Etienne is 75, one could assign a fuzzy truth value of 0.8 to the statement. As a fuzzy set, this would be expressed as:
μold (Etienne) = 0.8
From what we have seen so far, membership in a fuzzy subset appears to be very much like probability. Both the degree of membership in a fuzzy subset and a probability value have the same numeric range: 0–1. Both have similar values: 0.0 indicating (complete) nonmembership for a fuzzy subset and FALSE for a probability and 1.0 indicating (complete) membership in a fuzzy subset and TRUE for a probability. What, then, is the distinction?
Consider a natural language interpretation of the results of the previous example. If a probabilistic interpretation is taken, the value 0.8 suggests that there is an 80% chance that Etienne is old. Such an interpretation supposes that Etienne is or is not old and that we have an 80% chance of knowing it. On the other hand, if a fuzzy interpretation is taken, the value of 0.8 suggests that Etienne is more or less old (or some other term corresponding to 0.8).
To further emphasize the difference between probability and fuzzy logic, let's look at the following example.
Example 1.4
Let
L = the set of all liquids
P = the set of all potable liquids
Suppose that you have been in the desert for a week with nothing to drink, and you find two bottles (Figure 1.4).
Figure 1.4 Bottles of liquids – front side.
Which bottle do you choose?
Bottle A could contain wash water. Bottle A could not contain sulfuric acid.
A membership value of 0.9 means that the contents of A are very similar to perfectly potable liquids, namely, water.
A probability of 0.9 means that over many experiments that 90% yield B to be perfectly safe and that 10% yield B to be potentially deadly. There is 1 chance in 10 that B contains a deadly liquid.
Observation:
What does the information on the back of the bottles reveal?
After examination, the membership value for A remains unchanged, whereas the probability value for B containing a potable liquid drops from 0.9 to 0.0 (Figure 1.5).
Figure 1.5 Bottles of liquids – back side.
As the previous example illustrates, fuzzy memberships represent similarities to objects that are imprecisely defined, while probabilities convey information about relative frequencies.
1.9 Expert Systems
Expert systems are an outgrowth of postproduction systems augmented with various elements of probability theory and fuzzy logic to emulate human reasoning within constrained domains. The general structure of such systems is a decision‐making portion known as an inference engine associated with a hierarchical knowledge structure or knowledge base.
The knowledge base typically consists of the domain‐specific knowledge and at least one level of abstraction. This second level contains the knowledge about knowledge or metaknowledge for the domain. As such, this level of abstraction provides the inference engine with the criteria for making decisions or reasoning within the specific domain of application.
In some cases, these systems have performed with remarkable success. Most notables are Dendral, Feigenbaum (1969), Meta‐Dendral, Buchanan (1978), Rule‐Based Expert Systems: MYCIN Experiments, Shortliffe (1984), and Prospector, Duda (1978). In each case, there have been tens and perhaps hundreds of man‐years devoted to tailoring each to a specific task. Nonetheless, those could be integrated, with very little effort, as a brute‐force solution to reasoning. These systems have little ability to learn from previous experience and are extremely fragile at the boundaries of their knowledge.
It is clear that human understanding of human knowledge is in its infancy. Sundry schemes have been suggested and tried while attempting to explain and to emulate or simulate the human thought and reasoning processes we so casually take for granted. None has demonstrated, nor even proposed, a universal answer; they merely chip at small corners and suggest that the elephant is a thin line like a rope, round like a cylinder, or flat like a wall.
1.10 Summary
We began this chapter with a look at some of the early works in learning and reasoning. We have seen that vagueness and imprecision are common in everyday life. Very often, the kind of information we encounter may be placed into two major categories: statistical and nonstatistical. The former we model using probabilistic methods, and the latter we model using fuzzy methods.
We introduced and examined the concepts of monotonic, non‐monotonic, and approximate reasoning and several models of human reasoning and learning.
We examined the concepts of crisp and fuzzy subsets and crisp and fuzzy subset membership. We learned that the possible degree of membership μxF() (membership of the variable x in the set F) of a variable x in a fuzzy subset spans the range [0.0–1.0] and that when we restrict the membership function so as to admit only two values, 0 and 1, fuzzy subsets reduce to classical sets. We also introduced the membership graph as a tool for expressing membership functions.
Review Questions
1 1.1 In the chapter opening, we introduced the term learning. Describe what it means.
2 1.2 Identify the three laws of thought and briefly describe each of them.
3 1.3 The chapter identified what is termed Feigenbaum's five‐phase learning process. Identify and describe each of those phases.
4 1.4 Based on Feigenbaum's learning process, identify and describe each member of the corresponding six‐level taxonomy.
5 1.5 In the context of this chapter, discuss what the term fuzzy means.
6 1.6 Identify and describe the difference between a classical set and a fuzzy set.
7 1.7 What is a fuzzy membership function? What information does such a function give you?
8 1.8 Formulate and graph a membership function for the concept very tall.
9 1.9 Formulate and graph membership functions for Example s 1.2 and 1.3.
10 1.10 What kind of information are we expressing with a crisp membership graph? What kind of information does such a graph give you?
11 1.11 What kind of information are we expressing with a fuzzy membership graph? What kind of information does such a graph give you?
12 1.12 Why do you think that membership graphs have different shapes?
13 1.13 Identify and describe how crisp and fuzzy membership graphs differ.
14 1.14 The chapter identified two kinds of imprecision.