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I did find some reasonably good definitions on the net while I was
being outraged over the Mikulecky mischaracterization of my father's
definition/s:
Richard Gallagher and Tim Appenzeller have written several things
together, most of which I can't access because one needs to be a subscriber of
Science magazine apparently... But I found the following, by
them:
?Complex System?: one whose properties are not
fully explained by an understanding of its component parts.
and
at http://www.usyd.edu.au/su/hps/newevents/Auyang1.html
I found Sunny Y. Auyang's work, from which I excerpted:
4. Formal Definitions of Complexity and the Combinatorial Explosion There is no precise definition of complexity and degree of complexity in the natural sciences. I use "complex" and "complexity" intuitively to describe self-organized systems that have many components and many characteristic aspects, exhibit many structures in various scales, undergo many processes in various rates, and have the capabilities to change abruptly and adapt to external environments. Nevertheless, there are two definitions of complexity in the information and computation sciences that can help us to appreciate nonreductive strategy for studying complex systems. The idea of complexity can be quantified in terms of information, understood as the specification of one case among a set of possibilities. The basic unit of information is the bit. One bit of information specifies the choice between two equally probable alternatives, for instance whether a pixel is black or white. Now consider binary sequences in which each digit has only two possibilities, 0 or 1. A sequence with n digits carries n bits of information. The information-content complexity of a specific sequence is measured in terms of the length in bits of the smallest program capable of specifying it completely to a computer. If the program can say of a n-digit sequence, "1, n times" or "0011, n/4 times," then the bits it requires are much less than n if n is large. Such sequences with regular patterns have low complexity, for their information contents can be compressed into the short programs that specify them. Maximum complexity occurs in sequences that are random or without patterns whatsoever. To specify a random sequence, the computer program must repeat the sequence, so that it requires the same amount of information as the sequence itself carries. The impossibility to squeeze the information content of a sequence into a more compact form manifests the sequence's high complexity. The information content complexity belongs to the definite description of a specific system, thus it is not useful in science because science is usually not so much interested in specific systems than in classes of system that satisfy certain general criteria. It often happens that totally random systems, which have highest information content complexity, exhibit other types of regularity than can be characterized rather simply if we are willing to adopt some other criteria of classification, e.g., use the law of large numbers. We have the probability calculus for such systems, and usually systems susceptible to the calculus are regarded as not that complex. Here we have the first instance of the theme of this talk; the flexibility to choose different criteria is paramount in scientific research. The second definition of complexity describes not systems but problems. Suppose we have formulated a problem in a way that can be solved by algorithms or step-by-step procedures executable by computers, and now want to find the most efficient algorithm to solve it. We classify problems according to their "size"; if a problem has n parameters, then the size of the problem is proportional to n. We classify algorithms according to their computation time which, given a computer, translates into the number of steps an algorithm requires to find the worse-case solution to a problem with a particular size. The computation-time complexity of a problem is expressed by how the computation time of its most efficient algorithm varies with its size. Two rough degrees of complexity are distinguished: tractable and intractable. A problem is tractable if it has polynomial-time algorithms, whose computation times vary as the problem size raised to some power, for instance n2 for a size-n problem. It is intractable if it has only exponential- time algorithms, whose computation times vary exponentially with the problem size, for instance 2n. Exponential-time problems are deemed intractable because for sizable n, the amount of computation time they require exceeds any practical limit. And then, of course, there's Femke Reitsma, the writer of the masters thesis which had the offending mis-characterizations of my father's work. Aside from the fact that Femke didn't do the proper homework-- by going to original sources for quotes rather than relying on second-hand interpretation-- I thought the musings on complexity contained in what I read were going in the right direction. By "right", naturally I mean the same direction that my father had already gone in! (I'm sure that comes as no surprise to anybody?) The definition of a complex system here was "a whole that cannot be fully delineated through an analysis of its component parts". It is encouraging to see so many unrelated voices saying that reductionism cannot help us understand complexity, but each of these voices is talking about a mere piece of the larger framework that my father already developed. It's a shame to have so many people re-inventing the wheel when their time could be better spent using it to travel where my father never got to. Judith
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