Characterizations of fuzzy implications satisfying the Boolean-like law y I(x, y)

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1 Characterizations of fuzzy implications satisfying the Boolean-like law y I(x, y) Anderson Cruz, Benjamín Bedregal, and Regivan Santiago Group of Theory and Intelligence of Computation - GoThIC Federal Rural University of Semi-Árido - UFERSA , Angicos, RN, Brazil anderson@ufersa.edu.br Group of Logic, Language, Information, Theory and Applications - LOLITA Federal University of Rio Grande do Norte - UFRN , Natal, RN, Brazil {bedregal,regivan}@dimap.ufrn.br Abstract. Properties valid on the classical theory (Boolean laws) have been extended to fuzzy set theory and so-called Boolean-like laws. The fact that they are not always valid in any standard fuzzy set theory induced a wide investigation. In this paper we show the sufficient and necessary conditions that the Boolean-like law y I(x, y) holds in fuzzy logic. We focus the investigation on the following classes of fuzzy implications: (S,N)-, R-, QL-, D-, (N,T)- and h-implications. Keywords: Implications, Fuzzy Logics, Boolean-like Laws. 1 Introduction Classical logic, well as its inference notion were the first formal elucidation of how would be a correct reasoning. Joined with those concepts is the classical implication definition (called material implication). Thus, such implication was the first one to be defined and disseminated. This fact induces us to believe that material implication is the correct notion (common sense) of what actually is a logical implication. However, other Boolean implications, such as intuitionistic, quantum or para-consistent implications, do give acceptable inference models and they must be regarded to understand the meaning of a logical implication. In fuzzy theory, implication operator was originally used to define the relation between antecedent and consequent of IF-THEN rules in Fuzzy Rule Based Systems (see [37]). However, the truth value of implication operators do not necessarily satisfy the classical implication truth table (see [21] for example). In search of a classical implication generalization, Baldwin and Pilsworth in [7] and Blander and Kohout in [10], proposed a few basic properties that should be required by a fuzzy implication. Further, Trillas and Valverde in [32] gave the first fuzzy implication axiomatic. Nowadays, the lack of a consensus on Boolean implication meaning entails non-equivalent acceptable fuzzy implication

2 238 Anderson Cruz, Benjamín Bedregal, Regivan Santiago definitions (see [3], [16], [17], [19], [23], [28], [29] and [35] as examples) and fuzzy implication classes, such as [5], [6], [24] and [13]. Some fuzzy implication classes try to define an acceptable fuzzy implication meaning and others try to generalize the implication from distinct Boolean logics. The best-known classes are: (S,N)-implication which is a fuzzy extension of the material implication p q p q, where is the negation operator and is the disjunction operator; R-implication whose motivation is the residuation concept employed in intuitionistic logic which is founded by the relative pseudo-complement definition [25, pp.54]; and QL-implication which is the generalization of the quantum implication p q p (p q), where is the conjunction operator. Therefore, for a better understanding of what is a logical implication (in Boolean and fuzzy contexts) and also to characterize the approximate reasoning in accordance to the implication definition of a given logic, several Boolean laws have been generalized and studied as (in)equations in fuzzy logics in which t- norms, t-conorms, fuzzy negations and implications are used see [1], [2], [8], [9], [29], [33] and [34]. Although in classic-like fuzzy semantics [12] Boolean-like laws are valid, if, and only if, 1 their original Boolean laws are tautologies. A lot of Boolean laws do not remain valid when are generalized to the fuzzy setting. In this scenario, this paper deals with the functional inequation (1), where I is a fuzzy implication. (1) generalizes the relation υ(q) (υ(p) υ(q)), in which is the implication operator and υ(p),υ(q) {0,1} are the truth values of p and q, respectively. Such relation is a well-known Boolean law that entails other important Boolean rules, e.g. (υ(p) 1) = 1 which can be translated as If the consequent of an implication is true, then the implication is also true 2. y I(x,y), for all x,y [0,1] (1) This paper extends [15]. This one provided sufficient and necessary conditions under which (1), holds for (S,N)-, R- and QL-implications. In this current paper, the main concern is extending such investigation for D-, (N,T)-, and h-implications. But we will also demonstrate equivalences between fuzzy implication classes 3, well as demonstrate relations between (1) and other fuzzy implication properties. 1 Iff, for short. 2 (υ(p) 1) = 1 can be generalized to fuzzy logic as I(x,1) = 1 (denoted in this paper by (I8)). 3 Some results are known, but their demonstrations are required, since we will relax the D-implication definition in this paper.

3 C. of fuzzy implications satisfying the Boolean-like law y I(x, y) 239 The paper is organized as follows. Section 2 recalls some basic definitions and results about t-norms, t-conorms, fuzzy negations and properties relating those fuzzy operators. Section 3 focus on fuzzy implications I, in such section we define them and demonstrate results about relations between (1) and other fuzzy implication properties. Section 4 recalls (S,N)-, R-, QL-, D-, (N,T)- and h- implications definitions and some useful results about equivalences between those classes of fuzzy implications. Section 5 demonstrates, for those classes, under which conditions (1) holds. Beyond that, It is shown that a fuzzy implication I satisfies (1) iff I up to Φ-conjugation also satisfies (1). Finally, the final remarks of the paper are exposed in section 6. 2 Basic definitions In this section we mention the preliminaries definitions and previous results about t-norms, t-conorms and fuzzy negations that we will use in this paper. Definition 1. A function T : [0,1] 2 [0,1] is a t-norm if T satisfies commutativity (T1), associativity (T2), monoticity (T3) and 1-identity (T4). Remark 1. Considering the partial order on the family of all t-norms induced from the order on [0,1], T M (x,y) = min(x,y) is the greatest t-norm. Therefore for any t-norm T, T(x,y) T M (x,y) y, for all x,y [0,1]. Definition 2. A function S : [0,1] 2 [0,1] is a t-conorm if S satisfies commutativity (S1), associativity (S2), monoticity (S3) and 0-identity (S4). Remark 2. Considering the point-wise order on the family of all t-conorms induced from the order [0,1], S M (x,y) = max(x,y) is the least t-conorm. Therefore for any t-conorm S, S(x,y) S M (x,y) y. Moreover, by definition of S M, S M (x,1) = S M (1,x) = 1, so for any t-conorm S, S(x,1) = S(1,x) = 1. Definition 3. A function N : [0,1] [0,1] is a fuzzy negation, if N(0) = 1, N(1) = 0 (N1), and N is decreasing (N2). Beyond this, a fuzzy negation is called strong, if it is involutive, i.e., N(N(x)) = x, for all x [0,1]. As examples of fuzzy negations, we cite the greatest fuzzy negation N : N (x) = { 0, if x = 1 1, if x [0,1[. 2.1 Properties involving fuzzy operators In this subsection we address three properties: Distributivity of t-conorms over t-norms; Law of excluded middle; and N-duality.

4 240 Anderson Cruz, Benjamín Bedregal, Regivan Santiago Distributivity of t-conorms over t-norms In classical logic, the distributivity of disjunction over conjunction is a well-known property, its extension to fuzzy logic takes into account t-norms and t-conorms: A t-conorm S is distributive over a t-norm T if S(x,T(y,z)) = T(S(x,y),S(x,z)). (2) An important result about such property is the following: Proposition 1. [18, Proposition 2.22] Let T be a t-norm and S a t-conorm, then S is distributive over T iff T = T M. Law of excluded middle One of the fundamental Boolean laws of classical theory is the Law of Excluded Middle (LEM). As LEM in classical logic states that p p is always true, we have the following extension to fuzzy logic. Definition 4. Let S be a t-conorm and N a fuzzy negation, the pair (S,N) satisfies the LEM if S(N(x),x) = 1, for all x [0,1]. (LEM) Remark 3. 4 [4, ] S(N (x),x) = 1, for any t-conorm S, that is, (S,N ) satisfies (LEM), for any S. Moreover, if S is positive 5, (S,N) satisfies (LEM) only if N = N. N-duality For any t-conorm S there exists a t-norm T such that, S(x,y) = 1 T(1 x,1 y). Moreover, let T be a t-norm, S a t-conorm and N a fuzzy negation then S is said the N-dual of T, if S(x,y) = N(T(N(x),N(y))). (3) 3 Fuzzy implications Several non-equivalent fuzzy implications have been announced and nowadays there is no consensus to one specific fuzzy implication definition. Without causing any loss to the paper, we consider just the boundary conditions to define a fuzzy implication I. Definition 5. A function I : [0,1] 2 [0,1] is called a fuzzy implication if it satisfies the following boundary conditions. I1.: I(0,0) = 1. I2.: I(0,1) = 1. I3.: I(1,0) = 0. I4.: I(1,1) = 1. Some other potential properties for fuzzy implications are: I5. Left antitonicity: if x 1 x 2 then I(x 1,y) I(x 2,y), for all x 1,x 2,y [0,1]; 4 Some proofs will refer this remark. 5 S is positive iff, if S(x,y) = 1 then x = 1 or y = 1

5 C. of fuzzy implications satisfying the Boolean-like law y I(x, y) 241 I6. Right isotonicity: if y 1 y 2 then I(x,y 1 ) I(x,y 2 ), for all x,y 1,y 2 [0,1]; I7. Left boundary condition: I(0,y) = 1, for all y [0,1]; I8. Right boundary condition: I(x,1) = 1, for all x [0,1]; I9. Left neutrality: I(1,y) = y, for all y [0,1]; I10. Identity property: I(x,x) = 1, for all x [0,1]; I11. Exchange principle: I(x,I(y,z)) = I(y,I(x,z)), for all x,y,z [0,1]; I12. Ordering property 6 : x y iff I(x,y) = 1, for all x,y, [0,1]; I12a. Left ordering property: if x y then I(x,y) = 1, for all x,y, [0,1]; I12b. Right ordering property: if I(x,y) = 1 then x y, for all x,y, [0,1]; I13. Gen. of the first classical axiom: I(y,I(x,y)) = 1, for all x,y, [0,1]; I14. Contraposition: Let N be a fuzzy negation, I(x,y) = I(N(y),N(x)), for all x,y, [0,1]; I15. Continuity. There are some relations between above properties as exposed in [14], [4], [30] and [31]. We highlight the previous study about (1) by [14] and [30] where Bustince, Shi et al. investigated the interrelationship between some fuzzy implications properties. In this scenario we also expose some relations between (1) and above properties. See the following propositions and lemmas. Proposition 2. [14] If a fuzzy implication I satisfies (1) then I satisfies (I8). Lemma 1. [14, Lemma 1 viii] If a fuzzy implication I satisfies (I9) and (I5), then I satisfies (1). Adapting the results of [30, Remark 7.5] we have the next proposition. Proposition 3. Let I be a fuzzy implication: If I satisfies (I11) and (I12) then I satisfies (1); If I satisfies (I5), (I11) and I(x,0) = N I (x) then I satisfies (1); If I satisfies (I5), (I11) and (I15) then I satisfies (1). Proposition 4. If a fuzzy implication I satisfies (1) and (I12a) then I satisfies (I13). Proof. Straightforward. Proposition 5. If a fuzzy implication I satisfies (I13) and (I12b) then I satisfies (1). Proof. Straightforward. Corollary 1. If a fuzzy implication I satisfies (I12), then I satisfies (1) iff I satisfies (I13). Proof. Straightforward from Propositions 4 and 5. 6 Also called confinement property.

6 242 Anderson Cruz, Benjamín Bedregal, Regivan Santiago 4 Classes of fuzzy implications In this section we recall the definitions of the (S,N)-, R-, QL-, D-, (N,T)- and h-implications. We also present some results about those classes. Some of those results are concerned to explicit the equivalence among such classes of fuzzy implication. Lemmas and theorems contained in this section will be useful in the next section when we will demonstrate the necessary and sufficient conditions to (1) be held for those classes. 4.1 (S,N)-implications Definition 6. A function I : [0,1] 2 [0,1] is called an (S,N)-implication if there exist a t-conorm S and a fuzzy negation N, such that I(x,y) = S(N(x),y) (4) The (S,N)-implication generated by a t-conorm S and a fuzzy negation N is denoted by I S,N. 4.2 R-implications Definition 7. A function I : [0,1] 2 [0,1] is called an R-implication if there exists a t-norm T such that I(x,y) = sup{t [0,1] T(x,t) y} (5) The R-implication generated by a t-norm T is denoted by I T. Lemma 2. [4, Theorem 2.5.4] and [12, pp.359] Every R-implication satisfies (I1)-(I10) and (I12a). 4.3 QL-implications Definition 8. A function I : [0,1] 2 [0,1] is called a QL-implication if there exist a t-conorm S, a fuzzy negation N and a t-norm T, such that I(x,y) = S(N(x),T(x,y)) (6) The QL-implication generated by a t-conorm S, a fuzzy negation N a t-norm T is denoted by I S,N,T. Lemma 3. [4, Theorem 2.6.2] Every QL-implication satisfies (I1)-(I4), (I6), (I7) and (I9). Lemma 4. 7 If (S,N) satisfies (LEM) and T = T M, then a QL-implication I S,N,T satisfies (I10). 7 A similar result is found in [4, Proposition ].

7 C. of fuzzy implications satisfying the Boolean-like law y I(x, y) 243 Proof. Let I S,N,T be a QL-implication and T = T M iff S is distributive over T (Prop. 1), so: I S,N,T (x,x) = S(N(x),T(x,x)) by(6) = T(S(N(x),x),S(N(x),x))) by(2) = T(1,1) by(lem) = 1 by(t4). Proposition 6. Given a QL-implication I S,N,T and an (S,N)-implication I S,N, I S,N,T I S,N. Proof. By Def. 8 I S,N,T (x,y) = S(N(x),T(x,y)) and T(x,y) y (by Remark 1), so S(N(x),T(x,y)) S(N(x),y) = I S,N (x,y). 4.4 D-implications Definition 9. A function I : [0,1] 2 [0,1] is called a D-implication if there exist a t-conorm S, a t-norm T and a fuzzy negation N such that I(x,y) = S(T(N(x),N(y)),y) (7) The D-implication generated by a t-conorm S, a t-norm T and a fuzzy negation N is denoted by I S,T,N. 8 Lemma 5. Every D-implication satisfies (I1)-(I4), (I5) and (I9). Proof. Let I S,T,N be a D-implication, so I S,T,N satisfies (I1)-(I4) since: I S,T,N (0,0) = S(T(N(0),N(0)),0) = S(T(1,1),0) = S(1,0) = 1. I S,T,N (0,1) = S(T(N(0),N(1)),1) = 1. I S,T,N (1,0) = S(T(N(1),N(0)),0) = S(T(0,1),0) = S(0,0) = 0. I S,T,N (1,1) = S(T(N(1),N(1)),1) = 1. Now, assume that x 1,x 2,y [0,1] and x 1 x 2. Then, by (N2), N(x 1 ) N(x 2 ). By (T3), T(N(x 1 ),N(y)) T(N(x 2 ),N(y)), and by (S3) we have S(T(N(x 1 ), N(y)),y) S(T(N(x 2 ),N(y)),y). Hence I S,T,N (x 1,y) I S,T,N (x 2,y). Hence I S,T,N satisfies (I5). For any y [0,1], I S,T,N (1,y) = S(T(N(1),N(y)),y) = S(T(0,N(y)),y) and T(0,N(y)) = 0 (by Remark 1). Since S(0,y) = y, so I S,T,N (1,y) = y. Hence I S,T,N satisfies (I9). 8 D-implications are generally defined from a strong negation [22], [26] and [27]. However, D-implications defined from strong (or non-strong) fuzzy negations are also fuzzy implications (according to our fuzzy implication definition Definition 5). Due to this, we generalize the D-implication definition and define it from any fuzzy negation.

8 244 Anderson Cruz, Benjamín Bedregal, Regivan Santiago 4.5 (N,T)-implications The (N,T)-implications was firstly defined in [11] as an N-dual implication of a t-norm T. Since T(x,y) = N(I N,T (x,n(y))) and I N,T (x,y) = N(T(x,N(y))) for more details see the [11, Def. 2.7]. Definition 10. A function I : [0,1] 2 [0,1] is called a N-dual fuzzy implication of T ((N,T)-implication, for short) if there exist a t-norm T, and a negation N such that I(x,y) = N(T(x,N(y))) (8) The (N,T)-implication generated by a t-norm T and a fuzzy negation N is denoted by I N,T. Lemma 6. [11, Prop. 2.6] Every (N,T)-implication satisfies (I1)-(I6). Lemma 7. An (N,T)-implication I N,T satisfies (I9), if N is a strong negation. Proof. Since N is a strong negation, then I N,T (1,y)=N(T(1,N(y)))=N(N(y)) = y. 4.6 h-implications The h-implications were defined by Massanet et al. in [24] in a similar way realized by Yager in [36]. Definition 11. A function I : [0,1] 2 [0,1] is called an h-implication if there exist an e ]0,1[ and, a strictly increasing and continuous function h : [0,1] [,+ ] in which h(0) =, h(e) = 0 and h(1) = +, such that 1, if x = 0 I(x,y) = h 1 (x h(y)), if x > 0 and y e h 1 ( 1 x h(y)), if x > 0 and y > e. The function h is called an h-generator (with respect to e) of the function I defined above. The h-implication generated by a continuous and strictly increasing function h is denoted by I h. Lemma 8. [24, Prop. 1 and Theo. 5(i)] Let h be an h-generator w.r.t. a fixed e ]0,1[, then I h satisfies (I1)-(I6) and (I9). 9 So I h is a fuzzy implication. 9 Truly, in [24, Prop. 1 and Theo. 5(i)] is demonstrated that I h satisfies (I2) is trivially deduced from (I4) and (I5). Beyond that, we also can deduce straightforward (I7) from (I1) and (I6), and (I8) from (I4) and (I5).

9 C. of fuzzy implications satisfying the Boolean-like law y I(x, y) Equivalences among the fuzzy implications classes Proposition 7. [22], [23] Let N be a strong negation and, given a D-implication I S,T,N and a QL-implication I S,N,T. If I S,T,N or I S,N,T satisfies the contraposition (I14), then I S,T,N = I S,N,T. Lemma 9. [4, Proposition 4.2.2] Given an (S,N)-implication I S,N and a QLimplication I S,N,T. If T = T M and (S,N) satisfies (LEM), then I S,N,T = I S,N. Lemma 10. Given a D-implication I S,T,N and an (S,N)-implication I S,N. If T = T M and (S,N) satisfies (LEM), then I S,T,N = I S,N. Proof. By Proposition 1, T = T M iff (S,T) satisfies (2). So, for all x,y [0,1]: I S,T,N (x,y) = S(T(N(x),N(y)),y) by(7) = S(y,T(N(x),N(y))) by(s1) = T(S(y,N(x)),S(y,N(y))) by(2) = T(S(N(x),y),S(N(y),y))) by(s1) = T(S(N(x),y),1) by(lem) = S(N(x),y) by(t4) = I S,N (x,y) by(4). Theorem 1. Given a QL-implication I S,N,T, a D-implication I S,T,N and an (S,N)-implication I S,N. If T = T M and (S,N) satisfies (LEM), then I S,T,N = I S,N = I S,N,T. Proof. Straightforward from Lemmas 9 and 10. If we regard that N is a strong negation, then we got another result relating (S,N)-, QL- and D-implications. Proposition 8. [22, Proposition 6] Let N be a strong negation and given a QL-implication I S,N,T, a D-implication I S,T,N and an (S,N)-implication I S,N. If I S,T,N and I S,N satisfy (I1)-(I6), then the corresponding QL- and D-implication coincide and are given by: I S,N,T (x,y) = I S,T,N (x,y) = { 1, if x y I S,N (x,y), otherwise. Lemma 11. Let N be a strong negation. I S,N = I N,T iff S is N-dual of T. Proof. Straightforward. Theorem 2. Given a D-implication I S,T,N, an (S,N)-implication I S,N and a QL-implication I S,N,T. If T = T M, S is N-dual of T and (S,N) satisfies (LEM). Then I S,T,N = I S,N = I S,N,T = I N,T. Proof. Straightforward from Lemma 11 and Theorem 1.

10 246 Anderson Cruz, Benjamín Bedregal, Regivan Santiago 5 On the y I(x,y) of fuzzy logic In the sequel we show some results about (1) and, (S,N)-, R-, QL-, D-, (N,T) and h-implications. Theorem 3. Every (S,N)-implication I S,N satisfies (1). Proof. Straightforward, since S M is the least t-conorm. Theorem 4. Every R-implication I T satisfies (1). Proof. Straightforward from Lemmas 1 and 2. Theorem 5. Every D-implication satisfies (1). Proof. Let I S,T,N be a D-implication, by Def. 9, I S,T,N (y,x)=s(t(n(y),n(x)), x) and, by Remark 2, regardless of the value of T(N(y),N(x)), S(T(N(y),N(x)), x) x. Hence I S,T,N satisfies (1). Theorem 6. If N is a strong negation then I N,T satisfies (1). Proof. Straightforward by Lemmas 6 and 7, or by Lemma 11 and Theorem 3. Theorem 7. Let h be an h-generator w.r.t. a fixed e ]0,1[, then I h satisfies (1). Proof. Straightforward from Lemmas 1 and 8. We demonstrated that (S,N)-implications satisfy (1). Since there is an intersection between the (S,N)- and QL-implications classes, we verify the sufficient and necessary conditions in which the elements of such intersection satisfy (1). 10 Theorem 8. If (S,N) satisfies (LEM) and T = T M then a QL-implication I S,N,T satisfies (1). Proof. Straightforward from Theorem 3 and Lemma 9, we deduce the following theorem. The reader will note that Lemma 9 and Theorem 8 give the sufficient conditions for I S,N,T to satisfy (1). In the sequel we present results that give the necessary conditions. Lemma 12. If a QL-implication I S,N,T satisfies (1) then (S,N) satisfies (LEM). Proof. By (1), 1 I S,N,T (y,1), then I S,N,T (y,1) = 1 (i.e. I S,N,T satisfies (I8)). So S(N(y),T(y,1)) = 1, and by (T4) S(N(y),y) = 1. Hence (S,N) satisfies (LEM). 10 Following results, about QL-implications, were rewritten from [15] for a better reading.

11 C. of fuzzy implications satisfying the Boolean-like law y I(x, y) 247 The reciprocal of Theorem 8 is not true (see [15, Example 5.1]). Theorem 9. Let S be a strictly increasing in [0,1[ t-conorm. If a QL-implication I S,N,T satisfies (1) and (I10), then (S,N) satisfies (LEM) and T = T M. Proof. By Lemma 12, if I S,N,T satisfies (1), then (S,N) satisfies (LEM). Now, by (I10), S(N(x),T(x,x)) = 1, and since (S,N) satisfies (LEM), then for any x [0,1], S(N(x),T(x,x)) = 1 = S(N(x),x). Case x = 1 so, trivially, T(x,x) = x. Case x < 1, since S is strictly increasing in [0,1[, then S(N(x),T(x,x)) = S(N(x),x) implies T(x,x) = x. Therefore T = T M, since T M is the only idempotent t-norm [20, Theorem 3.9]. Corollary 2. Let S be a strictly increasing in [0,1[ t-conorm. Then the following statements are equivalent: 1. A QL-implication I S,N,T satisfies (1) and (I10); 2. (S,N) satisfies (LEM) and T = T M. Proof. Straightforward from Lemma 4 and Theorems 8 and 9. Note that, if x y iff I(x,y) = 1, then I(x,x) = x. In other words, if I satisfies (I12), then I satisfies (I10). Therefore, by Theorem 9, we deduce the following Corollary. Corollary 3. Let S be a strictly increasing in [0,1[ t-conorm. If a QL-implication I S,N,T satisfies (1) and (I12), then (S,N) satisfies (LEM) and T = T M. 11 The reciprocal of Corollary 3 is not true. Its counter-example is I S,T M,N (given below), since (S,N ) satisfies (LEM) and I S,T M,N satisfies (1), but I S,T M,N does not satisfy (I12). { 1, if x < 1 I S,T M,N (x,y) = y, if x = 1. 6 Final remarks This paper provided sufficient and necessary conditions under which the Booleanlike law x I(y,x), refereed by (1), holds for (S,N)-, R-, QL-, D-, (N,T)- and h-implications; beyond of analyzing the relations among fuzzy implication properties and (1). The property (1) was firstly studied in [14] where Bustince et. al. demonstrated, among other results, that, let I be a fuzzy implication which satisfies (I1)-(I6), if I satisfies (I9) then I satisfies (1). The main results of this paper are stated by Theorems 3 and 4, 5, 6 and 7, and Corollary 2. From those results, we can conclude that there is not a common set of properties which guarantees the satisfiability of (1) for all of those classes. But there are some similarities. 11 In which S is any strictly increasing in [0,1[ t-conorm.

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