On the Law of Importation. in Fuzzy Logic. J.Balasubramaniam, Member IEEE. Abstract

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1 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 1 On the Law of Importation (x y) z (x (y z)) in Fuzzy Logic J.Balasubramaniam, Member IEEE Abstract The law of importation, given by the equivalence (x y) z (x (y z)), is a tautology in classical logic. In A-implications defined by Turksen et al., the above equivalence is taken as an axiom. In this work we investigate the general form of the law of importation J(T (x, y), z) J(x, J(y, z)), where T is a t-norm and J is a fuzzy implication, for the three main classes of fuzzy implications, viz., R-, S- and QL-implications and also for the recently proposed Yager s classes of fuzzy implications, viz., f- and g-implications. We give necessary and sufficient conditions under which the law of importation holds for R-, S-, f- and g-implications. We also give necessary conditions under which QL-implications have the law of importation and investigate some specific families of QL-implications. Also we have investigated the general form of the law of importation in the more general setting of uninorms and t-operators for the above classes of fuzzy implications. Following this, we propose a novel modified scheme of CRI inferencing called the Hierarchical CRI that alleviates some of the drawbacks of Zadeh s CRI. We give sufficient conditions on the operators employed under which the inference obtained from the Zadeh s CRI and the Hierarchical CRI become identical, highlighting the significant role played by the law of importation. Index Terms Law of Importation, Fuzzy Implications, Compositional Rule of Inference, Fuzzy Inference, Approximate Reasoning, Rule Reduction. The author is with the Department of Mathematics and Computer Sciences, Sri Sathya Sai Institute of Higher Learning, Prasanthi Nilayam, A.P , India. jbala@ieee.org.

2 2 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic I. INTRODUCTION The equation (x y) z (x (y z)), known as the law of importation, is a tautology in classical logic. The general form of the above equivalence is given by J(T (x, y), z) J(x, J(y, z)), (LI) where T is a t-norm and J a fuzzy implication. In the framework of fuzzy logic, the law of importation has not been studied so far in isolation. In A-implications defined by Turksen et al. [46], (LI) with T as the product t-norm T P (x, y) = x y, was taken as one of the axioms. Baczyński [1] has studied the law of importation in conjunction with the general form of the following distributive property of fuzzy implications [x (y z)] [(x y) (x z)] (1) and has given a characterisation. Bouchon-Meunier and Kreinovich [10] have characterised fuzzy implications that have the law of importation as one of the axioms along with (1). They have considered T M (x, y) = min(x, y) for the t-norm T and claim that Mamdani s choice of implication min is not so strange after all. A. Motivation for this work Recently, there have been many attempts to examine classical logic tautologies involving fuzzy implication operators, both from a theoretical perspective and also from possible applicational value. There were a number of correspondences [9], [12], [13], [14], [16], [31], [44] on the equivalence (x y) z (x z) (y z), (1) and related distributive properties in fuzzy logic, in the context of Inference Invariant Complexity Reduction in Fuzzy Rule Based Systems (see [7], [8], [41], [48]). Daniel-Ruiz and Torrens have investigated the distributivity of fuzzy implications over uninorms in [39]. Fodor [18] has investigated the contrapositive symmetry of the three classes of fuzzy implications, viz., R-, S- and QL-implications with respect to a strong negation. Igel and Tamme [26] have dealt with chaining of implication operators in a syllogistic fashion. Thus the investigation of these properties of fuzzy implications has interesting applications in Approximate Reasoning. In this work we consider the general form of the law of importation in the setting of fuzzy logic and explore its potential applications in the field of approximate reasoning. Hence the present work should be seen in this context.

3 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 3 B. Main contents of the paper The three established and well-studied classes of fuzzy implications in the literature are the R-, S- and QL-implications. Recently, Yager [49] has proposed two new classes of fuzzy implications, viz., f- and g-implications, that can be seen to be obtained from the additive generators of continuous Archimedean t-norms and t-conorms, respectively. In this work, we consider the above law of importation for these classes of fuzzy implications, viz., R-, S-, QL-, f- and g-implications. We give necessary and sufficient conditions under which the above equivalence can be established for R-, S-, f- and g-implications. We also give necessary conditions underwhich QL-implications have the law of importation and investigate some specific families of QL-implications. Also we have investigated the general form of law of importation in the more general setting of uninorms and t-operators and show that they get reduced to t-norms. The Compositional Rule of Inference (CRI) proposed by Zadeh [52] is one of the earliest and most important inference schemes in approximate reasoning involving fuzzy propositions. Inferencing in CRI involves fuzzy relations that are multi-dimensional and hence is resource consuming - both memory and time. We propose a novel modified scheme of CRI inferencing called the Hierarchical CRI that alleviates the above drawback and also has many advantages. Subsequently, we give sufficient conditions on the operators employed in the Hierarchical CRI under which the inference obtained from Zadeh s CRI and the Hierarchical CRI become identical. The law of importation plays a significant role in the above equivalence. We illustrate the above concepts through some numerical examples. It is not uncommon to use a t-norm for relating antecedents to consequents in CRI, in which case we have shown that under even less restrictive conditions the equivalence between the inferences in Zadeh s CRI and Hierarchical CRI can be obtained. C. Outline of the paper In section II we review the basic fuzzy logic operators and their different properties. The main theoretical results of this work are contained in Section III - wherein we investigate the general form of law of importation for some well-known classes of fuzzy implications, and in Section IV - where we investigate the general form of law of importation in the more general setting of uninorms and t-operators. In Section V we propose a novel scheme of inferencing called the Hierarchical CRI and highlight its advantages. In Section VI we illustrate the applicational values of the theoretical results in Sections III and IV, by demonstrating the

4 4 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic significant role the law of importation plays in the equivalence of the outputs obtained from Zadeh s CRI and the Hierarchical CRI. Section VII gives a few concluding remarks. II. PRELIMINARIES To make this work self-contained, we briefly mention some of the concepts and results employed in the rest of the work. Definition 1: Let ϕ : [0, 1] [0, 1] be an increasing bijection. If f : [0, 1] [0, 1] is any function, then the ϕ-conjugate of f is given by f ϕ (x) = ϕ 1 (f(ϕ(x))), x [0, 1]. (2) If F : [0, 1] [0, 1] [0, 1] is any binary function, then the ϕ-conjugate of F is F ϕ (x, y) = ϕ 1 (F (ϕ(x), ϕ(y))), x, y [0, 1]. (3) A. Negations Definition 2 ( Fodor & Roubens [19], Definitions ): (i) A function N : [0, 1] [0, 1] is called a fuzzy negation if N(0) = 1, N(1) = 0 and N is non-increasing. (ii) A fuzzy negation N is called strict if, in addition, N is strictly decreasing and N is continuous. (iii) A fuzzy negation N is called strong if it is an involution, i.e., N(N(x)) = x for all x [0, 1]. Theorem 1 (cf. Trillas [43], Fodor & Roubens [19] Theorem 1.1, Pg 4): A function N : [0, 1] [0, 1] is a strong negation if and only if there exists an increasing bijection ϕ : [0, 1] [0, 1], such that N(x) = ϕ 1 (1 ϕ(x)). B. T-norms and T-conorms Definition 3 ( Cf. Schweizer & Sklar [40]; Klement et al., [27] Definition 1.1 Pg 4): A function T from [0, 1] 2 [0, 1] is called a triangular norm (shortly t-norm) if, for all x, y, z [0, 1], T (x, y) = T (y, x), T (x, T (y, z)) = T (T (x, y), z), (T1) (T2) T (x, y) T (x, z) whenever y z, (T3) T (x, 1) = x. (T4)

5 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 5 Definition 4 (Cf. Schweizer & Sklar [40]; Klement et al., [27] Definition 1.13 Pg 11): A function S : [0, 1] 2 [0, 1] is called a triangular conorm (shortly t-conorm) if, for all x, y, z [0, 1], it satisfies S(x, y) = S(y, x), S(x, S(y, z)) = S(S(x, y), z), (S1) (S2) S(x, y) S(x, z) whenever y z, (S3) S(x, 0) = x. If F is an associative binary operation on a domain X then by the notation x (n) F F (x,, x) for an x X and n 2. Also x (1) }{{} F = x. n times (S4) we mean Definition 5 ( Klement et al., [27] Definitions 2.9 Pg 26 & 2.13 Pg 28): A t-norm T (tconorm S resp.) is said to be Continuous if it is continuous in both the arguments; Archimedean if T (S resp.) is such that for every x, y (0, 1] (x, y [0, 1) resp.) there is an n N with x (n) T < y ( x (n) S > y ) ; Strict if T (S resp.) is continuous and stricly monotone, i.e., T (x, y) < T (x, z) ( S(x, y) < S(x, z) ) whenever x > 0 (x < 1 resp.) and y < z; Nilpotent if T (S resp.) is continuous and if each x (0, 1) is such that x (n) T = 0 ( x (n) S = 1 ) for some n N. Theorem 2 ( Klement et al., [27] Theorem 5.1 Pg 122): For a function T : [0, 1] 2 [0, 1] the following are equivalent: (i) T is a continuous Archimedean t-norm. (ii) T has a continuous additive generator, i.e., there exists a continuous, strictly decreasing function f : [0, 1] [0, ] with f(1) = 0, which is uniquely determined up to a positive multiplicative constant, such that for all x, y [0, 1] T (x, y) = f ( 1) (f(x) + f(y)), (4) where f ( 1) is the pseudo-inverse of f and is defined as: f 1 (x), if x [0, f(0)] f ( 1) (x) =. (5) 0, if x [f(0), ] If f(0) = then T is strict and if f(0) < then T is nilpotent. Proposition 1 (Klement et al.,[27], Propositions 5.9 & 5.10 pp ): Let T be a continuous t-norm.

6 6 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic (i) T is strict if and only if it is a conjugate of the product t-norm T P (x, y) = x y, i.e., there exists a strictly increasing bijection ϕ on [0, 1] such that T (x, y) = ϕ 1 (ϕ(x) ϕ(y)), x, y [0, 1]. (6) (ii) T is nilpotent if and only if it is a conjugate of the Łukasiewicz t-norm T L (x, y) = max(x + y 1, 0), i.e., there exists a strictly increasing bijection ϕ on [0, 1] such that T (x, y) = ϕ 1 (max{ϕ(x) + ϕ(y) 1, 0}), x, y [0, 1]. (7) Theorem 3 (Klement et al., [27] Corollary 5.5 Pg 125): For a function S : [0, 1] 2 [0, 1] the following are equivalent: (i) S is a continuous Archimedean t-conorm. (ii) S has a continuous additive generator, i.e., there exists a continuous, strictly increasing function g : [0, 1] [0, ] with g(0) = 0, which is uniquely determined up to a positive multiplicative constant, such that for all x, y [0, 1] we have S(x, y) = g ( 1) (g(x) + g(y)) where g ( 1) is the pseudo-inverse of g and is given by g 1 (x), if x [0, g(1)] g ( 1) (x) =. (8) 1, if x [g(1), ] Theorem 4 ( Cf. Fodor & Roubens [19]; Klement et al [27]): Let S be a continuous t- conorm. (i) S is strict if and only if S is a conjugate of the algebraic sum t-conorm S P (x, y) = x + y x y, i.e., there exists an increasing bijection ϕ of the unit interval [0, 1] such that S(x, y) = ϕ 1 (ϕ(x) + ϕ(y) ϕ(x) ϕ(y)), x, y [0, 1]. (9) (ii) S is nilpotent if and only if S is a conjugate of the Łukasiewicz t-conorm, S L (x, y) = min(x + y, 1), i.e., there exists an increasing bijection ϕ on [0, 1] such that S(x, y) = ϕ 1( min(ϕ(x) + ϕ(y), 1) ), x, y [0, 1]. (10) Theorem 5 (Fodor & Roubens [19] Theorem 1.8 Pg 14): A continuous t-conorm S satisfies S(N(x), x) = 1, for all x [0, 1] with a strict negation N if and only if there exists a strictly increasing bijection ϕ of the unit interval [0, 1] such that S has the representation in (10), and N has the following representation: N(x) ϕ 1 (1 ϕ(x)), x [0, 1]. (11)

7 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 7 C. Uninorms and t-operators Definition 6 ( Fodor et al., [20] Definition 1, Pg 411): A uninorm is a two-place function U : [0, 1] 2 [0, 1] which is associative, commutative, non-decreasing in each place and such that there exists some element e [0, 1] called the neutral element such that U(e, x) = x, x [0, 1]. If e = 0 then U is a t-conorm and if e = 1 then U is a t-norm. For any uninorm U, U(1, 0) {0, 1} (see [20] Corollary 1). A uninorm U such that U(1, 0) = 0 is called a conjunctive uninorm and if U(1, 0) = 1 it is called a disjunctive uninorm. Note that it can be easily shown that a uninorm U that is continuous on the whole of [0, 1] 2 is either a t-norm or a t-conorm. There are three main classes of uninorms that have been well studied in the literature, viz., 1) Pseudo-continuous uninorms (see [30]), i.e., uninorms U that are continuous on [0, 1] 2 except on the segments (0, e), (1, e) and (e, 0), (e, 1). These are precisely the uninorms such that both functions U(x, 1) and U(x, 0) are continuous except at the point x = e. The class of conjunctive uninorms having this property are usually referred to as U Min and the disjunctive ones by U Max (see [20] ). 2) Idempotent uninorms, i.e., uninorms U such that U(x, x) = x for all x [0, 1] (see [50], [4], [28], [38] ). 3) Representable uninorms that have additive generators and are continuous everywhere on the [0, 1] 2 except at the points (0, 1), (1, 0) (see [20]). Definition 7 (Mas et al., [29] Definition 3.1): A t-operator is a two-place function F : [0, 1] 2 [0, 1] which is associative, commutative, non-decreasing in each place and such that F (0, 0) = 0; F (1, 1) = 1. The sections F 0, F 1 : [0, 1] [0, 1] are continuous functions, where F 0 (x) = F (0, x) ; F 1 (x) = F (1, x), for all x [0, 1]. Theorem 6 ( Mas et al., [29] Theorem 3.2): Let F : [0, 1] 2 [0, 1] be a t-operator with F (1, 0) = k [0, 1]. Then If k = 0, F is a t-norm. If k = 1, F is a t-conorm.

8 8 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic If k 0, 1, there exist a t-conorm S and a t-norm T such that ( x ks k, y ), if 0 x, y k k ( x k F (x, y) = k + (1 k)t 1 k, y k ), if k x, y 1 1 k k, if min(x, y) k max(x, y) (12) D. Fuzzy Implications Definition 8 ( Fodor & Roubens [19] Definition 1.15, Pg 22): A function J : [0, 1] 2 [0, 1] is called a fuzzy implication if for all x, y, z [0, 1], it satisfies J(x, z) J(y, z), if x y, (J1) J(x, y) J(x, z), if y z, (J2) J(0, y) = 1, J(x, 1) = 1, (J3) (J4) J(1, 0) = 0. (J5) A fuzzy implication J is said to be Neutral if J(1, y) = y for all y [0, 1]. The following are the two important classes of fuzzy implications well-established in the literature: Definition 9 (Fodor & Roubens [19] Definition 1.16, Pg 24): An S-implication J S,N is obtained from a t-conorm S and a strong negation N as follows: J S,N (x, y) = S(N(x), y), x, y [0, 1]. (13) Definition 10 (Fodor & Roubens [19] Definition 1.16, Pg 24): An R-implication J T is obtained from a t-norm T as its residuation as follows: J T (x, y) = Sup {t [0, 1] : T (x, t) y}, x, y [0, 1]. (14) A t-norm T and J T the R-implication obtained from T are said to have the residuation principle if they satisfy the following: T (x, y) z iff J T (x, z) y, x, y [0, 1]. (RP) Remark 1: It is important to note that (RP) is a characterising condition for a left-continuous t-norm T (see [19] Pg 25, [25] Proposition 5.4.2). In this work we only consider R- implications J T obtained from t-norms T such that the pair (J T, T ) satisfy the residuation principle (RP), or equivalently T is a left-continuous t-norm. Also if T is a left-continuous t-norm then sup in (14) reduces to max.

9 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 9 Any R-implication J T obtained from a left-continuous t-norm T has the following properties (see for example, Trillas and Valverde [45] ) for all x, y, z [0, 1]: J T (x, J T (y, z)) = J T (y, J T (x, z)), Exchange Principle (EP) x y J T (x, y) = 1, Ordering Property (OP) J T (1, y) = y, Neutrality Principle (NP) An R-implication J T also satisfies the following inequality (see Pedrycz [35], [36]): J T (x, T (x, y)) y, x, y [0, 1]. (15) Since an Archimedean t-norm T that is left-continous is also continuous, and viceversa, (see Klement et al., [27] Proposition 2.16 Pg 30), the R-implications J T considered in the sequel all have the properties (EP) - (NP) and (15). Proposition 2 (Baczyński [2] Propositions 12, Pg. 271 & 21, Pg 276 ): Let ϕ : [0, 1] [0, 1] be an increasing bijection. (i) If J S,N is an S-implication obtained from a t-conorm S and strong negation N, then its ϕ-conjugate (J S,N ) ϕ is also an S-implication obtained from the t-conorm S ϕ and the strong negation N ϕ. (ii) If J T is an R-implication obtained from a left continuous t-norm T, then its ϕ-conjugate (J T ) ϕ is also an R-implication obtained from the left continuous t-norm T ϕ. A third important class of fuzzy implications studied in the literature is the following: Definition 11 (Fodor & Roubens [19] Pg 24): A QL-implication is obtained from a t- conorm S, t-norm T and a strong negation N as follows: J QL (x, y) = S(N(x), T (x, y)), x, y [0, 1]. (16) Remark 2: Again an S-implication J S,N (Definition 9) or a QL-implication J QL can be considered for non-involutive negations also, but in this study we only consider the stronger definition as given in [19]. It should also be noted that not all QL-implications satisfy the condition (J1) of Definition 8. Further, a QL-implication satisfies (J4) only if the strong negation N and the t-conorm S in Definition 11 are such that S(x, N(x)) = 1 for all x [0, 1]. All the above three classes of R-, S- and QL-implications are neutral. Tables I, II and III list few of the well-known S-, R- and QL-implications, respectively.

10 10 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic Name S(x, y) J S,N (x, y)y T (x, y) : N-dual of S Kleene-Dienes S M : max(x, y) J KD(x, y) = max(1 x, y) T M : min(x, y) Reichenbach S P : x + y x y J R(x, y) = 1 x + x y T P : x y Łukasiewicz S L : min(1, x + y) J L(x, y) = min(1, 1 x + y) T L : max(0, x + y 1) TABLE I SOME OF THE WELL KNOWN S-IMPLICATIONS WITH THEIR CORRESPONDING t-conorms, WHERE N IS THE STANDARD NEGATION 1 x Name T (x, y) J T (x, y) 8 Godel T M : min(x, y) >< 1, if x y J GD(x, y) = >: y, otherwise Łukasiewicz T L : max(0, x + y 1) J L(x, y) = min(1, 1 x + y) n Goguen T P : x y J GG(x, y) = min 1, y o x TABLE II SOME OF THE WELL-KNOWN R-IMPLICATIONS AND THEIR CORRESPONDING t-norms E. Yager s Classes of Fuzzy Implication Recently Yager [49] has proposed two new classes of fuzzy implication operators, viz., f- and g-generated implications, that cannot be strictly categorised to fall in the above classes. The recent work [3] discusses the intersection of the above classes of fuzzy implications with the classes of R- and (S, N)-implications. They can be seen to be obtained from additive generators of t-norms and t-conorms, whose definitions we give below along with few of their properties. Definition 12 ( Yager [49] Pg 196): An f-generator is a function f : [0, 1] [0, ] that is a strictly decreasing and continuous function with f(1) = 0. Also we denote its pseudoinverse by f ( 1) given by (5). Definition 13 (Yager [49] Pg 197): A function from [0, 1] 2 to [0, 1] defined by an f-generator as J f (x, y) = f ( 1) (x f(y)), (17) with the understanding that 0 = 0, is called an f-generated implication. It can easily be shown that J f is a fuzzy implication (see [49], Pg 197).

11 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 11 Name S(x, y) T (x, y) J QL(x, y) Zadeh S M : max(x, y) T M : min(x, y) J Z(x, y) = max(1 x, min(x, y)) - S P : x + y x y T P : x y J(x, y) = 1 x + x 2 y Łukasiewicz S L : min(1, x + y) T M : min(x, y) J L(x, y) = min(1, 1 x + y) Kleene-Dienes S L : min(1, x + y) T L : max(0, x + y 1) J KD(x, y) = max(1 x, y) TABLE III EXAMPLES OF QL-IMPLICATIONS WITH THEIR CORRESPONDING t-conorms AND t-norms, WHERE N IS THE STANDARD NEGATION 1 x Name f(x) f(0) J f (x, y) - log x y x s x 1 Frank ln ; s > 0, s 1 log s 1 + (s 1) 1 x (s y 1) x s 1 Trigonometric cos π 2 x 1 cos 1 x cos π 2 y Yager s class (1 x) λ ; λ > x 1 λ (1 y) TABLE IV EXAMPLES OF SOME J f IMPLICATIONS WITH THEIR f -GENERATORS Definition 14 (Yager [49] Pg 201): A g-generator is a function g : [0, 1] [0, ] that is a strictly increasing and continuous function with g(0) = 0. Also we denote its pseudo-inverse by g ( 1) given by (8). as Definition 15 (Yager [49] Pg 201): A function from [0, 1] 2 to [0, 1] defined by a g-generator J g (x, y) = g ( 1) ( 1 x g(y) ), (18) with the understanding that 0 =, is a fuzzy implication and is called a g-generated implication. Note that, since x 1, x f(y) f(y) f(0), we have f ( 1) = f 1 in (17). As can be seen from Theorems 2 and 3 the f- and g-generators can be used as additive generators for generating t-norms and t-conorms, respectively. We will use the terms f-generated (ggenerated resp.) implication and f-implication (g-implication resp.) interchangeably. It is easy to see that J f, J g are neutral fuzzy implications and a few examples from these two classes are given in Tables IV and V (see [49] Pp ).

12 12 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic g(x) g ( 1) (z) g(1) J g(x, y) log(1 x) 1 e z 1 (1 y) x 1 π tan 2 x 2 π tan 1 (z) π 1 π 2 tan 1 x tan 2 y x min(1, z) 1 min 1, y x x λ min 1, z λ 1 y ; λ (0, ) 1 min 1, x 1 λ TABLE V EXAMPLES OF SOME J g IMPLICATIONS WITH THEIR g-generators Lemma 1: Let J be (i) an f-implication J f obtained from an f-generator, or (ii) a g-implication J g obtained from a g-generator such that g(1) =. Then J(x, y) = 1 iff either x = 0 or y = 1. Proof: (i) Let J be an f-implication J f. Then the reverse implication is obvious. On the other hand, J f (x, y) = 1 = f ( 1) (x f(y)) = 1 = x f(y) = f(1) = 0, which implies either x = 0 or f(y) = 0, which by the strictness of f means y = 1. (ii) Let J be a g-implication J g obtained from a g-generator such that g(1) =. Again, the reverse implication is obvious. On the other hand, since g(1) =, we have g ( 1) = g 1 and ( ) 1 J g (x, y) = 1 = g 1 x g(y) = 1 = 1 g(y) = g(1) = x which implies either x = 0 or g(y) =, i.e., y = 1. Lemma 2: Let J be (i) an f-implication J f obtained from an f-generator with f(0) =, or (ii) a g-implication J g. Then 0, if x (0, 1] J(x, 0) = 1, if x = 0.

13 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 13 Proof: x 0, then If x = 0 then J f (x, 0) = J g (x, 0) = 1 since J f, J g are fuzzy implications. If (i) J f (x, 0) = f 1 (x f(0)) = f 1 (x ) = f 1 ( ) = 0, by definition of f. 1 (ii) x and since g is strictly increasing and g(0) = 0 we have J g(x, 0) = g ( 1) g ( 1) (0) = 0. ( ) 1 x g(0) = III. THE LAW OF IMPORTATION In the following sub-sections we discuss the validity of (LI) when J is an R-, S-, QL-, f- and a g-implication. A. R- and S-Implications and the Law of Importation (LI) The following result can easily be seen: Lemma 3: If x {0, 1} or y {0, 1} or z = 1 then (LI) holds for any neutral fuzzy implication J and any t-norm T. Since all the classes of fuzzy implications considered in this work, viz., R-, S-, QL-, f- and g-implications, are neutral implications, by Lemma 3 it suffices to consider the cases when x, y (0, 1) and z [0, 1). Theorem 7: An S-implication J S,N, obtained from a t-conorm S and a strong negation N satisfies (LI) with a t-norm T iff T is the N-dual of S. Proof: Let J S,N be an S-implication obtained from a t-conorm S and a strong negation N. (= :) Let J S,N satisfy (LI) with a t-norm T. If S is the N-dual t-conorm of T then LHS (LI) = J S,N (T (x, y), z) = S[N(T (x, y)), z] = S[S (N(x), N(y)), z] (19) RHS (LI) = J S,N (x, J S,N (y, z)) = S[N(x), J S,N (y, z)] = S[N(x), S(N(y), z)] (20) Since (19) = (20), taking z = 0 we have S (N(x), N(y) = S(N(x), N(y) for all x, y [0, 1]. Since N is a strong negation and hence both one-to-one and onto on [0, 1], for every

14 14 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic x, y [0, 1] there exists x, y [0, 1] such that x = N(x ); y = N(y ). Now it follows that S (x, y) = S(x, y) for all x, y [0, 1], and that S is the N-dual t-conorm of T. ( =:) If T is the N-dual of S, then S = S and (19) = (20), by associativity (S2) of S. Thus an S-implication J S,N and a t-norm T satisfy (LI) iff T is the N-dual of S. Theorem 8: An R-implication J T obtained from a left-continuous t-norm T satisfies (LI) with a t-norm T iff T = T. Proof: Let J T be the R-implication obtained as the residuation of the left-continuous t-norm T. Then J T (x, y) = max{t [0, 1] : T (x, t) y}, for all x, y [0, 1]. Since ( JT, T ) satisfy the residuation principle (RP), we have, T (x, y) z x J T (y, z). Also by (OP) we have J T (x, y) = 1 iff x y. ( =:) Let T = T. Now, since T is left-continuous we have LHS (LI) = J T (T (x, y), z) = J T (T (x, y), z) (= :) Let J T z = T (x, y). Now, = max{t [0, 1] : T (T (x, y), t) z} = max{t [0, 1] : T (T (t, x), y) z} [ Associativity (T2) of T = max{t [0, 1] : T (t, x) J T (y, z)} [ By (RP) = max{t [0, 1] : t J T (x, J T (y, z))} [ By (RP) = J T (x, J T (y, z)) = RHS (LI) satisfy (LI) for some t-norm T. Let x, y (0, 1) be arbitrary but fixed and J T (T (x, y), T (x, y)) = J T (x, J T (y, T (x, y))) = J T (x, J T (y, T (y, x))) [ Commutativity (T1) of T J T (x, x) = 1 [ From (15) & (OP) = T (x, y) T (x, y). [ From (OP) (21) On the other hand, for the above fixed x, y let z = T (x, y). Then we have, J T (T (x, y), T (x, y)) = 1 = J T (x, J T (y, T (x, y))) = 1 = x J T (y, T (y, x))) [ From (OP) = T (x, y) T (x, y). [ From (RP) (22)

15 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 15 From (21) and (22) and since x, y are arbitrary we see that T = T. In Table I the t-norms under the column T and in Table II the t-norms under the column T are the t-norms corresponding to the S-implications and R-implications ( under the columns J S,N and J T ), respectively, that satisfy (LI). From Theorems 7, 8 and Proposition 2 we have the following Corollary: Corollary 1: Let ϕ : [0, 1] [0, 1] be an increasing bijection. (i) The ϕ-conjugate of an S-implication J S,N satisfies (LI) iff T is the N ϕ -dual of S ϕ. (ii) The ϕ-conjugate of an R-implication J T satisfies (LI) iff T = Tϕ, where Tϕ is the ϕ-conjugate of the t-norm whose residuation J T is. B. QL-Implications and the Law of Importation Theorem 9: Let J QL be a QL-implication obtained from a t-conorm S, t-norm T and a strong negation N. If (LI) holds for a QL-implication J QL and a t-norm T then S (N (x), x) = 1, for all x [0, 1]. If S is continuous then (LI) implies that S and N have the representation as given in (10) and (11), respectively. Proof: Let J QL be a QL-implication obtained from a t-conorm S, t-norm T and a strong negation N and T be any t-norm. Consider the case when y = 0 and x, z [0, 1]. Then LHS (LI) = J QL (T (x, 0), z) = J QL (0, z) = 1 [ From (J3) RHS (LI) = J QL (x, J QL (0, z)) = J QL (x, 1) Since J QL satisfies (LI), from Remark 2 we see that J QL (x, 1) = 1 only if S (N (x), x) = 1, for all x [0, 1]. If S is a continuous t-conorm, then that S and N have the representation as given in (10) and (11), respectively, is an obvious consequence from Theorem 5. In the rest of this sub-section we consider only continuous t-conorms S and hence by Theorem 9 if the QL-implication J QL given by (16) satisfies (LI) then we know that S is nilpotent and hence from Theorem 4 the ϕ-conjugate of the Łukasiewicz t-conorm S L (x, y) = min(x + y, 1) for an increasing bijection ϕ of the unit interval [0, 1]. Let us consider the extreme case when N (x) = ϕ 1 (1 ϕ(x)) (from the same bijection ϕ), in which case from Theorem 1 we have that N is a strong negation. Also we consider T to be either a ϕ-conjugate of a continuous Archimedean t-norm or min. Theorem 10: Let S be a continuous t-conorm with representation (10) for an increasing bijection ϕ : [0, 1] [0, 1], N be the associated strong negation, i.e., N (x) = ϕ 1 (1 ϕ(x)).

16 16 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic Let J QL be a QL-implication obtained from S, T, N. (i) If the t-norm T is the ϕ-conjugate of the Łukasiewicz t-norm T L, then J QL satisfies (LI) with a t-norm T iff T = T M, the minimum t-norm. (ii) If the t-norm T is the ϕ-conjugate of the product t-norm T P, then J QL satisfies (LI) with a t-norm T iff T = T. (iii) If the t-norm T = T M, the minimum t-norm, then J QL satisfies (LI) with a t-norm T iff T is the ϕ-conjugate of the Łukasiewicz t-norm T L. Proof: By the choice of S, N we have that they satisfy S (N (x), x) = 1, for all x [0, 1]. Also from our assumptions on N we have ϕ(n (x)) = ϕ ϕ 1 (1 ϕ(x)) = 1 ϕ(x). (i) Let T be the ϕ-conjugate of the Łukasiewicz t-norm. Note also that T is the N -dual of S. Then it is immediate that T is a nilpotent t-norm and has a representation as in (7) and ϕ(t (x, y)) = max{ϕ(x) + ϕ(y) 1, 0}. Now, J QL (x, y) = S (N (x), T (x, y)) = ϕ 1 (min{ϕ(n (x)) + ϕ(t (x, y)), 1}) = ϕ 1 (min{1 ϕ(x) + max[ϕ(x) + ϕ(y) 1, 0], 1}) ϕ 1 (1 ϕ(x)), if ϕ(x) + ϕ(y) 1 = y, if ϕ(x) + ϕ(y) > 1 N (x), if y ϕ 1 (1 ϕ(x)) i.e., y N (x) i.e., J QL (x, y) = y, if y > ϕ 1 (1 ϕ(x)) i.e., y > N (x) = max{n (x), y} which is an S-implication with S = max. Hence by Theorem 7 we have that J QL satisfies (LI) iff T = min, which is the N-dual of max for any strong N. (ii) Let T be the ϕ-conjugate of the product t-norm. Then it is immediate that T is a strict t-norm and has a representation as in (6). Also ϕ(t (x, y)) = ϕ(x) ϕ(y) and J QL (x, y) = S (N (x), T (x, y)) = ϕ 1 (min{ϕ(n (x)) + ϕ(t (x, y)), 1}) = ϕ 1 (min{1 ϕ(x) + ϕ(x) ϕ(y), 1}) = ϕ 1 (1 ϕ(x) + ϕ(x) ϕ(y))

17 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 17 which is the ϕ-conjugate of the Reichenbach implication J R (x, y) = 1 x+x y which is an S-implication obtained from the algebraic sum t-conorm S P (x, y) = x + y x y and the strong negation N(x) = 1 x. Now by Corollary 1 (i) we have that J QL satisfies (LI) if and only if T is the N ϕ -dual of the ϕ-conjugate of the algebraic sum t-conorm (S P ) ϕ (x, y) = ϕ 1 (ϕ(x)+ϕ(y) ϕ(x) ϕ(y)) where N ϕ (x) = ϕ 1 (1 ϕ(x)) = N (x). Now by an easy verification we can see that T (x, y) = N ϕ [S ϕ (N ϕ (x), N ϕ (y))] = ϕ 1 (ϕ(x) ϕ(y)) = T (x, y). (iii) Let T = min. Then J QL (x, y) = S (N (x), min(x, y)) = ϕ 1 (min{ϕ(n (x)) + ϕ(min(x, y)), 1}) = ϕ 1 (min{1 ϕ(x) + min[ϕ(x), ϕ(y)], 1}) 1, if ϕ(x) ϕ(y) = ϕ 1 (1 ϕ(x) + ϕ(y)), if ϕ(x) > ϕ(y) = min{1, ϕ 1 (1 ϕ(x) + ϕ(y))} which is the ϕ-conjugate of the Łukasiewicz implication J L (x, y) = min(1, 1 x + y). Now by Corollary 1 (ii) we have that J QL satisfies (LI) if and only if T is the ϕ- conjugate of the Łukasiewicz t-norm T L, i.e., T (x, y) = ϕ 1 (max{ϕ(x)+ϕ(y) 1, 0}). C. Yagers f- and g-implications and the law of importation Theorem 11: Let J f be an f-generated implication. J f satisfies the law of importation (LI) with a t-norm T if and only if T = T P, the product t-norm. Proof: Since J f is a neutral fuzzy implication by Lemma 3 it suffices to consider (LI) only for the cases x, y (0, 1) and z [0, 1). ( =:) Let T be the product t-norm. Then for all x, y (0, 1) and z [0, 1), RHS (LI) = J f (x, J f (y, z)) = f ( 1) [x f(j f (y, z))] = f ( 1) [x f f ( 1) (y f(z))] [ f ( 1) = f 1 = f ( 1) [x (y f(z))] = J f (x y, z) = LHS (LI)

18 18 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic (= :) Let J f obey the law of importation (LI) with a t-norm T. Then for all x, y (0, 1) and z [0, 1), LHS (LI) = RHS (LI) = J f (T (x, y), z)) = J f (x, J f (y, z)) = f ( 1) [T (x, y) f(z)] = f ( 1) [x f f ( 1) (y f(z))] = [T (x, y) f(z)] = x y f(z) [ f ( 1) = f 1 (23) Now, if z (0, 1) then, since f is strictly decreasing, 0 < f(z) < and hence from (23) we have T (x, y) = x y for all x, y [0, 1]. Lemma 4: If a g-implication J g satisfies the law of importation (LI) with a t-norm T, then T (x, y) 0 for any x, y (0, 1). Proof: On the contrary, let T (x 0, y 0 ) = 0 for some x 0, y 0 (0, 1). Let z [0, 1). Then J g (T (x 0, y 0 ), z) = J g (0, z) = 1 by (J3). Since, J g satisfies the law of importation (LI) w.r.to T we have 1 = J g (x 0, J g (y 0, z)). In particular if z = 0 then J g (y 0, z) = J g (y 0, 0) = 0 and J g (x 0, J g (y 0, z)) = J g (x 0, 0) = 0 from Lemma 2 (ii) from which we have 1 = 0, a contradiction. Hence T (x, y) 0 for any x, y (0, 1). Theorem 12: Let J g be a g-generated implication with g : [0, 1] [0, ] such that g(1) =. J g satisfies the law of importation (LI) with a t-norm T if and only if T = T P, the product t-norm. Proof: Let J g be a g-generated implication obtained from a g-generator such that g(1) =. Then we have g ( 1) = g 1. Again from Lemma 2 (ii) J g (x, 0) = 0, for all x (0, 1] and hence (LI) holds for J g and any t-norm T when z = 0. Hence in the following we consider (LI) only for z (0, 1), which implies g(z) (0, ). ( =:) Let T be the product t-norm. Then for all x, y, z (0, 1) RHS (LI) = J g (x, J g (y, z)) ( ) 1 = g 1 x g [J g(y, z)] ( [ ]) 1 1 = g 1 x g g 1 y g(z) ( ) [ ] 1 = g 1 x y g(z) = g 1 1 T (x, y) g(z) = J g (T (x, y), z)) = LHS (LI)

19 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 19 (= :) Let J g obey the law of importation (LI) with a t-norm T. Then for all x, y, z (0, 1), we note that from Lemma 4 T (x, y) 0, and also [ ] J g (T (x, y), z) = J g (x, J g (y, z)) = g 1 1 T (x, y) g(z) i.e., 1 T (x, y) g(z) = 1 x y g(z) ( ) 1 = g 1 x g [J g(y, z)] from which we obtain T (x, y) = x y, for all x, y [0, 1], since g(z) (0, ). Toward proving the above result for the case when g(1) < we prove the following proposition: Proposition 3: Let g be a g-generator with g(1) < and let x, y, z (0, 1) such that 1 y g(z) < g(1). Then the following equality (24) holds if and only if T = T P, the product t-norm: ( ) ( ) 1 1 g ( 1) x y g(z) = g ( 1) T (x, y) g(z) (24) Proof: (= :) Clearly if T = T P then we have the equality (24). ( =:) We prove the converse by considering the following two cases. Let ( ) ( ) 1 1 d = g ( 1) y g(z) = g 1 y g(z), with y, z (0, 1) where 1 y g(z) < g(1). Note that g(d) = 1 y g(z). 1) Let x (0, 1) be such that 1 g(d) g(1). Then we claim that T (x, y) x y. Indeed, x since 1 x g(d) g(1) we have g( 1) ( 1 x g(d) ) g ( 1) g(1) = 1. Hence from the equality (24) we have ( ) 1 g ( 1) T (x, y) g(z) = 1 ( ) 1 T (x, y) g(z) g(1). Now, under the above assumptions on x, y, z let T (x, y) > x y. Then T (x, y) > x y = 1 x y > 1 T (x, y) = g(z) x y > g(z) T (x, y) = g ( 1) ( g(z) x y ) > g ( 1) ( g(z) T (x, y) a contradiction to the fact that the equality (24) holds. 1 2) Let x (0, 1) be such that g(d) < g(1). Then we claim T (x, y) = x y for x all x, y (0, 1). Infact, since 1 ( ) 1 x g(d) < g(1) we have by (8) g( 1) x g(d) = ),

20 20 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic g 1 ( 1 x g(d) ). Hence by equality (24) we have T (x, y) = x y, for all x, y (0, 1), since g(z) (0, ). Remark 3: Note that, in Proposition 3, in the proof of the case when 1 g(d) g(1) if x T (x, y) x y we still have the equivalence, since then T (x, y) x y = 1 x y 1 T (x, y) = 1 = g ( 1) ( g(z) x y g(z) = x y g(z) T (x, y) ) ( ) g(z) g ( 1) T (x, y) ( ) g(z) and hence g ( 1) = 1 and equality (24) holds. T (x, y) Theorem 13: Let g be a g-generator such that g(1) <. Then J g the g-generated implication satisfies the law of importation (LI) if and only if T = T P, the product t-norm. Proof: Let a g-generator be such that g(1) <. As noted earlier we only consider x, y, z (0, 1) and from Lemma 4 we know that T (x, y) 0. Firstly, we note the following: Since g(1) <, we have by (8), g ( 1) g = id but g g ( 1) (x) = x, only if x g(1). 1 x > 0 = 1 y y, for any y [0, ). x Since z (0, 1), g(z) (0, ). We have the following two cases. 1) Let y, z (0, 1) such that 1 g(z) g(1) and T be any t-norm. Then for all x (0, 1), y LHS (LI) RHS (LI) = 1. ( Indeed, ) since g is increasing and g ( 1) g = id, we have 1 1 g(z) g(1) = g( 1) y y g(z) = 1, and ( ( )) ( ) RHS (LI) = J g (x, J g (y, z)) = g ( 1) x g g( 1) y g(z) = g ( 1) x g(1) = 1. Now, let T be any t-norm. Then, T (x, y) y = 1 y 1 T (x, y) = g(1) g(z) g(z) y T (x, y) ( ) g(z) = g ( 1) g(1) g ( 1) T (x, y) = 1 = J g (T (x, y), z)) = LHS (LI)

21 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 21 Hence LHS (LI) RHS (LI) = 1, for any t-norm T. 2) Let us consider the other case when y, z (0, 1) such that 1 g(z) < g(1). Then y ( ) 1 1 g(z) < g(1) = g g( 1) y y g(z) = 1 y g(z). Thus we have, ( ( )) 1 1 RHS (LI) = J g (x, J g (y, z)) = g ( 1) x g g( 1) y g(z) ( ) 1 = g ( 1) x y g(z). (25) ( ) 1 LHS (LI) = J g (T (x, y), z)) = g ( 1) T (x, y) g(z). (26) We see that the equivalence of (25) and (26) under the above assumptions is nothing but the equality (24), which from Proposition 3 holds if and only if T = T P in (26). From the above cases it is clear that the g-implication J g obtained from a g-generator with g(1) < satisfies the law of importation (LI) if and only if T = T P, the product t-norm. IV. THE LAW OF IMPORTATION IN THE SETTING OF UNINORMS AND t-operators In this section we investigate the equivalence (LI) when T is either a uninorm or a t- operator. A. Law of Importation with a Uninorm U instead of a t-norm T In this section we consider the following equality, where U is a uninorm: J(U(x, y), z) = J(x, J(y, z)) (LI:U) Lemma 5: If a neutral fuzzy implication J satisfies (LI:U) then U(0, 1) 1, i.e. U is conjunctive. Proof: Let J be a neutral fuzzy implication, i.e., J(1, x) = x, x [0, 1] and U(0, 1) = 1. Letting x = 0, y = 1, z 1, we have, LHS (LI:U) = J(U(0, 1), z) = J(1, z) = z RHS (LI:U) = J(0, J(1, z)) = J(0, z) = 1 LHS (LI:U) = RHS (LI:U) = z = 1, a contradiction. Since all of the fuzzy implications considered in this work are neutral, in this section, we only consider conjunctive uninorms U for (LI:U).

22 22 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic Theorem 14: An S-implication J S,N, obtained from a t-conorm S and a strong negation N, satisfies (LI:U) with a conjunctive uninorm U only if the identity e of U is 1, or equivalently U = T, the t-norm that is N-dual of S. Proof: Let U be a conjunctive uninorm with the identity element e (0, 1] and J S,N be the S-implication obtained from a t-conorm S and a strong negation N. Letting x = e, y = 1, z = 0 we have LHS (LI:U) = J S,N [U(e, 1), 0] = J S,N (1, 0) = 0. RHS (LI:U) = J S,N [e, J S,N (1, 0)] = J S,N (e, 0) = N(e). LHS (LI:U) = RHS (LI:U) = N(e) = 0 = e = 1 = U = T, a t-norm. Now from Theorem 7 we have that U is the t-norm that is N-dual of S. From the proof of the above theorem and noting that J QL (x, 0) = S(N(x), T (x, 0)) = N(x) for all x [0, 1] we have that a QL-implication J QL satisfies (LI:U) with a conjunctive uninorm U only if the identity e of U is 1, i.e., U is a t-norm. Theorem 15: An R-implication J T obtained from a left-continuous t-norm T satisfies (LI:U) with a conjunctive uninorm U only if the identity e of U is 1, or equivalently U = T, the t-norm from which J T was obtained. Proof: Let U be a conjunctive uninorm with the identity element e (0, 1]. Let J T be an R-implication obtained from a left-continuous t-norm T. We know from (OP) that J T (x, x) = 1, for all x [0, 1]. Letting x = 1, y = z = e in (LI:U), LHS (LI:U) = J T [U(1, e), e] = J T (1, e) = e. RHS (LI:U) = J T [1, J T (e, e)] = J T (1, 1) = 1. Now, LHS (LI:U) = RHS (LI:U) = e = 1, i.e., U = T, a t-norm. Now from Theorem 8 we have that U = T is the t-norm from which J T was obtained. Theorem 16: An f-implication J f satisfies (LI:U) with a conjunctive uninorm U only if the identity element e of U is 1, or equivalently, U is the product t-norm T P. Proof: Let J f be an f-implication obtained from an f-generator. Let x = z = e, y = 1. Then LHS (LI:U) = J f (U(e, 1), e) = J f (1, e) = e; RHS (LI:U) = J f (e, J f (1, e)) = f 1 (e f(e)) J f satisfies (LI:U) only if f 1 (e f(e)) = e = e f(e) = f(e) which implies either e = 1 or f(e) = 0 or f(e) =. Now, f(e) = = e = 0, a contradiction to the fact that U is a

23 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 23 conjunctive uninorm and e (0, 1]. Hence we have e = 1. Now, from Theorem 11 we have that U is the product t-norm T P. Theorem 17: A g-implication J g satisfies (LI:U) with a conjunctive uninorm U only if the identity element e of U is 1, or equivalently U is the product t-norm T P. Proof: Let J g be a g-implication obtained from a g-generator. Let x = z = e, y = 1. Then LHS (LI:U) = J g (U(e, 1), e) = J g (1, e) = e while RHS (LI:U) = J g (e, J g (1, e)) = J g (e, e) = g ( 1) ( 1 e g(e) ). J g satisfies (LI:U) only if g ( 1) ( 1 e g(e) ) = e. Here again we consider the following two cases: Case 1: Let 1 e g(e) < g(1). Then g( 1) ( 1 e g(e) ) = g 1 ( 1 e g(e) ). Hence, g 1 ( 1 e g(e) ) = e = 1 e = 1 e g(e) = g(e) = 1 or g(e) = 0 or g(e) = = e = 1 or e = 0 Now, e = 0 implies U is a t-conorm, contradicting the fact that U is a conjunctive uninorm and so we have e = 1 and U = T, a t-norm. Case 2: Let 1 e g(e) g(1). Then g( 1) ( 1 e g(e) ) = 1 = e and U = T, a t-norm. Now, from Theorems 12 and 13 we have that U is the product t-norm T P. B. Law of Importation with a t-operator F instead of a t-norm T In this section we consider the following equality, where F is a t-operator: J(F (x, y), z) = J(x, J(y, z)) (LI:F) Lemma 6: Let J be a fuzzy implication operator such that J(x, y) = 1 iff either x = 0 or y = 1. Then J satisfies (LI:F) only if k = 0, or equivalently F = T, a t-norm. Proof: Let x = 0, y = 1, z [0, 1). Then LHS (LI:F) = J(F (0, 1), z) = J(k, z) and RHS (LI:F) = J(0, J(1, z)) = 1. Hence for J to satisfy (LI:F) we need J(k, z) = 1, which since z 1 by the hypothesis we have k = 0, or equivalently F = T, a t-norm. Lemma 7: Let J be a fuzzy implication operator such that J(x, 0) = 1 if and only if x = 0. Then J satisfies (LI:F) only if k = 0, or equivalently F = T, a t-norm.

24 24 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic Proof: Let x = 1, y = z = 0. Then LHS (LI:F) = J(F (1, 0), 0) = J(k, 0) and RHS (LI:F) = J(1, J(0, 0)) = 1. Hence for J to satisfy (LI:F) we need J(k, 0) = 1, and from the given condition on J we have k = 0, or equivalently F = T, a t-norm. Theorem 18: Let J be either (i) an S-implication J S,N, or (ii) an R-implication J T obtained from a left-continuous t-norm T, or (iii) an f-implication J f, or (iv) a g-implication J g. Then J satisfies (LI:F) only if the absorption element k of F is 0, i.e., F = T, a t-norm. Proof: Let F be a t-operator with the absorption element F (1, 0) = k. (i) Let J S,N be an S-implication obtained from a t-conorm S and strong negation N. Then J S,N (x, 0) = N(x), the strong negation and hence J S,N (x, 0) = 1 x = 0. Now, by Lemma 7 we have k = 0, i.e., F is a t-norm. (ii) Let J T be an R-implication obtained from a left-continuous t-norm T. We know, from the Ordering Property (OP), that J T (x, x) = 1, x [0, 1]. Now let 0 x < k. Then by definition F (1, x) = k. Let a = 1, b = x = c we have LHS (LI:F) = J T [F (1, x), x] = J T (k, x). RHS (LI:F) = J T [1, J T (x, x)] = J T (1, 1) = 1. Hence LHS (LI:F) = RHS (LI:F) = J T (k, x) = 1 = k x, a contradiction to the fact that x < k. Thus there does not exist any x such that 0 x < k = k = 0, i.e., F is a t-norm. (iii) Let J f be an f-implication. Then the result is obvious from the Lemmas 6 and 1. (iv) Let J g be a g-implication. Then the result is obvious from the Lemmas 7 and 2. Corollary 2: (i) An S-implication J S,N satisfies (LI:F) iff F is the N-dual t-norm of S; (ii) An R-implication J T obtained from a left-continuous t-norm T satisfies (LI:F) iff F = T ; (iii) An f-implication J f satisfies (LI:F) iff F is the product t-norm; (iv) A g-implication J g satisfies (LI:F) iff F is the product t-norm. Once again, from part (i) of the proof of the Theorem 18 and noting that J QL (., 0) = N, a strong negation, we have that a QL-implication J QL satisfies (LI:F) only if the absorption

25 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 25 element k of F is 0, or equivalently F = T, a t-norm. V. HIERARCHICAL CRI In this section we discuss the structure and inference in one of the most established methods of inferencing in Approximate Reasoning, viz., Zadeh s Compositional Rule of Inference (CRI) [52]. After detailing some of the drawbacks of CRI, we propose a novel inferencing scheme called the Hierarchical CRI, which is a modification of the CRI and has many advantages over CRI, most notably computational efficiency. We illustrate these concepts with some numerical examples. Definition 16: A fuzzy set on a non-empty set X, A : X [0, 1], is said to be a fuzzy singleton if there exists an x 0 X such that A has the following representation: 1, if x = x 0 A(x) = (27) 0, if x x 0 We say A attains normality at x 0 X. A. Structure and Inference in CRI The Compositional Rule of Inference (CRI) proposed by Zadeh [52] is one of the earliest implementations of Generalised Modus Ponens. Here, a fuzzy if-then rule of the form: If x is A T hen ỹ is B (28) is represented as a fuzzy relation R(x, y) : X Y [0, 1] as follows: R(x, y) = [A(x) B(y)] = J(A(x), B(y)) (29) where is any fuzzy implication J : [0, 1] [0, 1] [0, 1] and A, B are fuzzy sets on their respective domains X, Y. In this section, unless otherwise explicitly stated x X, y Y and z Z. Then given a fact x is A, the inferred output B is obtained as composition of A (x) and R(x, y), i.e., B (y) = A (x) R(x, y) (30) where is a Sup t-norm composition given by B (y) = A (x) R(x, y) = Sup x X {T [A (x), R(x, y)]} (31) where T can be any t-norm (see Wangming [47]).

26 26 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic In the case when the input A is a singleton (27) attaining normality at an x 0 X then B (y) = A (x) R(x, y) = Sup x X {T [A (x), R(x, y)]} = T [A (x 0 ), R(x 0, y)] = R(x 0, y) (32) In the case of Multi-Input Single-Output (MISO) fuzzy rule given by R : If x is A and ỹ is B T hen z is C. (33) the relation R is given by R(x, y; z) = J[ce(A(x, y)) ce(b(x, y)), C(z)] = J[A(x) B(y), C(z)] (34) where ce(a(x, y)) stands for the Cylindrical Extension of the fuzzy set A with respect to the universe of B, and viceversa, and is the antecedent combiner, which is usually a t-norm. Consider a single MISO rule of the type (33), denoted (A, B) C, for simplicity. Then given a multiple-input (A, B ) the infered output C, taking the Sup T composition, is given by, C = (A, B ) [(A, B) C] (35) C (z) = Sup x,y T {[A (x) B (y)], [A(x) B(y)] C(z)} (36) As in the SISO case, when the inputs are singleton fuzzy sets A and B attaining normality at the points x 0 X, y 0 Y, respectively, we have from (36) C (z) = [A(x 0 ) B(y 0 )] C(z) = R(x 0, y 0 ; z) (37) Once again, is the antecedent combiner which is usually any of the t-norms. Example 1: Let A = [ ], B = [1.6.8] and C = [.1.1.2], where A, B and C are defined on X = {x 1, x 2, x 3, x 4 }, Y = {y 1, y 2, y 3 } and Z = {z 1, z 2, z 3 }, respectively. Let J be the Reichenbach implication J R (x, y) = 1 x + x y and the antecedent combiner be the product t-norm T P (x, y) = x y. Then taking the cylindrical extensions of A and B with respect to T P, we have that A B = T P (A, B) =

27 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic 27 Now, with x y = 1 x + x y, T P (A, B) C from (34) will be given by R(A, B : C) = [C(z 1 ) C(z 2 ) C(z 3 )], where T P (A, B) C = T P (A, B) [z 1 z 2 z 3 ] = T P (A, B) [.1.1.2] and C(z i ) = T P (A, B) z i are given as follows: C(z 1 ) = C(z 2 ) = ; C(z 3 ) = Let A = [ ], B = [0 1 0] be the given fuzzy singleton inputs. Then A B = T P (A, B ) = (38) Taking the Sup min composition, we have C = T P (A, B ) [T P (A, B) C] = [ ] (39) B. Drawbacks of CRI Despite its prevalent use CRI is not without its drawbacks, chiefly: Computational Complexity: The calculation of the supremum in (36) is a time consuming process. When X = m and Y = n, the complexity of a single inference amounts to O(m n). Explosion in Dimensions: For an n-input 1-output system (A 1, A 2,..., A n ) B with the cardinality of the base sets X i of each of the inputs A i being n i, we have an n- dimensional matrix having n i=1 n i entries. Expensive Storage: As noted above, we need to store n-dimensional matrices for every fuzzy if-then rule. The many works proposing modifications to Zadeh s CRI can be broadly categorised into works that attempt to improve the accuracy of inferencing in CRI and those that intend to enhance the efficiency in its inferencing. Some works relating to the former approach are those of Ying [51] who investigated the reasonableness of CRI and proposed a suitable modification of the CRI, Wangming [47] who was one of the earliest to employ the Sup T composition and has studied the suitability of pairs (T, J) - of the t-norm T

28 28 BALASUBRAMANIAM: On the Law of Importation in fuzzy logic used for composition and the fuzzy implication J - that satisfy some appropriate criteria for Generalised Modus Ponens (GMP) and Generalised Modus Tollens (GMT). In [11] necessary and sufficient conditions on operators involved in CRI inferencing are investigated to fit the CRI into the analogical scheme. For works relating to computation of CRI and exact formulae see the works by Fullér and his group [22], [23], [21]. For some recent works see Moser and Navara [32], [33], [34]. In the case when there are more than two antecedents involved in fuzzy inference, Ruan and Kerre [37] have proposed an extension to Zadeh s CRI, wherein starting from a finite number of fuzzy relations of an arbitrary number of variables but having some variables in common, one can infer fuzzy relations among the variables of interest. As noted above, because of the multi-dimensionality of the fuzzy relation (34) inferencing in CRI (35) is difficult to perform. To overcome this difficulty, Demirli and Turksen [15] proposed a Rule Break-up method and showed that rules with two or more independent variables in their premise can be simplified to a number of inferences of rule bases with simple rules (only one variable in their premise). For further modification of this method see [24]. In the following we propose a modified but novel form of CRI to address the drawbacks noted above. C. Hierarchical CRI In the field of fuzzy control, Hierarchical Fuzzy Systems (HFS) hold a center stage, since, where applicable, they help to a large extent in breaking down the complexity of the system being modeled, both in terms of efficiency and understandability. For a good survey of HFS see Torra [42] and the references therein. In this section, we propose a new modified form of CRI termed Hierarchical CRI owing to the way in which multiple antecedents of a fuzzy rule are operated upon in the inferencing. A distinction should be made between the method proposed in this work - which uses the given MISO fuzzy rules as is - and typical inferencing in HFS which is dictated by the hierarchical structure that exists among the modeled system variables. It should be emphasised that the term Hierarchical CRI, as employed here, refers to the modifications in the inferencing procedure of CRI and does not impose a hierarchical architecture on the MISO fuzzy rules. From the law of importation (LI), we see that (34) can be rewritten as, using a t-norm T