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Boixader Ibáñez, Dionís

Recasens Ferrés, Jorge

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Transitive OpeningsD. Boixader1 J. Recasens21UPC, Spain, j.recasens@upc.edu2UPC, Spain, dionis.boixader@upc.eduAbstractThough the transitive closure of a reflexive and sym-metric fuzzy relation R is unique and there are sev-eral algorithms to calculate it, there can be manytransitive openings (maximal T -indistinguishabilityoperators among the ones smaller than or equal toR). This paper presents a method to calculate tran-sitive openings of a reflexive and symmetric fuzzyrelation R. It is worth noticing that apart form theminimum t-norm, this is the first algorithm that al-lows us to calculate them.Keywords: Similarity, Indistinguishability Opera-tor, Finite-Valued t-norm, Transitive Closure, Tran-sitive Opening1. IntroductionThe transitive closure of a reflexive and symmet-ric fuzzy relation R gives a T -indistinguishabilityoperator greater than or equal to R. In this caseit is possible to obtain the best upper approxima-tion since the infimum of T -indistinguishability op-erators is also a T -indistinguishability operator. Ifwe want a lower approximation, then the situationis more complicated since the supremum of indis-tinguishability operators is not such an operator ingeneral. What we can find is T -indistinguishabilityoperators maximal among the ones that are smallerthan or equal to a given reflexive and symmetricfuzzy relation. These relations are called transitiveopenings and in general they are not unique, butthere can be an infinite quantity of them, even insets of finite cardinality.There is no general method in the literature tocalculate them. In [5], an algorithm to find maxi-mal transitive openings of a given fuzzy relation isgiven, but the obtained openings are not symmetricin general and in the process of symmetrizing them,maximality can be lost. Heuristic methods to ob-tain T -indistinguishability operators smaller thanor equal to a given fuzzy relation close to maximalones have been proposed [3], but until now there isnot a general methodology to find them.The minimum t-norm is an exception because ofthe special behaviour of min-indistinguishability op-erators. In this case there are a number of algo-rithms to find at least some of the min-transitiveopenings of a given reflexive and symmetric fuzzyrelation. A classic method is the complete linkage.Other algorithms can be found in [4] [6].In this presentation a method of calculating allthe transitive openings of a reflexive and symmet-ric L-relation on a finite set X will be explained.This result will then be exploited to obtain tran-sitive openings with respect to the  Lukasiewicz t-norm of a given reflexive and symmetric fuzzy re-lation (valued in the unit interval). The methodis very effective when combined with heuristic algo-rithms such as the ones proposed in [3] (see Example3.8).2. PreliminariesThis section contains some definitions and resultson finite-valued t-norms that will be needed lateron the paper. The proofs of the results as well asmore information about finite-valued t-norms canbe found in [11]. The definitions of indistinguisha-bility operator, proximity and transitive closure arealso recalled.The study of operators defined on a finite chainL is of great interest, especially because reasoningis usually done by using linguistic terms or labelsthat are totally ordered. For instance, the size ofan object can be granularized in very small, small,average, big, very big. If an operator T is defined onthis set, then we will be able to combine these labelsin order to obtain for example T (average, very big).The calculations are simplified greatly by address-ing the problem of combining labels in this way,since there is no need to assign numerical values tothem or to identify them with an interval or with afuzzy subset.Finite chains are also useful in cases in which thevalues are discrete by nature or by discretization.On a customer-satisfaction survey, respondents maybe asked to describe their satisfaction with a serviceusing natural numbers from 0 to 5 or labels rangingfrom not at all satisfied to very satisfied.In this line, various authors have translated t-norms and t-conorms to finite chains ([10], [11]) ob-taining interesting theoretical results.Let L be a finite totally ordered set with mini-mum e and maximum u.Definition 2.1. A binary operation T : L×L→ Lis a t-norm if and only if for all x, y, z ∈ L1. T (x, y) = T (y, x)EUSFLAT-LFA 2011 July 2011 Aix-les-Bains, France© 2011. The authors - Published by Atlantis Press 4932. T (T (x, y), z) = T (x, T (y, z))3. T (x, y) ≤ T (x, z) whenever y ≤ z4. T (x, u) = x.The set of t-norms on a finite chain depends onlyon its cardinality. For this reason we will only con-sider the chains L = {0 = 0n , 1n , 2n , ..., nn = 1}.Example 2.2.1. The Minimum t-norm T on L is defined byT (i, j) = min{i, j}.2. The  Lukasiewicz t-norm T on L is defined byT (i, j) = max{i + j − 1, 0}.Definition 2.3.• A map f : L→ L is smooth if and only if0 ≤ f(i + 1)− f(i) ≤ 1nfor all i ∈ L, i < 1.• A map F : L× L→ L is smooth if and only ifit is smooth with respect to both variables.Smooth t-norms on finite chains are the equiva-lent of continuous ones defined on [0,1].Definition 2.4. A t-norm T on L is divisible if andonly if for all i, j ∈ L with i ≤ j there exists k ∈ Lsuch thati = T (j, k).Smoothness and divisibility are equivalent con-cepts for t-norms.Proposition 2.5. A t-norm on L is smooth if andonly if it is divisible.The next proposition characterizes all smooth t-norms on L as particular ordinal sums of copies ofthe t-norm of  Lukasiewicz.Proposition 2.6. A t-norm T on L is smooth ifand only if there exists J = {0 = i0 < i1 < ... <im = 1} ⊆ L such thatT (i, j) = max{ik, i + j − ik} if i, j ∈ [ik, ik+1]for some ik ∈ Jmin{i, j} otherwise.T is said to be an ordinal sum and is represented byT =< 0 = i0, i1, ...im = 1 >.Indistinguishability operators fuzzify the con-cepts of crisp equality and crisp equivalence rela-tion. They have been studied under different set-tings, mainly valued on [0,1] and with respect toa left continuous t-norm, though some generaliza-tions to more general structures like GL-monoidshave been carried on.Definition 2.7.• A fuzzy relation is a map X ×X → [0, 1]• An L-relation is a map X ×X → L.Definition 2.8. Given a t-norm T , a T -indistin-guishability operator (L-indistinguishability opera-tor) E on a set X is a fuzzy relation (an L-relation)on X satisfying for all x, y, z ∈ X1. E(x, x) = 1 (Reflexivity)2. E(x, y) = E(y, x) (Symmetry)3. T (E(x, y), E(y, z)) ≤ E(x, z) (T -transitivity).Definition 2.9. A fuzzy relation (L-relation) R ona set X is a proximity relation if and only if it isreflexive and symmetric.Definition 2.10. Let T be a t-norm on L (on [0, 1])and R a proximity relation on a set X. The T -transitive closure of R is the L-indistinguishabilityoperator (T -indistinguishability operator) E on Xsatisfying1. R ≤ E.2. If E′ is an L-indistinguishability operator (T -indistinguishability operator) on X such thatR ≤ E′ ≤ E, then E′ = E.3. Transitive OpeningsIn this section a method for calculating all the tran-sitive openings of a reflexive and symmetric L′-relation on a finite set X will be explained. Thisresult will then be exploited to obtain transitiveopenings with respect to the  Lukasiewicz t-normof a given reflexive and symmetric fuzzy relation(valued in the unit interval). The method is veryeffective when combined with heuristic algorithmssuch as the ones proposed in [3] (see Example 3.8).Definition 3.1. Let R be a proximity relation on aset X and T a t-norm on L. A T -indistinguishabil-ity operator (L-indistinguishability operator) R onX is a T -transitive opening (L-transitive opening)of R if and only if• R ≤ R• If E is another T -indistinguishability operator(L-indistinguishability operator) on X satisfy-ing E ≤ R, then E ≤ R.Proposition 3.2. Let R be a proximity relation ona finite set X = {r1, r2, ..., rs} of cardinality s and Ta t-norm. S is a T -indistinguishability operator (L-indistinguishability operator) smaller than or equalto R if and only if its entries satisfy the followingsystem of inequalities:0 ≤ S(ri, rj) ≤ R(ri, rj)for all i, j = 1, 2, ..., s.T (S(ri, rj), S(rj , rk)) ≤ S(ri, rk)for all i, j, k = 1, 2, ..., s.S(ri, rj) = S(rj , ri)for all i, j = 1, 2, ..., s.Proof. Trivial.494Example 3.3. Let us consider the reflexive andsymmetric L-relation R on X = {a, b, c} with L ={0, 13 , 23 , 1}.R =1 23 023 1230 23 1 .An L-relation S on X with matrixS =1 p qp 1 rq r 1is an L-indistinguishability operator smaller than orequal to R if and only if0 ≤ p ≤ 230 ≤ q ≤ 00 ≤ r ≤ 23T (p, q) ≤ rT (p, r) ≤ qT (q, p) ≤ rT (q, r) ≤ pT (r, p) ≤ qT (r, q) ≤ p.If T is the t-norm of  Lukasiewicz, then there are 8possible solutions:p = 0,13, q = 0, r = 0,13,23p =23, q = 0, r = 0,13.Among them, there are 2 L-transitive openings ofR. Namely1 13 013 1230 23 1 1 23 023 1130 13 1 .This exemplifies how the transitive openings of areflexive and symmetric L-relation can be obtained.The method is very greedy and in general wouldneed many calculations. Fortunately, better lowerbounds can be found for the entries of the matri-ces. For example, it is well known ([15]) that theinfimum of the L-indistinguishability operators gen-erated by the columns of a reflexive and symmetricL-relation R is always smaller than or equal to R.In the previous example, this infimum is1 13 013 1130 13 1and hence the first three inequalities can be replacedby13≤ p ≤ 230 ≤ q ≤ 013≤ r ≤ 23.Better lower bounds can be found using exist-ing heuristic methods (cf [3], for example) to cal-culate lower approximations of reflexive and sym-metric fuzzy relations by T -transitive ones. This isillustrated by Example 3.8.The previous results can be applied to obtaintransitive openings of reflexive and symmetric fuzzyrelations valued in the unit interval as will be provedin Theorem 3.7.Definition 3.4. For α ∈ [0, 1], let bαc be the great-est value of L satisfying bαc ≤ α.Lemma 3.5. Let R be a reflexive and symmetricL-relation on a finite set X. If R is an L-transitiveopening of R with respect to the  Lukasiewicz t-normT on L and S is a T -transitive opening of R asa fuzzy relation valued on [0, 1] with R ≤ S, thenR(x, y) = bS(x, y)c for all x, y ∈ X.Proof. bSc is an L-indistinguishability operatorgreater than or equal to R. Since R is a transitiveopening of R, R = bSc.Lemma 3.6. Let L = {0, 1n , 2n , ..., 1}, R be a re-flexive and symmetric L-relation on a finite set Xof cardinality s and R an L-transitive opening ofR with respect to the  Lukasiewicz t-norm T . Givena, b ∈ X such that R(a, b) < R(a, b) and 0 <  < 1n ,let us consider the following fuzzy relation R′:R′(x, y) = R(a, b) +  if (x, y) = (a, b)or (y, x) = (a, b)R(x, y) otherwise.Then the T -transitive closure R′ of R′ is not a tran-sitive opening of R in [0, 1].Proof. Let us consider the L-relation R′′ on X.R′′(x, y) = R(a, b) +1n if (x, y) = (a, b)or (y, x) = (a, b)R(x, y) otherwise.Since R is a transitive opening of R in L, thetransitive closure R′′ of R′′ is not smaller than orequal to R. Therefore there exist x, y ∈ X withR′′(x, y) > R(x, y).Since X is finite of cardinality s,R′′(x, y)= maxz1,...,zs−1T (R′′(x, z1), R′′(z1, z2), ..., R′′(zs−1, y))and this maximum is attained when it containsR′′(a, b). Then there exist z1, z2, ..., zs−3 ∈ X suchthatR′′(x, y)= T (R′′(x, z1), R′′(z1, z2), ..., R′′(zi, a), R′′(a, b),R′′(b, zi+1), R′′(zs−3, y))= T (R(x, z1), R(z1, z2), ..., R(zi, a), R(a, b)+1n,R(b, zi+1), R(zs−3, y)).495T (R(x, a), R(a, b) +1n,R(b, y))≥ T (R′′(x, z1), R′′(z1, z2), ..., R′′(zi, a), R′′(a, b),R′′(b, zi+1), R′′(zs−3, y))= R′′(x, y) > R(x, y)and thereforeT (R(x, a), R(a, b), R(b, y)) = R(x, y) = R(x, y).So,R′(x, y) ≥ T (R(x, a), R(a, b) + , R(b, y)) > R(x, y).and therefore R′ is not a transitive opening of R.Theorem 3.7. Let R be a reflexive and symmet-ric L-relation on a finite set X of cardinality sand R an L-transitive opening of R with respect tothe  Lukasiewicz t-norm T . Then R is also a T -transitive opening of R as a fuzzy relation.Proof. Let us consider a fuzzy relation S on X suchthatR < S.Then there exists a, b ∈ X with R(a, b) < S(a, b).The fuzzy relation R′ of the previous Lemma 3.6with  ≤ S(a, b)−R(a, b) satisfiesR < R′ ≤ S.Since the transitive closure R′ of R′ is greater thanR and the map that assigns the transitive closureto a given reflexive and symmetric fuzzy relationis non-decreasing, a fortiori S is greater than R.Therefore there are no T -indistinguishability oper-ators between R and R.The next example shows the effectiveness of theproposed method for finding transitive openingswhen combined with heuristic algorithms.Example 3.8. Let us consider the reflexive andsymmetric fuzzy relation R on a set of cardinality 5with matrixR =1.0 0.5 0.5 0.7 0.60.5 1.0 0.8 0.7 0.50.5 0.8 1.0 0.9 0.80.7 0.7 0.9 1.0 0.90.6 0.5 0.8 0.9 1.0 .In [3] an algorithm is used to calculate a T -indistinguishability operator S (T the  Lukasiewiczt-norm) which is not a transitive opening of R butsmaller than R and close to a transitive opening:S =1.0 0.5 0.5 0.6 0.60.5 1.0 0.8 0.6 0.50.5 0.8 1.0 0.8 0.70.6 0.6 0.8 1.0 0.90.6 0.5 0.7 0.9 1.0 .R (and S) can also be considered L′-relations whereL′ = {0, 110 , 210 , ..., 1}. There are only 2 intermedi-ate T -indistinguishability operators valued in L thatcan be obtained in the same way as after the Exam-ple 3.3: 1.0 0.5 0.5 0.6 0.60.5 1.0 0.8 0.6 0.50.5 0.8 1.0 0.8 0.70.6 0.6 0.8 1.0 0.90.6 0.5 0.7 0.9 1.01.0 0.5 0.5 0.7 0.60.5 1.0 0.8 0.6 0.50.5 0.8 1.0 0.8 0.70.6 0.6 0.8 1.0 0.90.6 0.5 0.7 0.9 1.0 .The second matrix is a transitive opening of R asan L-relation and therefore also as a fuzzy relation.4. Concluding RemarksA way to obtain all the transitive openings of a re-flexive and symmetric L-relation is provided thathas been used to obtain transitive openings of agiven reflexive and symmetric fuzzy relation.The algorithm for calculating the transitive open-ings of reflexive and symmetric fuzzy relationswith respect to the t-norm  L of  Lukasiewicz canbe used for any other continuous Archimedean t-norms. Indeed, if T is a continuous Archimedeant-norm with additive generator t and E is a T -indistinguishability operator on a finite set X, thenE′ = α−α◦ t◦E is an  L-indistinguishability opera-tor on X [2]. In this way we can transfer a transitiveopening with respect to the  Lukasiewicz t-norm toa transitive opening with respect to another contin-uous Archimedean t-norm.Acknowledgments.Research partially supported by project numberTIN2009-07235.References[1] D. Boixader, J. Jacas, J. Recasens (2000).Fuzzy Equivalence Relations: Advanced Ma-terial. In Dubois, Prade Eds. Fundamentals ofFuzzy Sets, Kluwer, 261-290.[2] D. Boixader, J. Recasens (2010). Indistinguish-ability operators with respect to different t-norms. Submitted.[3] B. De Baets, H. De Meyer (2003). Transitiveapproximation of fuzzy relations by alternatingclosures and openings. Soft Computing 7 210-219.[4] P. Dawyndt, H. De Meyer, B. De Baets (2005).The complete linkage clustering algorithm re-visited. 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