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Chapter 1 Fuzzy Sets and Fuzzy Logic

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Chapter 1 Fuzzy Sets and Fuzzy Logic CHAPTER 1 FUZZY SETS AND FUZZY LOGIC Fuzzy Sets and Fuzzy Logic 1,1 Introduction Even though fuzzy sets were introduced in their modern form by Zadeh [18] in 1965, the idea of a multi-valued logic in order to deal with vagueness has been around from the beginning of the century. Peirce was one of the first thinkers to consider vagueness seriously, he did not believe in the separation between true and false and believed everything in life is a continuum. In 1905 he stated : "I have worked out the logic of vagueness with something like completeness" [15]. Other famous scientists and philosophers probed this topic further. Russell claimed, "All language is vague" and went further saying, "vagueness is a matter of degree" (e.g., a blurred photo is vaguer than a crisp one, etc.) [16]. Einstein said that "as far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality" [2]. Lukasiewicz took the first step towards a formal model of vagueness, introducing in 1920 a three-valued logic based on true, false, and possible [9]. In doing this he realized that the laws of the classical two- valued logic might be misleading because they address only a fragment of the domain. A year later Post outlined his own three-valued logic, and soon after many other multi-valued logics proliferated (Godel, Von Neumann, Kleene, etc.). A few years later, in 1937 Black outlined his precursor of fuzzy sets [2]. He agreed with Peirce in terms of the continuum of vagueness and with Russell in terms of the degrees of vagueness. Therefore, he outlined a logic Chapter 1 2 based on degrees of usage, based on the probability that a certain object will be considered belonging to a certain class [2]. Finally, in 1965 Zadeh [18] elaborated a multi-valued logic where degree of truth (rather than usage) are possible. Fuzzy set theory generalizes classical set theory in that the membership degree of an object to a set is not restricted to the integers 0 and 1, but may take on any value in [0, 1]. By elaborating on the notion of fuzzy sets and fuzzy relations we can define fuzzy logic systems (FLS). FLSs are rule- based systems in which an input is first fuzzified (i.e., converted from a crisp number to a fuzzy set) and subsequently processed by an inference engine that retrieves knowledge in the form of fuzzy rules contained in a rule-base. The fuzzy sets computed by the fuzzy inference as the output of each rule are then composed and defuzzified (i.e., converted from a fuzzy set to a crisp number). A fuzzy logic system is a non-linear mapping from the input to the output space. [6, 7, 4, 13, 14]. This Chapter serves to provide the necessary background to understand the developments of the next chapters. The basic notion of fuzzy set will be introduced and discussed in the introductory section on fuzzy sets. Most of this material was edited from the brilliant introduction provided in Bezdek [1]. Fuzzy logic and fuzzy relations will be discussed as in Mendel [12, 13], finally leading to fuzzy logic and fuzzy logic systems and their principles of operation. As the reader will notice there are many possible choices in the design of a FLS, we will discuss the most common choices and present the formulation of the corresponding nonlinear mapping implemented by a FLS. Chapter 1 3 1.2 Fuzzy Set Theory Fuzzy sets were introduced by Zadeh [18] in 1965 to represent / manip­ ulate data and information possessing nonstatistical uncertainties. It was specifically designed to mathematically represent imcertainty and vagueness and to provide formalized tools for dealing with the imprecision intrinsic to many problems. Fuzzy sets serve as a means of representing and manipulating data that was not precise, but rather fuzzy. There is a strong relationship between Boolean logic and the concept of a subset, there is a similar strong relationship between fuzzy logic and fuzzy subset theory. In classical set theory, a subset A of a set X can be defined by its characteristic function XA as a mapping from the elements of X to the elements of the set {0,1}, XA:X-^{0,1}. This mapping may be represented as a set of ordered pairs, with exactly one ordered pair present for each element of X. The first element of the ordered pair is an element of the set X, and the second element is an element of the set {0,1}. The value zero is used to represent non-membership, and the value one is used to represent membership. The truth or falsity of the statement "x is in A" is determined by the ordered pair {X,XA{X)). The statement is true if the second element of the ordered pair is 1, and the statement is false if it is 0. Similarly, a fuzzy subset A of a set X can be defined as a set of ordered pairs, each with the first element from X, and the second element from the Cliaptcr I -i interval [0, 1], with exactly one ordered pair pr(\seiit for (^a.eh element of X. This defines a mapping /M : X ^ [0, 1] between elements of the set X and values in the interval [0, 1]. The value zero is used to represent complete non-membership, the value one is used to represent complete membership and values in between are used to represent intermediate degrees of membership. The set X is referred to as the universe of discourse for the hizzy subset A. Fi-eciuently, tlie mapping HA is described as a hniction, t,he meml)ersliip function of A. The degree to which tlie statement, ".r is in A" is true and is determined by finding the oj'dered pair {x, //,4(.;;)). The degree of truth of the statement is the second eknnent of the ordered pair [5]. Definition 1.1 [18] : Let X be a non-empty set. A fuzzy set A in X is characterized by its membership functtion /i.4 : X -^ [0,1] and fiA{x) is interpreted as the degree of membership of element x in fuzzy set A for each x ^ X. It is clear that A is completely determined by the set of tuples A = {ix,fiA{x)\xeX}. Definition 1.2 : Support. The support of a fuzzy set A is the set of all points with nonzero membership degree in A : support(.4) = {.X e X\fiA{x) > 0}. Clmptcr 1 5 Definition 1.3 : Core. The core of a fuzzy scf A is the set of iill points with unit nienib(;rship degree in A : core(^) = {x e X\IJ,A{X) - 1}. Definition 1.4 : Normality. A fuzzy set A is normal if its core is nonempty. In other words, we can always find at least a point x E X such that /i.4(.r) = 1. Definition 1.5 : Crossover points. A crossover point of a fuzzy set A is a point X G X at which I.IA{X) =0.5. Definition 1.6 : Fuzzy singleton. A fuzzy set whose support is a single point in X with I.IA{X) = 1 is called a fuzzy singleton. Definition 1.7 : a-cut, strong a-cut. The o-cut or a-level set of a fuzzy set A is a crisp set defined by /IQ = (x e A'|/f,4(:j;) > a). Strong a-cut or strong o-level set are defined similarly : < = {x e X|/x,4(3;) > ex). Using this notation, we can express the support and core of a fuzzy set A as support(v4) = A'Q and core(A) = A^. Definition 1.8 : Convexity. A fuzzy set A is convex if and only if V Xi,X2 G X and V A G [0,1], //^[Axi + (1 - A).X2] > min{/iA(xi),/./A(2;2)}. Alternatively, A is convex if all its a-level sets are convex. Note that the definition of convexity of a fuzzy set is not as strict as the common d(!finition Chapter 1 fi of convexity of a function. Indeed, it corresponds to the definition of ([uasi- concavity of a function. Definition 1.9 : Fuzzy numbers [5]. A fuzzy number A is a fuzzy set of tfie real line with a normal, (fuzzy) convex and continuous membership function of bounded support. The family of fuzzy numbers will be denoted by !F. Fuzzy subsets of the real line are called fuzzy quantities. To distinguish a fuzzy number from a crisp (non-fuzzy) one, tln^ former will sometimes be denoted with a tilde ~. Definition 1.10 : Quasi fuzzy number. A quasi fuzzy number A is a fuzzy set of the real line with a normal, fuzzy convex and continuous menilx^-ship function satisfying the limit conditions lim iJ.A{t) = 0 hm /'•/!(/) = O.^ t—^00 (,—»—OO In the literature the terms fuzzy number and quasi fuzzy number are often used interchangeably. It is easy to see that the membership function of a fuzzy number A has the following properties: (i) /UVA(^) = 0, outside of some interval [c, rf]; (ii) there are real numbers a and b,c < a < b < d such that /t,4(t) is monotone increasing on the interval [c, a] and monotone d(!creasing on [b,d]; (iii) iJ-A{t) = 1 for each x E [a, b]. Definition 1.11 : Triangular fuzzy number. A fuzzy set A is called triangular fuzzy number with peak (or center) a, left width a > 0 and right Chapter 1 7 width 6 > 0 if its membership fimction lias the following form { 1 - ^ if a - a<t< a - 1 - ^ if a<t<a + ft 0 otherwise and we use the notation A — (a, a, j3).
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