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Fuzzy Logic

In the February 1993 issue of Scientific American there is an article title "A Partly True Story". It appears in the 'Mathematical Recreations' column and was authored by the world renowned mathematician Ian Stewart.

The theme of this article is one of the logic problems of the ancient Greek philosophers. These are problems and apparent paradoxes of the type sometimes attributed to Zeno. In one form, twin statements are put forwards such as "All Cretans are Liars" and "I am a Cretan".
Maybe they were written to test students, a tutorial in faulty logic. Maybe as a lesson that makes clear the need for perceiving truth as declentive, in shades of pink rather than red and white.

The storyline in the article presents the following example of a double sided business card...


the Epimenides (mouseover) Business Card

However in attempting to more clearly define the shades of pink that lie betweem True and False some interesting mathematics has been proposed which can be used for a lot of lighthearted mathematical fun. It attempts to evaluate how truthful a statement might be on a scale that lies between 1 (completely truthful) and 0 (utterly false). For example a statement could be mathematically determined to have a truth value 0.15 ie be 15% true. (Or may be it is 85% false?)

If you want to understand more about fuzzy logic it would be better to read the article (a partly true story) or look elsewhere since this page is oriented, in the main, to the mathematics involved rather than the underlying philosophy. It suffices to say that the formulæ at play in what is refered to as the Chaotic Dualist can be used to produce the beautiful "Escape-Time" diagrams which appear on the fuzzy-logic pages of this website.

The fundamental logic is enshrined in these two statements:
X : X is as true as Y is true
Y : Y is as true as X is false

The dynamics are formularized as follows:
x <= 1 - |x - y|
y <= 1 - |y - (1 - x)|

The escape-time diagrams are generated from these two equations. Here quoted is an excerpt from the original article:-

"...To see what it does, you chose an initial pair of values, say, (x,y) = (0.2,0.9), and calculate successive pairs of values. Think of them as coordinates and plot them in the plane. You get a geometric shape called the 'attractor of the the dynamic system'. In this case, you get a triangle densly filled with points,[illustration included in original but not here]. This representation can be transformed into a beautiful and intricate image known as an escape-time diagram. To create it, temporarily relax the conditions that x and y lie between 0 and 1. The idea is to watch how far (x,y) moves from the origin (0,0) and to count how many calculation steps are needed before it goes beyond some threshold value. Then the point (x,y) is plotted in a color that depends on the number of steps required. To start, you should try a threshold value just larger than 1 ( comment: one will do nicely)..."(end of quote)

However the content of the article, (partly) true to its premise, only delivers a percentage of the truth and fails to give all the information neccessary to reproduce the diagram displayed. The programmer is left to figure much out for him/her self. For example the center of the escape-time diagram plots at (0.5,0.5)


circular escape-time sci-am
Circular Escape-Time Diagram 1

The quest was born to develop a program that would reproduce the escape time diagram as shown.

After much tinkering with the code the above diagram was produced, but it is not the only way to portray the escape-time diagram. The circular motif is produced by using a plain circular function that is integrated with the fundamental guts of the program. Something that is not apparent from the text.

On closer examination it is obvious that the dark blue perimeter of the circles in diagram 1 covers up a geometric pattern. A simple scheme was devised to trim the excess part of the dark blue top layer (and subsequent layers) and plot what remained of what lay hidden underneath. The result is a trimmed escape time diagram.

escape-time trimmed
Trimmed Circular Escape-Time Diagram 2

By removing the circular function and replacing it with a linear function, a different escape-time diagram, not needing to be trimmed, can be produced...

escape-time linear
Square Escape-Time Diagram 3

From the above it can be deduced that many variations of functions can be applied, producing a wide variety of motifs and styles within the constraints. Many of the results of this process can be seen on the fuzzy logic marquee web pages featured on this site

The next step was to see if other forms of the chaotic dualist existed and plot their escape-time diagrams. The article describes other formulæ that might be employed. In this instance a simple approach was taken and four formula pairs were derived making up a set through the equivalent of putting 'the cat among the pigeons,' a judicious application of the negative sign. This created three other pairs of formulæ as follows...

Formula pair 2...
  x <= 1 - |x - y|
-y <= 1 - |y - (1 - x)|

Formula pair 3...
-x <= 1 - |x - y|
  y <= 1 - |y - (1 - x)|

Formula pair 4...
-x <= 1 - |x - y|
-y <= 1 - |y - (1 - x)|

Each pair might represent a variation of the original logic statements, since they each produce a different escape-time diagram with a changed relative size. The pairs produce diagrams of size 3, size 5, size 7 and size 9 respectively. The process somewhat resembles an original form of cellular expansion. Pair 1 produces the pseudo-octagon as already shown.

5
Linear Escape-Time (pair 1,3) Diagram 4

Pair 2 produces a pseudo-octagon that appears to be stretched unequally in two directions (see diag-4).
5
Linear Escape-Time (pair 2,5) Diagram 5

The diagram produced by pair 3 is complex and is better looked at rather than described. It appears to be an overlapping doubled version of pair 2 see diagram (4). It somewhat resembles a cross-section of two kidneys or the two lobes of a brain. This diagram probably has an already decided upon scientific name, if you know please let me know. For now it could perhaps be called an "Octo-Lobogram" or some such name.
7
Linear Escape-Time (pair 3,7) Diagram 6

Pair 4 produces a psuedo-octagon, but now three times larger than that produced by the formulæ of pair 1...

7
Linear Escape-Time (pair 4,9) Diagram 7

These diagrams form a partly consistent sequence, in that in each case an arrangement of the motif was chosen that would better reveal what is happening.

For a much clearer understanding of fuzzy-logic it might well be useful to visit the website of Patrick Grim or take the link to the book he jointly authored, the Philosophical Computer.

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