Computing a Transitive Opening of a Reflexive and Symmetric Fuzzy Relation

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1 Computing a Transitive Opening of a Reflexive and Symmetric Fuzzy Relation Luis Garmendia and Adela Salvador 2 Facultad de Informática, Dpto. de Lenguajes y Sistemas Informáticos, Universidad Complutense of Madrid, Madrid, Spain lgarmend@fdi.ucm.es 2 E.T.S.I. Caminos Canales y Puertos, Dpto. de Matemática Aplicada, Technical University of Madrid, Madrid, Spain ma09@caminos.upm.es Abstract. There are fast algorithms to compute the transitive closure of a fuzzy relation, but there are only a few different algorithms that compute transitive openings from a given fuzzy relation. In this paper a method to compute a transitive opening of a reflexive and symmetric fuzzy relation is given. Even though there is not a unique transitive opening of a fuzzy relation, it is proved that the computed transitive opening closure is maximal. Introduction The transitivity property of fuzzy relations can be understood as a threshold of a degree of relation (for example, a degree of equality) between two elements, when a degree of relation between those elements with a third one in a universe of discourse is known. The classical concept of transitivity is generalised in fuzzy logic by the T- transitivity property of fuzzy relations. Fuzzy relations are useful to represent degrees of relations between elements of a universe, and can be used to obtain consequences from a set of premises by the use of the fuzzy compositional rule of inference. Some properties of fuzzy relations give a lot of information of how the consequences are going to be. For example, when an inference is done making a fuzzy composition of a fuzzy set with a reflexive and T-transitive fuzzy relation (called T-preorder), the output contains all the inferable information. The consequences C(A) drawn by making fuzzy inference from a fuzzy set A with T- preorders are Tarski consequences that verify the fuzzy inclusion, so A C(A), monotony, so if A B then C(A) C(B), and idempotence, so C(C(A)) = C(A). Similarities can be used to represent the concept of equality, neighbourhood, generalising the classical equivalence relations. In fact, the α-cut of a similarity is a classical equivalence relation for any value α. Some applications of similarities can be found in some classification and clusterization methods to distinguish and classify objects. Fuzzy relations on a finite set can also represent labelled directed graphs. Symmetric fuzzy relations can represent weighted complete undirected graphs where the set L. Godo (Ed.): ECSQARU 2005, LNAI 357, pp , Springer-Verlag Berlin Heidelberg 2005

2 588 L. Garmendia and A. Salvador of nodes is the universe of discourse and the weighs of the edges are the relationship degrees. Given a fuzzy relation, it is well known that a unique transitive closure exists. Some proposed algorithms to compute the transitive closure of a fuzzy relation are given in Dunn [974], Kander and Yelowitz [974], Larsen and Yager [990], Guoyao Fu [992], Lee [200], Naessens, De Meyer and De Baets [2002]. An algorithms to compute T-transitive openings of fuzzy relations for any t-norm T and any fuzzy relation is given by Garmendia and Salvador [2000]. Other algorithms are given by Baets [2003] and Dawyndt [2003]. There are transitive opening of a fuzzy relation, but in general the highest transitive opening cannot be found. This paper puts forward the existence of a maximal Mintransitive opening from a reflexive and symmetric fuzzy relations, which is not unique, but there is not a transitive fuzzy relation that contains the opening and is contained in the fuzzy relation. It is given an algorithm to compute it and it is proved that such transitive opening is maximal. 2 Preliminaries Let E = {a,..., a n } be a finite set. Given a fuzzy relation R: E E [0, ], let a ij be the value of the relation degree of the elements a i and a j in E. So a ij = R(a i, a j ). A fuzzy relation R is reflexive if a ii = for all i n. The relation R is symmetric if a ij = a ji for all i, j n. Definition 2.. A fuzzy relation R: E E [0, ] is transitive (or Min-transitive) if Min(R(a, b), R(b, c)) R(a, c) for all a, b, c in E. So Min(a ik, a kj ) a ij for all i, j n. Definition 2.2. A reflexive and symmetric fuzzy relation is called a proximity relation. A similarity is a reflexive, symmetric and min-transitive fuzzy relation. Definition 2.3. The relation A includes the relation B (A B) if a ij b ij for all i, j n. Definition 2.4. Given a t-norm T and a fuzzy relation B on a finite universe there exists a unique fuzzy T-transitive relation A, called the T-transitive closure of B, that includes B, and if a fuzzy T-transitive relation includes B then it also includes A. Definition 5. Given a reflexive and symmetric fuzzy relation A on a finite universe, the a transitive opening of A is a fuzzy similarity relation B satisfying: B is included in A (B A) If any fuzzy similarity relation H includes B and is included in A then it is B. (If H; B H A then H = B). Note that it can be several maximal transitive openings of a fuzzy relations, as it is shown in figure :

3 Computing a Transitive Opening of a Reflexive and Symmetric Fuzzy Relation 589 transitive closure T R R Other transitive aproximations of R not comparable with R by the set inclusion transitive openings R T... R T n R T i Fig.. Relation of the T-transitive closure, T-transitive openings and other T-transitive approximations not comparable by In this paper it is proven that in the case of reflexive and symmetric fuzzy relations and t-norm minimum, there exists at least a maximal transitive opening. It also provides an algorithm to compute the maximal transitive opening of a reflexive and symmetric fuzzy relation. Lemma 2.. Let π be a permutation on E. If A is a similarity then the fuzzy relation P π (A) is also a fuzzy similarity. Proof. It is obvious. P π (A) is reflexive and symmetric. If a ij Min{a ik, a kj } for all i, j, k then a rs = a π(i) π(j) Min{a π(i) k, a kπ(j) } = T{a rk, a ks } for all r, s, k Example. Let π be the permutation. An example of a similarity A and its permuted similarity P π (A) is the following: 0,5 0,6 0,6 0,5 0,5 0, 6 A =, P π (A) = 0, 6 0,5 0,5 0,5 0,5 0,6 0,6 0,5 0,6 0,6 0,5 0,5 0,5 0,5 A method to build a similarity of lower dimension is given. This method allows to have an easier understanding of the algorithm to compute a transitive closure given at the end of this paper.

4 590 L. Garmendia and A. Salvador As the permutations of similarities are also similarities, it is possible to sort the elements of the universe of discourse E to decompose a similarity in boxes of subsimilarities. 3 Construction of a Fuzzy Similarity from Subsimilarities of Lower Dimension Let C and D be two similarities with dim(c) = n and dim(d) = n 2. A similarity relation R(F; C, D) of dimension n + n 2 can be constructed with the following form: C R (F; C, D) = F F T D A method for giving the bridging values e ij in F, (when j n < i ) is the assignation of a unique value, f, in all the n n 2 values if F. This value must be chosen in an interval [0, a] where a = min{min(c), min(d) }. The values in F T are the symmetric values f of the computed F. So the computed values in F are equal and satisfy that f = e ij min{min(c), min(d)}. Lemma 3.. If C and D are fuzzy similarities, then R(f; C, D) is also a fuzzy similarity, f [0, min(min(c), min(d))]. Proof. The proof is in Lee [200]. Example 2. The similarity given in example is constructed from other subsimilarities. Let T = Min, let C = and D = () be two similarities. The construction of R(F; C, D) is given by assigning equal values to a 3 and a 32 in the interval [0, ]. Those values can be, for example, a 3 = a 32 = 0,6 = f. Then R(f; C, D) = 0,6 0,6 0,6 0,6 0,6 Now let C2 = 0,6 and D = (), then the new values in F must be chosen equal in the interval [0, 0,6]. If a 4 = a 42 = a 43 = 0,5 = f is chosen then 0,6 0,6 0,6 0,5 R2(f; C2, D) = 0,6 0,5, which is the similarity given in example. 0,6 0,6 0,5 0,5 0,5 0,5

5 Computing a Transitive Opening of a Reflexive and Symmetric Fuzzy Relation 59 The lowest similarity constructed from C and D is the following: 0 R3(f; C, D) = 0 and R4(f; R3, D) = The highest similarity constructed from C and D is the following: R5(f; C, D) = and R6(f; R5, D) = Lemma 3.2. If B n n is a fuzzy similarity, then there exists a decomposition such that B n n = P π (R(f; C n n, D n2 n2 )). Proof. The proof is in Lee [200]. 4 Generation and Decomposition of a Given Similarity A similarity with dimension greater than one can be generated from two subsimilarities. If those subsimilarities are also of dimension greater than one, it is possible to decompose them on other subsimilarities, and so on. The following method of reasoning generates a given similarity using this decomposability concept in a reverse order. A first similarity of dimension two is created by using the greatest non-diagonal values and more similarities keep adding in such a way that the desired similarity is obtained. Method to Generate a Given Similarity Let A be a similarity on an universe E. Let U(A) be the set of the upper triangular values of A sorted in a decreasing order. The method gives sub similarities B k on the elements of E with the highest values in A. First step: Let a ij be the highest value in the list U(A). The first dimension 2 similarity A is created on the Cartesian Product of {e i, e j }, forcing it to be reflexive and symmetric. So A = B = a ij a ij Step k: Let a ij k be the highest value in U(A) not already computed. Then a similarity B k+ and other similarity A k whose dimension n k depends on the position (i, j) of a ij k are created from B k. Such position defines a partition of the subset of natural numbers

6 592 L. Garmendia and A. Salvador E k = {, 2,..., n +n 2 + +n k } into two disjoint sets I and I in a way that the elements a ij in B k- verify that (i, j) I I, the elements of the new similarity box A k verify that (i, j) I I, and so the elements in the bridging box F are f ij = b ij, where (i, j) I I. B The step 2 makes a similarity B 2 with the shape A 2 And step k makes a similarity with the shape B k- A k B F B F k- F A F A T T k 2 or with the shape or with the shape The sets I and I are defined from the indexes (i, j) of the chosen highest not computed a ij k in U(A) as follows: I = {j; b rj is computed in B k- } and I = {i; b is is computed in B k- }. As the generated similarities B k must be always reflexive, it must be considered that all the values of the diagonal of B k are already computed. Then the elements of F and F T can be computed in every step as follows: Set b ij = b ji = min{a ij, where i I and j I }, for all (i, j) I I. Example 3. Let A be the similarity given by the following matrix: A = 0,3 0,3 0,7 0,3 0,7 0,3 0,3 0,3 0,3 0,3 To generate A (or to decompose A in similarities) the greatest element in U(A), which is = a 2 is chosen, so A = = B. In the second step the second greatest element in U(A), which is 0,7 = a 34 is chosen, so I = {j; b 3j is computed in the new B}={3} and I = {i; b i4 is computed in the new B}={4}. Then b 34 = b 43 = min {a ij, where i I and j I } = 0,7 for all (i, j) I I. So B 2 = 0,7 0,7

7 Computing a Transitive Opening of a Reflexive and Symmetric Fuzzy Relation 593 In the next step the next greatest element in U(A), which is = a 3 is chosen, so I = {j; b j is computed in B 2 }={, 2} and I = {i; b i3 is not computed in B 2 }={3, 4}. Then b ij = b ji = min {a ij, where i I and j I } = for all (i, j) I I. So B 3 = 0,7 0,7 In the next step, the greatest element in U(A) is 0,3 = a 5, so I={5} and I ={, 2, 3, 4} So B 4 = 0,3 0,3 0,7 0,7 0,3 0,3 0,7 F 0,3 0,3 = A. 0,3 0,3 () 0,7 F T = 5 Algorithm to Compute a Maximal Transitive Opening of a Reflexive and Symmetric Fuzzy Relation Input: a reflexive and symmetric fuzzy relation A = [aij] Output: a similarity B that is a transitive opening of A Step. Set B to be initially blank. Step 2. Sort the elements of U(A) in descendent order. Step 3. Set b ii = for i from to n. Step 4. While there is a blank in B do Let a rs be the highest value of the list U(A). If b rs is blank, Let I = {j; b rj is not blank} and I = {i; b is is not blank}. Let f = Min{a ij, i I, j I }. Set b ij = b ji = f where i I and j I. Delete the highest value from U(A).

8 594 L. Garmendia and A. Salvador Example 4. Given the following proximity fuzzy relation: 0,7 0,8 A = 0,7 0, 2 0,3 0,8 0, 2 0,7 0,3 0, 7 The algorithm is applied to compute a transitive opening B as follows. Step : Set B to be blank Step 2: Let U(A) be the set of elements of the upper triangular matrix of A sorted in descending order. U(A) = {; 0,8; 0,7; 0,7; 0,3; 0,2}. Step 3: Set b ii = for all i. Step 4: The greatest value of U(A), a 4 =, is taken. Let I = {j; b j that are not blank values in matrix B} = {} and let I = {i; b i4 that are not blank in matrix B} = {4}. The values b 4 = b 4 = a 4 = are computed in B. B = The next highest element in U(A) is 0,8 = a 3. I = {j; b j are not blank in B} = {, 4} and I = {i; b i3 is not blank in B} = {3} are defined and the values b 3, b 43 and its symmetric values, having b 3 = b 43 = Min{a ij, i I, j I } = Min{0,8; 0,7} = 0,7 are computed in B. B = 0,7 0,7 0,7 0,7 The next non-blank highest element in U(A) is 0,7 = a 2. I = {j; b j are not blank in B} = {, 3, 4} and I = {i; b i2 is not blank in B } = {2} are defined and the values b 2, b 32, b 42 and its symmetric values, having b 2 = b 32 = b 42 = Min{a ij, i I, j I } = Min{0,7; 0,3; 0,2} = 0,2 are computed in B. So B = 0,2 0,7 0,2 0,2 0,2 0,7 0,2 0,7 0,2 is a transitive opening of A. 0,7

9 Computing a Transitive Opening of a Reflexive and Symmetric Fuzzy Relation 595 An easier aspect of the similarity is shown making boxes of subsimilarities after 0,8 0, applying the permutation π =, then P π (A)= 0,7 0, ,8 0,7 0,2 0,7 0,3 0,2 and a maximal transitive opening is P π (B) = 0,7 0,2 0,7 0,7 0,7 0,2 0,2 0,2 0,2 0,2 Example 5. Given the following proximity fuzzy relation: 0, 0,2 0,5 A = 0, 0, 4 0, 0, 2 0, 4 0, 0, 5 0, 0, The algorithm is applied to compute a transitive opening B. Step : Set B to be blank Step 2: Let U(A) be the set of elements of the upper triangular matrix of A, sorted in descending order. U(A) = {0,5; ; 0,2; 0,; 0,; 0,}. Step 3: Set b ii = for all i. Step 4: The highest value of U(A), a 4 = 0,5 is taken t. Let I = {j; b j that are not blank values in matrix B} = {} and let I = {i; b i4 that are not blank in matrix B} = {4}. The values b 4 = b 4 = a 4 = 0,5 are computed in B. B = 0,5 0,5 The following highest element in U(A) is = a 23. I = {j; b 2j are not blank in B} = {2} and I = {i; b i3 is not blank in B} = {3} are defined. It is computed in B the value b 23 and its symmetric value, having b 23 = a 23 =. B = 0,5 0,5

10 596 L. Garmendia and A. Salvador The following non-blank highest element in U(A) is 0,2 = a 3. I = {j; b j are not blank in B} = {, 4} and I = {i; b i3 is not blank in B } = {2, 3} are defined. The values b 2, b 3, b 42, b 43 and its symmetric values, having b 2 = b 3 = b 42 = b 43 = Min{a ij, i I, j I } = Min{0,; 0,2; 0,; 0,} = 0, are computed in B. B = 0, 0, 0,5 0, 0, 0, 0, 0,5 0, 0, An easier view of the similarity is shown after applying the permutation π = 0,5 0,2 0, 2 3 4, then P π (A)= 0,5 0, 0, and a maximal transitive opening is ,2 0, 0, 0, P π (B) = 0,5 0, 0, 0,5 0, 0, 0, 0, 0, 0, The following lemmas show that the previous algorithm gives a maximal transitive opening of a reflexive and symmetric fuzzy relation. Lemma 5.. The output of the Algorithm applied to a reflexive and symmetric fuzzy relation is a fuzzy similarity relation Proof. The proof is trivial from lemma 3. and lemma 3.2. Lemma 5.2. Let A be a reflexive and symmetric fuzzy relation, and let B be the output of the previous algorithm applied to A. If any fuzzy similarity relation H includes B and is included in A then it is B (for all similarity H, if B H A then H = B. Proof. Let H = (h ij ) be a fuzzy similarity relation such that B H A. So b ij h ij a ij for all i, j. If B H then (r, s) such that b rs < h rs a rs. (2) Let I, I be the set of indexes given by the algorithm in the step in which b rs is generated. Then (r, s) I I. As B is computed by the algorithm, b rs is generated from the value of some a kl. (k, l) I I such that b kl = a kl = f = Min{a ij, i I, j I } = b rs. (3) H is transitive, so h kl max{min{h k, h l },..., min{h kn, h nl }}

11 Computing a Transitive Opening of a Reflexive and Symmetric Fuzzy Relation 597 B H A, and the values in the indexes I I are always lower than the values in I I and I I, so h kl max{min{h k, h l },..., min{h kn, h nl }} = max{h kj, h il } i I, j I For i = r it is held that h kl h rl, so h rl max{min{h r, h l },..., min{h rn, h nl }} = max{h rj, h il } i I, j I In particular, for j = s, it is held that h kl h rl h rs. But H A, so h rs h kl a kl = b kl = b rs. This is contradictory to (2). Thus, any fuzzy similarity H such that B H A verifies that H = B. Therefore, the algorithm computes a maximal similarity from a reflexive and symmetric fuzzy relation, which is a transitive opening. Lemma 5.3. A maximal transitive opening of a reflexive and symmetric fuzzy relation can be computed in O(n 2 log n) time in the worst case. Proof. The computational complexity of the time consumed by the given algorithm is analysed as follows: Step 2 sorts n 2 n 2 values, so it takes O(n 2 log n) time. The loop in step 4 iterates at most n- times, so it iterates O(n) times in the worst case. When a dimension box is added in an iteration, the maximum number of computed bridging elements is n-, so the computation of the bridging elements takes O(n) time in the worst case. Hence the total time spent in step 4 when all new boxes are of dimension one is O(n 2 ) time in the worst case. When the new box has dimension k, then the computation is on (n-) k values in the worst case, which have the same computational complexity than computing sorting at most (n-) values k times, so the total computational complexity of step 4 is O(n 2 ) time in the worst case. Therefore, the complexity of the time consumed by the algorithm is O(n 2 log n) time. 6 Conclusions Giving a proximity relation on a finite universe, there exists a unique transitive similarity (called transitive closure) that contains it and that is contained in any similarity that contains the proximity relation. It is well know how to compute the transitive closure of a fuzzy relation, but there exists several maximal similarities that are contained in the original proximity relation, that are called transitive openings. An open problem is the computation of transitive openings, but in general there is not a unique solution, and so in general it is not possible to find the highest transitive opening a given fuzzy relation. In this paper it is proven that in the case of reflexive and symmetric fuzzy relations (proximity relation) that there exists at least a transitive opening (a maximal similarity relation). An O(n 2 log n) time algorithm to compute a transitive opening of a prox-

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