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Topic: Richard Johns: Dynamical Complexity and Regularity
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posted 01. February 2002 12:05
Dynamical Complexity and Regularity
by Richard A. Johns
johns@interchange.ubc.ca
ABSTRACT—The aim of this paper is to provide a mathematical basis for the plausible idea that regular dynamical laws can only produce (quickly and reliably) regular structures. Thus the actual laws, which are regular, can only produce regular objects, like crystals, and not irregular ones, like living organisms.
1. The dynamical complexity of an object is defined. This is something like its algorithmic information content, but I use a dynamical system in place of a universal Turing machine. A dynamically-complex object is, roughly speaking, one that the dynamical laws have little or no tendency to produce from a random initial state. A "GIGO" theorem is proved, that an object with dynamical complexity n bits requires time 2^n to be generated, so that highly-complex objects effectively cannot be generated spontaneously.
2. The term "regular dynamical law" is defined, in such way that the known dynamical laws of physics are regular. The irregularity of an object is defined, and then I try to show that the dynamical complexity of an object s, with respect to a regular law, always exceeds the irregularity of s. It seems clear that living organisms are highly irregular, in the sense defined, so (if this result holds) they must be dynamically complex as well. It would then follow that living organisms could not have been produced, from a random initial state, by the known dynamical laws.
To read the entire paper, please click here
[ 05 May 2002, 15:00: Message edited by: Moderator ]
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William A. Dembski
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posted 07. February 2002 18:35
Richard Johns was kind enough to send me this paper last summer, but I got swamped with things and have only now read it. I enjoyed it very much, and think he's on to something with trying to tease apart various types of complexity, notably a dynamical form of complexity which concerns how difficult it is for things to happen in the course of natural history and a more static, descriptive complexity characterizing the intrinsic complexity of a thing as it is now (the analogue in computational complexity is between time and space complexity). Although I find Johns's article richly suggestive, I'm not sure it succeeds. As he himself notes in the conclusion, establishing that irregularity (static complexity) supplies a lower bound for dynamic complexity can be demonstrated only on the basis of assumptions that design critics are not likely to grant (his Global Blindness of Natural Laws principle, much less his symmetry principle). I think these ideas need more work. Perhaps the place to start would be by providing a user friendly plain English version of his paper. I'm afraid that even with my math background I haven't worked through all the details. Once such a version is in place, with clear bridges preferably to biological examples (though we'll need toy examples as intuition pumps), I would like to encourage the biologists of ISCID to come in and see (1) whether Johns's notion of irregularity can be given biological content; and if so (2) whether it can legitimately form a lower bound for a characterization of dynamic complexity that is directly relevant to assessing the power of naturalistic mechanisms for generating biological complexity.
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Richard A. Johns
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posted 11. February 2002 01:09
I generally agree with Bill’s comments on my paper. He’s right that there’s a lot more work to be done! My aim in putting this paper online was not to present a fait accompli but rather to make contact with other mathematicians who may be interested in this topic, so that together we may solve the remaining problems more quickly.I see the remaining tasks to be as follows: 1. Concerning cellular automata: Prove the symmetry principle from the assumptions of locality and translation/rotation invariance. (GBRL and the Regularity principle will follow, as I already have relative proofs of them.) 2. Generalize the results to an arbitrary dynamical system, in continuous time and space. Solve the problem of what should count as an “object”. 3. Attempt to estimate the irregularity of an actual organism. (It will probably be easier to work with the genome rather than the phenotype.) Some input from biologists will no doubt be necessary here. Of course research is never predictable, and things might not work out as planned. But that’s what I hope will happen. I don’t see any point at present in writing a non-math version, as I think the mathematical work has to be finished before any biologists get involved. Until (1) and (2) are complete, there’s really nothing for biologists to do.
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John Bracht
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posted 26. February 2002 13:46
Richard,I found your paper to be very intriguing, and it left me with a few questions. quote:
The question now is: What kinds of object are produced by (have high salience relative to) such regular laws? The short answer to this question is: "regular objects," where object regularity is defined in terms of regular laws. (p 13)
It is my understanding that you define regularity it terms of a simple, repeating pattern (and irregularity as the lack of such a pattern). Correct me if I'm wrong on these definitions. Now, here is my question: What do you do when regular laws produce irregular objects, as when the cell produces DNA in the process of cell division? Certainly, the operation of proteins and DNA are fully controlled by regular laws. They produce irregular DNA with very high salience (low dynamical complexity), which seems to contradict your thesis that dynamical complexity is always greater than the irregularity. I also suspect that a measure of dynamical complexity should rate DNA as very high on the scale. Here is a case where global laws are indeed blind to the overall structure, but they manage to transmit an overall structure in space and time. The overall structure is fully consistent with natural laws, is transmitted by them, and yet is not determined by those laws. This seems to be a case not dealt with by your model (please correct me if I'm wrong). Furthermore, I wonder what your model does with random phenomena. For instance, consider the precise state of air molecules in this room. Certainly they have no pattern and are thus irregular. However, one cannot argue that their positions is the result of design rather than regular laws, for regular laws did indeed produce that configuration. Does this imply high salience and thus low dynamical complexity? I am not sure how to evaluate the salience of the configuration of air molecules in the room, but I wonder whether they may actually have low salience considering the vast number of alternative conformations. This would imply that the air molecules have high dynamical complexity, again a result which seems counterintuitive. These counterexamples suggest to me that the definitions of regularity and irregularity might need some adjustment. I suspect that some account of patterns apart from simple, repetitive ones may be needed. Additionally, some account for information flow via regular laws might be a useful addition. Indeed, if one wants to model dynamical complexity, I see the most important application being to trace the dynamics of information flow via law-governed interactions in time and space. John Bracht
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Richard A. Johns
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posted 07. March 2002 00:07
Reply to John Bracht
Thank you for your reply, which brings up some interesting issues. I think, however, that the cases you mention are not counterexamples to my definitions.
1. DNA produces DNA, under the operation of regular laws.
In my paper I’m only interested in what regular laws can produce from a random initial state. DNA molecules, I assume, are unlikely to exist in a randomly-chosen state, so I simply didn’t consider that case.
It seems obviously true (although I haven’t investigated this) that regular laws can preserve existing complex structures, as well as transform existing complex structures into related ones. Note that the GIGO theorem allows a complex object to appear in a very short time, with the operation of a suitable “program”, i.e. a restriction of the initial state. The presence of DNA constitutes a very strict constraint on the initial state.
2. Is a random arrangement of particles complex? Is it irregular?
On my definitions, the precise state of the air molecules in a given room will have very low salience. There are so many ways to arrange those molecules, and they are all equally likely, more or less, so each arrangement is monstrously improbable. Thus the state has high dynamical complexity. If this seems counterintuitive, then you probably have a different notion of “complexity” in mind – there are so many! This one, which just measures the difficulty of producing the state, seems to suit my purpose. It has nothing to do with order, for example. It doesn’t distinguish between English prose and random gibberish.
The state of the air molecules is also (likely to be) highly irregular. (Almost all states are highly irregular.) But even so, it doesn’t look like a counterexample to the regularity principle, as the complexity should be equally large.
3. Extra comments
The idea of studying information flow is interesting. In my paper I only study the production of information, from a random initial state, but it does seem important to understand how information can get transferred from one system to another. It doesn’t seem relevant to my problem, on the other hand, as naturalism holds that the cosmos is a closed system, and receives no input from outside. I’m trying to see if this is consistent with biological complexity.
I should also point out that my definition of irregularity doesn’t appeal to pattern, or repetition. It’s defined (roughly) as the number of bits of information needed to specify the state, given its local structure. To be honest, I’m not exactly sure which states end up being regular, on this definition. I do know that periodic structures like crystals are highly regular in this sense, but there may be other kinds of regular structure. In particular I should look at fractal patterns, but I haven’t done that yet.
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