[few simple programs per natural phenomenon] - A New Kind of Science: The NKS ForumA New Kind of Science: The NKS Forum
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few simple programs per natural phenomenon
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Posted by: Philip Ronald Dutton
When thinking in terms of nks simple programs in relation to biological process (or natural processes in general), I am sort of assuming that (if nks is the true way the universe is working) we should not find more than a few simple programs per natural process. It just seems like a safe assumption.
Thoughts?
Posted by: Jason Cawley
That would seem to be what we'd mean by "per natural process". When we can find a simple rule that captures an aspect of what is going on, we abstract that aspect out of the mass of phenomena, and analyse the resulting gob of data down to the underlying rule we found, and perhaps some external inputs (initial or boundary conditions, we typically say).
For example, locally inhibited crystal growth can account for the basic shapes of snowflakes. Along each arm, might there also be additional sub-processes that introduce minor variation? Sure. If the model is right, those won't account for the basic shapes one sees. But they might account for a fuzz of variation around those basic shapes.
Sometimes one will find, rather than a single simple rule per se, a simple rule with parameters. An example is phyllotaxis, where local depletion can account for "golden ratio" spirals, and varying the rate (or "damping") can account for a variety of related patterns without any additional "subrule". See pages 408-412 -
http://www.wolframscience.com/nksonline/page-408
Some background may be helpful here, because NKS is a significant extension of the sort of simple rules one can find, compared to traditional mathematics models. NKS methods are better suited to modeling a different kind of interrelation among system parts, one in which synthesis or how the system elements are put together matters more.
Traditional mathematics (by which I primarily mean equations and especially calculus based methods, differential equations and the like) tends to provide good models where the underlying phenomena "analyse easily", meaning interact in a particular way.
By analyse I mean break down into smaller parts, each simple. If the way the interaction or re-synthesis works is smooth, additive, or similar across most points, then traditional mathematics deals with it well. Linearly additive phenomena are what the whole approach was originally designed for, and what such mathematics handle best.
Those methods can be pushed to a relatively limited set of cases that don't fit that criterion (especially where various kinds of symmetry are present), but it gets harder and harder to do as the synthesis side gets more and more involved. Another extension is statistical methods, which basically work by averaging things enough to get them to fit that linear additive model. Local variation is left unpredicted while one looks for traditional mathematical patterns satisfied by a parameter like a mean or a standard deviation.
Analysis "wants" variation to be local. It wants there to exist some scale below which everything going on is completely simple, and then it wants to add up lots of those simple, local things in smooth regular ways. When this condition is satisfied, those methods can be extremely general; that is what makes them powerful as modeling tools. By that I mean, anything smooth enough you can fit a function to. More, there are multiple families of "orthogonal" functions that can act as a "basis", to describe any smooth enough pattern (e.g. polynominals and series expansions, or trig functions and fourier analysis).
NKS methods don't need this smoothness condition, which amounts to a certain sort of simplicity in the casual interrelation of the system's parts. They can handle very "messy" casual interrelations among parts - like the dependence of the middle of the rule 30 pattern at step 500, on some value back at step 300 off on the upper right.
But NKS methods are looking for a different sort of simplicity. A stability in element relations, either locally or in time. The same rule applied many times, either at many places or on many steps. You then have a wide choice of rules to find being repeatedly applied. One is looking for a pattern, still, which can be there or fail to be there - repeated application of some definite algorithym.
To be able to model anything efficiently, at some level it must follow definite rules. There is explanatory gain when we find something much simpler than the overall phenomenon, from which we can generate the basic pattern seen in the phenomenon. If it isn't much simpler, we haven't explained, we've just reported the data again. (It is true a "transform" on data may allow us to see patterns we wouldn't have seen otherwise. But just doing a transform without seeing a pattern in the results doesn't much help).
Philosophically one can speculate, will every phenomena either show the local independence and resulting large scale smoothness of traditional mathematical methods, or the casual interdependence and resulting computation-like character of discrete algorithyms? In principle it need not happen, by which I mean one can imagine cases that lack either the smoothness wanted by mathematical analysis or the repeated rule pattern of NKS simple programs.
But in practice, natural systems are made of simpler parts which follow rules which are both stable and widespread. Making it feasible to pick up the patterns those rules necessarily leave behind. Will a "residual" be left? Probably. We can imagine getting the exact rule the natural system is using and thus leaving no residual. In practice we are quite satisfied if we can get most of the variation seen, whatever method we use to get that much.
So how do we think about a system we are looking at, that may have such patterns in it, when we haven't yet found them? If preliminary investigation (or just looking at it) shows there is interdependent messiness going on, we expect to quickly reach a point of diminishing returns with traditional mathematical analysis. Then we want to look for NKS style rules. We pick out aspects of the phenomenon that look likely to yield to that method, and try different mappings of system elements to formal rule elements of this rule type or that rule type.
We'd call such an aspect "a natural process", and would justify the singular by pointing to the model rule we found that captures most of what is going on. Otherwise we'd be inclined to think there is more than one natural process going on, and that their interaction must be tracked. We'd then ask, is that interaction, on that other level of analysis, a smooth enough one to have averages that follow traditional mathematics? Or an interdependent one? If the second, can we find an NKS style rule governing that higher level interaction?
And so we go. Scanning the levels of analysis, scanning the kinds of element interdepence the phenomenon actually shows, looking for patterns, comparing analysis patterns where there is smoothness and independence enough for them to apply, comparing NKS rule patterns where there are signs of repeated application, by location or through time, of some definite rule. In all of which, we first of all need to have in our heads the kinds of things each of these formal systems can do. That is what we mean by "developing our intuition" for a given method.
I hope this helps.
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