Editor | On 04, Feb 2008

By Abram Teplitskiy

Classic logic comes down to the law of the excluded third – A or not A, white or not white. In ancient Greece they started to consider paradoxes. Engineering professor Bart Kosko calls this a “mismatch” problem, “The world is gray, but science is black and white.” An electron cannot be assigned an exact position and velocity simultaneously – one of the founders of quantum mechanics, Werner Heisenberg, named this the uncertainty principle, which shows that “fuzziness” is a principle of physics. Three-valued logic came, in which statements could be true, false or indeterminate.

Fuzzy Worlds

An optimist estimates the degree of “emptiness” of a glass as half-full while a pessimist estimates it as half-empty. Engineers found two other options: “For system ‘glass-water’ the provision of safety is two” and “empty half of glass is without any use.” The fuzzy interpretation considers it as a midpoint phenomenon. Statements that describe them are literally half-truths. They are true 50 percent of the time, not 100 or zero percent. This is real state of the world. Can the fuzzy logic approach help solve problems?

Adjust Assumptions

Practice fuzzy logic by solving the 9-dot problem. Fuzzy dots have different diameters from zero. In fuzzy geometry, lines may not pass exactly through the center of the dots. With this fuzzy geometry, find a way of using a line to touch the dots with only three strokes. The solution is shown in Figure 1 – draw one wide vertical line, simultaneously marking all three vertical columns with only one stroke. Or, use a small line and go up and down beyond the dots to connect them. Notice that these options are not prohibited in the instructions.

Figure 1: Solutions to the Nine Dot Problem Courtesy Misha Teplitskiy

Challenge the “flat paper” assumption and consider spatial solutions in a fuzzy world. Figure 2 shows other means by which nine dots can be connected using no more than three lines. Roll the paper into a tube and connect the dots with a spiral – when unrolled, the spiral becomes a straight line. Try ripping the paper; poke a hole through each of the dots and all dots become connected. Another (incorrect, in this case) assumption is that the lines cannot extend beyond the edge of the paper. If the length of a line is not limited, it could travel around the earth and return to the other side of paper. A line that circumnavigates the planet twice will also solve the 9-dot problem.

Figure 2: Spatial Solutions to the Nine Dot Problem Courtesy Misha Teplitskiy

Fuzzy Justifications

Another example of using fuzzy logic principles is demonstrated by solving the problem of how to fairly share pieces of a pie. One person has to move a knife along the pie, while another participant directs where to make the cuts. In this example, participants represent an expert system. Using their joint expertise, they develop a “dividing” decision to fit both of their interests. (Complicate this problem by increasing the number of people getting pieces of pie.) Figure 3 shows a formula for fair cutting, but be careful – the resulting pieces could be fuzzy-justified: half the pie, a huge piece, an average slice or just a sliver.

Figure 3: Dividing Pieces of Pie Courtesy Merle and Kelly Cunningham

Traffic is an important fuzzy system. Imagine the millions of cars that are in motion every day facing fuzzy characteristics: weather conditions, the number of cars, road construction, etc. And, unfortunately, accidents can complicate the traffic.

Figure 4: Fuzzy Traffic Situations Courtesy Merle and Kelly Cunningham

Consider this fuzzy problem: how to create and develop the system for traffic controlling, which will direct drivers, prevent accidents and otherwise keep the system safe. To solve this problem you have to equip all highways and roads with sophisticated sensors, systems of car identification, databases, decision-making systems, weather prediction and evaluation equipment, ability to quickly/accurately divert cars when problems occur, a system for automatically verifying speed, etc.

Conclusion

The whole world is a fuzzy system in principle. Some simplification of the world can be described using classic and dialectic logic. But the most common system is a fuzzy system, which accepts and tolerates fuzzy solutions like contradictions. Sometimes, when the rules are fuzzy, you never get a precise solution. (Maybe humans are fuzzy people?)

Fuzzy Logic Exercises

1. Ask people to write a sentence that contains the name of any color as a word. Than ask them identify color of the ink they used and record the time it takes to answer. With a lot of responses, the dependencies can be analyzed.
2. The “measuring” and/or “evaluating” of taste or odor is very fuzzy! Propose ideas to evaluate taste, beauty and odor of different fuzzy objects – food, fashion, etc.
3. Three wise people were arguing about who is the wiser. A stranger tried to help them. He settled them opposite to each other, blindfolded them and put domes on their heads. Then he removed the blindfolds and asked each wise person to guess the color of the dome on his own head. How did the first wise man solve this problem? (Hint. In a new interpretation, the “wisest” man guesses that all domes have the same color, because otherwise it will be a simple problem, with a known long time solving algorithm.)
4. Finally, consider the “Protagoras paradox.” Protagoras, an ancient Greek philosopher, gave law lessons to a student who agreed to pay for the lessons after winning his first case. But the student did not get clients, so Protagoras sued him. Protagoras believes he will win because: 1) if the court sides with him, the student will have to pay and 2) if he loses, his student will have won his first case and will have to pay anyway. The student believes the opposite: 1) if the court sides with him, then he will not have to pay and 2) if the court sides with Protagoras, he would not win his first case, and, therefore, will not have to pay.Who will win – and why?

Happy inventing!