Interval Valued Fuzzy Sets from Continuous Archimedean. Triangular Norms. Taner Bilgic and I. Burhan Turksen. University of Toronto.

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1 Interval Valued Fuzzy Sets from Continuous Archimedean Triangular Norms Taner Bilgic and I. Burhan Turksen Department of Industrial Engineering University of Toronto Toronto, Ontario, M5S 1A4 Canada 1 Introduction Interval valued fuzzy sets are suggested in (Turksen 1986) to model the situations where linguistic connectives as well as the variables are fuzzy. They are dened using the discrepancy of conjunctive and disjunctive Boolean Normal Forms in the fuzzy case. The discrepancy is due to relaxing some of the axioms of classical logic. In Section 2 we briey investigate the basic operations in the unit interval. Specically the literature on representing the negation functions and triangular norms is recalled. Archimedean triangular norms are investigated as possible candidates for the logical connective AND. De Morgan triples are constructed utilizing a general result for negations. Two broad families of De Morgan triples are identied; strict and strong, which are neither distributive nor idempotent. Section 3 introduces the concept of an interval valued fuzzy set and presents the main results of the paper, namely for strict and strong De Morgan triples interval valued fuzzy sets are well de- ned. The paper is technical in nature and extends some results obtained in (Turksen 1986) to more general settings using generator functions. Appeared in the Proceedings of FUZZ-IEEE '94, Orlando, pp. 1142{ Basic operations on the unit interval This section summarizes some basic operations on the unit interval. A continuous, strictly increasing function ' : [0; 1] 2! [0; 1] satisfying boundary conditions '(0) = 0 and '(1) = 1 is called an automorphism of the unit interval. Note that the inverse, '?1, is also increasing. A continuous, strictly decreasing function n : [0; 1]! [0; 1] satisfying boundary conditions n(0) = 1 and n(1) = 0 is called a strict negation. A strict negation which satises n(n(x)) = x for every x 2 [0; 1] is called a strong negation and is denoted by N. A standard example of a strong negation is given by N(x) = 1? x called the pseudo-complement. As for the representation of a negation function we have the following theorem (Trillas 1979). Theorem 2.1 n is a negation function if and only if there exists a continuous strictly increasing function : [0; 1]! Re such that (0) = 0, (1) < +1 and n(x) =?1 ((1)? (x)): The representation of the above theorem can also be stated in terms of a continuous strictly decreasing function for strong negation functions as in the following 1

2 Theorem 2.2 n is a negation function if and only if there exists a continuous strictly decreasing function g : [0; 1]! Re such that g(1) = 0, g(0) < +1 and n(x) = g?1 (g(0)? g(x)): A similar result gives a representation in terms of automorphisms (Ovchinnikov and Roubens 1991). Theorem 2.3 Any strong negation N can be represented by an automorphism ' of [0; 1] as N(x) = '?1 (1? '(x)) Triangular norms (t{norms) are developed as tools to use in probabilistic metric spaces (cf. Schweizer and Sklar (1983)). Weber (1983) proposed to use them as connectives in fuzzy set theory. Although, in general, t{norms are not necessarily in [0; 1] all continuous t{norms are in [0; 1]. In this study only continuous t{norms are considered. A continuous t{norm is dened as a symmetric, associative, nondecreasing and continuous function, T : [0; 1] 2! [0; 1], satisfying boundary condition T (1; x) = x for all x 2 [0; 1]: Denition 2.1 A t{norm T (a) is Archimedean if T (x; x) < x for all x 2 (0; 1), (b) has zero divisors if T (x; y) = 0 for some positive x and y, (c) is strict if it is continuous on [0; 1] 2 and strictly increasing in each place on (0; 1] 2. A typical example of a continuous t{norm with zero divisors is the Lukasiewicz t{norm or the bold intersection: TB(x; y) = maxfx+y?1; 0g A typical continuous strict t{norm is the algebraic product: TA(x; y) = xy A symmetric, associative and nondecreasing function S : [0; 1] 2! [0; 1] is called a t{conorm if it satises the boundary condition S(0; x) = x for every x 2 [0; 1]. A t{conorm can be obtained from a t{norm by: S(x; y) = n?1 (T (n(x); n(y))): (1) Note that since t{norms are associative, T (x; y; z) = T (T (x; y); z) = T (x; T (y; z)) is well dened. 2.1 Generators of Continuous Archimedean t{norms In this section we briey present additive generators of continuous Archimedean t{norms. For more details see Schweizer and Sklar (1983). The following is a representation theorem of Ling (1965). (See Schweizer and Sklar (1983) for historical comments on this representation.) Theorem 2.4 A t{norm T is continuous and Archimedean if and only if there exists a continuous and strictly decreasing function g : [0; 1]! Re + with g(1) = 0 and T (; ) = g?1 (minfg() + g(); g(0)g): (2) where g [?1] () = g?1 (minf; g(0)g) for 2 Re + and is called the quasi{inverse of g. A t{norm, T, which satisfy the hypotheses of Theorem 2.4 is said to be additively generated by g and g is called the additive generator of T. In terms of Theorem 2.4 a t{norm is strict if and only if g(0) = +1 and has zero divisors if and only if g(0) < +1. Since all continuous t{norms are in [0; 1] it is interesting to nd representations for continuous Archimedean t{norms in terms of automorphisms of the unit interval. The following result is proved in Schweizer and Sklar (1983) Theorem 2.5 Any continuous, strict t{norm T can be represented as a '{transform of algebraic product, TA as T (x; y) = '?1 ('(x)'(y)) (3) where ' is an automorphism of the unit interval. This result shows that any continuous, strict t{norm is isomorphic to the algebraic product. A similar result is valid for t{norms with zero divisors, (Ovchinnikov and Roubens 1991), stating that any continuous t{norm with zero divisors is isomorphic to the Lukasiewicz t{norm. 2

3 2.2 De Morgan Triples Denition 2.2 If T is a continuous t{norm, n is a strict negation and (1) holds, then the triple ht; S; ni is called a De Morgan triple. Denition 2.3 If ht; S; N i is a De Morgan triple such that T has zero divisors, N is a strong negation, then ht; S; N i is called a strong or Lukasiewicz like De Morgan triple. In this case using Theorems 2.2 and 2.4, T (; ) = g?1 (minfg() + g(); g(0)g) (4) S(; ) = g?1 (maxfg() + g()? g(0); 0g) (5) N() = g?1 (g(0)? g()) (6) Denition 2.4 If ht; S; N i is a De Morgan triple such that T is strict, N is a strong negation and both are generated by the same automorphism ', then ht; S; Ni is called a strict De Morgan triple. In this case, T (x; y) = '?1 ('(x)'(y)) (7) S(x; y) = '?1 ('(x) + '(y)? '(x)'(y))(8) N(x) = '?1 (1? '(x)) (9) 3 Interval{Valued Fuzzy Sets from Boolean Normal Forms In this section, interval{valued fuzzy sets as de- ned by Turksen (1986) are introduced. Basic denitions are given in Section 3.1 and then it is shown that interval{valued fuzzy sets are well de- ned for De Morgan triples built from continuous Archimedean t{norms in Section Basic Denitions The Boolean Disjunctive and Conjunctive normal forms (DNF and CNF, respectively) are equivalent in classical logic. In many{valued logics however, DNF may not be equal to CNF in general. Establishing that DNF representations of the concepts do not coincide to their CNF representations in many{ valued logic, and for certain t{norm families DNF is included in the corresponding CNF, Turksen (1986) proposed to dene the interval{valued fuzzy set (IVFS) as follows: IV F S() = [DN F (); CN F ()] Assume that a De Morgan triple ht; S; Ni is used to model conjunction, disjunction and complement respectively, Table 2 shows the canonical Boolean normal forms in the membership domain for the concepts given in Table 1. The lowercase letters denote membership function values in the unit interval (i.e., x = X()). 3.2 Interval{Valued Fuzzy Sets from Continuous Archimedean t{norms It should be observed that for a given De Morgan triple, the sixteen t{normed representations given in Table 2 can be partitioned as f1; 2g; f3; : : :; 10g; f11; 12g; f13; : : :; 16g, i.e., (1) and (2) are equivalent, (3){(10) are equivalent, (11) and (12) are equivalent and (13){(16) of Table 2 are equivalent to each other in form. This fact is rst realized in (Piaget 1949) for two{valued logic. Dubois and Prade (1980) discuss it for fuzzy sets without going into normal forms. Turksen (1984) establishes the equivalence of (1) and (2), (3){(6) and (13){(14) as examples of the equivalence in terms of a group structure. Therefore in order to show DN F () CN F () for all sixteen combined concepts with a particular De Morgan triple, it is sucient to show that the following are satised: S[T (x; y); T (x; N(y)); T (N(x); y)] S[T (x; y); T (N(x); N(y))] S(x; y) (10) T [S(x; N(y)); S(N(x); y)] (11) S[T (x; y); T (x; N(y))] T [S(x; y); S(x; N(y))] (12) 3

4 Table 1: List of Combined Concepts for X and Y Number Concept Combination 1 Complete Armation True 2 Complete Negation False 3 Disjunction X OR Y 4 Conjunctive Negation NOT X AND NOT Y 5 Incompatibility NOT X OR NOT Y 6 Conjunction X AND Y 7 Implication (IF.. THEN) NOT X OR Y 8 Non-Implication X AND NOT Y 9 Inverse Implication X OR NOT Y 10 Non-inverse Implication NOT X AND Y 11 Equivalence X IFF Y 12 Exclusion X XOR Y 13 Armation X 14 Negation NOT X 15 Armation Y 16 Negation NOT Y Note that for rows (1) and (2) of Table 2, DN F () CN F () is always satised for all t{norms and it is sucient to show that (Turksen 1986) S(T (x; y); T (x; N(y))) x (13) holds for all x 2 [0; 1] in order (10) to be satised. The main result of this paper is that, for strong and strict De Morgan triples, interval valued fuzzy sets are well dened. Theorem 3 of Turksen (1986) establishes the conditions under which the premise should hold for dierent families of connectives. Here, we establish the result for strict and strong De Morgan triples using their generating functions. Theorem 3.1 If ht; S; Ni is a strong De Morgan triple then DNF() CNF() for the sixteen combined concepts. Proof: The proof is given in terms of the generating functions. Recall that the strong De Morgan triple, ht; S; Ni is given by (4),(5) and (6) respectively. We will show that (11), (12) and (13) are satised for a strong De Morgan triple, ht; S; Ni. First we write (11), (12) and (13) in terms of (4),(5) and (6). S(T (x; y); T (x; N(y))) = g?1 (maxfminfg(x) + g(y); g(0)g + minfg(x)? g(y); 0g; 0g) (14) S(T (x; y); T (N(x); N(y))) = g?1 (maxfminfg(x) + g(y); g(0)g + minfg(0)? g(x)? g(y); 0g; 0g) (15) T (S(x; N(y)); S(N(x); y)) = g?1 (minfmaxfg(x)? g(y); 0g + maxfg(y)? g(x); 0g; g(0)g) (16) T (S(x; y); S(x; N(y))) = g?1 (minfmaxfg(x) + g(y)? g(0); 0g + maxfg(x)? g(y); 0g; g(0)g) (17) Now our aim is translated to showing (14) x, (14) (17), and (15) (16). There are 4 cases to consider: 1. g(0) g(x) + g(y) and g(x)? g(y) > 0. 4

5 Table 2: T -normed representation of Boolean Normal Forms N o: DNF CNF 1 S[T (x; y); T (x; N (y)); T (N (x); y); T (N (x); N (y))] T [S(x; y); S(x; N (y)); S(N (x); y); S(N (x); N (y))] 3 S[T (x; y); T (x; N (y)); T (N (x); y)] S(x; y) 4 T (N (x); N (y))] T [S(x; N (y)); S(N (x); y); S(N (x); N (y))] 5 S[T (x; N (y)); T (N (x); y); T (N (x); N (y))] S(N (x); N (y))] 6 T (x; y) T [S(x; y); S(x; N (y)); S(N (x); y)] 7 S[T (x; y); T (N (x); y); T (N (x); N (y))] S(N (x); y) 8 T (x; N (y)) T [S(x; y); S(x; N (y)); S(N (x); N (y))] 9 S[T (x; y); T (x; N (y)); T (N (x); N (y))] S(x; N (y)) 10 T (N (x); y) T [S(x; y); S(N (x); y); S(N (x); N (y))] 11 S[T (x; y); T (N (x); N (y))] T [S(x; N (y)); S(N (x); y)] 12 S[T (x; N (y)); T (N (x); y)] T [S(x; y); S(N (x); N (y))] 13 S[T (x; y); T (x; N (y))] T [S(x; y); S(x; N (y))] 14 S[T (N (x); y); T (N (x); N (y))] T [S(N (x); y); S(N (x); N (y))] 15 S[T (x; y); T (N (x); y)] T [S(x; y); S(N (x); y)] 16 S[T (x; N (y)); T (N (x); N (y))] T [S(x; N (y)); S(N (x); N (y))] 2. 2g(0) g(x) + g(y) > g(0) and g(y)? g(x) g(0) 3. 2g(0) g(x) + g(y) > g(0) and g(y)? g(x) < 0 4. g(0) g(x) + g(y) and g(x)? g(y) 0 The details are straightforward. 2 An immediate consequence is the following Corollary 3.1 If '(z) = z for all z 2 [0; 1] then DNF() CNF() for Lukasiewicz triples. The same result can be shown for strict De Morgan triples. Theorem 3.2 If ht; S; Ni is a strict De Morgan triple then DNF() CNF() for the sixteen combined concepts. Proof: Recall that a strict De Morgan triple, ht; S; Ni is given by (7),(8) and (9) respectively. We will show that (11), (12) and (13) are satised for a strict De Morgan triple, ht; S; Ni. First we write (11), (12) and (13) in terms of (7),(8) and (9). S(T (x; y); T (x; N(y))) = '?1 ('(x)'(y) + '(x) '(y)? '(x) 2 '(y) '(y)) (18) S(T (x; y); T (N(x); N(y))) = '?1 ('(x)'(y) + '(x) '(y)? '(x)'(y) '(x) '(y)) (19) T (S(x; N(y)); S(N(x); y)) = '?1 (('(x) + '(y)? '(x) '(y)) ( '(x)'(y)? '(x)'(y))) (20) T (S(x; y); S(x; N(y))) = '?1 (('(x) + '(y)? '(x)'(y)) ('(x) + '(y)? '(x) '(y))) (21) Note that '(x) = 1? '(x) is used to simplify the notation. Now our aim is translated to showing (18) x, (18) (21), and (19) (20). The rest of the proof is straightforward. 2 Note that one does not have to resort to dierentiability in order to prove Theorem 3.2. An immediate consequence is the following Corollary 3.2 If '(z) = z for all z 2 [0; 1] then 5

6 DNF() CNF() for Algebraic triples. References Dubois, D. and Prade, H. (1980). Fuzzy sets and systems : theory and applications, Academic Press, New York. Ling, C. H. (1965). Representation of associative functions, Publicationes Mathematicae Debrecen 12: 189{212. Ovchinnikov, S. and Roubens, M. (1991). On strict preference relations, Fuzzy Sets and Systems 43: 319{326. Piaget, J. (1949). Traite de logique: essai de logistique operatoire, A. Colin, Paris. Schweizer, B. and Sklar, A. (1983). Probabilistic Metric Spaces, North-Holland, Amsterdam. Trillas, E. (1979). Sobre functiones de negation en la teoria de conjunctos diusos, Stochastica 3: 47{59. Turksen, I. B. (1984). Klein groups in fuzzy inference, Proceedings of the 1984 American Control Conference, American Automatic Control Council, pp. 556{560. Turksen, I. B. (1986). Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20: 191{210. Weber, S. (1983). A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms, Fuzzy Sets and Systems 11: 115{134. 6

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