[NKS and Constructal] - A New Kind of Science: The NKS Forum

A New Kind of Science: The NKS Forum

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NKS and Constructal

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Posted by: Sean Lynch

I recently stumbled upon Adrian Bejan's constructal theory. Although I haven't had the time to study it in depth, I've had the feeling that it contains some ideas that are similar to NKS (In one case I even saw a picture of Rule 110 but the article was in French so I couldn't read it).

I did a couple of searches to see if anyone has studied the constructal theory from an NKS perspective but I've found nothing. And a search on this forum also turns up nothing.

Is there a connection between the two? Has anyone done any research on the constructal theory from an NKS perspective?

Thanks


Posted by: Jason Cawley

I can't claim to be all that familiar with it, but from what I know of it the basic idea is to explain complexity as a result of constrained optimization, and to use formal techniques of constrained optimization in engineering. From my perspective, the second of those is more sensible than the first. But it is worth exploring, and there are a few places where the NKS book touched on the subject, others where more has been done on it since the book.

Wolfram argues that optima are unlikely to be actually reached if the constraints are complicated enough. Iterative improvement dynamics just get stuck. This suggests that it is better to look directly at the dynamics of any iterative scheme than to solve for optima and try to deduce end states from that. That is, you are more likely to find some fractal-ee creature grew from a rule and seed in NKS form, than that it exactly maximizes this measure or minimizes that one.

Now, clearly there are cases where you can just solve for optima. We do it all the time using calculus based methods. From single extrema of simple functions we can generalize to variational principles. Constraints can give us regions in which to apply those. Or in simple enough cases we can just come up with a linear program form and readily find overall optima under a constraint. Which are fundamental in economics and in practical applications of it.

But general discrete problems with dynamics rarely have such forms.

So much the book already said. Since then a few people, including myself and some starting at the first summer program at Brown (with economists overrepresented among those interested in this question) looked at the related issue of control of NKS systems. By control I mean the classic idea of feedback, that some form of systematic "nudging" of a simple program system might drive it toward some "sought" state.

The two issues are related because presumably the mechanism that is supposed to push some system to its overall optimum under some measure or other of optimum, is some feedback or reinforcement scheme, that formally or functionally depends on distance from a goal or stable state.

What we found is that standard schemes suggested by classic linear feedback typically fail in class 3 systems, in particular. Other schemes can get a certain distance but face rapidly diminishing returns. The reason is, the spreading impacts from each "nudge" tend to reinforce or interfer largely randomly, once one is talking about effects far enough in space and time from the nudge itself. The cumulative effect of lots of nudges then mimics "rerolling" a new random initial condition. But since those typically give class 3 randomness, that is what you get after lots of nudging. Basically, class 3 systems do what they were going to do anyway, or can be thought of as "thermodynamically stable", in a suitably loose sense. Even though in detail they are arbitrarily sensitive.

Now, part of that is a result of using objectives or measures that are average property or statistical themselves, and unlikely in a purely statistical sense. If say all you want to do is make {x, t} position be a 1 not a 0, clearly that is a target you can hit easily by disturbing the system a few times. But "more black on average" may be much harder to swing, if the rule naturally tends to this or that concentration of black cells (or whatever other local system state measure etc).

The idea of linking the evolution of a simple program system to some global state variable has been looked at in other ways, as well, and the two have been "crossed". GCAs were the first simple example of this. An optimization or iterative improvement feedback can in principle operate in local bottom up ways (the previous case, above) or top down. The form of the rule or the program followed might be X in this "region" or regime, and Y in that one, with the two distinguished by the global measure - for example. That top down case is what the GCA idea was set up to explore.

Looking at them for a while, I found their natural generalization is the ICA (the "machine with input" form of a CA), in which the form of rule applied at each step can vary generally. Then getting the sequence of these allowed "interaction condition" variations from some global variable of the system state, gives the GCA case. They aren't equivalent because the interaction series may not be a deterministic function of a single global measure, so the GCAs are a proper subset of the ICAs.

Relating these to optimization, I looked at fiddling an ICA's interaction condition to "drive" a system towards some desired goal or measure. This works and does so pretty straightforwardly, though how far one can get depends on the rules being mixed in the ICA (aka the range of behaviors available) and remains within limits set by the natural tendencies of the rules etc. Simple and linear dependencies are rare, though, and generally only happen if one of the allowed behaviors is always superior for your desired measure. When the optimum involves sequenced mixing, the best is a complicated "key" in the space of possible sequences and is found by brute force. But will be missed by simpler procedures, for much the same reasons Wolfram discussed in his section on constraint systems.

As an aside, I extended the "optimize an ICA" idea to multiple controllers with different objectives as ICA games.

One might also note that the idea of setting up an objective function and then rejiggering a rule to find cases that get closer to it is widely used by other "iterative improvement" schemes, like simulated annealing or genetic algorithms. On which there is an extensive literature.

All of this was exploratory and abstract. Given how useful straightforward constrained optimization is in general engineering, I have no doubt this school can find problems on which they can say meaningful and useful things. Continuity favors variational methods, for one thing, so it might be natural for all sorts of fluid problems. I just doubt it is the leading or a leading cause of natural complexity, particularly in cases that turn on complex interaction among discrete components.

For what it is worth.




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