Registered: Oct 2003
Posts: 167

While I do do bursts of detailed experiments from time to time, my overwhelming interest is in big picture questions. From that perspective, it was obvious by the late 1980s that the edge of chaos-border of order was where anything interesting happens, thanks in a large part to Stephen Wolfram's identification of Class 4 cellular automata.

Then over a decade, the prevailing wisdom in the complex systems community drifted to the point of regarding that edge/border as at best elusive and at worst unimportant. This retreat has always seemed wrong to me, to the point where I put an essay about it, Beating the Retreat on my Memes site. (Link updated following site reorganisation.)

I don't want to revisit much of that essay here, save for drawing attention to one point which I now see as possibly the heart of the issue:

While Class 1 through 4 processes are readily recognised, there are no Class 4 outcomes. Class 1/2 have regular/cyclic/nested outcomes. Class 3 has random outcomes. Class 4 can exhibit either kind of outcome, so the fact that a process is Class 4 cannot be determined from an outcome in isolation.

Now that the specific experiments I did with "Life in a Tube" are several months behind me, I see that the main thing that kept my attention through a sizable survey were the occasional instances where it did not become clear for many thousands of generations as to whether the outcome would be cyclic/nested or random. (For those who need to know, these experiments were done using Andrew Trevorrow's excellent LifeLab for Macintosh which allows you to run Conway's Life (and other 1D/2D CAs) on an arbitrary grid. Setting Options->Join Edges and using Edit->Tile With Selection makes it easy to "roll out" a few copies of a narrow cylindrical Life world of arbitrary (small) circumference. Seeding a Tube with a single row (circumference) of live cells (superstring) plus some arbitrary thing else produces a vast range of outcomes.)

Those instances whose outcome was not determined for many thousands of generations built up complex structures which finished up supporting very long cycles or what I came to call "metabolic" cycles where a network of "reactions" included at least one apparently random interaction (at the active boundary) and consequently produced an outcome with a random component. The long cycles could arise at any stage provided a certain "phase change" had not previously occurred at the active edge, without which an "ice breaker" configuration could arise at any time at the edge and thereafter propagate at C leaving an outcome with period 2**n for n often reasonably large (> 10). (There will be pictures and a lot more explanation when I finally get around to putting up a Life in a Tube page.)

For now my concern is just to emphasise that, having had more than a year to digest NKS from an extreme generalist perspective, that there is a lot more to be gained by going back to the edge of chaos/border of order (Class 4) and looking harder armed with ever improving tools and understanding.

I became particularly concerned about this after wading through a self published manuscript of a well intentioned and credentialed friend of a (deceased) friend who called on the Principle of Computational Equivalence to support his claim that the "internal" world of a human is equivalent (in complexity) to the universe. The trouble is I do understand all too well where he is coming from and even see that there is a kind of correspondence between mind and world, but run into problems where he extends that into a claim that the social is inherently less than the individual. Such a view collides head on with my own view that we should grant our social structures ethical status as representatives of the (possible) future.

Beyond returning focus and adding legitimacy to the field, I now see that the most useful idea to come out of NKS is the likely role of Class 3 randomness in underpinning the resilience that is such a pervasive feature of the world we find ourselves is (but not of our simple models of that world). While it was not the way that Wolfram did most of his research, the focus on PCE and universality is likely to just serve to invite more effort to be invested on the kind of unproductive couplings of simple mechanisms with highly engineered data which have too often dominated the field (just check any collection of Life patterns). If our ultimate goal is to improve our understanding of the natural universe (physical, biological and social) we should make sure we are looking for both mechanisms and data which have a prospect of a believable (pre-)history. The evidence I see points to Class 4 as the place to look for this.

Last edited by Tony Smith on 11-04-2004 at 09:33 AM

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Registered: Oct 2003
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... at least in the universe defined by Life in a Tube. The animated example that accompanies that post is highly symmetric, as are many of the more appealing examples that I have found.

The Tube geometry causes symmetry to arise more commonly than in less narrowly constrained 2D Life worlds. From many thousands of experiments, I have only discovered two basic forms of persistent asymmetry, with nothing really profound arising from either of them.

Eight months after completing the core logic to produce animated GIFs from Life in a Tube patterns, I have finally gotten around to incorporating that logic into a sufficiently neat CGI to start routine production of such GIFs in preparation for a long promised web "paper" more comprehensively exploring Life in a Tube.

Meanwhile, I thought I should at least share the attached GIF which shows more interesting details than any other asymmetric Tube I have found to date. It is basically period 32, although the leading edge is period 16 for almost 150 cells/generations before doubling. As a circumference 29 Tube, it is also at the wider end of the main range of interest. Asymmetric leading edges take longer to stabilise, but, typically for any periodic leading edge, after nearly another 200 generations it leaves a trail, each 32 generations, of three common Life forms: a block, a glider and a fleet, the latter two being only able to occur unpaired in asymmetric Tubes. The glider means that first trail is not quite stable, but the near stability is made more conspicuous because the fleet forms as close as possible to the path of the approaching glider for it to survive without immediate disruption. The diagonal path of the glider and the closeness of the period (32) to the circumference (29) then conspire to allow the glider a surprisingly long flight path which sees 14 gliders in flight and 10 fleets persisting at any time before the 15th glider finally disrupts the 11th fleet. Then after nearly another 400 generations of asymmetric disorder, another period 32 trail of common Life forms finally fully stabilises.

The attached GIF is 1178 pixels high because that's what it takes to show the whole long stabilisation process. It is 87 pixels wide to show the circumference 29 Tube rolled out three times for easier viewing. And it cycles through 33 frames, the first and last being held for longer to show that they are identical save for being displaced by 32 cells.

Tony Smith has attached this image:

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Last edited by Tony Smith on 02-23-2004 at 08:28 AM

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While I'm at it, I should also post an even larger Life in a Tube animation which is just across the chaos side of the border. This particular animation could easily be reduced to a multi colour 1D rule in which each colour maps onto a specific configuation of 16 longitudinal cells across 16 generations, but that is because it is the simplest example of a phenomenon which is quite common within the universe of Life in a Tube. Other similar results, which I call "metabolisms", do not simplify to the same extent, although they are all driven by a region which follows Rule 22 which in turn often corresponds to Rule 90 at every 4th cell and 4th generation, as well as at higher doublings.

This animation is necessarily large (290K) as it takes 400 generations to give a reasonable hint of the way that each new binary "result" (either a correspondingly positioned blinker, or the same blinker accompanied by a pair of blocks) is not able to be determined by anything less than running that number of generations, or at least the equivalent 1D rule. I have run it for 8000 generations (equivalent in this case to 500 1D generations) to produce 158 random binary bits, but this GIF is uncomfortably big already.

The point of these two animations is to show how Life in a Tube can produce cyclic or chaotic outcomes. Life in a Tube also produces nested outcomes or seemingly complete chaos until it suddenly "halts". (The Rule 22 region stops interacting with the debris.) Both of those as well as much larger cycles and much more complex "metabolisms" are all common enough outcomes, but are often only revealed over thousands of generations and thousand of longitudinal cells.

Tony Smith has attached this image:

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