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A Simple CA Model for our Universe
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Posted by: Jesse Nochella
Attached at the bottom of this post is a notebook containing a small program that I've written.
This is a very simple program. It works in a nearly identical fashion as the CAEvolveList function on NKS page 865. What it does involves elementary CA rules 110 and 30. I do not think that these particular rules are in fact fundamental, as I'll discuss later, but for reasons of simplicity in implementation and instruction, I use them as the fundamental rules in illustrating the model.
Now, basically what happens is quite easy to comprehend -- what the graphic represents is the successive insertion of the digits of the center column of rule 30 being computed by rule 110. Starting from the initial condition of some table of all white cells, cellular automaton rule 110 is applied. To the result of this step, the first digit of the center column of rule 30 replaces some cell (the center cell in this case). Rule 110 is applied one step to this modified set of conditions, and to the next step of evolution, the second digit of the center column of rule 30 replaces the cell to the immediate right of the one that it replaced the step before. This routine continues so on for the amount of steps specified and at the end of it all what one sees is the regular patterns and localized structures generated by rule 110 -- but instead of just expanding to the left, the rule expands both ways, with the regular pattern of rule 110 from a single black cell appearing on the left, and a rough, pitted edge, coresponding exactly to the diagonal layout of the center column of rule 30 appearing on the right. In the middle a continually growing field of interacting localized structures persists on down the page forever.
I think that this process can be used to model the earliest begginings of our universe, right down to the very first moments of the big bang, perhaps it can model events even before the big bang, as a sort of causal precursor for our own universe. This may all sound somewhat strange or counterintuitive -- first of all, nobody ever said anything about pre-bang events ever occurring, or even needing to occur.. that I've heard of anyway; and second, this is of course a CA model apparently taking the role of a fundamental mechanism. How can this be? Isn't a CA model too rigid to model causal interactions?
I will try to explain my reasoning in all of this. My intent is simply to create a model that is as easy to implement and understand as possible. From my understanding, the basic idea behind what constitutes a universe is rather easy to comprehend, but how the actual physical phenomena that we observe by living in one is harder to explain. However, if one starts with a model that is sufficiently simple and delivers it in a straightfoward manner perhaps some headway can be made in understanding general properties of a system like our universe.
So, the basic idea behind behind what constitutes a universe seems to revolve around the remarkably simple idea of class 4 universality, and class 3 randomness somehow being combined. This idea sounds simple enough that one might suspect that it has already been well understood. Perhaps it is not mentioned frequently because live input cellular automata are ultimately not dissimilar in overall bahavior with ordinary cellular automata; much in the same way that cellular automata laid out on a network don't produce any more complicated behavior than their linear counterparts. Indeed, any live automaton can just represent a sort of slice (a very loosely termed slice I might add) of a larger automaton functioning under one single rule that produces the visualized rule output, as well as the "live" input data. And in fact for the program in the provided notebook I believe a rather simple example of one of these exists.
Although I have not confirmed this personally, I think that a simple 2-dimensional rule involving only a few colors, needing only 4 neighbors starts from a single cell and can emulate the program in the attached notebook precisely. The basic idea for this rule is much like how one would idealize one-dimensional cellular automata evolving in 2 dimensional space. 1d CA's in 2d space can either evolve step after step in one single line, or have a shifting effect as they evolve. This basic phenomenon happens when a rule is in effect limited to certain dimensions of allowable replcements when it is applied.
My idea for a 2d model shows rule 110 evolving from a single cell and expanding horizontally to the left, all of one particular color. Rule 30's implementation is vertical where the left is downward and right is upward, and is of a diffirent color than the cells that compose rule 110's evolution. The initial condition is a single cell of the color that composes rule 30's evoultion. When the cellular automaton rule is applied, the initial 30 cell creates the next step of rule 30 emulation on the cell to the immediate right and changes its own color to the one corresponding to rule 110's cells. 110's cells do not interact with any cells other than ones of its own color, and do not interact with any cells vertically. The output of this cellular automaton I can imagine looks quite interesting. A sort of sideways 90 Degree triangle growing in all directions, with a definite tip on the left exibiting exactly 2 types of repetitive patterns, continuing from the left, one sees various patterns copuled with a broken C-like curve ending the repetive pattern on the left. from here on, there is layers of all kinds of difinite localized structures moving back and forth, some staying still. This continues all the way up to the edge, or bottom of our visualized triangle, where apparently random behavior is all that seems to occur. The interacting structures for the most part seem quite random, but for the majority of the bottom half there may be more repetition. The attached program and picture by default correspond to the middle horizontal slice of this 2-dimensional model, and can easily be changed to any horizontal slice by adding a third arguement to the CPCAEvolveList function that specifies, relative to the center column, which vertical slice of rule 30 is to be used.
As far as I can tell, The actual density of interactions sparsen at a slow rate. This is because of the limiting effect that the repetitive pattern that occurs on the left. Perhaps this effect can be generalized to account for rates of inflation. when one changes the rule applied from 110 to 110's reverse, 124, there is increasing sparseness further away from the input line at essentially the same rate that localized structures in rule 110 or its analogues from random initial conditions thin out. Notice the fact that structures not moving left or right are just as likely to collide as ones that do.
This overall phenomenon of heirarchial persistent structures seems to also be related to phenomena like brain activity, and may have some applications to the effect of, say, extracting exact information out of random sensory input. But before that, lets get even more general and return to the idea that a program like this can be used to model the universe, as I'll discuss in my next post.
Posted by: Jesse Nochella
In chapter 9 of NKS, on pages 541 and 542, Wolfram talks some about systems like the one considered here and makes a very important point that opens up a way to the abstraction that I want to make in explaining my reasoning. He says:
"Nevertheless, a potentially important point is that it is in some ways misleading to think of particles in a network as just interacting according to some definite rule, and being perturbed by what is in essence a random background. For this suggests that there is in effect a unique history to every particle interaction--determined by the initial conditions and the configuration that exists in the random background.
But the true picture is more complicated. For the sequence of updates to the underlying network can be made in any order--yet each order in effect gives a different detailed history for the network. But if there is causal invariance, then ultimately all these different histories must in a sense be equivalent. And with this constraint, if one breaks some process into parts, there will typically be no simple way to describe how the effect of these parts combines together."
I want to point out that what Wolfram says about why the true picture is more complicated is correct even for ordinary cellular automata of the type I discuss here. To see this, one must take the model presented to a higher level of abstraction than CAs are typically done.
Before we do this, it's perhaps important recognize just how far one can go with the idea of the universe being the simple cellular automaton rule 110 purturbed by the center column of rule 30. Here we have what seems like elementary particles with particle-like interactions that have uncanny similarity to Feynman diagrams, space expands, and it is conceivable that over a long time there can occur all sorts of complicated structures that are made of these particles. There are also basic fundamental limits like the speed of travel through the system. There is discreteness, in both space and in time as well. Also, there is a sort of computational irreducibility for the events going on throughout the evolution of the system for both sentient beings as well as outside observers. The systems involved combine the idea of a sequence that in all its evolution will never repeat, in effect generating every possible string of information, and the idea of universal computation which, in a way equivalent to our own thinking process, takes the random background and gives it meaning. has potential to create any conceivable system throughout its evolution. Already we have a model that seems to explain some of the thoughest questions, such as why space is configured in exactly the way it is and no other way. And above all this, the simplicity of the rule seems almost by default to give fundamental meaning to it.
But given all this, there are still all sorts of details about our universe that this rule insofar does not account for at all. I think that this rule however can indeed be abstracted further to a level that once done lets one easily understand just how a rule like this can in fact create our universe in every detail. A system that can be abstracted in this way need not the vast majority of features explained above. And in fact only the interaction of intrinsically generated randomness and universality are needed. I suspect that this can happen in a less seperated way than the model described here, but the crucial idea and the point that I want to make is that no matter the implementation, the outcome is always essentially the same so long as randomness and universality are combined in some way.
Essentially, the abstraction I want to make is a suggestion of a kind of general invariance for any randomly perturbed system capable of universal computation.
So, lets shift our perceptions a bit. Instead of trying to make the leap from that single cell at the top to the current state of our universe, which presumably is on some given step and is running from that step by applying the same rule over and over again, lets take a different road in seeing what's possible.
When modeling systems with cellular automata, there is usually a difficulty finding any real-life system that looks exactly like the output from a given rule. But in some ways the attempt to find a CA that can be said to create the universe in every detail we are trying to do essentially just that. The only difference is that instead of looking at the system from the outside, we are looking at it from within, and instead of making the analogies from, say, a CA that resembles boiling water or fluid turbulence to the actual systems with similar behavior, we are in effect trying to make the analogy from the behavior of particular rule to an exact state of our perception.
But how does this change things? Ultimately I don't think it does, but what it implies is that there can never be a meaningful model of our universe set up where there is undecidability. And so what one must do guarantee that a given rule is in fact the rule for our universe is to guarantee that there will be no such undecidability whether our exact state will be visited or not.
So how can one guarantee such a rule to visit our state? The answer, I believe, can be found by taking a system like the cellular automaton rule suggested above and abstracting it from a mechanical perspective to an analytical one. By doing this, we are not looking what the general properties of the system are in a certain time frame, but instead are looking at what the system will do over an infinite amount of time.
Such an analysis on systems like cellular automata usually have very little meaning, for it is usually impossible to find out what kinds of specific features a cellular automaton with complex behavior will have if run for a given number of steps in any shorter time than it takes to simply apply the rule over and over again. But the crucial point that I make here is that for systems like the one discussed above there are at least some features that that are guaranteed to persist. And indeed, the complexity alone of these persistent features is what leads me to the suspicion that at least on some fundamental level, they are the basic mechanisms at work.
Posted by: Jesse Nochella
In such a system as the one discussed above, there is the persistent property of randomness affecting some computationally universal process. Because we know for a fact that given space rule 30 will run forever producing random irreducible output that never repeats itself, and that in the case of the rule discussed this output dictates directly the exact conditions under which rule 110 computes anything, we can then deduce the fact that the actual computations that rule 110 performs will have a sort of infinite variety. This implies that over an infinite amount of time, every possible computation will in fact be performed.
I've put some thought into this fact and its potential implications but haven't come up with any more definite conclusions. What do you guys think?
Posted by: Jesse Nochella
Today while working towards implementation of
the 2-dimensional CA mentioned in the first
post, I encountered an error in my speculation.
The CA mentioned in the first post actually has
a requirement of all 8 neighbors to create what
I had said it would.
A simpler implementation is what I ended up
with. Requiring only 4 neighbors, the output of
rule 30 does not travel, and instead just stays
in the same place and expands, in this case
vertically.
Previously I had not liked this version because
the random pertubation overwrites black cells.
And I thought that somehow this was not as
good as other ways of iserting information. But
as I look at it more, I am starting to see how it
is fundamentally simpler than the original
model that I had proposed. And because of its
greater simplicity, it may give more
characteristics of the systems which I am trying
to explain.
I explicitly constructed the rule itself using a
great program called MCell. its 3-state
Neumann binary number is:
3000000220110110000110110000000010000110110000110110000220000000000000000000000000110110000100100000100100000110110000100100000100100000000000000000000000000000000220000000000000000000000000000000000000000000000000000220000000000000000000000000
It does not work the same with solid or
wrapped edges and I suppose could be improved
to that degree. It is a mix between chaos and
computability that is graranteed to last forever
and have infinite variety.
The notebook at the beginning of this thread
has been updated do display slices of the rule
specefied above.
Posted by: Jesse Nochella
In this post I want to talk some about the 2-dimensional CA in the previous post. Some of it will be slightly altered reiteration of some of the ideas discussed above, but I also want to bring the idea itself further and shine a few lights on the more far out implications of the rule and generalities related to it.
I guess one of the most important aspects of the cellular automaton described above is its ability to capture essential features of any kind. Its actually a very simple and general feature itself, but somehow it hasn't seemed to have been looked at in the same way that I seem to be doing now.
All that is required for a system to have this interesting property is the interaction of a universal computation with a computation that is categorized as class 3. In the 2d CA example given in the previous post A computation corresponding to elementary cellular automaton 110 interacts with another computation that happens to correspond to elementary cellular automaton 30.
When I first discovered this idea for myself, I thought that its behavior could be seen as a sort of causal precursor for any given system. That is, given the causal network for any given computation, this precursor rule along its course of evolution will eventually always produce it.
And in fact, for every single case where unbounded universal behavior for sure interacts with unbounded class 3 behavior in absolutely any way for an infinite amount of time, it is certain that eventually, somewhere the exact causal network, no matter which one you specify, will emerge.
This is really what defines a causal precursor. It acts sort of like a bootstrap for the computational universe. Really rather exiting stuff.
So, this seems to have at least some implications for us. One obvious implication is that there is no reason to believe we couldn't all be a part of a system like this. It seems that no matter what our universe actually is, it is indeed true that the causal origins of it are just as likely to have come from one of these systems I discuss as from any other.
And perhaps it is even true, that any causal network should come from a rule such as this, no matter what its implementation. For these implementations must come from somewhere. And of all concievable scenarios it may be the case that one involving a rule such as the one I've been discussing is the most probable.
Insofar this all seems to be getting more and more of metaphysical sorts. Well What I'm really saying is that certain rules are guaranteed to create our universe, and so when we try to explain where we come from, we need not go beyond those rules. Sort of a shortcut to what we all usually consider an impossible destination. Quite a tall status to give to such a simple rule.
And it may seem that after I explain this once or twice, it still all kind of seems so obvious that really it's useless, because no problems are actually being solved, just being thought of in a different way. But I do not think so.
I don't think so because the idea that I'm talking about is not just some theory, spoken from the ground up with just words and imagery. It's actually a real phenomenon that occurs in certain simple programs. The fact that these programs are so simple immediately implies that they can easily be generated by nature. And I believe that no matter which way you look at it, the fact remains that in nature this phenomenon is quite common. And indeed, I would not be suprised that if one day we find that rule for our universe, it would show in its behavior that at a fundamental level, this phenomenon is really all there is.
Posted by: Jesse Nochella
I made another mistake. This one came as a suprise – but now I'm not suprised at all. The systems that I've been discussing along this whole post are indeed mentioned in the book. I just missed them. On page 947, almost the exact picture you get with AltCPCAEvolveList[110, steps] is shown in the bottom of the left column. The note on the top of the right column titled "Continual injection of randomness" talks about exactly the phonomena that I've been talking about. I didnt mean to leave this note out. Honestly, I never actually knew that it was there untill now. Not suprised that NKS continues to kick my ass after all this time. And I dont think it's going to stop just yet. There's a note on page 855 about this.
Posted by: Jesse Nochella
this post is going to be somewhat speculative. I want to remain as general as possible but talk about specific known properties of our universe. My intent is to allude to similarities between the systems I've discussed here and what we know about our own.
First of all, one of the most obvious features is that there are localized structures which just are too similar to what we know about elementary particles. there is a seemingly certain amount of types of particles. They collide in predictable ways. Sometimes the same structures can interact in different ways depending on what cycle they are on when they collide. This has a strange similarity to particle interactions that have certain probabilities of behaving in different ways. Larger particles are rarer and more likely to be unstable, but still sometimes have the potential to be. Some structures have undecidable behavior, and it is not impossible that this undecidability could be related to decay rates of radioactive elements.
There is also general phenomena about space that we observe. In chapter 9 of NKS wolfram explains how 3 dimensional space can emerge as an overall property from a network . This explains quite a lot. Somewhere I remember hearing him saying that observed speed might have something to do with sampling different amounts of nodes. And this setup happens to have exactly the consequences that Special Relativity implies
It does not seem obvious how this could be implemented in a system like the one on page 947. I did try in my second post and it worked somewhat, but I'm still not really satisfied. I think that a network system having the same properties as the ones I discuss would show how 3 dimensional space emerges from randomness.
One well known feature about our universe is that space expands. It is seemingly obvious how space can be shown to expand with a system like the one on page 947, although it is somewhat trivial. But in my experience, as little as it is, I have noticed that as long as the trivial examples are simpler, they are usually the ones that turn out to be correct. Perhaps in the end inflation may be caused by a phenomenon as simple as the growth of a cellular automaton.
But one thing that doesn't seem to be explained by this. Space has evidenced that the rate of inflation has changed over time and that now It may be speeding up. How can this happen in a model where there is only linear growth?
I think that our answer is in the fact that the model that I present acts as a framework for causal networks and thus the causal network itself that is generated by the random input. This network in general will expand, but can vary over time sort of as the automaton on page 393 does.
Perhaps this kind of growth accounts for the rapid "bang" in a different way than, say, a literal "bang". When people first learn that the universe began with a sort of literal bang, one of the most common questions that follow in them is about the environment of the bang. I must say that if one answers this question in the ways that I've heard then the idea that you end up with is some lame overcomplicated mess (and for some strange reason a bunch of people who think it's funny that you're taking the explanation literally... weird). With our model of a causal network being sustained by random input, there is a phenomenon with uncanny similarity to what people call entropy increase that creates new events in the causal network. I haven't talked about this much insofar and should expand later. The basic idea is that because the sequence that interacts with our universal system never repeats, the computations never end up repeating, so there is always something to do.
It may be now, all of a sudden surprisingly obvious to us how our universe can be in just one single configuration out of the seemingly infinite possible ones. Perhaps we're just computing things like rule 30. It seems obvious why space looks essentially the same no matter where you are in it. None of this n-dimensional crap, just a simple network of nodes. No frills.
All of the phenomena described above are not just features of some specific universe. They are the features of class 4. It is the computational universe we see through the clear night sky. All of these features are inevitable as they are the features of computation itself.
And in fact I would not be surprised if in some way our exact state inevitable. Which sort of completes the abstraction that I had attempted in my second post. That we are not the absolute tearing edge of just one of these so called precursor rules, but in fact all of them, because all of them in the end yield exactly the same result.
The reason why causal precursors are so handy is because they serve as a model as well as a mechanism. they show you just what class 4 is capable of, as well as providing an implementation for whatever you want to specify.
If one can prove causal invariance in precursor rules then they have proved that our universe must be the way it is, and that we're all just part of the evolution of computations with properties like the one on page 947.
Posted by: Jesse Nochella
"With our model of a causal network being sustained by random input, there is a phenomenon with uncanny similarity to what people call entropy increase that creates new events in the causal network. I haven't talked about this much insofar and should expand later."
I'm sure there could be more technical explanations for this phenomenon. I've waited long enough and will just give a basic summary of what I think about this idea.
Attached below is a notebook with some code for taking repeating cell sequences and feeding them into cellular automata the way I've been discussing so far. You can generate some nice pictures with it.
These are not precursor rules, as their input stream is not from class 3 behavior; it repeats (one thing to note is that these are raw input conditions, so inevitably some strings when repeated produce the same pattern).
There seems to be a vast overall trend for the system to behave in complicated ways for some amount of time. Sometimes the period is short, but sometimes, it can be remarkably long even with a simple pattern (as in condition 5), but eventually they always seem to "terminate" into a repeating pattern. I think I know why this is, but it's a little hard for some reason to verbalize. There is a distinct diffirence between this phenomenon and ordinary difussion, despite it's parallels. I really don't know exactly what analogy to make, but there should be a general well known name for it out there somewhere.
I suspect there are may be cases where a repeating sequence produces a complex result, though I haven't found any yet, and I'm not too sure whether they can emerge at all as anything unique.
The point there is that even if one were to find a seemingly complex result, effectively an emulation of some complex rule evolution, the conditions would always have to be finite. The process has to start at some finite time, and so therefore cannot have infinite initial conditions.
So the only thing that a repeating sequence can do is to emulate any ordinary program with a finite initial condition. The only thing we can get out of this is either repetition, unbounded repetitive growth, eventual repetition and eventual unbounded repetition, class 3, and last, unbounded complex growth.
Well, unbounded complex growth is exactly what defines a causal precursor. This is basically why I don't think there's really any other explanation behind inflation other than that of a causal precursor. The only other scenario is the extremely unlikely one where it is still undecidable whether the universe will end up repeating or not. Now, really the only other option for a rule with such undecidability is to in fact become a causal precursor, because it is the only thing that guarantees never to repeat.
But what about the actual rate of inflation itself?
Ultimately I just think that it has to do with the intrinsic randomness in causal precursors. Only they can make entirely new events. It's sort of like what happens in brains, if the same input is fed in continuously, some optimum is in general reached and there is no further expansion. But for irreducible input, there is always something to work on.
And from this we can see how the rate of inflation can change. It's exactly the same kind of phenomenon as is discussed on page 398 regarding biological evolution and the history of technology:
A typical pattern--remarkably similar, as it happens, to what occurs in the history of technology--is that at some point in the fossil record some major new capability or feature is suddenly seen. At first there is then rapid expansion, with many new species trying out all sorts of possibilities that have been opened up. And usually some of these possibilities get quite ornate and elaborate. But after a while it becomes clear what makes sense and what does not. And typically things then get simpler again..
The universe can fluctuate for essentially the same reason that the stock market fluctuates; or why styles go in and out of fashion. Its computational irreducibility. The general trend is what we see we see, and the behavior can never ultimately be predicted.
Posted by: Jesse Nochella
With inital condition {1,1} in a repetitive background {1,0,0,1}, running totalistic rule 1599 for 1500 steps we get the picture attached below.
This is the first one dimensional causal precursor candidate that I've found. The inital conditions are not as simple as I would like, but it at least serves the purpose of suggesting that causal precursors can exist at least in the k=3 r=1 domain from a single cell and uniform background.
I've looked though about 6000 rules so far with my own eyes looking for one that has the same features as the picture here. No luck so far, but I think it might be out there.
Is the pattern on the left class 3?
*Note: It's been found that the pattern on the left is not class 3, every fourth cell corresponds to elementary rule 105 evolution
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