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Fig. 11b. Alternative modes of mm-precisiation of “approximately a”, “a, within the framework of fuzzy logic.

Figure 11 b. Alternative modes of mm-precisiation of “approximately a”, “a, within the framework of fuzzy logic.

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Fig. 1. Principal facets of fuzzy logic (FL). The core of FL is graduation/granulation, G/G. tion may be viewed as a form of inrormation compression Of variables and input/output relations. An important concept which is related to the concept of a linguistic variable is the concept of a granular value [107]. More specifically, consider a variable, X, which takes values in U. Let u be a value of X. Infor- mally, if wis known precisely, then w is referred to as a singular (point) value of X. If X is not known precisely, but there is some information which constrains possible values of u, then the constraint on uw defines a granular value of X, (Fig. 3). For example, if what is known about uw is that it is contained in an interval [a, b], then [a, b] is a granular value of X. A granular variable is a variable which takes granular values. In this sense, a linguistic variable is a granular variable which carries linguistic labels. It should be noted that a granular value of Age is not restricted to young, middle-aged or old. For example, ‘“‘not very young” is an admissible granular value of CO IRIE gg EEE SETS SRE HERESIES gies 1 EES An instance of granulation i is s the concept of a linguistic variable —a concept which was introduced i in my 1973 paper “Outline of a new approach to the analysis of complex systems and decision processes” [79]. A simple example of a linguistic variable is shown in Fig. 2. Today, the concept of a linguistic variable is used in almost all applications of fuzzy logic, especially in the realms of control and consumer products. Granula- tion may be viewed as a form of information compression of variables and input/output relations. i Furthermore, in fuzzy logic everything is or is allowed to be granulated, with a granule being a clump of attribute-values drawn together by indistinguishability, similarity, proximity or functionality. Graduated granulation, or equivalently fuzzy granulation, is a unique feature of fuzzy logic. Graduated granulation is inspired by the way in which humans deal with complexity and imprecision [87,97,98].
Fig. 1. Principal facets of fuzzy logic (FL). The core of FL is graduation/granulation, G/G. tion may be viewed as a form of inrormation compression Of variables and input/output relations. An important concept which is related to the concept of a linguistic variable is the concept of a granular value [107]. More specifically, consider a variable, X, which takes values in U. Let u be a value of X. Infor- mally, if wis known precisely, then w is referred to as a singular (point) value of X. If X is not known precisely, but there is some information which constrains possible values of u, then the constraint on uw defines a granular value of X, (Fig. 3). For example, if what is known about uw is that it is contained in an interval [a, b], then [a, b] is a granular value of X. A granular variable is a variable which takes granular values. In this sense, a linguistic variable is a granular variable which carries linguistic labels. It should be noted that a granular value of Age is not restricted to young, middle-aged or old. For example, ‘“‘not very young” is an admissible granular value of CO IRIE gg EEE SETS SRE HERESIES gies 1 EES An instance of granulation i is s the concept of a linguistic variable —a concept which was introduced i in my 1973 paper “Outline of a new approach to the analysis of complex systems and decision processes” [79]. A simple example of a linguistic variable is shown in Fig. 2. Today, the concept of a linguistic variable is used in almost all applications of fuzzy logic, especially in the realms of control and consumer products. Granula- tion may be viewed as a form of information compression of variables and input/output relations. i Furthermore, in fuzzy logic everything is or is allowed to be granulated, with a granule being a clump of attribute-values drawn together by indistinguishability, similarity, proximity or functionality. Graduated granulation, or equivalently fuzzy granulation, is a unique feature of fuzzy logic. Graduated granulation is inspired by the way in which humans deal with complexity and imprecision [87,97,98].
Fig. 2. Granulation of Age; young, middle-aged and old are linguistic (granular) values of age. Fig. 3. Singular and granular values.
Fig. 2. Granulation of Age; young, middle-aged and old are linguistic (granular) values of age. Fig. 3. Singular and granular values.
Fig. 4. Granulation of a function. S (small), M@ (medium) and L (large) are fuzzy sets. The granuland of f, “f, may be viewed as a summary of f.
Fig. 4. Granulation of a function. S (small), M@ (medium) and L (large) are fuzzy sets. The granuland of f, “f, may be viewed as a summary of f.
Fig. 5. Granulation and interpolation of a probability distribution.
Fig. 5. Granulation and interpolation of a probability distribution.
Fig. 6. Granular vs. granule-valued distributions.
Fig. 6. Granular vs. granule-valued distributions.
Fig. 7. The cornerstones of a nontraditional view of fuzzy logic.
Fig. 7. The cornerstones of a nontraditional view of fuzzy logic.
Fig. 10. Disambiguation of P: Most tall Swedes. Fig. 9. Modalities of precisiation. mh-precisiation = nonmathematical precisiation of meaning; mm-precisiation = mathematical precisiation of meaning.
Fig. 10. Disambiguation of P: Most tall Swedes. Fig. 9. Modalities of precisiation. mh-precisiation = nonmathematical precisiation of meaning; mm-precisiation = mathematical precisiation of meaning.
Fig. 8. Basic concepts relating to precisiation and cointension.
Fig. 8. Basic concepts relating to precisiation and cointension.
Fig. 1la. Alternative modes of mm-precisiation of “approximately a”, “a, within the framework of bivalent logic. A precisiend may be precisiated in a large, perhaps unbounded, number of ways. As an illustration, Figs. lla and 11b show some of the simpler mm-precisiands of the predicate “approximately a,” or “a for short. Note that the simplest mm-precisiand is a. This very stmple mode of mm-precisiation is widely employed in many fields of science. Probability theory is a case in point. Most real-world probabilities are based on per- ceptions rather than on measurements. Perceptions are intrinsically imprecise. As a consequence, so are most
Fig. 1la. Alternative modes of mm-precisiation of “approximately a”, “a, within the framework of bivalent logic. A precisiend may be precisiated in a large, perhaps unbounded, number of ways. As an illustration, Figs. lla and 11b show some of the simpler mm-precisiands of the predicate “approximately a,” or “a for short. Note that the simplest mm-precisiand is a. This very stmple mode of mm-precisiation is widely employed in many fields of science. Probability theory is a case in point. Most real-world probabilities are based on per- ceptions rather than on measurements. Perceptions are intrinsically imprecise. As a consequence, so are most
Fig. 13. Fuzzy graph. The fuzzy graph constraint is defined as the possibilistic constraint
Fig. 13. Fuzzy graph. The fuzzy graph constraint is defined as the possibilistic constraint
In relation to GCL, PCL (primary constraint language) and SCL (standard constraint language) are subsets of GCL which are generated, respectively, by primary and standard constraints (Fig. 14). Note. GCL is more than a language — it is a language system. A language has descriptive capability. A lan- guage system has dedutive capability in additiont to descriptive capability. GCL has both capabilities. The concept of a generalized constraint plays a pivotal role in fuzzy logic. In particular, it serves to prec- isiate the concepts of information and meaning. More specifically, the fundamental thesis of fuzzy logic is that information may be represented as a generalized constraint.
In relation to GCL, PCL (primary constraint language) and SCL (standard constraint language) are subsets of GCL which are generated, respectively, by primary and standard constraints (Fig. 14). Note. GCL is more than a language — it is a language system. A language has descriptive capability. A lan- guage system has dedutive capability in additiont to descriptive capability. GCL has both capabilities. The concept of a generalized constraint plays a pivotal role in fuzzy logic. In particular, it serves to prec- isiate the concepts of information and meaning. More specifically, the fundamental thesis of fuzzy logic is that information may be represented as a generalized constraint.
Fig. 15. Structure of the extension principle.
Fig. 15. Structure of the extension principle.
Fig. 16. Fuzzy graph extension principle. B = “f(A). In this instance, the extension principle may be expressed as follows:
Fig. 16. Fuzzy graph extension principle. B = “f(A). In this instance, the extension principle may be expressed as follows:
In these rules, NS and PS denote Negative Small and Positive Small, respectively; NM and PM denote Neg- ative Medium and Positive Medium, respectively, NL and PL denote Negative Large and Positive Large, respectively; ZR denotes zero; and y is the stabilizing input. Such rules can be formulated without any knowl- edge of dynamics or fuzzy logic. It is this remarkable ability of fuzzy if-then rules modeling language to bypass the need for bivalent-logic-based models that explains why fuzzy logic is employed so widely in applications ranging from industrial control to consumer products.
In these rules, NS and PS denote Negative Small and Positive Small, respectively; NM and PM denote Neg- ative Medium and Positive Medium, respectively, NL and PL denote Negative Large and Positive Large, respectively; ZR denotes zero; and y is the stabilizing input. Such rules can be formulated without any knowl- edge of dynamics or fuzzy logic. It is this remarkable ability of fuzzy if-then rules modeling language to bypass the need for bivalent-logic-based models that explains why fuzzy logic is employed so widely in applications ranging from industrial control to consumer products.
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