Extensions of a Resemblance Relation J.C. Cubero (*), J.M. Medina, O. Pons, M.A. Vila
Dpto. Ciencias de la Computacion e Inteligencia Arti cial Universidad de Granada. 18071 Granada, Spain. Abstract
We consider an arbitrary crisp domain where a resemblance relation is de ned, and we are going to extend it to work with fuzzy values. This will lead us to the concepts of weak and strong resemblance. The extension may be carried out with or without the help of precision levels for the fuzzy values. We show certain equivalence between both extensions, and apply this study to the de nition of fuzzy dependencies1 .
Keywords: Resemblance relation, weak resemblance, strong resemblance, fuzzy dependency Section one introduces the notation followed in this paper, as well as the concept of a resemblance relation de ned in a crisp domain. In the second section we study the extension to the fuzzy case of a resemblance relation using precision levels, and without them in section three. This will lead us to de ne a weak and a strong resemblance measure. Finally, we shall prove that the rst approach (using precision levels) can be considered as a special case of the second one. The paper concludes with an application study on the problem of de ning a fuzzy functional dependency in the framework of a fuzzy relational database.
1 Introduction The study of similarity measures between fuzzy values is of prime interest in a wide range of fuzzy application areas. Some authors de ne similarity between fuzzy values regarding only the membership degrees of each fuzzy set without taking into account any similarity between the crisp elements of the domain (recent studies appear in [4, 6, 7]). A more general case deals with the problem of extending a similarity relation de ned on a crisp domain to compare fuzzy values, giving a degree of resemblance between two fuzzy values ([1, 3, 8, 9, 10, 12, 13]). We are going to consider the second case because of its generality. Usually, re exivity and symmetric properties are needed, whereas other studies require some kind of transitivity. We now proceed to introduce the notation followed in this paper. From now on, a crisp set will be denoted by D, whereas the set of fuzzy subsets de ned in D will be called P (D). We shall use for fuzzy inclusion, _ for fuzzy union and ^ for fuzzy intersection. These symbols will also be used to denote maximum and minimum. To denote the kernel of a fuzzy value F, we shall use ker(F), and F for the ?cut of F. De nition 1 Let D be a crisp domain. A resemblance relation is a binary relation: R : D D ! [0; 1] satisfying re exive and symmetric properties:
R(x; x) = 1 8 x 2 D ; R(x; y) = R(y; x) 8 x; y 2 D De nition 2 Let D be a crisp domain, and R a resemblance relation de ned in D D. Two crisp values x; y 2 D are resemblant at level , if and only if: R(x; y) Now, we proceed to extend these de nitions to work with fuzzy sets. First, we shall use precision levels over the fuzzy values. Work partially supported by Research Project TIC94-1347 Corresponding author. e-mail:
[email protected]
1
(*)
1
2 An Approach Using Levels of Precision In [1] we proposed an extension of de nition 1 using precision levels. We are going to review and complement this work. The basic idea consists of xing levels of precision and comparing the crisp elements belonging to the cuts of the fuzzy values (these measure were elicited from an expert in medicine as a degree of resemblance between two fuzzy values) Let us consider a domain D, and associated to it, a set of levels of precision f0 j 1gj =1:::h and a set of levels of resemblance f igi=1:::h0 . For simplicity's sake, we assume h = h0 , i.e, the same number of precision and resemblance levels.
2.1 Weak Resemblance
De nition 3 Let D be a crisp domain, R a resemblance relation de ned in D D, and two fuzzy values F 2 P (D); F 0 2 P (D). F and F 0 are weakly resemblant at the level of precision i , and level of resemblance i, denoted by
F ii F 0
if and only if
R i (F; F 0) i
where
R i : P (D) P (D) ?! [0; 1] 4 R i (F; F 0) =
sup
x2F i ; y2F 0 i
R(x; y)
(1)
4 reads as is de ned by. The symbol should include the resemblance relation R which Remark. The symbol = is beeing extended (for instance R ); nevertheless, we omit it for simplicity's sake in our notation. To introduce a coherent de nition we must force next restriction:
j > j 0 ) j j 0
(2)
Example 1 Let us consider = 0:8, = 0:7, = 1, = 0:8 and the trapezoidal fuzzy numbers F and F 0, 1
2
1
2
given by the membership functions of gure 1. Let us consider a distance measure R. Therefore R 1 (F; F 0) = R(20; 24) and R 2 (F; F 0) = R(21; 23). Let us suppose R(20; 24) = 0:6 and R(21; 23) = 0:8. Then R 1 (F; F 0) 6 1 and R 2 (F; F 0) 2. We conclude that F and F 0 are weakly resemblant at the level of precision 2 and the level of resemblance 2 (but not for 1 and 1).
Proposition 1 If F ii F 0 then F jj F 0 8 j i 8 j i Proof. Let us suppose j i . Then, F i F j ; F 0i F 0j )
)
sup
x2F j ;y2F 0 j
R(x; y)
sup
x2F i ;y2F 0 i
R(x; y) i
On the other hand j i , and we can apply equation 2; therefore j i. Chaining these terms together we obtain: sup 0 R(x; y) j x2F j ;y2F j
2
2
2.2 Strong Resemblance
De nition 4 Let D be a crisp domain, R a resemblance relation de ned in D D, and two fuzzy values G 2 P (D); G0 2 P (D). G and G0 are strongly resemblant at the level of precision i , and the level of resemblance i , denoted by
G ii G0
if and only if
Ri (G; G0) i
where
Ri : P (D) P (D) ?! [0; 1] 4 R(x; y) inf Ri (G; G0) = x2G ; y2G0 i
i
(3)
Again, the restriction j > j 0 ) j j 0 should be imposed.
Example 2 Let us consider = 0:8, = 0:7, = 1, = 0:7 and the trapezoidal fuzzy numbers given by 1
2
1
2
the membership functions of gure 2. For a distance measure R, R1 (G; G0) = R(61; 63) and R2 (G; G0) = R(60:8; 63:2). Let us suppose R(61; 63) = 0:8 and R(60:8; 63:2) = 0:6. Then R1 (G; G0) 1 and R2 (G; G0) 6 2 . Therefore we conclude that G and G0 are strongly resemblant at the level of precision 1 and the level of resemblance 1 (but not for 2 and 2 ). This example shows that two strongly resemblant fuzzy values (applying de nition 4) at levels (1 ; 1) may not be resemblant at levels (2 ; 2), which is not a good behaviour. Therefore, although these measures were elicited by a (non fuzzy) expert, we want to model them from a di erent point of view, without using precision levels. This is carried out in the next section, whereas in section four we see the needed conditions to relate both approaches.
3 Approach without Using Levels of Precision 3.1 Weak Resemblance
De nition 5 Let us consider a domain D of crisp values, and a resemblance relation R, de ned in D D. Two fuzzy values F and F 0 2 P (D) , are weakly resemblant at level , and will be denoted by: F F 0 if and only if where
R(F; F 0) R : P (D) P (D) ?! [0; 1] 4 sup (R(x; y) ^ (x) ^ 0 (y)) R(F; F 0) = F F x;y
(4)
This measure gives us the extent to which some element in F is resemblant to some element in F 0. The next properties can be easily proved:
Proposition 2 Let D be a crisp domain, R a resemblance relation de ned on D D, F , F 0 fuzzy values in P (D), and an arbitrary threshold. In this situation, the following properties hold: i) R (F; F 0) = supy (F 0 (y) ^ R (F; y)) ii) F F 0 ) F F 0 8 3
iii) R (F; F 0) = R(F 0 ; F). Therefore F F 0 , F 0 F iv) If x0 2 D exists, such that F (x0 ) , then F F If F is normalized, then R (F; F) = 1 i) gives another way to work with equation 4, ii) is a basic property that a de nition of resemblance must verify, and iii), iv) says that weak resemblance maintains the symmetric and re exive properties of R. The following
propositions study the relation of weak resemblance and inclusion between fuzzy sets:
Proposition 3 Under the hypotheses of proposition 2, the following properties are veri ed: i) If F F 0 and F H then F 0 H ii) If F F 0, H H 0 and F H , then F 0 H 0 iii) Let us suppose 9 x 2 D such that F (x ) . Then: iii.a) F F 0 8 F; F 0 such that F F 0 iii.b) If F F and F F then F F 8 0
1
0
2
1
2
Proof. i) F F 0 ) F (x) F 0 (x) 8 x R (F 0; H) = sup(F 0 (x) ^ H (y) ^ R(x; y)) sup(F (x) ^ H (y) ^ R(x; y)) = R(F; H) ) x;y
x;y
) R (F 0; H) ) F 0 H
ii) is a direct consequence of i) iii.a) R is re exive, and therefore R(x0; x0) = 1 ) F (x0 ) ^ R(x0 ; x0) . Now, F F 0 ) F (x)
F 0 (x) 8 x and thus: F (x0 ) ^ F 0 (x0) ^ R(x0; x0) ) R (F; F 0) iii.b) follows in the same way Concerning the union between fuzzy sets, we have:
2
Proposition 4 Under the hypotheses of proposition 2, this property is veri ed: If it is F _ F 0 H , then F H or F 0 H Proof. Let us suppose sup F (x) ^ H (y) ^ R(x; y) < and sup F 0 (y) ^ H (y) ^ R(x; y) < x;y
x;y
By applying the hypothesis,
sup F _F 0 (x) ^ H (y) ^ R(x; y) x;y
is satis ed. Now, the rst term is equal to: supf(F (x) _ F 0 (x)) ^ H (y) ^ R(x; y)g = x;y
= sup max fF (x) ^ H (y) ^ R(x; y) ; F 0 (x) ^ H (y) ^ R(x; y)g x;y max sup F (x) ^ H (y) ^ R(x; y) ; sup F 0 (x) ^ H (y) ^ R(x; y) < x;y
x;y
and this is a contradiction. 2 Remark. Part iv) of proposition 2, part i) of proposition 3, and proposition 4 can be easily extended to work with arbitrary t-norms and co-norms (see for instance [3]). 4
3.2 Strong Resemblance
De nition 6 Let us consider a domain D of crisp values, and a resemblance relation R de ned in D D. Two fuzzy values G and G0 2 P (D) , are strongly resemblant at level , and will be denoted by: G G0 if and only if where
R(G; G0) R : P (D) P (D) ?! [0; 1] 4 inf fR(x; y) _ (1 ? (x)) _ (1 ? 0 (y))g R (G; G0) = G G x;y
(5)
Strong resemblance gives us the extent to which all the elements in G are resemblant to all the elements in G0. But this is true only when G and G0 are normalized fuzzy values. To illustrate this, let us consider G = a=0:3, G0 = b=0:2 and R(a; b) = 0 ) R (G; G0) = f7 _ 1 _ 0g^f1 _ 0:8 _ 0g = 1 ) G G0 8 , and obviously, it is contradictory with the previous interpretation. Therefore, from now on, the strong measure will be applied only to normalized fuzzy sets. Nevertheless, in this section, we shall require normalization whenever it is absolutely necessary to prove the results. The measures given in de nitions 5 and 6 of weak and strong resemblance, both of them were introduced by Prade and Testemale in [8], in the context of querying a fuzzy database. We are going to complement this work introducing some useful properties related to these de nitions.
Proposition 5 Let D be a crisp domain, R a resemblance relation de ned in D D, G, G0 fuzzy values in P (D), and an arbitrary threshold. Then, the following properties are satis ed: i) R (G; G0) = inf y ((1 ? G0 (y)) _ R (G; y)) ii) G G0 ) G G0 8 iii) R (G; G0) = R(G0 ; G). Therefore G G0 , G0 G i) gives another way to work with equation 5, ii) is a basic property that a de nition of resemblance must verify, and iii) says that strong resemblance maintains the symmetric property of R. Now, re exivity is not attained
with strong resemblance, because it may be the case that G 6 G (consider for instance R a distance measure and two values x; y 2 ker(G) with R(x; y) = 0). Thus, we shall derive good properties for a strong measure just for fuzzy values verifying what we call the level of granularity:
De nition 7 Let D be a crisp domain, R a resemblance relation de ned in D D, and a resemblance threshold. Under these hypotheses, a fuzzy value G 2 P (D) satis es the level of granularity if and only if: G G (6) Remark. We suppose for the sake of simplicity that is a xed threshold associated to D. Therefore, we de ne the level of granularity without specifying the level . Otherwise, we would talk about a -level of granularity. Granularity restriction for G states that G can not be too fuzzy, in the sense that all the crisp values in G must be resemblant enough. The next proposition studies the relation of strong and weak resemblance: it states that strong resemblance is a more restricted measure than weak resemblance.
Proposition 6 Under the same conditions as proposition 5, let us consider two normalized fuzzy sets G and G0. Then, the following properties are satis ed: i) R (G; G0) R(G; G0)
5
ii) If G G0 then G G0
Proof. ii) is a direct consequence of i). To prove i), we can apply the normalization hypothesis:
9 x0 ; y0 such that G (x0) = 1; G0 (y0 ) = 1 ) ) minfG (x0); G0 (y0 ); R(x0; y0)g = R(x0; y0) ) ) sup minfG (x); G0 (y); R(x; y)g R(x0; y0) , R (G; G0) R(x0; y0) Now,
x;y
1 ? G (x0) = 0 ; 1 ? G0 (y0 ) = 0 ) ) maxf1 ? G (x0 ); 1 ? G0 (y0 ); R(x0; y0 )g = R(x0; y0 ) ) ) inf maxf1 ? G (x); 1 ? G0 (y); R(x; y)g R(x0; y0) , R (G; G0) R(x0; y0) x;y
2 Remark. In proposition 6 we required normalization for G and G0, but we could relax it with the existence of elements in D with a membership degree greater than or equal to the resemblance threshold. In this case we would obtain this property: We prove the result by chaining the latter terms.
If > 0:5 and 9 x0; y0 such that G (x0 ) ; G0 (y0 ) then R (G; G0) ) R (G; G0) The following results study the relation between strong resemblance and fuzzy inclusion and union. We shall assume the same conditions as in proposition 5.
Proposition 7
9 G0 satisfy the level of granularity = ; ) G H G; H G0
Proof. G G0 ) 1 ? G (x) 1 ? G0 (x) 8 x H G0 ) 1 ? H (x) 1 ? G0 (x) 8 x Therefore:
R (G; H) = inf (R(x; y) _ (1 ? G (x)) _ (1 ? H (y))) x;y
inf (R(x; y) _ (1 ? G0 (x)) _ (1 ? G0 (x))) = R (G0 ; G0) x;y
G0 satisfy the level of granularity ( ), then:
R (G0 ; G0) ) R (G; H)
2 A direct consequence of this proposition is:
Proposition 8 If G G0 and G0 satisfy the level of granularity, then: i) G satis es the level of granularity. ii) G G0
6
Proposition 9 i) If G G and G G , then G G _ G ii) If G G , G G , G G and G G , then G _ G G _ G 1
2
1
2
1
1
3
3
1
4
2
2
4
3
3
1
4
2
3
Proof. i) R (G1; G2 _ G3) = inf x;y ((1 ? G1 (x)) _ (1 ? G2 _G3 (y)) _ R(x; y))
By applying
1 ? G2 _G3 (y) = min(1 ? G2 (y); 1 ? G3 (y))
we obtain: R (G1; G2 _ G3 ) = inf f(1 ? G1 (x)) _ [(1 ? G2 (y)) ^ (1 ? G3 (y))] _ R(x; y)g = x;y = inf f[(1 ? G1 (x)) _ (1 ? G2 (y)) _ R(x; y)] ^ [(1 ? G1 (x)) _ (1 ? G3 (y)) _ R(x; y)]g x;y
R(G1 ; G2) ^ R (G1; G3)
Now:
R (G1; G2) and R (G1; G3) ) R (G1; G2) ^ R (G1 ; G3) ) R (G1; G2 _ G3)
ii) We have proved in i) the following equality:
R (G1; G2 _ G3) = R(G1 ; G2) ^ R (G1; G3) Then, replacing G1 by G1 _ G4 : R (G1 _ G4; G2 _ G3) R (G1 _ G4 ; G2) ^ R(G1 _ G4; G3)
R (G1 ; G2) ^ R(G4 ; G2) ^ R (G1; G3) ^ R (G4; G3) Each term is greater than or equal to , and therefore R (G1 _ G4; G2 _ G3)
2
Proposition 10 If G and G0 satisfy the level of granularity, and G G0, then: i) (G _ G0) G ii) (G _ G0) G0 iii) (G _ G0) (G _ G0), i.e, G _ G0 satis es the level of granularity. Proof. By applying proposition 9 we nd: G (G _ G0) and G0 (G _ G0 ) By applying proposition 9 again with G1 = G _ G0 we obtain: (G _ G0 ) (G _ G0)
2 7
Proposition 11 If G G0 , H G and H 0 G0 then H H 0 Proof. R (H; H 0) = inf (R(x; y) _ (1 ? H (y)) _ (1 ? H 0 (x))) x;y But H G and H 0 G0 )
8 y H (y) G(y); 8 x H 0 (x) G0 (x) )
R (H; H 0) inf (R(x; y) _ (1 ? G (y)) _ (1 ? G0 (x))) = R (G; G0) x;y
2 Let us see now the inverse of proposition 9:
Proposition 12 G G _ G , G G and G G . 1
2
3
1
2
1
3
Proof. To show )) we apply proposition 11: 9 G1 G2 _ G3 = ) G1 G2 G2 G2 _ G3 ; We obtain the same result for G3 . The other part () is given by proposition 9
2
4 Equivalence Between Previous Approaches Now, we proceed to study the relation between the fuzzy extensions of resemblance, i.e, using precision levels (de nitions 3 and 4) or not (de nitions 5 and 6). First, we characterize weak and strong resemblance under the hypotheses of continuous membership functions and continuous resemblance relations when the domain is not discrete.
4.1 Characterization of Weak Resemblance First, we are going to prove that supremum in equation 1 can be achieved using just one cut, which is the same level of resemblance. We need this lemma:
Lemma 1 Let D be a compact set, R a continuous resemblance relation de ned in D D, a resemblance threshold and F 2 P (D) with a continuous membership function satisfying F = 6 ;. Then, the following is satis ed:
y F , max (x) ^ R(x; y) , xmax (x) ^ R(x; y) x F 2F F
Proof. y is xed, so we can denote F (x) ^ R(x; y) by H(x). Applying the hypotheses, F is a compact set, and H(x) is a continuous function. Then: sup H(x) = xmax H(x) ; sup H(x) = max H(x) x 2F
x2F
x
Thus we must prove the following: max H(x) , xmax H(x) x 2F 8
))
max H(x) ) 9 x0 2 D such that H(x0 ) x H(x0 ) ) F (x0) ; R(x0 ; y) F (x0) ) x0 2 F ) R(x0; y) xmax R(x; y) 2F
Therefore maxx2F R(x; y) () F D ) maxx2F H(x) maxx H(x) ) maxx H(x)
2
Remark. In the discrete case, the supremum is always achieved at one point in the domain, so the proof can be easily extended.
Theorem 1 Let us consider two fuzzy values F and F 0 under the hypotheses of lemma 1. Then, the following is satis ed:
R(x; y) F F 0 , y2Fmax ; x2F 0
Proof.
)) Applying the hypothesis we obtain: max (F (y) ^ R (F 0; y)) y Now, R (F 0; y) = maxx2F 0 F (x) ^ R(x; y) is a continuous function in y. Then, following the same steps as in lemma 1: max (F (y) ^ R(F 0 ; y)) , ymax R (F 0; y) ) y 2F
)
9 y0 2 F such that R (F 0; y0 )
Now, applying lemma 1:
)
R (F 0; y0) , xmax R(x; y0) ) 2F 0 R(x; y0) ) ymax 9 y0 2 F such that xmax max R(x; y) 2F 0 2F x2F 0
() Can be proved following the same reasoning as in lemma 1.
2
4.2 Characterization of Strong Resemblance Working with strong resemblance, we must distinguish between the discrete and continuous cases.
Lemma 2 Let us consider D and R under the hypotheses of lemma 1, and a resemblance level. Let G 2 P (D), with a continuous membership function satisfying G ? 6= ;. Then, the following is satis ed: 1
y G , min (1 ? G (x)) _ R(x; y) , x2min (1 ? G (x)) _ R(x; y) x G1? Proof. Let us denote (1 ? G(x)) _ R(x; y) by H(x). G1? is a compact set and H is a continuous function, so that: H(x) H(x) = x2min inf H(x) = min H(x) ; x2inf x x G1? G1? So, we must prove the following:
min H(x) , x2min H(x) x G1? 9
This can be easily done using the following decomposition: H(x) ; x62inf min H(x) = min x2min H(x) x G1? G 1?
If it is the case that
(7)
min H(x) x
then, applying equation 7, it will be true
min H(x)
x2G1?
Therefore we must now prove: min H(x) ) min H(x) x
x2G1?
We are going to show that the last term in 7 is always greater than or equal to :
8 x 62 G1? ) G (x) < 1 ? ) 1 ? G (x) > ) (1 ? G (x)) _ R(x; y) > , H(x) > 8 x 62 G1? ) x26 inf H(x) G1? But it is true that minx2G1? H(x) , and therefore min x2min H(x) ; inf H(x) G x62G 1?
Now, by applying equation 7, we nd
1?
min H(x) x
2
Lemma 3 Under the hypotheses of lemma 2, the following is satis ed: y G , x2min R(x; y) G1? Proof. Applying lemma 2, we must prove that: min H(x) , x2min R(x; y) G1?
x2G1?
()
min R(x; y) ) R(x; y) 8 x 2 G1? )
x2G1?
) (1 ? G (x)) _ R(x; y) 8 x 2 G1? ) x2min H(x) G1? )) Let us suppose that:
min R(x; y) < ; x2min H(x) G
x2G1?
1?
Let us consider 0 = minx2G1? R(x; y). Then:
8 x 2 G1? R(x; y) 0
(8)
Now, 0 < ) 1 ? 0 > 1 ? . But G is a continuous function, therefore:
9 x0 2 D such that 1 ? 0 > G (x0) > 1 ? 10
(9)
By joining equations 8 and 9 together: 9 x0 2 G1? ) R(x0; y) 0 < = ; ) 1 ? G (x0 ) <
) 9 x0 2 G1? such that (1 ? G (x0 )) _ R(x0 ; y) <
But applying the hypothesis, we nd:
min H(x) )
x2G1?
8 x 2 G1? (1 ? G (x)) _ R(x; y) 2
which is a contradiction. When D is discrete, equation 9 can not be applied. In this case, the equivalence in lemma 3 becomes:
Lemma 4 Let D be a discrete crisp domain, R a resemblance measure de ned in D D, a level of resemblance and G 2 P (D) with G ? 6= ;. Let be that value verifying 1 ? = minfG (x) such that G (x) > 1 ? g 1
Then
y G , x2min R(x; y) G1?
Proof. First
G1? 6= ; ) 9 such that 1 ? = minfG (x) such that G (x) > 1 ? g Let us denote (1 ? G (x)) _ R(x; y) by H(x). Now, we consider these sets: G1? = fx 2 D such that G (x) > 1 ? g ) ( H(x) ; min min H(x) = min min > H(x) < H(x) ; x2min x G= x2G1?
1?
The de nition of strong resemblance states that:
x2G1?
(10)
y G , min H(x) x Therefore minx H(x) , if and only if each term is greater than or equal to . By applying the hypothesis we obtain G>1? = G1? 6= ;. When the other sets are empty, the proof is nished. Thus, we suppose G=1? 6= ; or G>1? 6= ;.
{ Let us suppose x 2 G
