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A parametric representation of linguistic hedges in Zadeh's fuzzy logic

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A parametric representation of linguistic hedges in Zadeh's fuzzy logic
  A Parametric Representation of LinguisticHedges in Zadeh’s Fuzzy Logic V.N. Huynh a , b , ∗ , T.B. Ho a , Y. Nakamori a a School of Knowledge Science, Japan Advanced Institute of Science and Technology, Tatsunokuchi, Ishikawa, 923-1292, Japan  b Department of Computer Science, Quinhon University, Vietnam  Abstract This paper proposes a model for the parametric representation of linguistic hedgesin Zadeh’s fuzzy logic. In this model each linguistic truth value, which is generatedfrom a primary term of the linguistic truth variable, is identified by a real number r  depending on the primary term. It is shown that the model yields a method of efficiently computing linguistic truth expressions accompanied with a rich algebraicstructure of the linguistic truth domain, namely De Morgan algebra. Also, a fuzzylogic based on the parametric representation of linguistic truth values is introduced. Key words:  Linguistic hedges, linguistic variable, distributive lattice, De Morganalgebra, fuzzy logic, approximate reasoning 1 Introduction In 1970s, L. Zadeh introduced and developed the theory of approximate rea-soning based on the notions of linguistic variable and fuzzy logic [19–23].Informally, by a linguistic variable we mean a variable whose values are wordsin a natural or artificial language. For example,  Age  is a linguistic variablewhose values are linguistic such as  young ,  old ,  very young  ,  very old  ,  quite young  ,  more or less young  ,  not very young and not very old  , etc. As is well-known, the values of a linguistic variable are generated from primary terms(e.g.,  young  and  old  in the case of linguistic variable  Age ) by various linguistichedges (e.g.,  very ,  more or less  , etc.) and connectives (e.g.,  and, or, not  ). ∗ Corresponding author. Email address:  (V.N. Huynh). International Journal of Approximate Reasoning   30  (2002) 202–223.  In Zadeh’s view of fuzzy logic, the truth-values are linguistic, e.g., of theform “ true  ”, “ very true  ”, “ more or less true  ”, “  false  ”, “ possible false  ”, etc.,which are expressible as values of the linguistic variable  Truth  , and the rulesof inference are approximate rather than exact. In this sense, approximatereasoning (also called fuzzy reasoning) is, for the most part, qualitative ratherthan quantitative in nature, and almost all of it falls outside of the domainof applicability of classical logic (see Zadeh [2,22,23]). The primary aim of the theory of approximate reasoning is to mimic human linguistic reasoningparticularly in describing the behaviour of human-centered systems.Throughout this paper, by a fuzzy logic we mean a fuzzy logic in the senseof Zadeh, that is, its truth-values are linguistic values of the linguistic truthvariable, which are represented by fuzzy sets in the interval [0 , 1] . According to Zadeh’s rule for truth qualification [23], a proposition such as“Lucia is very young” is considered as being  semantically equivalent   with theproposition “Lucia is young is very true”. This semantic equivalence rela-tion plays an important role in approximate reasoning. In fuzzy set basedapproaches to fuzzy reasoning [7,22,23,2], the primary linguistic truth-valuessuch as  true  and  false  are correspondingly assigned to fuzzy sets defined overthe interval [0 , 1] ,  which are designed to interpret the meaning of these primaryterms. The composite linguistic truth-values are then computed by using thefollowing procedure: •  Linguistic hedges 1 , for example  very  and  more or less  , are defined as unaryoperators on fuzzy sets, for example CON, DIL, respectively; •  The logic connectives such as  and ,  or ,  not  and  if ...then  are defined gen-erally as operators such as  t - norm ,  t - conorm ,  negation , and  implication respectively.As is well known, one of inherent problems in a model of fuzzy reasoningis that of linguistic approximation, i.e., how to name by a linguistic term aresulted fuzzy set of the deduction process. This depends on the shape of theresulted fuzzy set in relation with the primary fuzzy sets and the operators.Based on two characteristics of linguistic variables introduced by Zadeh (namely,the context-independent meaning of linguistic hedges and connectives, andthe universality of linguistic domains), and the meaning of linguistic hedges innatural language, Nguyen and Wechler [15,16] proposed an algebraic approachto the structure of linguistic domains (term-sets) of linguistic variables. It isshown in [12–14] that the obtained structute is rich enough for the investi-gation of some kinds of fuzzy logic. Furthermore, the approach also providesa possibility for introducing methods of linguistic reasoning that allow us tohandle linguistic terms directly, and hence, to avoid the problem of linguistic 1 also called linguistic modifiers [6]. 2  Fig. 1. Membership functions of truth-values by Lascio’s model approximation [10,11].It is of interest that in [6], Lascio et al. have proposed a model for repre-sentation of linguistic terms satisfying the hypotheses imposed on linguistichedges introduced by Nguyen and Wechler [15]. In their model, each linguisticterm of a linguistic variable is characterised by three parameters and can beidentified by only a positive real number. It is shown that the set of liguisticterms of the linguistic truth variable in Lascio’s model exhibited interestingsemantic properties justified by intuitive meaning of linguistic hedges, whichwere axiomatically formulated in the terms of hedge algebras [15]. However,going back to the membership function representation, Lascio’s model doesnot give a good interpretation at the intuitive level on logical basis behind theshape of membership functions of linguistic truth values (see Fig. 1).It is important to note that in the conventional approach to fuzzy reasoning,fuzzy logic, which a method of fuzzy reasoning bases on, can be viewed as afuzzy extension of a underlying multi-valued logic (i.e.,  base logic  ), in whichthe truth-values are fuzzy sets of the truth-value set of the base logic (see,e.g., [2,22,23]). Although membership functions of primary terms such as  true or  false  are defined subjectively, it will be natural to hope that a fuzzy logicshould meet the base logic at the limited cases. For example, for membershipfunction of the  unitary  truth-value  u - true  [23], that is  µ u - true ( v ) =  v  for v  ∈  [0 , 1] ,  and the linguistic hedge  very  defined by CON operation, we have very n true  tends to  Absolutely true   as  n  tends to infinity, where  Absolutely true  is identified with 1 as a nonfuzzy truth-value, see Fig. 2. Unfortunately, thisis not the case for Lascio’s model, again see Fig. 1.In this paper, we introduce a new representation model for linguistic termsof the linguistic truth variable in fuzzy logic. In this model, each linguistictruth value generated from a primary term of the linguistic truth variable isidentified by a real number  r  depending on the primary term. It will be shownthat the model not only satisfies the interesting semantic properties justifiedby intuitively meaning of linguistic hedges as Lascio’s model, but also meets3  Fig. 2. Membership functions of unitary truth-values [4,23] in the special cases the well-known models in the literature.The paper is organised as follows. In Section 2, we briefly present some pre-liminary notions on linguistic variables, the fuzzy set based interpretation of linguistic hedges, as well as the related work in the literature. A new rep-resentation model for linguistic terms of the linguistic truth variable will beintroduced in Section 3. The model allows to represent two ordered sets of linguistic terms generated from two primary terms  true  and  false ; each lin-guistic truth value is associated with a real number depending on the primaryterm from which it is generated. Section 4 introduces a fuzzy logic based onthis model in comparison with the models already known in literature. Finally,some concluding remarks will be given in Section 5. 2 Preliminaries 2.1 Linguistic variables  In this subsection, we briefly recall the notion of linguistic variables and thefuzzy set theoretic interpretation of linguistic hedges introduced by Zadeh in1970s. More details can be referred to [5,19–21,23].Formally, a  linguistic variable   is characterised by a quintuple ( X  ,T  ( X  ) ,U,R,M  ) , where: X   is the name of the variable such as age variable  Age , truth variable  Truth , etc.; T  ( X  ) denotes the  term-set  of   X  ,  that is, the set of linguistic values of thelinguistic variable  X  ; U   is a universe of discourse of the base variable;4
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