from which it evolved an internal description of the concept it was to be learning. The knowledge acquired was incorporated into a semantic network where all of the interrelationships among the constituent elements were described.
Critical to the effective use of Winston's algorithm are the ideas that positive training instances are evolutionary rather than revolutionary. In Winston's algorithm, negative training instances are those that reflect only minimal differences from the concept being investigated; thus, no learning occurs.
1.4.4 Analogical or Metaphorical Learning
Analogical or metaphorical learning is probably one of the more common methods by which human beings acquire new knowledge. As Winston points out in his work Learning and Reasoning by Analogy, with such an approach, once again, the learner's contribution to the process is increased. When learning by example, the learner is presented with positive and negative instances of the concept to be learned. With an analogy, the student has only closely related instances from which to extract the desired concept.
Jaime Carbonell identifies transformational and derivational as the two principal methods of reasoning by analogy. When learning by transformational analogy, the line of reasoning proceeds incrementally from some old or known solution to the new or desired solution through a series of mappings means‐ends called transform operators. The operators are applied using a means‐ends paradigm until the desired transformation is achieved.
Knowledge acquisition by derivational analogy achieves learning by recreating the line of reasoning that resulted in the solution to the problem. The reconstruction includes both decision sequences and attendant justifications.
1.4.5 Learning by Problem Solving
Learning by problem solving can easily be viewed as subsuming all other forms of learning discussed. However, such a technique has sufficient merit in its own right that it deserves individual consideration. With this approach, the knowledge to be imparted is embedded in a problem or sequence of problems. The objective is for the learner to acquire that knowledge by solving the problem. The most serious difficulty with such an approach is the intolerance of individual method.
Consider the question: “What is the sum of ¼ and ¼?” Although a response of 2/4 would be completely correct, that answer may be considered wrong since it did not match the “correct” answer of ½.
1.4.6 Learning by Discovery
Learning by discovery is the antithesis of rote learning. In this paradigm, the learner is the initiator in all five phases of learning discussed earlier. There is no new knowledge in the world since all knowledge already exists and is merely waiting some clever individual to discover it.
According to Carbonell, two basic methods of acquiring knowledge by discovery are available: observation and experimentation. Observation is considered to be a passive approach because the learner collects information by watching a particular event and then later forms a theory to explain the phenomenon. In contrast, experimentation is viewed as active. Here, the process generally involves the learner postulating a new theory about the existence of a particular piece of knowledge and verifying that theory by experiment. In neither case is the possibility of serendipitous discovery excluded.
Discovery, Carbonell proposes, is a three‐step process begun by hypothesis formation. The hypothesis may be either data driven, as is the case for observation, or theory driven as with experimentation. The initial step is followed by a refinement process in which partial theories are merged and boundary conditions are established. Finally, what has been learned and created is extended to new instances.
Clearly, any theory or model of human (or machine) learning must include aspects of each member of the established taxonomy. Today, no comprehensive and unified theory of human learning exists. Only partial theories that attempt to explain portions of the whole of human learning have been developed. Looking back over each of the ideas, we can take away the notion that learning and problem solving are effectively used interchangeably.
1.5 Crisp and Fuzzy Logic
As we move forward, we will explore, study, and learn two forms of logic and reasoning called crisp logic or reasoning and fuzzy logic or reasoning. The two graphical diagrams shown in Figure 1.1 suggest the difference between these two forms.
Figure 1.1 Crisp and fuzzy circles.
On the left is a crisp, precise circle and on the right is one in which the shape is less precise but can still be viewed in many contexts as a kind of circle.
1.6 Starting to Think Fuzzy
Over the years, fuzzy logic has been found to be extremely beneficial and useful to people involved in research and development in numerous fields including engineers, computer software developers, mathematicians, medical researchers, and natural scientists. As we begin, with all those people involved, we raise the question: What is fuzzy?
Originally, the word fuzz described the soft feathers that cover baby chicks. In English, the word means indistinct, imprecise, blurred, not focused, or not sharp. In French, the word is flou and in Japanese, it is pronounced “aimai.” In academic or technical worlds, the word fuzz or fuzzy is used in an attempt to describe the sense of ambiguity, imprecision, or vagueness often associated with human concepts.
Revisiting an earlier example, trying to teach someone to drive a car is a typical example of real‐life fuzzy teaching and fuzzy learning. As the student approaches a red light or intersection, what do you tell him or her? Do you say, “Begin to brake 25 m or 75 ft from the intersection?” Probably not. More likely, we would say something more like “Apply the brakes soon” or “Start to slow down in a little bit.” The first case is clearly too precise to be implemented or executed by the driver. How can one determine exactly when one is 25 m or 75 ft from an intersection? Streets and roads generally do not have clearly visible and accurate millimeter‐ or inch‐embedded gradations. The second vague instruction is the kind of expression that is common in everyday language.
Children learn to understand and to manipulate fuzzy instructions at an early age. They quite easily understand phrases such as “Go to bed about 10:00.” Perhaps with children, they understand too well. They are adept at turning such a fuzzy expression into one that is very precise. At 9:56, determined to stay up longer, they declare, “It's not 10:00 yet.”
In daily life, we find that there are two kinds of imprecision: statistical and nonstatistical. Statistical imprecision is that which arises from such events as the outcome of a coin toss or card game. Nonstatistical imprecision, on the other hand, is that which we find in instructions such as “Begin to apply the brakes soon.” This latter type of imprecision is called fuzzy, and qualifiers such as very, quickly, slowly or others on such expressions are called hedges in the fuzzy world.
Another important concept to grasp is the linguistic form of variables. Linguistic variables are variables with more qualitative rather than numerical values, comprising words or phrases in a natural or potentially an artificial language. That is, whether simple or complex, such variables are linguistic rather than numeric.
