CHAPTER 1 THE FUZZY WORLD What’s the process of parallel parking a car? First you line up your car next to the one in front of your space. Then you angle the car back into the space, turning the steering wheel slightly to adjust your angle as you get closer to the curb. Now turn the wheel to back up straight and—nothing. Your rear tire’s wedged against the curb. OK. Go forward slowly, steering toward the curb until the rear tire straightens out. Fine—except, you’re too far from the curb. Drive back and forth again, using shallower angles. Now straight forward. Good, but a little too close to the car ahead. Back up a few inches. Thunk! Oops, that’s the bumper of the car in back. Forward just a few inches. Stop! Perfect!! Congratulations. You’ve just parallel-parked your car. And you’ve just performed a series of fuzzy operations. Not fuzzy in the sense of being confused. But fuzzy in the real-world sense, like “going forward slowly” or “a bit hungry” or “partly cloudy”—the distinctions that people use in decision-making all the time, but that comput- ers and other advanced technology haven’t been able to handle. 1 Fuzzy Logic A Practical Approach by F. Martin McNeill and Ellen Thro Chapter 1: The Fuzzy World 2 What kind of problems? For one, waiting for an elevator at lunch hour. How do you program elevators so that they pick up the most people in the least amount of time? Or how do you program elevators to minimize the waiting time for the most people? Suppose you’re operating an automated subway system. How do you program a train to start up and slow down at stations so smoothly that the passengers hardly notice? For that matter, how can you program a brake system on an automo- bile so that it works efficiently, taking road and tire conditions into account? Perhaps you have a manufacturing process that requires a very steady temperature over a many hours. What’s the most efficient and reliable method for achieving it? Or, suppose you’re filming an unpredictable and fast-moving event with your camcorder—say, a birthday party of 10 three-year-olds. What kind of a camera lets you move with the action and still end up with a very nonjerky image when you play it back? Or, take a problem far from the realm of manufacturing and engineer- ing, such as, how do you define the term family for the purposes of inclusion in health insurance policy? Do all these situations have something in common? For one thing, they’re all complex and dynamic. Also, like parallel parking, they’re more easily characterized by words and shades of meaning than by mathematics. In this book you’ll be immersed in the fuzzy world, not an easy process. You’ll meet the basics, manipulate the tools (simple and complex), and use them to solve real-world problems. You can make your experience interactive and hands on with a series of programs on the accompanying disk. (See the Preface for an explanation of how to load it onto your hard disk.) To make the trip easier, you’ll be following in the many footsteps of our fuzzy field guide, Dr. Fuzzy. The good doctor will be on call through Help menus and will show up in the book chapters with hints, further information, and encouraging messages. -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - The real world is up and down, constantly moving and E-MAIL changing, and full of surprises. In other words, fuzzy. FROM Fuzzy techniques let you successfully handle real- DR. FUZZY world situations. -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Fuzzy Logic A Practical Approach by F. Martin McNeill and Ellen Thro Chapter 1: The Fuzzy World 3 APPLES, ORANGES, OR IN BETWEEN? As the fiber-conscious Dr. Fuzzy has discovered, one of the easiest ways to step into the fuzzy world is with a simple device found in most homes—a bowl of fruit. Conventional computers and simple digital control systems follow the either-or system. The digit’s either zero or one. The answer’s either yes or no. And the fruit bowl (or database cell) contains either apples or oranges. Take Figure 1.1, for example. Is this a bowl of oranges? The answer is No. How about Figure 1.2? Is it a bowl of oranges? The answer in this case is Yes. This is an example of crisp logic, adequate for a situation in which the bowl does contain either totally apples or totally oranges. But life is often more complex. Take the case of the bowl in Figure 1.3. Someone has made a switch, Figure 1.1: Is this a bowl of oranges? Figure 1.2: Is this a bowl of oranges? Fuzzy Logic A Practical Approach by F. Martin McNeill and Ellen Thro Chapter 1: The Fuzzy World 4 Figure 1.3: “Thinking fuzzy” about a bowl of oranges. Figure 1.4: Fuzzy bowl of apples. Figure 1.5: Fuzzy bowl of apples (continued). swapping an orange for one of the apples in the Yes—Apple bowl. Is it a bowl of oranges? Suppose another apple disappears, only to be replaced by an orange (Figure 1.4). The same thing happens again (Figure 1.5). And again (Figure 1.6). Is the bowl now a bowl of oranges? Suppose the process continues Fuzzy Logic A Practical Approach by F. Martin McNeill and Ellen Thro Chapter 1: The Fuzzy World 5 Figure 1.6: Fuzzy bowl of apples (continued). Figure 1.6: Fuzzy bowl of apples (continued). Fuzzy Logic A Practical Approach by F. Martin McNeill and Ellen Thro Chapter 1: The Fuzzy World 6 (Figure 1.7). At some point, can you say that the “next bowl” contains oranges rather than apples? This isn’t a situation where you’re unable to say Yes or No because you need more information. You have all the information you need. The situation itself makes either Yes or No inappropriate. In fact, if you had to say Yes or No, your answer would be less precise that if you answered One, or Some, or A Few, or Mostly—all of which are fuzzy answers, somewhere in between Yes and No. They handle the actual ambiguity in descriptions or presentations of reality. Other ambiguities are possible. For example, if the apples were coated with orange candy, in which case the answer might be Maybe. The complex- ity of reality leads to truth being stranger than fiction. Fuzzy logic holds that crisp (0/1) logic is often a fiction. Fuzzy logic actually contains crisp logic as an extreme. -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Really want to think fuzzy apples and oranges? They have less distinct boundaries than you might think. Both apples and oranges are spheres, and both are about the same size. Both grow on trees that reproduce simi- larly. You can make a tasty drink from each. They even go to their rewards the same way, by being eaten and digested E-MAIL by people, or by being composted by my relatives, near and FROM distant. If the apples are red, even the colors are related— DR. FUZZY red + yellow = orange And don’t neglect the bowl. Both fruits nestle the same way in the same kind of bowl, and they leave similar amounts of unoccupied space. -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - With fuzzy logic the answer is Maybe, and its value ranges anywhere from 0 (No) to 1 (Yes). Fuzzy Logic A Practical Approach by F. Martin McNeill and Ellen Thro Chapter 1: The Fuzzy World 7 -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - E-MAIL FROM Crisp sets handle only 0s and 1s. DR. Fuzzy sets handle all values between 0 and 1. FUZZY -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Crisp No Yes Fuzzy No Slightly Somewhat SortOf A Few Mostly Yes, Absolutely Looking at the fruit bowls again (Figure 1.8), you might assign these fuzzy values to answer the question, Is this a bowl of oranges? -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Characteristics of fuzziness: E-MAIL (cid:176) • Word based, not number based. For instance, hot; not 85 . FROM • Nonlinear changeable. DR. FUZZY • Analog (ambiguous), not digital (Yes/No). -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - If you really look at the way we make decisions, even the way we use computers and other machines, it’s surprising that fuzziness isn’t considered the ordinary way of functioning. Why isn’t it? It all started with Aristotle (and his buddies). IS THERE LIFE BEYOND MATH? The either-apples-or-oranges system is known as “crisp” logic. It’s the logic developed by the fourth century B.C. Greek philosopher Aristotle and is often called Arisfotelian in his honor. Aristotle got his idea from the work of an earlier Greek philosopher, Pythagoras, and his followers, who believed that matter was essentially numerical and that the universe could be defined as numerical relationships. Their work is traditionally credited with providing Fuzzy Logic A Practical Approach by F. Martin McNeill and Ellen Thro Chapter 1: The Fuzzy World 8 Figure 1.8: Fuzzy values. Fuzzy Logic A Practical Approach by F. Martin McNeill and Ellen Thro Chapter 1: The Fuzzy World 9 the foundation of geometry and Western music (through the mathematics of tone relationships). Aristotle extended the Pythagorean belief to the way people think and make decisions by allying the precision of math with the search for truth. By the tenth century A.D., Aristotelian logic was the basis of European and Middle Eastern thought. It has persisted for two reasons—it simplifies think- ing about problems and makes “certainty” (or “truth”) easier to prove and accept. Vague Is Better In 1994 fuzziness is the state of the art, but the idea isn’t new by any means. It’s gone under the name fuzzy for 25 years, but its roots go back 2,500 years. Even Aristotle considered that there were degrees of true-false, particularly in making statements about possible future events. Aristotle’s teacher, Plato, had considered degrees of membership. In fact, the word Platonic embodies his concept of an intellectual ideal—for instance, of a chair—that could be realized only partially in human or physical terms. But Plato rejected the notion. Skip to eighteenth century Europe, when three of the leading philoso- phers played around with the idea. The Irish philosopher and clergyman George Berkeley and the Scot David Hume thought that each concept has a concrete core, to which concepts that resemble it in some way are attracted. Hume in particular believed in the logic of common sense—reasoning based on the knowledge that ordinary people acquire by living in the world. In Germany, Immanuel Kant considered that only mathematics could provide clean definitions, and many contradictory principles could not be resolved. For instance, matter could be divided infinitely, but at the same time could not be infinitely divided. That particularly American school of philosophy called pragmatism was founded in the early years of this century by Charles Sanders Peirce, who stated that an idea’s meaning is found in its consequences. Peirce was the first to consider “vagueness,” rather than true-false, as a hallmark of how the world and people function. The idea that “crisp” logic produced unmanageable contradictions was picked up and popularized at the beginning of the twentieth century by the flamboyant English philosopher and mathematician, Bertrand Russell. Fuzzy Logic A Practical Approach by F. Martin McNeill and Ellen Thro Chapter 1: The Fuzzy World 10 He also studied the vagueness of language, as well as its precision, conclud- ing that vagueness is a matter of degree. -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Crisp logic has always had fuzzy edges in the form of para- doxes. One example is the apples-oranges question earlier E-MAIL in the chapter. Here are some ancient Greek versions: FROM • How many individual grains of sand can you remove from DR. FUZZY a sandpile before it isn’t a pile any more (Zeno’s paradox)? • How many individual hairs can fall from a man’s head before he becomes bald (Bertrand Russell’s paradox)? In ancient, politically incorrect mainland Greece they said, “All Cretans are liars. When a Cretan says that he’s ly- ing, is he telling the truth?” The logical problem: How sta- ble is the idea of truth and falsity? In the early twentieth century, Bertrand Russell (who seemed to be amazingly interested in human fuzz) asked: A man who’s a barber advertises “I shave all men and only those who don’t shave themselves.” Who shaves the barber? The down-home illustration involved this logical question: Can a set contain itself? -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - The German philosopher Ludwig Wittgenstein studied the ways in which a word can be used for several things that really have little in common, such as a game, which can be competitive or noncompetitive. The original (0 or 1) set theory was invented by the nineteenth century German mathematician Georg Kantor. But this “crisp” set has the same shortcomings as the logic it’s based on. The first logic of vagueness was developed in 1920 by the Polish philosopher Jan Lukasiewicz. He devised sets with possible membership values of 0, 1/2, and 1, later extending it by allowing an infinite number of values between 0 and 1. Later in the twentieth century, the nature of mathematics, real-life events, and complexity all played roles in the examination of crispness. So did the amazing discovery of physicists such as Albert Einstein (relativity) and Werner Heisenberg (uncertainty). Einstein was quoted as saying, ”As far Fuzzy Logic A Practical Approach by F. Martin McNeill and Ellen Thro