Transitive closure and betweenness relations

Fuzzy Sets and Systems 120 (2001) 415–422

www.elsevier.com/locate/fss

Transitive closure and betweenness relations Dionis Boixader a , Joan Jacas b , Jordi Recasens a ; ∗ a

Sec. Matem atiques i Informatica, E.T.S.A.V., Universitat Politecnica de Catalunya, Pere Serra 1-15, 08190 Sant Cugat, Prov. Barcelona, Spain b Sec. Matem atiques i Informatica, E.T.S.A.B. Avda. Diagonal 649, 08028 Barcelona, Spain Received 23 July 1998; received in revised form 9 July 1999; accepted 13 July 1999

Abstract Indistinguishability operators fuzzify the concept of equivalence relation and have been proved a useful tool in theoretical studies as well as in di0erent applications such as fuzzy control or approximate reasoning. One interesting problem is their construction. There are di0erent ways depending on how the data are given and on their future use. In this paper, the length of an indistinguishability operator is de2ned and it is used to relate its generation via max-T product and via the representation theorem when T is an Archimedean t-norm. The link is obtained taking into account that indistinguishability c 2001 Elsevier Science operators generate betweenness relations. The study is also extended to decomposable operators.
B.V. All rights reserved. Keywords: Indistinguishability; Dimension; Archimedean t-norm; Betweenness relation; Transitive closure

1. Introduction Indistinguishability operators are an important theoretical tool that generalizes the concepts of equivalence relation and equality to the fuzzy context. They are also called similarities [13] and fuzzy equalities [7], and measure the degree of indistinguishability or similarity between the objects of a given universe of discourse. They seem an interesting concept in order to provide a sound theoretical framework in many 2elds and have been proved useful in di0erent problems [1,5,7].

∗

Corresponding author. E-mail addresses: [email protected] (D. Boixader), jacas@ ea.upc.es (J. Jacas), [email protected] (J. Recasens).

A challenging question in their study is the diverse ways to generate them. They depend on the way the data are given and on the use you want to make of them. One of the most common tools used in the construction of indistinguishability operators is the maxT product, which allows us to construct the transitive closure of a given re>exive and symmetric fuzzy relation (also called tolerance relation). Another way to generate indistinguishability operators is by using the Representation Theorem 2:12 that allows the construction of these relations starting on from a family of fuzzy sets. For instance, if we have a set X of objects to be classi2ed and some fuzzy subsets of X expressing the degree of ful2llment of these objects to some criteria, the theorem provides a method to generate an indistinguishability operator on X in a natural way.

c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 9 9 ) 0 0 1 3 3 - 5

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A third way to generate such operators is constructing a decomposable relation from a given fuzzy subset (De2nition 2.15), an interesting way related to Mamdami’s approach to fuzzy control. This paper relates these generating methods, when the t-norm T is Archimedean, developing the fact that, in this case, indistinguishability operators generate metric betweenness relations in the sense of Menger [8]. The structure of these relations depends on the length of the operator (De2nition 4.1) and on its dimension (De2nition 2.13). Since the length and the dimension of a T indistinguishability operator are related with its generation via Max-T and via the Representation Theorem 2:12, respectively, betweenness relations are an ideal tool to link both methods. For instance, in Section 4 they are used to prove the equivalence between one dimensionality and maximality of the length. This result can be generalized in the following rule of thumb: “The greater the dimension the smaller the length, and vice versa”. Also in Section 4, the relation with decomposable indistinguishability operators is studied. 2. Preliminaries In this section we recall some concepts related to indistinguishability operators. Denition 2.1. Given a t-norm T , a T -indistinguishability operator E on a set X is a fuzzy relation on X that satis2es • E(x; x) = 1 ∀x ∈ X (re>exivity), • E(x; y) = E(y; x) ∀x; y ∈ X (symmetry), • T (E(x; y); E(y; z))6E(x; z) ∀x; y; z ∈ X (T -transitivity). E separates points when x = y ⇒ E(x; y) = 1. Denition 2.2. Let T be a t-norm and R; S two fuzzy relations in a set X . The max-T (or sup-T ) product R ◦ S of R and S is the fuzzy relation on X de2ned by (R ◦ S)(x; y) = sup T (R(x; z); S(z; y)) ∀x; y ∈ X: z∈X

Assuming T continuous, due to the associativity of max-T product, we can de2ne for each n ∈ N the

power Rn of a fuzzy relation R recursively: R1 = R; Rn+1 = R ◦ Rn

∀n ∈ N:

Denition 2.3. The T -transitive closure (or T -closure) RK of a fuzzy relation R on a set X is de2ned by RK = sup Rn : n∈N

Proposition 2.4. If R is a re6exive and symmetric fuzzy relation on a 7nite set X of cardinality n; then RK = Rn−1 . Proposition 2.5. Let R be a re6exive and symmetric fuzzy relation on a set X . R = RK if and only if ∀x; y; z ∈ X; T (R(x; y); R(y; z))6R(x; z). Therefore, the transitive closure RK of a re>exive and symmetric fuzzy relation is a T -indistinguishability operator. Moreover, it is straightforward to prove that K RK is a relation greater or equal than R (R¿R) and it can be shown [12] that if E is a T -indistinguishability K In other operator greater or equal than R, then E¿R. K words, the transitive closure R of R is the smallest T -indistinguishability operator that contains R and is therefore the best upper approximation of R by means of such operators. In fact, the following proposition can be proved. Proposition 2.6. Given a re6exive and symmetric fuzzy relation R on a set X; let A be the set of T indistinguishability operators on X greater than or equal to R and RK its transitive closure. Then; for any x; y ∈ X; K y) = inf {E(x; y)}: R(x; E∈A

In [12] a representation theorem for T -indistinguishability operators is proved. This theorem states that every T -indistinguishability operator on a set X can be generated by a family of fuzzy subsets of X and, conversely, every family of fuzzy subsets of X generates a T -indistinguishability operator on X in a natural way. These fuzzy subsets can be understood as the degrees of similarity of the elements of X to some prototypes, and this point of view makes the

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theorem useful e.g. in Cluster Analysis and Pattern Recognition.

guishability operator if and only if there exists a family {hj }j∈J of fuzzy subsets of X such that

Denition 2.7. Given a continuous t-norm T , its quasi-inverse or residuum Tˆ is de2ned by

E = inf Ehj :

Tˆ (x|y) = sup{ ∈ [0; 1] | T (; x)6y}

Denition 2.13. The in2mum of the cardinalities of indices of all the generating families of a T -indistinguishability operator E in the sense of the preceding theorem is called the dimension of E. T indistinguishabilities of low dimension are specially desirable, since all their information can be stored in a few fuzzy subsets, and it is worth noting that in [4,5] algorithms are developed that allow the calculation both of the dimension and of such minimal generating families of fuzzy subsets.

for any x; y ∈ [0; 1]: Denition 2.8. Given a continuous t-norm T , its ↔ ↔ symmetrized biresiduum T is de2ned by T (x; y) = min(Tˆ (x|y); Tˆ (y|x)) = Tˆ (max(x; y); min(x; y)) for any x; y ∈ [0; 1]. In the setting of fuzzy logic, Tˆ can be viewed as the residuated implication associated to T , and it is ↔ very common to note Tˆ (x|y) by x → T y and T (x; y) by x ↔T y (the natural equivalence or biimplication) [3,7]. Lemma 2.9 (Valverde [12]). Given a fuzzy subset h of a set X; the fuzzy relation Eh on X de7ned by ↔

Eh (x; y) = T (h(x); h(y)) is a T -indistinguishability operator on X . Example 2.10. (1) If T = L is the Lukasiewicz t-norm (T (x; y) = max{x + y − 1; 0}), then Eh (x; y) = 1 − |h(x) − h(y)|: (2) If T =  is the product t-norm, then      min h(x) ; h(y) if h(x) = h(y); h(y) h(x) Eh (x; y) =  1 otherwise: (3) If T = Min; then  1 if h(x) = h(y); Eh (x; y) = min{h(x); h(y)} otherwise: Lemma 2.11. If {Ej }j∈J is a family of T -indistinguishability operators on a set X; then E = inf j∈J Ej is a T -indistinguishability operator on X . From Lemmas 2.9 and 2.11 it follows: Theorem 2.12 (Representation Theorem, Valverde [12]). A fuzzy relation E on a set X is a T -indistin-

j∈J

A third way to build indistinguishability operators, is by generating decomposable relations by means of a fuzzy set [6,7]. Proposition 2.14. Given a t-norm T and a fuzzy set h on a set X; the fuzzy relation on X de7ned by  T (h(x); h(y)) if x = y; E(x; y) = 1 otherwise is a T -indistinguishability operator. Denition 2.15. A T -indistinguishability operator E on a set X is called decomposable if and only if there exists a fuzzy subset h on X such that E can be expressed as in the preceding proposition. In the sequel, unless otherwise stated, we will assume that T is an Archimedean t-norm and E a separating T -indistinguishability operator on a set X such that for any x; y ∈ X ⇒ E(x; y)¿0. 3. Betweenness relations In this section, it will be proved that, if T is an Archimedean t-norm, every T -indistinguishability operator E that separates points on a set X de2nes a metric betweenness relation on X , and its relation with the dimension and decomposability of E will be stated.

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The notion of metric betweenness relation appears in [8] and it is de2ned as follows: Denition 3.1. A (metric) betweenness relation on a set X is a ternary relation B on X (i.e. B ⊂ X 3 ) satisfying ∀x; y; z; t ∈ X 1. (x; y; z) ∈ B ⇒ x = y = z = x, 2. (x; y; z) ∈ B ⇒ (z; y; x) ∈ B, 3. (x; y; z) ∈ B ⇒ (y; z; x) ∈= B; (z; x; y) ∈= B, 4. (x; y; z) ∈ B; (x; z; t) ∈ B ⇒ (x; y; t) ∈ B; (y; z; t) ∈ B. If (x; y; z) ∈ B, then y is said to be between x and z. If given any three elements of B, one of them is always between the other two, then the betweenness relation is called linear or total. The idea from metric betweenness appeared in the study of metric spaces: If d is a distance de2ned on a set X , the relation “y is between x and z when d(x; y) = d(y; z) + d(x; z)” satis2es the axioms of a betweenness relation and Menger used these relations in the study of isometric embeddings of metric spaces. The fact that indistinguishability operators separating points de2ne betweenness relations (when T is Archimedean) provides them with a metric >avour that allows their study from a metric point of view [5,10]. Theorem 3.2. Let E be a T -indistinguishability operator. The ternary relation B on X de7ned by (x; y; z) ∈ B if and only if x = y = z = x and T (E(x; y); E(y; z)) = E(x; z) is a betweenness relation on X . Proof. 1. Trivial. 2. Follows from the commutativity of T and the symmetry of E. 3. Let us prove for instance that, if (x; y; z) ∈ B, then (y; z; x) ∈= B: If T (E(y; z); E(z; x)) = E(y; x), then T (T (E(y; z); E(z; x)); E(y; z))

Since E(x; z) = 0, this implies that T (E(y; z); E(y; z)) = 1 and also E(y; z) = 1 contradicting the separability of E. 4. Let us prove, for instance, that, if (x; y; z) ∈ B and (x; z; t) ∈ B, then (y; z; t) ∈ B: E(x; t) = T (E(x; z); E(z; t)) = T (T (E(x; y); E(y; z)); E(z; t)) = T (E(x; y); T (E(y; z); E(z; t))) and since T (E(y; z); E(z; t))6E(y; t); monotonicity of T assures that E(x; t) ¿ T (E(x; y); E(y; t)) ¿ T (E(x; y); T (E(y; z); E(z; t))) = E(x; t) and therefore T (E(x; y); E(y; t)) = T (E(x; y); T (E(y; z); E(z; t))) = 0 and since E(x; y)¿0 E(y; t) = T (E(y; z); E(z; t)): The structure of the betweenness relation generated by a T -indistinguishability operator re>ects its combinatorial complexity expressed by its dimension. Theorem 3.3 (Jacas and Recasens [5]). Let E be a T -indistinguishability operator. E is one dimensional if and only if the betweenness relation B determined by E on X is linear. It is worth noting that in [1] a characterization theorem of one-dimensional T -indistinguishability operators for general continuous t-norms is proved that generalizes this result.

and therefore

Corollary 3.4. Let E be a T -indistinguishability operator on a 7nite set X of cardinality n. E is one  dimensional if and only if the cardinality of B is 2 n3 .

T (T (E(y; z); E(y; z)); E(x; z)) = E(x; z):

Proof. Trivial.

= T (E(y; x); E(y; z)) = E(x; z)

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The tight link between the betweenness relation de2ned by a T -indistinguishability operator and its dimension is also shown in the next theorem. Theorem 3.5 (Recasens [10]). Let E be a T -indistinguishability operator on a 7nite set X of cardinality n. If E is bidimensional; then the cardinality of B is greater than or equal to 2T (n; 5; 3) where T (n;  5;3) is a Turects the behaviour of centralized systems where there is a central kernel that controls all information and relations in the system. It turns out that these radial relations characterize decomposable indistinguishability operators (Theorem 3.7). Denition 3.6. A betweenness relation B on a set X is called radial if and only if there exists an element a ∈ X such that a is between any other two elements of X and these are the only betweennesses of B (i.e. (x; y; z) ∈ B i0 y = a). The element a is called the center of the betweenness relation. Theorem 3.7. Let E be a T -indistinguishability operator and B the betweenness relation generated on X by E. E is descomposable if and only if B is radial or E can be extended to a T -indistinguishability EK on XK = X ∪ {a} with a ∈ X in such a way that the betweenness relation BK generated on XK is radial with center a. Proof. (⇒) Let E(x; y) = T (h(x); h(y)) if x = y. (i) If there exists an x0 ∈ X such that h(x0 ) = 1, then we are in the 2rst case, since (a) given x; y ∈ X with x = y = x0 = x, T (E(x; x0 ); E(x0 ; y)) = T (T (h(x); h(x0 )); T (h(x0 ); h(y))) = T (h(x); h(y)) = E(x; y) and, therefore, x0 is between any other elements of X .

419

(b) If z = x0 and x = z = y = x, then T (E(x; z); E(z; y)) = T (T (h(x); h(z)); T (h(z); h(y))) = T (T (h(z); h(z)); T (h(x); h(y))) = T (T (h(z); h(z)); E(x; y)): Taking into account that h(z)¡1 since E separates points, E(x; y)¿0 and T is Archimedean, the last expression is smaller than E(x; y) and, therefore, z is not between x and y. (ii) If there exists no x0 ∈ X with h(x0 ) = 1, then we can de2ne the fuzzy subset hK of XK by K = h(x) ∀x ∈ X and h(a) K = 1 and consider the h(x) K relation EK on XK generated by h. (⇐) If B is radial with center c, then the fuzzy subset h of X de2ned by h(x) = E(c; x) ∀x ∈ X generates E. Indeed: if x = c = y = x then E(x; y) = T (E(x; c); E(c; y)) = T (h(x); h(y)): In the second case, E is generated by the fuzzy subset K a) ∀x ∈ X . h of X de2ned by h(x) = E(x;

4. The length of a T-indistinguishability operator In this section, the notion of length of a T indistinguishability operator is introduced and its relation with its dimension and decomposability is studied. Denition 4.1. Given a t-norm T and a T -indistinguishability operator E on a set X , the length of E is the maximum k ∈ N (if it exists) such that there exists a re>exive and symmetric fuzzy relation R on X with Rk−1 = Rk = E and length E = ∞ otherwise. Note that length E¿1, since E 1 = E, and if X is 2nite of cardinality n, then length E6n − 1 (Proposition 2.4). In order to simplify the notation and due to the associativity of t-norms, we can de2ne T (a1 ; a2 ; : : : ; an ) inductively by T (a1 ) = a1 , T (a1 ; a2 ; : : : ; an ) = T (T (a1 ; a2 ; : : : ; an−1 ); an ). Lemma 4.2. Let R be a fuzzy relation on a 7nite set X; a; b ∈ X; T a continuous t-norm and k a positive

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integer

From this fact it follows that E(xi ; xj ) = Rj−i (xi ; xj )

Rk (a; b) =

sup

x2 ; :::; xk ∈X

T (R(a; x2 ); R(x2 ; x3 ); : : : ; R(xk ; b)) :

Rn+1 (a; b) = (Rn ◦ R)(a; b)

xn+1 ∈X

sup

for some y2 ; y3 ; : : : ; yk ∈ X . Replacing R(xi ; xi+1 ); : : : ; R(xj−1 ; xj ) by R(xi ; y2 ); R(y2 ; y3 ); : : : ; R(yk ; xj ) in (1) we would have

T (R(a; x2 ); : : : ;

R(xn ; xn+1 )) ; R(xn+1 ; b)

Rm (x1 ; xl+1 )¿Rl (x1 ; xl+1 )

= sup T (R(a; x2 ); : : : ; R(xn+1 ; b)) : x2 ; :::; xn+1

The next lemma will be the cornerstone to relate the length with the dimension and the decomposability. Lemma 4.3. Let E be a T -indistinguishability operator on a 7nite set X and B the betweenness relation de7ned by E. If length E = l; then there exist x1 ; x2 ; : : : ; xl+1 ∈ X such that (xi ; xj ; xk ) ∈ B if i¡j¡k. Proof. If length E = l, then there exists a re>exive and symmetric fuzzy relation R on X such that E = Rl ¿Rl−1 . Therefore, there exist x1 ; xl+1 ∈ X such that Rl (x1 ; xl+1 )¿Rl−1 (x1 ; xl+1 ). On the other hand, by Lemma 4.2, Rl (x1 ; xl+1 ) = T (R(x1 ; x2 ); R(x2 ; x3 ); : : : ; R(xl ; xl+1 )) (1) for some x2 ; x3 ; : : : ; xl ∈ X . Moreover, Rj−i (xi ; xj ) = T (R(xi ; xi+1 ); R(xi+1 ; xi+2 ); : : : ; R(xj−1 ; xj ))

for all j¿i;

since Rj−i (xi ; xj ) = T (R(xi ; yi+1 ); R(yi+1 ; yi+2 ); : : : ; R(yj−1 ; xj )) ¿T (R(xi ; xi+1 ); R(xi+1 ; xi+2 ); : : : ; R(xj−1 ; xj )) for some yi+1 ; : : : ; yj−1 ∈ X would contradict (1).

and therefore,

¿ T (R(xi ; xi+1 ); : : : ; R(xj−1 ; xj ))

x2 ;:::; xn ∈X

E(xi ; xj ) = Rk (xi ; xj )¿Rj−i (xi ; xj )

Rk (xi ; xj ) = T (R(xi ; y2 ); R(y2 ; y3 ); : : : ; R(yk ; xj ))

= sup T (Rn (a; xn+1 ); R(xn+1 ; b)) = sup T

(3)

Since, if this was false, there would exist k¿j − i such that

Proof. The assertion is trivially true for k = 1; 2. Let us suppose the result true for k = n

xn+1 ∈X

∀j¿i:

(2)

contradicting the maximality of l. Now, from (2) and (3) the result follows easily: If i¡j¡k, then T (E(xi ; xj ); E(xj ; xk )) = T (R(xi ; xi+1 ); : : : ; R(xj−1 ; xj ); R(xj ; xj+1 ); : : : R(xk−1 ; xk )) = Rk−i (xi ; xk ) = E(xi ; xk ) and therefore, (xi ; xj ; xk ) ∈ B. Lemma 4.4. Given an Archimedean t-norm T; with t as an additive generator [11]; m ∈ N and a ∈ ]0; 1]; there exists b ∈ [0; 1[; such that T (a; b; :m: : ; b) = 0. Proof. For any x ∈ [0; 1]; T (a; x; :m: : ; x) = t [−1] (t(a) + mt(x)). If mt(x)¡t(0) − t(a), then T (a; x; :m: : ; x) = 0. Therefore, taking b¿t −1 ([1 − t(a)]=m) the lemma follows. As a consequence of these lemmas we can prove the following results. Theorem 4.5. If E is a T -indistinguishability operator on a set X of cardinality n; then E is one dimensional if and only if length E = n − 1. Proof. (⇒) Let us order the elements x1 ; x2 ; : : : ; xn ∈ X in such a way that (xi ; xj ; xk ) ∈ B i0 i¡j¡k or

D. Boixader et al. / Fuzzy Sets and Systems 120 (2001) 415–422

k¡j¡i; where B denotes the total betweenness relation on X generated by E. Let a = E(x1 ; xn ). Due to Lemma 4.4, there exists b ∈ [0; 1[ such that T (a; b; n−2 : : : ; b) = 0. Let us de2ne a re>exive and symmetric fuzzy relation R on X by R(xi ; xj )  if i = j; 1 if |i − j| = 1; = E(xi ; xj )  T (E(xi ; xj ); b; k−1 : : : ; b) if |i − j| = k¿1: R satis2es Rn−2 = Rn−1 = E. (⇐) From Lemma 4.3, the betweenness relation de2ned by E is linear and Theorem 3.3 assures the one dimensionality of E. Theorem 4.6. Let E be a T -indistinguishability operator on a set X of cardinality n and length E = n − 2. Then E is bidimensional. Proof. From Lemma 4.3, all the elements of X but one (a ∈ X ) form a chain x1 ; x2 ; : : : ; xn ∈ X such that (xi ; xj ; xk ) ∈ B if i¡j¡k. Let us prove that the fuzzy subsets h1 and h2 of X de2ned by h1 (x) = E(x1 ; x) and h2 (x) = E(a; x) ∀x ∈ X generate E: Given xi ; xj ∈ X with i; j = 1; 2; : : : ; n − 1 and i¡j, T (E(x1 ; xi ); E(xi ; xj )) = E(x1 ; xj ) or T (h1 (xi ); E(xi ; xj )) = h1 (xj ). Therefore, since h1 (xi )¿h1 (xj ); ↔

E(xi ; xj ) = Tˆ (h1 (xi )|h1 (xj )) = T(h1 (xi ); h1 (xj )): On the other hand, given xi ∈ X i = 1; 2; : : : ; n − 1; ↔

E(a; xi ) = h2 (xi ) = T(h2 (a); h2 (xi )): So, the dimension of E is less than or equal to 2. If E were one dimensional, then the length of E would be n − 1 from the preceding theorem. Therefore, the dimension of E is 2. In the same way, the following result can be proved: Theorem 4.7. Let E be a T -indistinguishability operator on a set X of cardinality n and length E = k. Then the dimension of E is less or equal than n − k.

421

Theorem 4.8. Let E be a T -indistinguishability operator and B the betweenness relation de7ned by E then; length E = 1 if and only if B = ∅. Proof. Lemma 4.3. Theorem 4.9. If E is a decomposable T -indistinguishability operator on a 7nite set X; then length E62. Proof. From Theorem 3.7 it follows that the betweenness relation B associated to E is empty or radial. If the betweenness relation B generated by E is empty, then length E = 1 by Theorem 4.8. If B is radial, then by Lemma 4.3 length E62. 5. Concluding remarks In this paper, the length of a T -indistinguishability operator E is introduced and it is used as a tool to relate di0erent ways to generate such operators when the t-norm is Archimedean using the fact that, in this case, they generate betweenness relations. In particular, it is proved that the maximality of the length is equivalent to one dimensionality, and that decomposable indistinguishability operators are of length at most 2. It is interesting to note that these results cannot be generalized to non-Archimedean t-norms, since in this case indistinguishability operators do not de2ne betweenness relations. For instance,     1 0:5 0:2 1 0:2 0:2  0:5  0:2 1 0:2  ; 1 0:2  0:2 0:2 1 0:2 0:2 1 are two Min-indistinguishability operators of length 2 and the 2rst one is one dimensional while the second one has dimension two [4]. References [1] D. Boixader, ContribuciNo a l’estudi dels mor2smes entre operadors d’indistingibilitat. AplicaciNo al raonament aproximat, Ph.D. Dissertation, UPC, 1997. [2] P. ErdPos, On the combinatorial problems which I would most like to see solved, Combinatoria 1 (1981) 25–42. [3] S. Gottwald, H. Bandemer, Fuzzy Sets, Fuzzy Logic, Fuzzy Methods, Wiley, New York, 1996.

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