The development from fuzzy set theory to fuzzy technology during the first half of the 1990s was very fast. Therefore, this book is limited to the theory and application of fuzzy set theory.
1 INTRODUCTION TO FUZZY SETS
Crispness, Vagueness, Fuzziness, Uncertainty
Therefore, we should say that the formalization of the concept of approximate constructive necessary satisfaction is the main task of the semantic study of models in the empirical sciences. Due to the lack of information, the future state of the system may not be fully known.
Fuzzy Set Theory
However, I am not sure if it can (still) be considered the most important goal of fuzzy set theory. Most of the discussion will proceed along the lines of the early concepts of fuzzy set theory.
I FUZZY MATHEMATICS
Basic Definitions
FUZZY SETS-BASIC DEFINITIONS
Then the fuzzy set "comfortable type of house for a family of four" can be described. A fuzzy set is represented only by stating its membership function [e.g., Negoita and Ralescu 1975].
Basic Set-Theoretic Operations for Fuzzy Sets
That is, accepting the truth of the statement "S and T" requires more, and accepting the truth of the statement "S or T" less than accepting only S or T as true. However, these requirements are not sufficient to uniquely determine the mathematical form of the complement.
Types of Fuzzy Sets
3 EXTENSIONS
Further Operations on Fuzzy Sets
Xn•The Cartesian product is then a fuzzy set in the product space XI x ..XK; with the membership function. The bounded sum and the algebraic sum have been proposed as alternative models for the union of fuzzy sets.
4 FUZZY MEASURES AND MEASURES OF FUZZINESS
Fuzzy Measures
Measures of fuzziness, unlike fuzzy measures, attempt to indicate the degree of fuzziness of a fuzzy set. Others [Kaufmann 1975] suggested an index of blurriness as a normalized distance, and others [Yager 1979; Higashi and Klir 1982] base their concept of a degree of fuzziness on the degree of distinction between the fuzzy set and its complement. He therefore suggests that any degree of vagueness should be a measure of the lack of distinction between A and ¢A or Il,j(x) and.
The reader should realize that the complement of a fuzzy set is not uniquely defined [see Bellman and Giertz 1973; Dubois and Prade 1982a; Lowen 1978].
The Extension Principle
THE EXTENSION PRINCIPLE AND
Operations for Type 2 Fuzzy Sets
The extension principle can be used to define set-theoretic operations for type 2 fuzzy sets as defined in definition 3-1. The u and Vj are membership degrees of type 1 fuzzy sets and the Iluj(x) and Ilvix, respectively, their membership functions. 318] show that type 2 fuzzy sets as defined above are idempotent, commutative and associative and satisfy DeMorgan's laws.
Example 5-2 is a good indication of the computational effort involved in Type 2 fuzzy set operations.
Algebraic Operations with Fuzzy Numbers
The reader should be aware that in this example the degrees of membership of only one element of a type 2 fuzzy set are calculated. If we want to use fuzzy sets in applications, we will have to deal with fuzzy numbers, and the extension principle is one way to extend algebraic operations from clear to soft numbers. If M, NOT F(IR) with /lNCx) and /l";(x) continuous membership functions, then using the expansion principle for the binary operation *:IR ® IR ~ R.
Since addition is an increasing operation according to Theorem 5-1, we obtain for extended addition EB of fuzzy numbers f(N,M).
Extended operations with fuzzy numbers involve fairly extensive computations as long as no restrictions are placed on the type of membership functions allowed. The generality is not significantly limited by restricting the extended operations to fuzzy numbers in the LR representation or even to triangular fuzzy numbers [van Laarhoven and Pedrycz 1983], and the computational effort is greatly reduced. The reader should also understand that extended operations based on min-max cannot be directly applied to "fuzzy numbers" with discrete supports.
Calculate ~ Auii and Il4AforA, iJas in Example 5-2. Which of the following fuzzy sets are fuzzy numbers?
6 FUZZY RELATIONS AND FUZZY GRAPHS
Fuzzy Relations on Sets and Fuzzy Sets
Let RandZ be the two fuzzy relations defined in Example 6-2. The union of R and Z, which can be interpreted as "x significantly greater or very close to y", is then given by . The max-min composition is now the best known and most used. If * is an associative operation that is monotonic non-decreasing in every argument, then the max-* composition essentially corresponds to the max-min composition.
If RI and k are reflexive soft relations, then the max-min composition RI orRz is also reflexive.
Fuzzy Graphs
The properties mentioned in Remarks 6-1 and 6-2 apply to the max-min composition. For the composition max-prod, property 3 of Remark 6-2 is also true, but not properties 1 and 3 of Remark 6-1 or property 1 of Remark 6-2. For their composition, properties 1 and 3 of remark 6-1 as well as properties 1 and 3 of remark 6-2 apply at most. A fuzzy graph is «forest if there are no cycles; that is, it is a fuzzy acyclic graph.
If the soft forest is connected, it is called a tree. (A fuzzy graph that is a forest is to be distinguished from a fuzzy graph that is a fuzzy forest.) The latter will not be discussed here [see Rosenfeld 1975, p.
Special Fuzzy Relations
A fuzzy relation that is (max-min) transitive and reflexive is called a fuzzy pre-order relation. A fuzzy relation that is (min-max) transitive, reflexive and antisymmetric is called a fuzzy order relation. Some properties of the special fuzzy relations defined in this chapter are summarized in Table 6-1.
In example 6-2, two relations are defined without specifying for which numerical values of [x.}, {y;} the relations are good interpretations of the verbal relations.
7 FUZZY ANALYSIS
- Fuzzy Functions on Fuzzy Sets
- Extrema of Fuzzy Functions
- Integration of Fuzzy Functions
- Fuzzy Differentiation
Let j(x) be a fuzzy function from X to IR, defined over a crisp and finite domain D. The fuzzy maximum of j(x) is then defined as. We must now consider a fuzzy functionj', according to Definition 7-2, which must be integrated over the sharp interval [a, b). The fuzzy function j'(x) is assumed to be a fuzzy number, that is to say a piecewise continuous convex normalized fuzzy set to R. Let Aa be the a-level set of the fuzzy setA. The support SeA)ofA is then SeA)= U An.
The derivative can be the same for some belonging to [a,b] . Therefore, the probability of j'(Xo) is defined [Zadeh 1078] to be the maximum of the probability values of j'(x)=t ,xE [a,b].
8 UNCERTAINTY MODELING
Application-oriented Modeling of Uncertainty
But the kind of information available is not enough to deterministically describe the situation. The situation of "certainty" normally assumes an absolute or at least a cardinal level of scale of the available information. The properties of the type of information clearly differ from those of numerical information or information in a formal language.
Therefore, if symbolic information is provided, the information is only as valid as the definitions of the symbols, and the type of processing of the information must also be symbolic and neither numerical nor linguistic.
Possibility Theory
To stay consistent with the common symbol of possibility theory, we will denote the probability distribution by 1tx rather than "it", even though it is a fuzzy set. Let Abe be a fuzzy set in the space U and let 7tx be the probability distribution associated with the variable X, which takes the value in U. The possibility measure,7tiA) of A is then defined by On the other hand, IfA is assumed to be the fuzzy set of "non-small integers" defined as.
The issue of how to derive conditional probability distribution functions from the joint probability distribution function is not yet fully resolved.
Probability of Fuzzy Events
Zadeh [1968] therefore defined the probability of a fuzzy event A(Le., a fuzzy set A with membership function 1lA:(x» as follows. In Zadeh [1968] the correspondence between the probability of fuzzy events and the probability of sharp events is The truth of the proposition "the probability Ais at least w" is defined as the fuzzy set Pj(A) with the membership function.
We then calculate the probability of the sharp events Aa and give the intervals of w for which P(Aa ) ~ w.
Possibility vs. Probability
It has been mentioned that possibility theory is more than the min-max version of fuzzy set theory. At the end of the chapter we will examine the relationship between possibility theory and probability theory. Now we are ready to compare "fuzzy sets" with "probabilities," or at least one certain version of fuzzy set theory with one of probability theory.
While fuzzy set theory has quite a few "degrees of freedom" regarding intersection and union operators, types of fuzzy sets (membership functions), etc., the latter two theories are well developed and uniquely defined in terms of function and structure.
II APPLICATIONS OF FUZZY SET THEORY
Linguistic Variables
FUZZY LOGIC AND APPROXIMATE
Fuzzy Logic
In Boolean logic, truth values can be 0 (false) or 1 (true), and through these truth values, the vocabulary (operators) is defined through truth tables. 101] is an extension of set-theoretic multi-valued logic in which the truth values are linguistic variables (or terms of the linguistic variable truth). 109] proposes truth tables for determining truth values for operators using a four-valued logic, including true, false, undecided, and unknown truth values.
If the number of truth values (expressions of the linguistic variable truth) increases, one can still "tabulate" the truth table for operators using Definition 9-6 as follows: Let us assume that the z For the latter, a well-accepted axiomatic system such as that of Smets and Magrez [1987] can be used. Such an inference presupposes or necessitates knowledge of changes in the premises and their consequences (for example, knowledge that an increase in "redness" indicates an increase in "maturity" [Dubois and Prade 1984b, p. In 1973, Zadeh proposed that compositional inference rule for the above type of fuzzy conditional inference. Then the composition rule states that the solution of the relational assignment equations (see definition 8-1)R(x)=AandR(x ,y)=8 is given byR(y) = A0 8, where A0 8 is the composition of A and8 . PRUF is an intentional language, i.e., an expression in PRUF is supposed to convey the intended meaning rather than the literal meaning of the corresponding expression in a natural language. In this case, the definition of a quantity uses cardinality or relative cardinality, as defined in Definition 2-5. More in line with fuzzy set theory is to consider the truth of a proposition as a fuzzy number. Let a proposition be of the form "NisF" and let the reference proposition be, r ==Nis G, where F and Gare are subsets of U. Then the truth of r, ofp with respect to r is defined as the compatibility of r with p, that is, . Methods can be used to determine near-optimal weights and fuzzy sets in the body of the rules based on a dataset of examples [Baldwin 1994]. The membership of elements in the discrete fuzzy set is given to the right of the colon. The term in the body of the rule (height of X tall) matches (length of John tall) with X instantiated for John. In terms of the example above, Jeffrey's rule is Pr{(shoe_size of John large)}=. Introduction to Expert SystemsApproximate and Plausible Reasoning
Fuzzy Languages
Support Logic Programming and Fril .1 Introduction
10 FUZZY SETS AND EXPERT SYSTEMS
