Registered: Mar 2005
Posts: 46

Consider a simple CA system which successively multiplies by a factor of 3.
Also consider a second simple CA system which successively multiplies by a factor of 5.
(edit) these CA systems are represented in binary cell sequences that evolve 2D down the page, other considerations have just not been considered yet.

Next consider a more complex CA system which has a set of initial conditions at the Initial Object position of the computation.
This CA applies a rule to the initial conditions and explicitly evolves it down the page. Also, we consider that the Initial Object (the set of initial conditions) is continually re-expressing, which means that it is actively applying the rule again and again at each successive step, sending an invisible clone down the page.

Next consider that we map the two simple CA systems, A and B to the initial conditions of C, the more complex CA.
We do this by taking the the current indexed values for both A and B and sum it together.. From this Sum set map towards the initial conditions of C in a sequencing ration.
An example ratio that i came up with was a sequencing of ABABABABABABABA,
You could do many sequence relations.

Let {AB...BA} = BILLY, a empty ordering of class dependent cells in a similar matrix which is the initial conditions for a 2 dimensional CA, C, which has the properties stated above.

Assumption...
If a cell connected to A or B in the similar matrix of C has no out put from A or B, then it is considered non existent. Successive additions of occupied (identified) cells into previously empty cells is called dimensionally expanding the initial conditions of C. This happens as the number of Cells in A and B required to represent the current value at their step increases towards infinity.
(EDIT-2)
Consider the fact that A and B map directly to Initial Conditions of C, but that A and B map through C into the space of C.

Now, consider a factor T, the timing function for the "run" of the CA's.
In this specific example i choose to allow them to start all at once..

I can only imagine in my head the CA which is produced because i do not have Mathematica. But i will recall the properties of the more complex CA C, which is a composition of A and B.

C has an Initial Object Position within a initial condition row of two empty cells, beyond it lies a continuous expanse of identity space in the X and Y directions. Each of the empty cells correspond one to A, one to B in the sequence pattern, BILLY={AB...BA}.

When C is ran, it will produce undefined unless it receives a value for A and B, the empty cells are considered undefined until this occurs.

This implies that the CA C is dependent upon the CA's A and B.
And also that initial row starts out undefined.

When we run all three CA's simultaneously, the number of cells require to produce the current step of A and B together becomes the number of initial condition cells which are considered at the initial Object position for C.
Recall that C both explicitly evolves down the page and that it also continually "re-emits" the "first up date" out side of the initial conditions..

Consider a vertical measure of cells, J.
Arbitrarily i call J a "vertically nesting measure". I assume that within the functionality of C there will be a J near the begging of the CA which has more "noise" from the changing initial conditions.

I also assume that there will be a much larger (perhaps infinite) J that extends beyond this first J.. Within this second J exists a subclass of other J which have what ever it is the relations you identify are... This second J i guess can create localized evolving structures which move down the page.

So, as A and B increase their size, the size of C is 2(n)... as the initial conditions are changing, simultaneously as a rule evolves down the page, new structures are introduced from the top of the CA due to the RE-EMITTING suggestion.

Last note, there is no consideration of the rule which unfolds the CA.

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Last edited by Enexseenge on 05-02-2007 at 05:00 PM

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Registered: Feb 2004
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Originally posted by Enexseenge
Consider a simple CA system which successively multiplies by a factor of 3.
Also consider a second simple CA system which successively multiplies by a factor of 5.
(edit) these CA systems are represented in binary cell sequences that evolve 2D down the page, other considerations have just not been considered yet.
....
...
...

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Registered: Mar 2005
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Since each row has a sequence of black and white squares then each row can be interpreted as a binary number.

Just to try to clarify.
C has initial conditions which are dependent on A and B.
The first row of C is empty until A and B have evolved and developed some structure.

"We do this by taking the the current indexed values for both A and B and sum it together.. From this Sum set map towards the initial conditions of C in a sequencing ratio.
An example ratio that i came up with was a sequencing of ABABABABABABABA,"

This was horribly written to say the least.
What was meant to be conveyed is that we take the state of a cell in A and use that as the left most initial condition for C. Then we take a cell in B and use it for the next initial condition for C, the cell one unit to the right. I have no idea why i wrote "sum" them together and then use that sum and represent it as ABABA in C...

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Originally posted by Enexseenge
Since each row has a sequence of black and white squares then each row can be interpreted as a binary number.
.....

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So you do have a binary digit sequence, but one makes a choice to read this from left or right or right to left. Within the CA it self the direction of read is ambiguous in a sense.

I wonder if there can be anything gained by looking at the CA as a collection of binary numbers, one for each column of the grid. Then the "initial conditions" would be the first number in the binary sequence and successive members of the sequence would be obtained on each update in terms of columns, not rows..

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Your idea for column style binary numbers from CA output would be an interesting study. This scheme requires some kind of decision about when to stop the CA automaton. Once you do stop it, you could also build an array of integer values and then sort the array. Maybe you could find some already known sequences?

I talked a little about sorting perceived binary numbers from ECA output here:
[url=http://forum.wolframscience.com/showthread.php?s=&threadid=1423]
sorted 1D ECA output[url]

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