| Yuri
Tarnopolsky eSSAYS
17. On Complexity
complexity. unified picture of the world. Ulf Grenander. pattern theory. |
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First, let us examine a block:
This is a description of the block: This block is a square with four connection points shown as red and blue dots, according to the picture. The block has its top and bottom, as the fill shows. Here is another block:  It differs from the first in the location of the dots. Here is yet another block:
This is how two blocks can be connected:
These are rules of connection: The blocks connect by
touching with
the dots of the same color.
Here is a combination of all three blocks :
Here are some other
combinations:
The blocks and the rules of connection define a space for all possible combinations and form a kind of a creative system for producing combinations. We will call the system of the above three blocks and the rules of connection SYSTEM 1. It has a certain complexity that we don't know how to measure. We can compare, however, two systems that do not differ much, i.e., one is produced from the other in a small step. This is how it could be done. Let us form a new system by adding four new blocks, without changing the rules. 
Although we add more blocks than there were initially, this is a small
step. If it seems big, we can add them one by one. If we changed the
blocks
and rules at the same time, it would be a bigger step. But we can
always
divide a change into minimal steps.
The new blocks (not the system yet) are more complex than the previous three because they have more kinds of dots: three colors instead of two. Therefore, the new system, let us call it SYSTEM 2, is more complex. SYSTEM 2 with seven blocks is more complex than the system with either three or four blocks because it uses more different types of blocks, some of them more complex. This is just
one example
of what we can make of the seven blocks.
Naturally, the number of combinations in SYSTEM 2 is larger than in SYSTEM 1. Now let us try something quite new: a mutation of the shape. We add two identical triangular blocks: ......................... There is only one way we can connect such triangle with only one of the seven:
This SYSTEM 3 of nine blocks is more complex than SYSTEM 2 because it has more types of blocks: square, as well as triangular. SYSTEM 4 comes next, in which the rule of matching colors is relaxed and the following combination, for example, is possible (it does not use all nine blocks):  SYSTEM 4 is less complex than SYSTEM 3 because it has less rules, even though it generates more combinations. If dots of all colors are equally connectable, then they can be reduced to just one color, which makes the blocks much simpler. We can have more combinations by allowing rotation of the blocks in plane (SYSTEM 5). We will require, however, the sides of the blocks be approximately parallel. The next two
combinations
by the rules of SYSTEM 5 look rather complex:
In fact, with the relaxed rules of connection, we can eliminate not only the colors, but also the fill that distinguishes between the top and bottom of the squares, and even the dots, so that the above combinations look very trivial:
A completely chaotic system, without any rules and with simplest blocks, is very simple. The billiard balls form such a system within the pool table. One can try
designing various
Lego-like systems and studying their complexity. The Microsoft Draw,
which
is a Microsoft Word function, is very convenient for this. It allows
for
a rather high complexity of combinations. I drew a picture.... To summarize, instead of trying to evaluate complexity of a single object, we do it for the system that generates it. All the objects generated by the system are of the same complexity. It seems to contradict our intuitive concept of complexity because large combinations look more complex than a single block or a couple of blocks. We should not mix up size and complexity, however. Still, we can take the number of different blocks in combinations within the same system as a measure of a partial complexity within the same system. The type of
connection can
vary, too. For example, it can have a direction and require two
different
types of connecting points, which would add to the complexity of the
system
:
I don't believe it is possible to find a universal numerical measure of complexity for everything in the world. Moreover, it is not necessary. Instead of measuring complexity of an individual object by a number, we compare any pair of systems and by transforming one into another, we can trace the number of steps that reduce or increase complexity. We may not have an exact measure in complicated cases, but we can still have a good idea about the difference between two systems. We might have even a scale of complexity by selecting a zero point. In this concept of complexity I use the same principle that Confucius used to quantify human virtues: by comparing two selected individuals and thus establishing a partial order in moral values (see Essay 13. On Numbers ). It is the same approach as with the beauty contest. There is no absolute numerical measure of beauty but we still can run a beauty pageant by ordering the contestants. This concept
of complexity,
which I would call pattern (or chemical) complexity, is based on
fundamental
ideas of Pattern Theory developed by Ulf Grenander. It is not limited
to
static structure and can be applied to dynamic systems where
transitions
from one static structure to another take place according to a separate
set of dynamic rules.
One way to solve a problem is to appoint a czar. A town meeting is another one. The third one is to ask a sage. NOTES 1.
Researchers involved in measuring complexity can compare the pattern
complexity
with Kolmogorov’s complexity and its controversial implications
concerning
random sequences. In pattern complexity, anything without rules is
inherently
simple.
2.
In authoritarian society some blocks have many copies and are
interchangeable,
and others have a few or one. In liberal democracy, ideally, all blocks
are unique, and the function of bureaucracy is to keep the uniqueness
down.
3. The following is the list of the individual blocks used in this Essay. They don't have either names or numbers. They are their own symbols, like pictograms in hieroglyphic script. By writing particular terms on the blocks and specifying rules, we can construct particular systems in various fields of knowledge, from poetry to molecules.         
4. Pattern Theory: Ulf Grenander, Elements of Pattern Theory, 1996. Baltimore and London: Johns Hopkins University Press.5. Aztec gods had colors associated with them. |
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