Registered: Mar 2005
Multiple grid cell dependency
Notes on Multi-grid dependency systems.
Generally, a multi-grid dependency system is a class of systems which contain two or more evolutions of computational rules on separate grids connected together through a dependency correspondence.
In the most simple of cases, a grid B which has a cell dependency within a region of it’s computational space towards the values of a computational space within grid A, illustrates the dependency correspondence. From this very general relational scheme one can create the actual rules of the dependency correspondence in more detail- creating a third rule.
One can consider two very simple 1d CA which are each running separately.
Through the dependency correspondence we perform a computational bridge between the two rule systems. This bridge is a rule which is outside of the either of the rules which we are running in the separate CA systems. Outside of the rule, it need not be differnt in every case.
To define this rule consider the most primitive form of relations within a two color 1d CA system.
Grid B is running rule 110, grid A is running rule 110,
both of these rules are starting from one initial condition.
Grid B has it’s initial cell color 0, corresponding to the first color,
which is the color that produces 0 upon the nearest neighbor rules (0,0,0) ,(1,0,0) and (1,1,1).
Grid A has it’s initial cell color 1, corresponding to the second color, and this color produces 1 upon the nearest neighbor rules (1,1,0), (1,0,1), (0,1,1), (0,1,0), (0,0,1).
Grid B’s initial cell is dependent upon the cell directly below the initial cell in grid A.
In the simplest cases it is a direct computational bridge, where the color in the independent cell of A is the color of the dependent cell in B. So after one evolution of rule 110 within grid A, the dependent cell in grid B becomes the color of that evolutionary step. So the question then becomes, how does one arrange this sequence between the two grids? How does one apply the correspondence rule?
Now that the independent cell in grid A has changed color the dependent cell in B will have to change in order for there to be a dependency correspondence. But one cannot ignore the difference in sequence between the two grids. What does grid B do during the computation within A which produces the first step. And when the independent cell within A becomes the color 1, then how “quickly” does this become represented in B?
It’s a simple question of how the steps are sequenced together. Must a “Step” happen to account for this third rule which “updates” the dependent cell? Or are we simply to consider another continuous entity.
So how do we represent the sequencing of steps between these three rules?
I don’t think that there is really any choice about which way we can choose as “correct” because there are a variety of different ways which we could choose to use.
We could say that the step in which the independent cell within A becomes updated is the same step in which the computational bridge rule is applied, and that step 0 (initial cell, dependent cell) of B is corresponded in sequence to step 2 of A, and 1 of B is corresponded to step 3 of A.
Or we could interpret the computational bridge in terms of difference in sequence, and that grid A evolves one step in the time it takes for the bridge to update the dependent cell in B. Then 0 of B would correspond in sequence to 3 of A, and 1 of B would correspond in sequence to 4 in A. We could also set up an arbitrarily vast difference in sequence.
For example, we may have 100 steps of A be comparable to the single update of the bridge which colors the dependent cell in B. It helps to note that these systems don’t have to be “run by the same computer” and that there is no “correct” sequence scheme to use.
The only things that must be fulfilled is that one cell in a grid must have some type of dependent correspondence to another cell in a separate grid.
An interesting question to ask is what exactly does grid B do while grid A is still on step 0. Do we imagine that as A moves from step 0 to step 1 that B moves from step zero to step 1 as well? But since it’s rule dictates that it will produce a homogeneously 0 colored computational space upon the initial conditions present at step 0 then we observe no noticeable computation occurring. What if we are to imagine that at each step another step is “entered” into the system from the top. This would cause the rule to reapply over what it has already produced, creating the exact structure that already embedded within the computational space. So could we consider that grid B behaves this way and that it may be possible that all 1d CA, and possible more systems, behave this way?
Could we set up some type of dependent correspondence for the initial cell of B which says that it is dependent on more then one cell within A so that if we place these dependent cells at different points in A, then we will see B’s initial condition change as these corresponded cells in A are changed through it's evolution. For example B’s initial cell may depend on both the cell directly below the initial cell in A, and the cell in the second column to the left, down to the fourth step. This would cause the initial condition to change at step 4 of A, but this is only if one assumes that the cell dependency correspondence only ever manifests when an active computation happens on a cell. If this were not so, then the initial cell in B would change right back to white after the first step of B since the way we are used to looking at a CA shows us that the "squares up ahead" are all colored 0 (white in this case with reference to NKS).
So two concepts have been talked about, multi-grid dependency systems and the “Step interjection” idea where a step is entered into the CA from the top at each step. Perhaps these systems are of no real interest, but I lack the ability to prevent my self from considering them.
A great revolution is at hand, but this is just a metaphor.
Last edited by Enexseenge on 06-26-2007 at 07:20 AM
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