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© Copyright 1993 Mario Hilgemeier

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 
and no type, and learning paradigms abound. Sadly, I can't go 
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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