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International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol 7, No (1999) 347-361 © World Scientific Publishing Company Int J Unc Fuzz Knowl Based Syst 1999.07:347-361 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 02/03/15 For personal use only H E D G E A L G E B R A S , LINGUISTIC-VALUED LOGIC A N D THEIR A P P L I C A T I O N TO FUZZY R E A S O N I N G NGUYEN CAT HO TRAN DINH KHANG The National Institute of Information Technology, Center for Natural Sciences and Technology of P.O.Box 626, Bo Ho, 10000 Hanoi, Vietnam E-mail: { ncho, tdkhang} @ioit nest ac Vietnam, HUYNH VAN NAM Department of Mathematics, Quinhon University of Pedagogy, 170-Nguyen Hue, Quinhon, Binhdinh, Vietnam NGUYEN HAI CHAU Mathematics-Mechanics-Informatics Faculty, College of Natural Sciences, Hanoi National University Received 15 January 1999 Revised 21 June 1999 People use natural languages to think, to reason, to deduce conclusions, and to make decisions Fuzzy set theory introduced by L A Zadeh has been intensively developed and founded a computational foundation for modeling human reasoning processes The contribution of this theory both in the theoretical and the applied aspects is well recognized However, the traditional fuzzy set theory cannot handle linguistic terms directly In our approach, we have constructed algebraic structures to model linguistic domains, and developed a method of linguistic reasoning, which directly manipulates linguistic terms In particular, our approach can be applied to fuzzy control problems In many applications of expert systems or fuzzy control, there exist numerous fuzzy reasoning methods Intuitively, the effectiveness of each method depends on how well this method satisfies the following criterion: the similarity degree between the conclusion (the output) of the method and the consequence of an if-then sentence (in the given fuzzy model) should be the "same" as that between the input of the method and the antecedent of this if-then sentence To formalize this idea, we introduce a "measure function" to measure the similarity between linguistic terms in a domain of any linguistic variable and to build approximate reasoning methods The resulting comparison between our method and some other methods shows that our method is simpler and more effective Keywords: Linguistic-valued fuzzy logic, linguistic variable, fuzzy reasoning, hedge algebra 63 347 348 H C Nguyen et al Int J Unc Fuzz Knowl Based Syst 1999.07:347-361 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 02/03/15 For personal use only Computational approach to human reasoning Fuzzy set theory was introduced in 1965 as a method of modeling human reasoning From the mathematical viewpoint, the main idea of this approach is to embed the finite linguistic domain of linguistic variables into the set of all functions T(U, [0,1]) defined on a universe U Based on the rich computational structure of T(U, [0,1]), ones can create several methods for reasoning This embedding has led to numerous successes, but it also has a problem: there are only finitely many linguistic terms and infinitely many functions, so, although we can represent each term by a function, most of the functions not have direct linguistic meaning ' 10 In our paper, we will try to recover an algebraic structure of linguistic domains or, algebraically, to embed these domains into natural algebraic structures, which only contain linguistic terms (and no other elements) Then, we will introduce a linguistic reasoning method for directly handling linguistic terms By equipping the resulting algebra with a metric, we can analyze different methods of fuzzy multiple conditional reasoning This paper is an overview of our research results; for more details, we refer the reader to ' ' ' ' 1 Hedge algebras as algebraic models of linguistic domains Mathematical structures on a given set of truth values play an important role in studying the corresponding logics Let us therefore find an appropriate mathematical structure of a linguistic domain of a linguistic variable As an example, let us consider the linguistic variable TRUTH with the domain dom(TRUTH) = {True, False, Very True, Very False, More-or-less True, More-orless False, Possibly True, Possibly False, Approximately True, Approximately False, Very Possibly True, Very Possibly False, } This domain is a partially ordered set (poset), with a natural ordering a < b meaning that b describes a larger degree of truth From the algebraic viewpoint, this set is generated from the basic elements (generators) C = {True, False} by using hedges, i.e., unary operations from a set H = {Very, Possibly, Approximately, More-or-less, } So, this domain can be described as an abstract algebra X_ = (X, C,H, x for all x We say that two operations h,k £ H are converse if Vx £ X (hx < x iff kx > x), and compatible if V# £ X (hx < x iff kx < x) These relations divide the set H into two subsets H+ and H~ so that every operation in H+ is converse w.r.t any operation from H~ and vice-versa, and operations belonging to the same subsets are compatible with each other 64 Int J Unc Fuzz Knowl Based Syst 1999.07:347-361 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 02/03/15 For personal use only Hedge Algebras, Linguistic- Valued Logic and Their Application to 349 If h modifies the linguistic terms stronger than x, i.e., if \/x G X (hx < kx < x or hx > kx > x), then we denote it by h < k It is reasonable to assume that the sets (H+, P, is recursively defined by the following equations: a) sign(c~~) = —1? sign{c+) = -f-1, b) sign(h'hx) = —sign{hx) c) sign{h'hx) = sign(hx) if h! is negative w.r.t h, and if h! is positive w.r.t h Proposition Every measure function v has the following properties: 1) < v(%) < for any x £ PC 2) for any x, y G X, x < y implies that v(x)