What is Complex and What is Not?

Posted on April 23, 2008
Filed Under Complex Systems |

When is a system complex?

When trying to decide if a given system is complex, it is important to be clear about some of the distinctions I made in the definition of complexity (yesterday’s post).

First, the strict definition of a complex system is that it has some observable behavior that can only be explained by a theory that is too large for us to discover (and possibly there is no explanation at all, except for simulating the entire system).  So the most basic criterion for complexity is the size of the theory that “explains” the system’s behavior.

Now before you jump on this idea, I need to add that in general you cannot know if a given system is complex!  To know for sure, you have to know the size of the theory that explains the system, but it only takes a moment’s thought to realize that this reduces to two possibilities, one of which is trivial and one of which is impossible.  If the system is understood (somebody has previously explained how the behavior is related to the mechanisms that drive the system), then we don’t even need to look at the size of the theory, because the very fact that theory exists tells us that the system is not complex.  That is trivial.  But if we do not have a theory of the system yet (if we cannot explain how the high-level behavior is explained by the low-level mechanisms), then we are completely in the dark.  We don’t know if a good theory is going to turn up tomorrow, or if scientists will struggle to understand this system for the next million years, and still not get anywhere.  This, of course, is an impossibly difficult situation to be in: strictly speaking, we can’t say anything because we do not know the size of the theory that explains the system (you can’t say anything about the size of something that you haven’t discovered yet).

So if someone brings up an example of a real world system (the F-14 fighter jet, or a craps table, for example) and wants to know if this fits my definition of “complexity”, my first reaction is to say we already understand how the system works, then the answer is a trivial “no”.  But if we do not understand (cannot explain) the system, then the strict answer is “we don’t know”.

That is not, of course, the end of the story (I would not talk about complex systems so much if it were), but it is very important to be clear about that first step.  Without that idea, endless confusion can result.  The definition of “complex” is an abstract one, and when it is taken in isolation it leads nowhere — the only person who can know for sure which systems are complex and which are not is someone outside the universe (the Divine Mother, as I put it before) who knows the size of all theories.

For this reason, then, I prefer not to be asked whether System X is an example of something that is complex, because if the person asking the question has not already taken on board the above concept then any attempt by me to answer the question can lead to tangled confusion.

Okay, now I am going to move on and assume that the basic concept is understood.

Focus on the ‘regularity’

I need to confront a small side issue that has to do with which part, or aspect of a system you want to ask questions about.

It is almost always possible to drill down into a system and find some complexity somewhere, so it is very important to focus on the “behavior” (or, in the terminology that I prefer, the “regularity”) of the system that is important to you.  What exactly is it, in this system, that is supposed to be in need of explanation?  Once you focus on that, it becomes easier to ask whether that behavior is understood (and, if it is not understood, whether it is complex).

For example, in the question about throwing dice at a craps table, are we asking for an explanation of exactly how the dice fall on a particular occasion, or are we asking about some statistical bias that we observe in many throws of the dice?  If we want to explain a particular throw, this is not really a regular behavior that is of any interest … the system has not shown a pattern of behavior that is non-random, because we are talking about a single event, and a single event is not a pattern.  Put it this way:  would a curious scientist say that the single event was a fascinating, inexplicable thing, and would she then stop and devote a couple of decades of serious research to trying to “explain” this single event?  Of course not:  she would just say that the trajectory of that one throw was too hard to measure, and she would add that there is nothing interesting in a single event anyway.

I hope this makes it a little clearer why we need to be careful to say exactly what is the regularity that we suspect might be complex.  Not the system in general, but the regularity (the behavior) that the system is exhibiting.  The complex system concept is about whether the things we observe can be explained

To put it more vividly, it does not make sense for someone to hold up an apple and say “Explain this.”  That is not a regularity or a behavior, it is a thing.  On the other hand, it does make perfect sense to hold up an apple and say “Explain how this can grow from a seed.”  In a similar fashion, it does not make much sense to present a system of some kind and simply ask if the system is a complex system.

The example of an F-14 fighter jet suffers from a similar problem.  At a low level, there are aspects of the flight behavior of this jet that are extremely unstable and, quite possibly, “complex” in the sense that they cannot be explained fully.  But what happens in an F-14 is that the computer works extremely hard to control the intrinsic instability of the machine.  Crucially, however, we notice that the engineers who designed the jet were able to write the software well enough to compensate for the instability (the complexity) of the underlying system.  So whatever that complexity was, it was simple and predictable enough that the control software could actually be written and the complexity could be cancelled out.  That means that the complexity was not really an important part of the design of the plane, it was more like a noise signal that the engineers managed to cancel out.  And, of course, the resultant system of [plane] plus [control software] was not a complex system at all:  the complete system was understood well enough to make it fly straight.  If it is understood, it is not complex.  One aspect of it might be complex, but the complexity in this case was deliberately nullified.

The lesson to be learned from this F-14 example is that there are two perspectives on the system.  The high level involves no complexity.  Down at a deeper level there are things going on that are possibly complex.  Which of these answers you give depends on which aspect of the system you want to discuss.

The only problem with this careful distinction I have just made between a system and a regularity displayed by the system, is the fact that I (like everyone else) will violate the distinction all the time and talk about “the system” when what I really mean is a particular regularity displayed by the system.  Unfortunately, we often need to use the shorthand form and just say “system”.  Just bear in mind that his almost always means a particular regularity (a particular behavior) of the system.

Systems that are not yet understood

Now back to the dilemma I laid out at the beginning.  If a system (which means a particular behavior of that system) is not already understood, how do we decide whether or not the system is complex?

What I am going to do at this point is to simplify the problem drastically.  If the system is on the borderline, I am just not interested. There is no point arguing about whether a fringe case is complex or not.  It only makes sense to target the really extreme cases, because there the situation is clear enough that we can say something meaningful.

This is why I talked about properties of systems that involve “untouchable” mathematics.  If the mechanisms that drive the system are so grossly ugly that they are not just intractable, but utterly unpleasant to a mathematician’s delicate sensibilities, then we can make a hypothesis that these mechanisms may never be solved.

So, for example, we can point to these characteristics:

- Memory.  Does the mechanism use stored information about what it was doing fifteen minutes ago, when it is making a decision about what to do now?  An hour ago?  A million years ago?  Whatever:  if it remembers, then it has memory.

- Development.  Does the mechanism change its character in some way over time?  Does it adapt?

- Identity.  Do individuals of a certain type have their own unique identities, so that the result of an interaction depends on more than the type of the object, but also the particular individuals involved?

- Nonlinearity.  Are the functions describing the behavior deeply nonlinear?

These four characteristics are enough. Go take a look at a natural system in physics, or an engineering system, and find one in which the components of the system interact with memory, development, identity and nonlinearity.  You will not find any that are understood.

Now go and make up some artificial systems that have all these properties, and look at their overall behavior.

Notice that it is not always random.  Notice that sometimes it is highly structured.  Notice that sometimes the stability (in some sense of ’stability’) is dependent on the local mechanisms, but we cannot explain exactly how.

Notice, above all, that no engineer has ever tried to persuade one of these artificial systems to conform to a pre-chosen overall behavior.  This is widely thoght to be an insanely impossible thing to try to do.

Then, finally, notice that the one example that we have of a really bad, seriously overachieving example of a system with all these properties is these things that we call “intelligent systems”.

And I ask you the question:  do we have any good reason to believe that, even though all four of these mischievous properties are present in intelligent systems, they are nevertheless not complex, and that therefore we can just treat them as if they are ordinary engineering systems like all the others that do not have these nasty properties?  I don’t want a hunch that everything is okay, I want a good reason why we can be sure that complexity is not, in fact, present in sufficient quantities to cause trouble.

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