,0,0,1,1,1,1,1,1,1, 0,0,1,0,0.718253,0,0,0,0,1, 0,1,0,1,1,0,1,1,0,1, The End Reverse Rule: --Result-- 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0.361423,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0.684988,0,0,0,0, 0,0.481429,0,0,0,0,0,0,0,0, 0.499924,0,0,0,0,0,0,0,0,0.274972, 0,0,0,0,0,0,0,0.927732,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0.437887,0, 0,0,0,0,0,0,0,0,0,0, 0.369152,0,0,0,0.823237,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0.32445,0,0,0,0,0,0,0,0, 0,0.393147,0,0,0,0,0,0,0,0, 0,0.918577,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0.601306,0,0, 0,0,0,0,0,0,0,0,0,0, 0.42394,0,0,0,0,0.36436,0.274377,0,0,0, 0,0,0,0,0,1.01328e-05,0,1.01328e-05,0,0, 0,0,0,0,0,0,1.01328e-05,0,0.707755,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0.224128,0,0,0,0,0,0, 0,0.747917,4.76837e-06,0,4.76837e-06,0,0,0,0,0, 0.422814,0,0.422814,4.76837e-06,0,0,0,0,0,0, 0,0.427485,0,0.00865763,0,0,0,0,0,0.00865763, 0.00865763,0,0.00865763,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0.814692, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0.0300302,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0.946532,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0.0984527,0,0.151189, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0.352367,0,0,0,0,0,0,0,0,0, 0,0,0,0,0.718253,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, The End Seed: --Result-- 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0.361423,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0.481429,0,0,0,0,0,0,0,0, 0.499924,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0.437887,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0.420797,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0.32445,0,0,0,0,0,0,0,0, 0,0.393147,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0.42394,0,0,0,0,0.36436,0.274377,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, The End

Reversible Cellular Automata Part 1

Reversible Cellular Automata Part 1

Part 2 Changing the Future state

Re: Reversible (Fuzzy) Cellular Automata Part 3

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