From Chaos to Robots

```We see that a string can be the description of the state of a
system. If the system is a plant, then  L-systems may be the
appropriate string rewriting systems (SRS) to model development
patterns. Other dynamical systems, in  celestial mechanics for
instance, are described by a set of numbers (three dimensional
coordinates and impulses) and the equations governing the relations
of the bodies (gravitation). The next set of numbers is derived from
the previous by iteration because gravitational systems of more than
two celestial bodies are not exactly solvable by symbolic
differential equations.

If a state is characterized by n numbers, it can be
represented as a point in n-dimensional space, the so-called phase
space. As the system evolves, the point hops through phase space,
not always very smoothly (large jumps happen). Although the
description of the dynamical system by differential equations yields
continuous movement, for reasons of computability the representation
is transformed into a discrete dynamical system where iteration is
used. But the introduction of iteration leads to discrete "jumps" of
the system state (quantization).

Watching only one coordinate of the phase space is like
measuring any parameter (e.g. the rainfall per day) of a dynamical
system (say, the weather). What you get is a time-series, sometimes
a fractal curve (this depends on the type of dynamical system). If
you enlarge portions of a fractal curve, they look "like the whole
curve;" you find the same patterns after each renewed magnification.

Any iteration of the GZR can be seen as a time-series. Figure
8 shows an example for the 49th iteration. How I arrived at this
curve is shown in pseudocode 3. This curve approaches
self-similarity because of the iteration rule of the GZR. But the
self- similarity is impure, because the line segments are finite.

Figure (7.)8

When you examine the phase space for a given dynamical system,
you find that certain regions will never be visited by the moving
system state. In other regions however, the point density is very
high - these are preferred states, visited often by the system.
Phase spaces with two or three dimensions can be represented
graphically very easily. Plot a point for each iteration of a
nonlinear dynamical system. The geometrical shapes that evolve may be
peculiarly beautiful or fractal [8, 9]. If the shapes are fractal,
they are called strange attractors. Many readers may be familiar
with the beautiful fractal images from the theory of dynamic
systems[11, 14].

The length of time for which predictions are valid for
nonlinear dynamical systems (weather, solar system) depends on our
(limited) knowledge of the initial conditions. Because system states
that are very close in phase space can get farther and farther apart
in the future, our prediction ability is as limited as our knowledge
of the initial conditions. Systems of orbiting bodies in celestial
mechanics that have rational proportions of orbital periods become
unstable sooner or later. Therefore it has been said that the
stability of the solar system crucially depends on the irrationality
of the proportions of the orbital periods. Well, for the foreseeable
future, no disaster looms, as far as computer models go today.
Computers often model states of the real world. By computation,
limited prediction and explanation of the phenomena of the world
becomes possible.

```

Robotics

```      A robot must have an internal model of the world around it -
at least of those qualities of the environment that it needs for its
function. For example, a washing machine is concerned with laundry
type, water, temperature, detergent, time and the like. The model of
the environment of the robot is represented in the tiny computer by
a set of numbers and rules, hence we have met another kind of phase
space here. The robot's state has a position in this
many-dimensional phase space. The robot's internal representation of
its "position in the world" moves in phase space. If it is to be a
successful robot, it will learn what the best actions are in certain
regions of the phase space (e.g. "if the is water too cold for
linen, start heater"). Of course, if the robot is to learn by
trial-and-error, it must somehow evaluate which actions were
successful.

We've entered the world of artificial intelligence (AI). AI is
concerned with reasoning, learning and making analogies in new
situations. The kinds of logic used are not always of the simple yes
further into this for now. If you want to read more, see the
references [15, 16]. The point I wanted to make here is that
fractals may have something to do with (machine) intelligence.
Iterated systems (IS), such as computers, can show chaotic (fractal)
behavior.

Many questions arise. Here are few: Does the robot's state in
phase space move toward a strange attractor? What are the
consequences? For what purpose can we use the dynamics of learning
systems if they are chaotic? Is there a danger in the low
predictability of chaotic systems? Engineers have just begun to
exploit nonlinear chaotic systems. Some chaotic systems can be
controlled by a chaotic signal and forced to express the desired
behavior [5]. Can chaotic systems be used to generate "creativity"
of robot artists? Or even for the automatic generation of unique
stories, films, architectural designs, virtual reality scenarios or
other cultural artifacts? In Birkhoff's aesthetic theory, art is
defined as something not boring, but at the same time not too
surprising [12]. Fractals surely fit this simple criterion. So they
may be useful for everyday design (and - in the hand of humans - for
works of art).

```
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