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Tony Smith
Meme Media
Melbourne, Australia

Registered: Oct 2003
Posts: 167

Irreversibility precedes reversibility

NKS added legitimacy to me getting back into some long adjourned CA projects, now aided by more powerful tools. The particular result I want to mention here came after a few months work on a new diversion, "Trapper", which itself only gained attention after a few months resumed exploration of a much older diversion "Life in a Tube". I will return to Trapper after introducing the broader conclusion:

Irreversibility precedes reversibility

One of the traditional big questions for science is how to get the time irreversibility we see in many aspects of the natural world when what are thought to be the most basic scientific theories appear to be time reversible. While I needed to do tens of millions of CA experiments before I started to realise what I was looking at, once I saw it I soon also saw why it should have been obvious all along that irreversibility precedes reversibility, if only my view had not been being blinkered by the traditions of the old kind of science.

Given that space time matter energy as we know them might be best seen as emergent properties of a Planck scale causal network, even "fundamental" particles are huge relative to the scale of the underlying simple mechanisms. What is now obvious to me is that such fundamental particles are almost guaranteed to arise from a wide range of initial conditions.

As a thought experiment, this can be tackled in the context of the notion of state space. Given that any finite state space X0 has a successor state space X1 determined by the underlying simple mechanism, and given that some small subset of state spaces form loops of size n>0 where X0= Xn, all state spaces which are not members of such loops must eventually evolve to states which are members of such loops and each loop, being a loop, is at least in some sense reversible.

So we start out with an irreversible mechanism at the smallest scale but this can readily evolve into stable/reversible entities at the first level of aggregation. Most of us will be familiar with this from Conway's Game of Life though it's dominance by small aggregations with very small n hides the implication I have drawn from Trapper.

The state space argument can be pushed to higher levels to help us understand why everything from basic biology to the motions of astronomical objects achieve persistence through cyclic repetition. Cycles are Class 1 outcomes and thus very easy to achieve. The first essay linked from my home page makes a point about Class 4 mechanisms producing a mixture of Class 1-3 outcomes.

Trapper is a small region of Rule 22 blockaded at each end which is equivalent to several 1D 3 colour symmetric neighbourhood rules and also equivalent to a configuration which arises frequently in Life in a Tube with any tube circumference up to ten cells. By a combination of tweaking Perl scripts for performance and analytically determined short cuts, I have been able to study the evolution of traps with widths up to 130 cells, exhaustively for width<24 and through random samples decreasing from 1000000 to 10 as the width increased from 24 to 130.

Rule 22 is Class 3 so behaviour of a trap most often initially appears random, but, as noted in NKS, certain configurations of Rule 22 emulate Rule 90 which is Class 2 and by extending the state space argument we can see that random Rule 22 configurations will most likely encounter a Rule 90 configuration before (or often when) they enter a loop, and thus the loops detected by Trapper are overwhelmingly Rule 90, or some close variant thereof which are another story.

As also reported in NKS, Wolfram et al in 1983 analysed the repetitions for cyclically confined Rule 90, and thus identified the basis of many of the cycles detected in my Trapper experiments. There is already a cycle of 4(2^30-1)=4294967292 generations around width 120, such large cycles being relatively easy to determine and detect through the range studied, although they get harder exponentially, with the number of possible states doubling for every increase of one cell in trap width.

Alongside the vast numbers of these largely predictable Rule 90 and related cycles, my study has revealed a small number of pure Rule 22 cycles which do not pass through any near Rule 90 states. Old science assumes that an atomic model will account for everything at higher levels of aggregation. However, if fundamental particles are the obvious product of a simple Planck scale mechanism, their existence may not preclude other unconnected phenomena arising from the same network.

Here, I am particularly reminded of my own early work on "Pattern Breeder" (Scientific American, September 1986), a generalisation of the Fredkin 2D CA which showed how an arbitrary pattern could be propagated at c in all directions. Could this provide a possible model for Rupert Sheldrake's much maligned notion of "morphic resonance"? If space time energy matter are emergent properties of a Planck scale causal network, might not our conservation laws be likewise and might there not thus be other things going on which are unconnected to those laws?

For me, the study of complex systems over the past two decades has convinced me that there are a set of general systems principles only just becoming visible that are a lot more fundamental than the laws of physics.

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10-22-2003 03:06 PM

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Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

First of all let me say welcome, and thanks for an outstanding post. This is good stuff. I have several somewhat disconnected comments to offer.

You say -

all state spaces which are not members of such loops must eventually evolve to states which are members of such loops

For a finite system run for a long enough time this is true. All eventually must repeat, though the repetition period can get abritrarily long rather quickly. It seems to me the more fundamental fact is that some state space configurations which are not members of such loops can evolve to states which are members of such loops, and can moreover do so quickly. It is not necessary that this be exhaustive, just that it can happen. You will hit some stable cycles. The NKS book talks about this in the section on class 2 behavior and systems of limited size.

One thing I've noticed repeatedly in NKS experiments is how number theoretic effects can crop up, related to things like system size. Not just large scale effects, 1000 wide being different from 10 wide because the possibility space is enourmously larger. But tiny ones, 359 wide being different from 360 wide, because 360 has many divisors and therefore many possible sub-cycles, while 359 is prime. You are more likely to hit short repetition periods in a 360 wide system - or any other local maximum of divisor sigma - than in the 359 system - or any other prime number width.

More basically, you are right that local reversibility can arise in pockets of class 1 or 2 behavior, that it can be an apparent property of subsystems. What I find fascinating is the possibility we may have sifted for just such subsystems with previous formal methods, based on continuous mathematics and geometrical thinking. Symmetry is a particularly useful analytic tool in those formal methods. Are essentially all our basic physical laws - except of course the Second - symmetric because only such laws somehow arise, or merely because such symmetries are particularly noticable to traditional mathematical methods?

It happens I've been reading the pragmatist CS Pierce. In a piece entitled "Causation and Force" (an interesting piece, incidentally, particularly for its day) he noticed the following about common sense ideas on causality compared to the physics of his day (classical mechanics - this was around 1900).

The grand principle of causation...involves three propositions to which I beg (your) particular attention. The first is, that the state of things at any one instant of time is completely and exactly determined by the state of things at one other instant. The second is that the cause, or determining state of things, precedes the effect or determined state of things in time. The third is that no fact determines a fact preceding it in time in the same sense in which it determines a fact following it in time. These propositions are generally held to be self-evident truths; but it is further urged that whether they be so or not, they are indubitably proved by modern science. In truth however all three of them are in flat contradiction to the principles of mechanics.

He goes on to explain what he means, and surprisingly enough he is right. Even with classical mechanics. Mechanics was reversible. It moreover involves second derivatives, and as such more than one preceding instant in time (that is, it requires not only positions but also velocities to be specified, and velocity involves a time spannedness). The mathematical relations involved do not favor the time relations our intuitions about casuality lay down. In field-like theories, causes do not precede effects but are instead instantaneous relations. Locality is made to do all the work, not precedence.

Pierce noticed that our common sense ideas of causality do correspond well to our psychological experience, and also to our experience of large scale aggregates, which of course involve the action of the Second Law. But do not correspond to reversible classical mechanics.

One can bend over backwards to find apparent irreversibility in underlying rules that are reversible, provided their behavior is complicated enough. The NKS book shows some ways to do that in the physics chapter. But you are correct that it is in many respects much easier to find reversibility as an emergent phenomenon of limited sub-systems or cases, within rules that are in principle irreversible.

And if that were really the case, it would mean (pace Pierce above) that much of our intuition about casuality might be saved. The simple programs NKS considers are typically set up to obey the three criteria of causality he lays down.

If it were so, then we would suspect the existence of laws relating to unsymmetric behaviors that previous mathematical formalisms have failed to spot. The second law is the leading known example, and it may be useful to consider why it is known. It deals with a different form of symmetry - a statistical one over initial conditions rather than a time related one within a single case.

A stable attractor is found as the evolutionary end point of a whole ensemble of possible previous states. It is "many to one"-ness of the paths (to an average) that produces irreversibility. Our mathematical methods spot it because there is a symmetry between time changes within the system and a different possible initial condition - both characterised by the same mathematical "object", a continuous distribution.

So some questions would be - can traditional mathematical methods spot laws that aren't symmetric, one way or another? And, can simple program models do so, more readily? If so to the last, what are some examples?

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10-22-2003 05:24 PM

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Tony Smith
Meme Media
Melbourne, Australia

Registered: Oct 2003
Posts: 167

Methodology melting pot

I must have said "Yes!" many many times while reading the NKS book, but the loudest "Yes!" was within four pages of the end of the notes (p.1193R):

Methodology in this book. Most of what is presented in this book comes from systematic enumeration of all systems of particular types. However, sometimes I have done large searches for systems ... I have occasionally explicitly constructed systems that show particular features.

With my more limited resources, that is exactly what I too try to do.

This isn't the place for me to try to substantiate all the reasons that I think the evidence I have found from "Trapper" to be usefully suggestive, especially after factoring out obvious areas in which it, like all other CAs, is far from a realistic model of the natural world. But first and foremost of those reasons is that Trapper turns up relatively frequently in systematic enumerations and in large searches. So for the purposes of discussion it certainly should not be seen as a highly contrived system.

How long is "eventually"? With respect to finite systems eventually finishing up in loops, you say:

For a finite system run for a long enough time this is true. All eventually must repeat, though the repetition period can get arbitrarily long rather quickly. It seems to me the more fundamental fact is that some state space configurations which are not members of such loops can evolve to states which are members of such loops, and can moreover do so quickly. It is not necessary that this be exhaustive, just that it can happen. You will hit some stable cycles.

In Trapper, while the number of states grows as 2^width, the largest loop grows as 2^(width/4). Not only that but just about every other indicative statistic I have been able to obtain, such as maximum and average "branch" lengths, grows closer to the lower rate. Still fast enough to test my aging G4, but not fast enough to frighten my interpretation.

It is useful to remember that the Planck time is of order 10^-43 seconds. If only we could run our CA generations half as fast. Since writing my original post it has occurred to me that most of the interesting (Class 4) systems have a ground state, even humble Rule 110* where, interestingly, the ground state is not uniform empty cells the way it is in Life and many other familiar CAs. It is now widely accepted that what we see as empty space is anything but empty, even though its uniformity makes it appear that way to us. The relative dominance of those ground states tends to isolate the areas in which something interesting might be happening, especially, I hypothesise, in a truly viable universe.

One thing I've noticed repeatedly in NKS experiments is how number theoretic effects can crop up, related to things like system size.

Such effects certainly show up in Trapper, a lot of information on the dominant and often "monster" Rule 90 based loops being revealed by the results of a short Perl script:

for ($m = 2; $m < 33; $m++) {
$x = 2;
while ($x > 1) {
print "$x ";
$x *= 2;
$x = 2 * $m + 1 - $x if $x > $m;
}
print "$x\n";
}

at least one characteristic of the results of which I expect can be determined using Mathematica's MultiplicativeOrder function, as per the NKS book (p.257) but I have not yet managed to work that connection through. To me one of the interesting things is the fraction of behaviours which do not succumb to such analyses.

Are essentially all our basic physical laws - except of course the Second - symmetric because only such laws somehow arise, or merely because such symmetries are particularly noticeable to traditional mathematical methods?

I have to plead guilty to being a "reformed" mathematician in the sense of a reformed smoker. Nowadays I treat mathematics and the scientific method itself as tools which produce useful results rather than as the reliable arbiter of right and wrong that the naive realists would have them be. (This did not stop me playing a driving role in the initial introduction of Mathematica to the recalcitrant Australian market, but that is another story.) So basically my answer is to expect that our mathematical methods blinker us, that they hide much truth and understanding from us. Computer simulations of complex systems can be a powerful tool to help us break out of that straightjacket.

So some questions would be - can traditional mathematical methods spot laws that aren't symmetric, one way or another? And, can simple program models do so, more readily? If so to the last, what are some examples?

The causal network that underpins our shared reality creates additional nodes every tick of the Planck time clock. I still can't see an easy way to do something equivalent in the more human perception friendly world of CAs.

*This was originally wrongly refered to as "Rule 30" but has now been corrected in the light of Jason's observation below.

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10-23-2003 12:16 PM

Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

A very minor point - you said in one passage

most of the interesting (Class 4) systems have a ground state, even humble Rule 30 where, interestingly, the ground state is not uniform empty cells

I think you must have meant rule 110. Rule 30 certainly doesn't produce uniform empty cells, but it does not have the repetitive background rule 110 does, and is a class 3 rule rather than class 4. Just a correction.

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11-06-2003 10:33 PM

Tony Smith
Meme Media
Melbourne, Australia

Registered: Oct 2003
Posts: 167

Rule 73 looks like producing similar results

A new post in Applied NKS directed my attention to Rule 73 which I very quickly established is likely to produce data comparable to my Trapper experiment.

Using Andrew Trevorrow's LifeLab it is very quickly obvious that two black cells embedded between two white cells form an impenetrable barrier with evolution towards cyclic repetition in the gaps between such barriers.

My second test run was with a 1% random seed on a 2520 cell square universe which soon became divided into 16 always odd width regions separated by those barriers. The narrowest gap was 15 cells wide between barriers and almost immediately settled into a period 22 cycle.

But it was the marginally second narrowest, at 47 cells wide, which exemplified the patterns I found in my investigation of Trapper which led to the ideas expressed in this thread. That 47 cell gap takes over 700 generations to stabilise into a period 12 cycle. The attached GIF shows that stabilisation.

The wider conclusion is that for this kind of system, there can be many more confined configurations which are not on cycles than which are, so what we finish up with after sufficient time is only those which are on cycles, and that set can thus be considered to be qualitatively distinct from the much larger set of all possible local configurations.

Tony Smith has attached this image:

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05-15-2006 02:03 AM

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