A New Kind of Science: The NKS Forum > NKS Way of Thinking > Rule Space.
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Enexseenge

Kingston WA

Registered: Mar 2005
Posts: 46

Rule Space.

Can we say that the cells them selves hold the rules? or is the rule something broader that exists outside of the cells?

It seems that computationally the rules have to be held in a memeory and then "re-called" at each step and that there is nothing inherently innate about the cells to the rules themselves.

There might be a "rule space" which holds rules and a "cell space" where the expression of these rules may manifest.

For example, in a computer program the memory of the computer holds the "rule space" for recall where as the processor preforms a function at each step to evolve the C.A. The rule exists independent of the grid of cells where the rule may express it self.

And in a physical system, the held (or current) state of matter represents the cell space, where as the rule space is represented by the range of transformational possibility which the matter has.

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05-04-2007 05:14 PM
Enexseenge

Kingston WA

Registered: Mar 2005
Posts: 46

Updated in parelell?

In a typical 1d CA which evolves down the page how can we say that each square in a row is updated in parallel when the computer which is running the program must actually find the results for each individual square.
Of course after it finds what each of the squares should be it then presents a result which creates an illusion of parallel update, but is it really a parallel update?

Are simple programs merely meant to act as a model of the universe, rather then claim that at the base level the universe operates as a simple program?

From what i get out of computational equivalence is that the only way for a modeling system to create all the details of a natural systems is for that modeling system to be the natural system it self.

If the universe at it's core works as a simple program then there must be an exchange which takes place that allows state to be updated, the computation. But when i try to consider what this exchange or computation could be beyond a model of a universe within a computer i am left in confusion.

At some level there is an exchange between discrete components which is responsible for the observable system which we are presented with.
What is this exchange?
can it be considered a signal?

An edge in a casual network seems to indicate the path of a signal in a sense, but does it give one information about the underlying rule, the rule which determines the results of a particular exchange?

What really exists within that edge?

If space is a collection of nodes then are we to assume that, for what ever configuration of nodes we have, as the universe has evolved in time that new nodes were created? or did all nodes exist prior to the universes existence in time and the casual relations between these nodes is the evolution of the universe?

Do we start with a grid of nodes, void of connections, and then begin drawing connections from one node to another?
Or do we start with connections approaching the first node, and from there we draw connections from that node outward until we decide to draw another node which receives these connections?

where is the computation?

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06-20-2007 08:08 PM
Tony Smith
Meme Media
Melbourne, Australia

Registered: Oct 2003
Posts: 167

Node persistence

If space is a collection of nodes then are we to assume that, for what ever configuration of nodes we have, as the universe has evolved in time that new nodes were created? or did all nodes exist prior to the universes existence in time and the casual relations between these nodes is the evolution of the universe?
Tackling this question in isolation my 2004 TickTock experiment showed that you do not need individual nodes to persist for local structure to persist. The TickTock rule is just the simplest of many that continuously regenerate a (graph-theoretic) network based on local rules. In conjunction with other results it is easy to see that local configurations which cycle can maintain the node count or can increase it exponentially and ultimately to recognise that there might be even more configurations which either produce or consume adjacent nodes. (Those configurations which quickly disappear/"die" or otherwise consume each other quickly filter themselves out of contention.)

[On an unrelated note, yesterday I tried my first "practical" application of the TickTock rule to quickly characterise the maximum connectivity of a random network.]

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06-21-2007 12:12 AM
Enexseenge

Kingston WA

Registered: Mar 2005
Posts: 46

On an unrelated note, yesterday I tried my first "practical" application of the TickTock rule to quickly characterise the maximum connectivity of a random network.

...

Tony
I'm a ecstatic about hearing this..
As soon as your results are complete i am very very interested in obtaining a copy.

I am interested in interpreting a continuous plane in terms of a random network, so you can see how these result would be of much value to me.

good day..

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06-21-2007 01:26 AM
Enexseenge

Kingston WA

Registered: Mar 2005
Posts: 46

Perspective and the nature of identity.

I am also interested in your input to the following ideas.

For any system which may exist within a larger system, but does not manifest until the proper conditions are met within the large system, we can consider in terms of an evolving graph. This system has a starting node which acts as the membrane between the larger system and our system of interest. You can consider that this embedded system has a "differnt" set of rules for connectivity within it's membrane, yet the initial inputs for it's starting node come from a system which at large need not share the same connective patterns.

Let us draw a boundary around this system which will appear at a later time in the evolution of our larger system, this serves merely as our arbitrary projection for conceptual aid.
Let us put a number on the outside of this boundary corresponding to the "ticks" within the larger system.

Now, what we do is allow the proper conditions to be met which allows an "input" into our embedded system. Prior to this input this system does not exist for there is no initial connective element which is allowed to spread from the membrane into the system. The nodes within this system cannot spontaneously generate connections but must be brought into existence through the reception of an incoming connection.

Now, we let this number outside of our box "tick" like a clock, and upon each tick we see that in our larger system there is an evolution of the #nodes and #connections for each tick. We find a rough pattern of replacement which for every x nodes at one tick there are x+n nodes at the next tick. This tick corresponds to the larger system's update speed.

After our secondary system receives a start up connection it begins to evolve.
We notice that for every update of our larger system there are many more updates within the embedded system, effectively creating a difference in time in a sense. So for every single evolution of our embedded system we may see that, in a metaphorical sense only a fraction of a single update has occurred in the larger system.
Eventually this embedded system produce connections which emanate out from the membrane and interact with the larger system.

Any ways, the idea is to make two systems (the larger system and the embedded system) which are part of an even larger third system which contains them both which can identify the difference in ticks, if ya get what i mean.

For example, in the embedded system there is a set "tick" speed wherein updates happens per tick (one update per tick?). But within the larger system, for every one tick corresponding to that larger system we see many more updates (more ticks) occur within the embedded system then within the larger. But still there is a passage of ticks which happen within the smaller system, a finer time in a sense, and both of these individual ticks for each respective system compare to each other within a difference in terms of the tick which you are taking as your unit. You have to choose one of them, don't you?

So if i take per tick within the smaller system (use it's time as my unit) i get a difference which is less then one. If i use the tick of the larger system as my one, then i get a difference which is greater then one.
But the third system can isolate each of their ticks and arbitrarily choose either one or the other, or both, as the unit. Allowing comparison

not sure what there is to respond to, just needed to write.

one more thing,
what does it mean when the difference in ticks is irrational?

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06-21-2007 02:12 AM
Enexseenge

Kingston WA

Registered: Mar 2005
Posts: 46

"quickly filter themselves out of contention."

Where do they go?

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06-21-2007 06:38 AM
Tony Smith
Meme Media
Melbourne, Australia

Registered: Oct 2003
Posts: 167

All of the above

Enexseenge, I spent more of the day than I maybe should have following the Tick Tock evolution of the random graph seed downloadable below.

The results, translated into 3D wire frames, are at http://www.twistet.com/ticktock/mystery.html. (The last couple of frames might take a while to load as the node count/file size grows exponentially but, once loaded, they rotate responsively on the relative luxury of a 24 inch iMac running Firefox.)

I need to emphasise that Tick Tock is just the simplest possible model of an as yet undefined class. It's nodes and links are minimalist and effectively ephemeral, though even while a complete new set of nodes and links are generated each tick, larger patterns persist.

Getting a ticking clock to emerge rather than be assumed is straddling the ravine between TODO and too hard. Tick Tock doesn't address clock variation and I don't have any coherent thoughts in that area.

Tick Tock links definitely cannot carry messages. There is no way for anything to influence anything it is not directly connected to. Change propagates only through the growth of the local network.

Your later question may be due to me conversationally assuming a couple of ideas which I've only started to document. "... filter themselves out of contention" is a translation of "cease to exist and therefore don't matter any more". As much as I find the thought discomforting, I'm now pretty much convinced by the idea that our cosmos is a bubble of (roughly) conserved space within a greater chaotically inflating universe, largely because an evolving graph where the only information is in the topology does not allow leakage into such bubbles. And, albeit in a different context, I'm presenting a paper later in the year on my notion that Aristotle's "final cause" can be non-teleologically updated into a notion that things (that matter) are finally caused post hoc by the uses to which they are put. It's only that which persists/is (roughly) conserved that can have any role in the future.

Tony Smith has attached this image:

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06-21-2007 11:32 AM
Enexseenge

Kingston WA

Registered: Mar 2005
Posts: 46

I need to emphasise that Tick Tock is just the simplest possible model of an as yet undefined class. It's nodes and links are minimalist and effectively ephemeral, though even while a complete new set of nodes and links are generated each tick, larger patterns persist.

Sounds very similar to a type of system i have been considering.
This system has 2 base categories for 4 different elements. These 4 elements are then broken up into two categories, the fixed and the fluid. I know it sounds cheesy and lame but i was trying to encompass the nature of the difference between the two different types of objects within those categories. The elements of this system interact in much the same way that an evolving graph does. They start at an initial seed point and then they evolve a connective pattern which produces a variety of possible structures.????

Very similar to your tick tock rule, but more complex, I have this concept for this 4 element system which says that in order for any structure within the network to exist it must display active (persistent) connections within an iterative application of connective rules within that network. I develop this concept because i wanted to create this powerful force of decay, something which would cause any element within the system to "disappear" in a sense, i wanted a return to nothing. But i also wanted a something which has persistent properties, but only under the condition that these somethings were able to "resist the decay" through sharing connective properties.

But these structures are not nodes / connection patterns in the typical sense, for within the system i have been considering there are two types of "nodes", these 2 nodes correspond to the fixed category of elements.
The first type of node is called a "structure node". It is a node which has no active connections to it. As a very rough metaphor you can consider the cells within a CA which are "ahead" of the CA evolution as "structure" type nodes.

The second type of node is called an "affected node".
An affected node is a structural node which has a connective element to another node within frozen reference frame which is obtained from out of the active application of connective rules upon the system which evolve within the iterative process.
This is very key, for there is a time period which is entirely dependent upon the "speed" of a tick (only comparable to a system which has a different tick (or set of ticks for a larger system), an observer within a specific tick system would notice no time for which they are "frozen and not moving" but would be entirely fluid within that speed of ticks). Only within this time period can we say that a node is a "affected node", for upon the next iteration we may find that this node no longer has any connections, and thus it would be a structural node. The key property of affect nodes is that if during one tick it exists, then naturally at the next tick the affect node must decay into a structural node. But it does release element number 3, which is discussed below.
But as you have shown, individual nodes need not persist in order for structure to persist, and that is very key to explain how a structural pattern within a network may "move forward" (non dependence to individually inhabited nodes at different ticks) and resist the decay, persist.
But i cant yet be as quick to abandon previously connected nodes. Perhaps my opinions could change in time.

Any ways, the structural node and affected node represent one of the categories within this system, the ridged aspects, or "fixed" aspects. Next we will talk about the fluid category of elements, expression and reception, which could possibly be represented by edges.

The third type of element within these systems is called an "expression".
An expression is a connective element which only has a connection to one node and it will persist within the next tick. iIn essence it's an open connection which has not yet become received by another node to establish a connection. The expression is what is left over after an affect node decays into a structural node. If within a frozen reference point of time (in-between ticks) for a system we find that we have an affected node, then it will inevitably have an “expression” extending outward from it. But this expression does not yet have a connection to another node. Upon the next tick the affected node decays into a structural node but the “expression” persists until it makes a connection with a node.

And that is when we get the fourth element of the system and the second fluid element, the reception.
A reception exists when a connection is formed to a structural node at a step, and it will decay upon the next tick. If an affect node is a structural node which has a connection to another node, then the reception is the connection between that prior node and the current node.

The system works like this.

An expression approaches a structural node.
An expression is received into a structural node.
The affect of this reception releases another expression.
The affect decays into a structure and the expression persists.

So for these 4 steps then what would be the state of our system?

Lets consider a system with one structural element.

System a.
E--> (--S--). t
(R-->A-->E ) t+1
(--S--) E--> t+2

Two structural elements.

System b.
E--> (--S--) [distance of one tick] (--S--) t
(R-->A-->E) (--S--) t+1
(--S--) (R-->A-->E) t+2
(--S--) (--S--) E--> t+3

Where S and A represent nodes and E and R represent edges.

Now, complexity I am assuming starts to appear when you connect up systems with different “tick speeds”. And the topic of different tick speeds requires more explanation.

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06-21-2007 09:49 PM
Enexseenge

Kingston WA

Registered: Mar 2005
Posts: 46

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.

For example.
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.

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06-26-2007 03:56 AM
Enexseenge

Kingston WA

Registered: Mar 2005
Posts: 46

Delay based systems

Attempting to develop upon the amateurly described suggestions in the previous posts I build upon the ideas of the “rule interjection” and the speak more about “multi grid dependency systems”.

First note that I do not have sufficient background in mathematical literature to navigate territory with much rigor but for some reason, I find my self considering systems in the way I do.

I got the idea for a multi grid dependency system when I was considering the idea that a collection of independent simple behaviors can produce a system which has very complex behavior if each of these simple behaviors could somehow be allowed to interact with eachother without effecting each of their independent grids. So I was not as interested in the rules being simple and producing complex behavior but for the behavior to be simple and then combined somehow with other simple behavior to produce a system which was complex. I thought that If I could arrange a set of separate CA rules which had regions of connective dependency to a main gird (the system which combines the simple behaviour into a complex behavior) that I could gain some insight into why I would even waste my time thinking about these things, since I have only one year of calculus and can barely make a nth root function in Lisp.

So there I was the observer viewing the separate CA grids… I noticed that each one evolved at a different speed and that shocked me and I questioned where and what exactly was produced when these separate grids are combined… Unfortunately I have to take the assumption that my viewing of the separate CA sets somehow represents the complex system which is produced from the collection of simple behaviors. I imagined that for each separate, actively evolving CA (region B) there was a connection to another CA (region A) which was a rule it self and this connection described how a region of cells in A is dependent on a region of cells in B. I then imagined that for any set of separately evolving network systems there could be a larger system which was connected to each one of those separate networks and could produce only what I can describe right now as “the sets behavior as a unit” (attachment).

So, for any system there is a rule which produces it’s behavior and a space in which that behavior must be displayed.
Separate systems, even ones of the same rule, can share between each other a dependency correspondence which is defined by a special connective rule which has “space elements” in each of the connected systems.
Attached is a crude schematic of this process.

In the case of the schematic space 1 and 2 would correspond to the “separate grids”, and space 3 would corresponds to a connection to of cells of 1 and 2. R1 would be the rule for the system of R1 and Space 1, R2 would be the rule for the system of R2 and Space 2, R3 would be the “dependency correspondence” rule which shows the connection between the two separate systems (R1,Space1) and (R2,Space2). Space 3 is not a separate gird it self, it is shown merely to indicate that the space in which Rule 3 operates on is contained in both space 1 and 2. In order for there to actually be a dependency correspondence then a region in space 1 has to be dependent on a region in space 2. This does not mean that space 2 cannot have a region which is dependent on a region in space 1. In face you could set up a dependency correspondence on just one grid to it self. This is where I consider the “step injection” idea where a rule continually injects a “front” of evolution on a space at each step.

So for any rule active in a space you have step one, then step two, step three and so on.
At each step we consider active cells, these active cells move down the page updating the space and leaving behind a frozen region. This idea is to interject new “steps” into the CA from various spots. Most simply one could consider interjecting new “steps” from the initial conditions so that as long as the initial conditions stayed the same, the same behavior would remain in the space. Imagine then setting up a dependency correspondence between certain cells in the initial conditions to cells within an entirely separate CA system. Then the “new steps” would produce different new behavior on the screen.

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07-11-2007 08:15 AM

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