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Comments on “Fuzzy data dependencies and implication of fuzzy data dependencies”
Fuzzy Sets and Systems 148 (2004) 153 – 156 www.elsevier.com/locate/fss
Discussion
Comments on “Fuzzy data dependencies and implication of fuzzy data dependencies” Thanh Ha Danga;∗ , Dinh Khang Tranb a
Laboratoire d’Informatique de Paris 6, Pˆole IA, 8 rue du Capitaine Scott, Paris 75015, France b Faculty of Information Technology, Hanoi University of Technology, Hanoi, Viet Nam Received 9 July 2003; received in revised form 18 November 2003; accepted 26 January 2004
Abstract In this paper we discuss the results given by Wei-Yi Liu in the article entitled “Fuzzy data dependencies and implication of fuzzy data dependencies” (Fuzzy Sets and Systems 92 (1997) 341) and we get some di9erent conclusions. c 2004 Elsevier B.V. All rights reserved. Keywords: Fuzzy data dependency; Implication of data dependencies; Semantic proximity
In “Fuzzy data dependencies and implication of fuzzy data dependencies” [1], Wei-Yi Liu has given the concept of semantic proximity between two fuzzy attribute values which uses the interval number description. Based on the semantic proximity, he has introduced the deAnitions of fuzzy functional dependencies (FFDs), fuzzy multivalued dependencies (FMVDs) and the inference rules of FFDs and FMVDs. There are some results in that paper: 1. The degree of proximity between two fuzzy values f1 ; f2 has been described by the SP(f1 ; f2 ) (06SP(f1 ; f2 )61). The following properties ought to be satisAed by SP(f1 ; f2 ): Let f1 = [a1 ; b1 ]; f2 = [a2 ; b2 ]; g1 = [c1 ; d1 ]; g2 = [c2 ; d2 ]. (i) (ii) (iii) (iv)
SP(f1 ; f2 ) = 1 if and only if a1 = b1 = a2 = b2 . SP(f1 ; f2 ) = 0 if and only if [a1 ; b1 ] ∩ [a2 ; b2 ] = . If a1 = a2 ; b1 = b2 ; c1 = c2 ; d1 = d2 and |d1 − c1 |¿|b1 − a1 | then SP(f1 ; f2 )¿SP(g1 ; g2 ). If |a2 − b2 | = |a1 − b1 | and f1 ∩ g1 ¿f2 ∩ g1 then SP(f1 ; g1 )¿SP(f2 ; g1 ).
∗
Corresponding author. Tel.: +33-1-44-27-87-51; fax: +33-1-44-27-70-00. E-mail address: [email protected] (T.H. Dang).
c 2004 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter doi:10.1016/j.fss.2004.01.007
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T.H. Dang, D.K. Tran / Fuzzy Sets and Systems 148 (2004) 153 – 156
A concrete instance is SP(f1 ; f2 ) =
f1 ∩ f2 f1 ∩ f2 − ; f1 ∪ f2
where h is the length of interval h, 0; ; h = |b − a|;
;
h = ; h = [a; a]; h = [a; b] and a = b; h=∞
and is a given coeJcient about the universe of the discourse, ¿f1 ∪ f2 . is a relatively small number. Example. Let = 300; = =10000. We have SP([10; 10]; [10; 10]) = 1, SP([2; 8]; [2; 8]) = 6=6 − 6=300 = 0:98, SP([2; 5]; [6; 7]) = 0, SP([2; 4]; [3; 6]) = 1=4 − 1=300 = 0:2467. 2. Denition 2 (Wei-Yi Liu [1]). A FFD X ∼ → Y with X; Y ⊂ U holds in a fuzzy instance r on U , if for all ti and tj ∈ r we have SP(ti [X ]; tj [X ])6SP(ti [Y ]; tj [Y ]). 3. Denition 3 (Wei-Yi Liu [1]). Let X; Y ⊂ U and Z = U − XY . A FMVD X ∼ → ∼ → Y holds in a fuzzy instance r on U if, for any two tuples ti ; tj ∈ r with SP(ti [X ]; tj [X ]) = there exists a tuple t in r with SP(t[X ]; ti [X ])¿ , SP(t[Y ]; ti [Y ])¿ and SP(t[Z]; tj [Z])¿ . Theorems 3 and 4. The following inference rules are sound and complete: FA1: If X ⊇ Y then X ∼ → Y . FA2: If X ∼ → Y then XW ∼ → YW . FA3: If X ∼ → Y , and Y ∼ → Z then X ∼ → Z. FA4: If X ∼ → ∼ → Y then X ∼ → ∼ → (U − XY ). FA5: If X ∼ → ∼ → Y , and V ⊆ W then XW ∼ → ∼ → YV . FA6: If X ∼ → ∼ → Y , and Y ∼ → ∼ → Z then X ∼ → ∼ → (Z − Y ). FA7: If X ∼ → Y then X ∼ → ∼ → Y . FA8: If X ∼ → ∼ → Y; Z ⊆ Y , Y ∩ W = and W ∼ → ∼ → Z then X ∼ → Z. There is an implied premise in the deAnition of semantic proximity: SP(f; f) = 1. For example, John’s height is [180 cm; 185 cm] (i.e., hjohn = [180; 185]). We know that an individual has the same height as himself, though hjohn is an ambiguous datum. But by the deAnition of the semantic proximity we And that is does not satisfy the reMexivity property: SP(f; f) = 1, where f = [a; b] and a = b, and it can cause some mistakes in the results above.
T.H. Dang, D.K. Tran / Fuzzy Sets and Systems 148 (2004) 153 – 156
155
We shall discuss the following examples: 1. Consider a relation R1 : R1 X Y Z t1 [0; 0] [1; 1] [1; 9] t2 [0; 0] [1; 1] [1; 8] R1 satisAes X ∼ → Y because SP(t1 [X ]; t2 [X ])6SP(t1 [Y ]; t2 [Y ]) = 1, but R1 does not satisfy X ∼ → ∼ → Y . Consider two tuples t1 and t2 , we have SP(t1 [X ]; t2 [X ]) = 1. For all t3 ∈ R1 (not necessary distinct from t1 ; t2 ), SP(t3 [X ]; t1 [X ]) = 1; SP(t3 [Y ]; t1 [Y ]) = 1, SP(t3 [Z]; t2 [Z])¡1 because SP([1; 9]; [1; 8])¡1, SP([1; 8]; [1; 8])¡1; SP([1; 9]; [1; 9])¡1. Consequently, rule FA7 is not sound. 2. We consider the next relation (using the SP of Wei-Yi Liu with = 300; = =10000): R2 X Y Z t1 [0; 6] [0; 6] [0; 6] t2 [3; 6] [6; 12] [6; 12] t3 [0; 3] [0; 3] [9; 12] t4 [3; 9] [6; 9] [0; 3] It is easy to verify that X ∼ → ∼ → Y : 3 For t1 ; t2 ∈ R2 ; SP(t1 [X ]; t2 [X ]) = 36 − 300 = 0:49, there exists t3 ∈ R2 such that SP(t3 [X ]; t1 [X ]) = 0:49, SP(t3 [Y ]; t1 [Y ]) = 0:49 and SP(t3 [Z]; t2 [Z]) = 0:49; there exists t4 ∈ R2 such that SP(t4 [X ]; t2 [X ]) = 0:49; SP(t4 [Y ]; t2 [Y ]) = 0:49 and SP(t4 [Z]; t1 [Z]) = 0:49. 3 For t1 ; t3 ∈ R2 ; SP(t1 [X ]; t3 [X ]) = 36 − 300 = 0:49, there exists t3 ∈ R2 such that SP(t3 [X ]; t1 [X ]) = 0:49; SP(t3 [Y ]; t1 [Y ]) = 0:49 and SP(t3 [Z]; t3 [Z]) = 0:99; there exists t1 ∈ R2 such that SP(t1 [X ]; t3 [X ]) = 0:49; SP(t1 [Y ]; t3 [Y ]) = 0:49 and SP(t1 [Z]; t1 [Z]) = 0:99. 3 For t1 ; t4 ∈ R2 ; SP(t1 [X ]; t4 [X ]) = 39 − 300 ≈ 0:323, there exists t1 ∈ R2 such that SP(t1 [X ]; t1 [X ]) = 0:98; SP(t1 [Y ]; t1 [Y ]) = 0:98 and SP(t1 [Z]; t4 [Z]) = 0:49; there exists t4 ∈ R2 such that SP(t4 [X ]; t4 [X ]) = 0:98; SP(t4 [Y ]; t4 [Y ]) = 0:99 and SP(t4 [Z]; t1 [Z]) = 0:49. For t2 ; t3 ∈ R2 ; SP(t2 [X ]; t3 [X ]) = =6 − =300 = 0:0049, there exists t2 ∈ R2 such that SP(t2 [X ]; t2 [X ]) = 0:99; SP(t2 [Y ]; t2 [Y ]) = 0:98 and SP(t2 [Z]; t3 [Z]) = 0:49; there exists t3 ∈ R2 such that SP(t3 [X ]; t3 [X ]) = 0:99; SP(t3 [Y ]; t3 [Y ]) = 0:99 and SP(t3 [Z]; t2 [Z]) = 0:49. 3 For t2 ; t4 ∈ R2 ; SP(t2 [X ]; t4 [X ]) = 36 − 300 = 0:49, there exists t4 ∈ R2 such that SP(t4 [X ]; t2 [X ]) = 0:49, SP(t4 [Y ]; t2 [Y ]) = 0:49 and SP(t4 [Z]; t4 [Z]) = 0:99; there exists t2 ∈ R2 such that SP(t2 [X ]; t4 [X ]) = 0:49; SP(t2 [Y ]; t4 [Y ]) = 0:49 and SP(t2 [Z]; t2 [Z]) = 0:98. For t3 ; t4 ∈ R2 ; SP(t3 [X ]; t4 [X ]) = =9 − =300 ≈ 0:00324, there exists t1 ∈ R2 such that SP(t1 [X ]; t3 [X ]) = 0:49, SP(t1 [Y ]; t3 [Y ]) = 0:49 and SP(t1 [Z]; t4 [Z]) = 0:49; there exists t2 ∈ R2 such that SP(t2 [X ]; t4 [X ]) = 0:49; SP(t2 [Y ]; t4 [Y ]) = 0:49 and SP(t2 [Z]; t3 [Z]) = 0:49. 3 But X ∼ → ∼ → Z is not satisAed. We have SP(t1 [X ]; t2 [X ]) = 36 − 300 = 0:49. We search for a tuple t in such a way that SP(t[X ]; t1 [X ])¿0:49; SP(t[Z]; t1 [Z])¿0:49 and SP(t[Y ]; t2 [Y ])¿0:49.
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T.H. Dang, D.K. Tran / Fuzzy Sets and Systems 148 (2004) 153 – 156
If t = t1 : SP(t1 [Y ]; t2 [Y ]) = =12 − = = 0:0024¡0:49. If t = t2 : SP(t2 [Z]; t1 [Z]) = =12 − = = 0:0024¡0:49. If t = t3 : SP(t3 [Z]; t1 [Z]) = 0¡0:49. 3 If t = t4 : SP(t4 [X ]; t1 [X ]) = 39 − 300 ≈ 0:323¡0:49. In fact, there does not exist any tuple t. Therefore the rule FA4 is wrong. To correct these mistakes, we can choose another degree of semantic proximity which satisAes the reMexivity property, and the deAnition of FMVD should be modiAed as follows by adding SP(t[X ]; tj [X ])¿ in the old deAnition: Denition. Let X; Y ⊂ U and Z = U − XY . A FMVD X ∼ → ∼ → Y holds in a fuzzy instance r on U if, for any two tuples ti ; tj ∈ r with SP(ti [X ]; tj [X ]) = there exists a tuple t in r with SP(t[X ]; ti [X ])¿ , SP(t[X ]; tj [X ])¿ , SP(t[Y ]; ti [Y ])¿ and SP(t[Z]; tj [Z])¿ . This way, rules FA4 and FA7 are correct. Moreover, it seems that the role of fuzzy sets is omitted in Wei-Yi Liu’s paper. The use of an interval number for describing a fuzzy value may not be a good idea. A fuzzy value is more complex than an interval number. However, an interval number could be used for an imprecise value. Acknowledgements The authors would like to thank the area editor and the referees for their valuable comments on the original version of this paper. References [1] W.-Y. Liu, Fuzzy data dependencies and implication of fuzzy data dependencies, Fuzzy Sets and Systems 92 (1997) 341–348.
Discussion
Comments on “Fuzzy data dependencies and implication of fuzzy data dependencies” Thanh Ha Danga;∗ , Dinh Khang Tranb a
Laboratoire d’Informatique de Paris 6, Pˆole IA, 8 rue du Capitaine Scott, Paris 75015, France b Faculty of Information Technology, Hanoi University of Technology, Hanoi, Viet Nam Received 9 July 2003; received in revised form 18 November 2003; accepted 26 January 2004
Abstract In this paper we discuss the results given by Wei-Yi Liu in the article entitled “Fuzzy data dependencies and implication of fuzzy data dependencies” (Fuzzy Sets and Systems 92 (1997) 341) and we get some di9erent conclusions. c 2004 Elsevier B.V. All rights reserved. Keywords: Fuzzy data dependency; Implication of data dependencies; Semantic proximity
In “Fuzzy data dependencies and implication of fuzzy data dependencies” [1], Wei-Yi Liu has given the concept of semantic proximity between two fuzzy attribute values which uses the interval number description. Based on the semantic proximity, he has introduced the deAnitions of fuzzy functional dependencies (FFDs), fuzzy multivalued dependencies (FMVDs) and the inference rules of FFDs and FMVDs. There are some results in that paper: 1. The degree of proximity between two fuzzy values f1 ; f2 has been described by the SP(f1 ; f2 ) (06SP(f1 ; f2 )61). The following properties ought to be satisAed by SP(f1 ; f2 ): Let f1 = [a1 ; b1 ]; f2 = [a2 ; b2 ]; g1 = [c1 ; d1 ]; g2 = [c2 ; d2 ]. (i) (ii) (iii) (iv)
SP(f1 ; f2 ) = 1 if and only if a1 = b1 = a2 = b2 . SP(f1 ; f2 ) = 0 if and only if [a1 ; b1 ] ∩ [a2 ; b2 ] = . If a1 = a2 ; b1 = b2 ; c1 = c2 ; d1 = d2 and |d1 − c1 |¿|b1 − a1 | then SP(f1 ; f2 )¿SP(g1 ; g2 ). If |a2 − b2 | = |a1 − b1 | and f1 ∩ g1 ¿f2 ∩ g1 then SP(f1 ; g1 )¿SP(f2 ; g1 ).
∗
Corresponding author. Tel.: +33-1-44-27-87-51; fax: +33-1-44-27-70-00. E-mail address: [email protected] (T.H. Dang).
c 2004 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter doi:10.1016/j.fss.2004.01.007
154
T.H. Dang, D.K. Tran / Fuzzy Sets and Systems 148 (2004) 153 – 156
A concrete instance is SP(f1 ; f2 ) =
f1 ∩ f2 f1 ∩ f2 − ; f1 ∪ f2
where h is the length of interval h, 0; ; h = |b − a|;
;
h = ; h = [a; a]; h = [a; b] and a = b; h=∞
and is a given coeJcient about the universe of the discourse, ¿f1 ∪ f2 . is a relatively small number. Example. Let = 300; = =10000. We have SP([10; 10]; [10; 10]) = 1, SP([2; 8]; [2; 8]) = 6=6 − 6=300 = 0:98, SP([2; 5]; [6; 7]) = 0, SP([2; 4]; [3; 6]) = 1=4 − 1=300 = 0:2467. 2. Denition 2 (Wei-Yi Liu [1]). A FFD X ∼ → Y with X; Y ⊂ U holds in a fuzzy instance r on U , if for all ti and tj ∈ r we have SP(ti [X ]; tj [X ])6SP(ti [Y ]; tj [Y ]). 3. Denition 3 (Wei-Yi Liu [1]). Let X; Y ⊂ U and Z = U − XY . A FMVD X ∼ → ∼ → Y holds in a fuzzy instance r on U if, for any two tuples ti ; tj ∈ r with SP(ti [X ]; tj [X ]) = there exists a tuple t in r with SP(t[X ]; ti [X ])¿ , SP(t[Y ]; ti [Y ])¿ and SP(t[Z]; tj [Z])¿ . Theorems 3 and 4. The following inference rules are sound and complete: FA1: If X ⊇ Y then X ∼ → Y . FA2: If X ∼ → Y then XW ∼ → YW . FA3: If X ∼ → Y , and Y ∼ → Z then X ∼ → Z. FA4: If X ∼ → ∼ → Y then X ∼ → ∼ → (U − XY ). FA5: If X ∼ → ∼ → Y , and V ⊆ W then XW ∼ → ∼ → YV . FA6: If X ∼ → ∼ → Y , and Y ∼ → ∼ → Z then X ∼ → ∼ → (Z − Y ). FA7: If X ∼ → Y then X ∼ → ∼ → Y . FA8: If X ∼ → ∼ → Y; Z ⊆ Y , Y ∩ W = and W ∼ → ∼ → Z then X ∼ → Z. There is an implied premise in the deAnition of semantic proximity: SP(f; f) = 1. For example, John’s height is [180 cm; 185 cm] (i.e., hjohn = [180; 185]). We know that an individual has the same height as himself, though hjohn is an ambiguous datum. But by the deAnition of the semantic proximity we And that is does not satisfy the reMexivity property: SP(f; f) = 1, where f = [a; b] and a = b, and it can cause some mistakes in the results above.
T.H. Dang, D.K. Tran / Fuzzy Sets and Systems 148 (2004) 153 – 156
155
We shall discuss the following examples: 1. Consider a relation R1 : R1 X Y Z t1 [0; 0] [1; 1] [1; 9] t2 [0; 0] [1; 1] [1; 8] R1 satisAes X ∼ → Y because SP(t1 [X ]; t2 [X ])6SP(t1 [Y ]; t2 [Y ]) = 1, but R1 does not satisfy X ∼ → ∼ → Y . Consider two tuples t1 and t2 , we have SP(t1 [X ]; t2 [X ]) = 1. For all t3 ∈ R1 (not necessary distinct from t1 ; t2 ), SP(t3 [X ]; t1 [X ]) = 1; SP(t3 [Y ]; t1 [Y ]) = 1, SP(t3 [Z]; t2 [Z])¡1 because SP([1; 9]; [1; 8])¡1, SP([1; 8]; [1; 8])¡1; SP([1; 9]; [1; 9])¡1. Consequently, rule FA7 is not sound. 2. We consider the next relation (using the SP of Wei-Yi Liu with = 300; = =10000): R2 X Y Z t1 [0; 6] [0; 6] [0; 6] t2 [3; 6] [6; 12] [6; 12] t3 [0; 3] [0; 3] [9; 12] t4 [3; 9] [6; 9] [0; 3] It is easy to verify that X ∼ → ∼ → Y : 3 For t1 ; t2 ∈ R2 ; SP(t1 [X ]; t2 [X ]) = 36 − 300 = 0:49, there exists t3 ∈ R2 such that SP(t3 [X ]; t1 [X ]) = 0:49, SP(t3 [Y ]; t1 [Y ]) = 0:49 and SP(t3 [Z]; t2 [Z]) = 0:49; there exists t4 ∈ R2 such that SP(t4 [X ]; t2 [X ]) = 0:49; SP(t4 [Y ]; t2 [Y ]) = 0:49 and SP(t4 [Z]; t1 [Z]) = 0:49. 3 For t1 ; t3 ∈ R2 ; SP(t1 [X ]; t3 [X ]) = 36 − 300 = 0:49, there exists t3 ∈ R2 such that SP(t3 [X ]; t1 [X ]) = 0:49; SP(t3 [Y ]; t1 [Y ]) = 0:49 and SP(t3 [Z]; t3 [Z]) = 0:99; there exists t1 ∈ R2 such that SP(t1 [X ]; t3 [X ]) = 0:49; SP(t1 [Y ]; t3 [Y ]) = 0:49 and SP(t1 [Z]; t1 [Z]) = 0:99. 3 For t1 ; t4 ∈ R2 ; SP(t1 [X ]; t4 [X ]) = 39 − 300 ≈ 0:323, there exists t1 ∈ R2 such that SP(t1 [X ]; t1 [X ]) = 0:98; SP(t1 [Y ]; t1 [Y ]) = 0:98 and SP(t1 [Z]; t4 [Z]) = 0:49; there exists t4 ∈ R2 such that SP(t4 [X ]; t4 [X ]) = 0:98; SP(t4 [Y ]; t4 [Y ]) = 0:99 and SP(t4 [Z]; t1 [Z]) = 0:49. For t2 ; t3 ∈ R2 ; SP(t2 [X ]; t3 [X ]) = =6 − =300 = 0:0049, there exists t2 ∈ R2 such that SP(t2 [X ]; t2 [X ]) = 0:99; SP(t2 [Y ]; t2 [Y ]) = 0:98 and SP(t2 [Z]; t3 [Z]) = 0:49; there exists t3 ∈ R2 such that SP(t3 [X ]; t3 [X ]) = 0:99; SP(t3 [Y ]; t3 [Y ]) = 0:99 and SP(t3 [Z]; t2 [Z]) = 0:49. 3 For t2 ; t4 ∈ R2 ; SP(t2 [X ]; t4 [X ]) = 36 − 300 = 0:49, there exists t4 ∈ R2 such that SP(t4 [X ]; t2 [X ]) = 0:49, SP(t4 [Y ]; t2 [Y ]) = 0:49 and SP(t4 [Z]; t4 [Z]) = 0:99; there exists t2 ∈ R2 such that SP(t2 [X ]; t4 [X ]) = 0:49; SP(t2 [Y ]; t4 [Y ]) = 0:49 and SP(t2 [Z]; t2 [Z]) = 0:98. For t3 ; t4 ∈ R2 ; SP(t3 [X ]; t4 [X ]) = =9 − =300 ≈ 0:00324, there exists t1 ∈ R2 such that SP(t1 [X ]; t3 [X ]) = 0:49, SP(t1 [Y ]; t3 [Y ]) = 0:49 and SP(t1 [Z]; t4 [Z]) = 0:49; there exists t2 ∈ R2 such that SP(t2 [X ]; t4 [X ]) = 0:49; SP(t2 [Y ]; t4 [Y ]) = 0:49 and SP(t2 [Z]; t3 [Z]) = 0:49. 3 But X ∼ → ∼ → Z is not satisAed. We have SP(t1 [X ]; t2 [X ]) = 36 − 300 = 0:49. We search for a tuple t in such a way that SP(t[X ]; t1 [X ])¿0:49; SP(t[Z]; t1 [Z])¿0:49 and SP(t[Y ]; t2 [Y ])¿0:49.
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T.H. Dang, D.K. Tran / Fuzzy Sets and Systems 148 (2004) 153 – 156
If t = t1 : SP(t1 [Y ]; t2 [Y ]) = =12 − = = 0:0024¡0:49. If t = t2 : SP(t2 [Z]; t1 [Z]) = =12 − = = 0:0024¡0:49. If t = t3 : SP(t3 [Z]; t1 [Z]) = 0¡0:49. 3 If t = t4 : SP(t4 [X ]; t1 [X ]) = 39 − 300 ≈ 0:323¡0:49. In fact, there does not exist any tuple t. Therefore the rule FA4 is wrong. To correct these mistakes, we can choose another degree of semantic proximity which satisAes the reMexivity property, and the deAnition of FMVD should be modiAed as follows by adding SP(t[X ]; tj [X ])¿ in the old deAnition: Denition. Let X; Y ⊂ U and Z = U − XY . A FMVD X ∼ → ∼ → Y holds in a fuzzy instance r on U if, for any two tuples ti ; tj ∈ r with SP(ti [X ]; tj [X ]) = there exists a tuple t in r with SP(t[X ]; ti [X ])¿ , SP(t[X ]; tj [X ])¿ , SP(t[Y ]; ti [Y ])¿ and SP(t[Z]; tj [Z])¿ . This way, rules FA4 and FA7 are correct. Moreover, it seems that the role of fuzzy sets is omitted in Wei-Yi Liu’s paper. The use of an interval number for describing a fuzzy value may not be a good idea. A fuzzy value is more complex than an interval number. However, an interval number could be used for an imprecise value. Acknowledgements The authors would like to thank the area editor and the referees for their valuable comments on the original version of this paper. References [1] W.-Y. Liu, Fuzzy data dependencies and implication of fuzzy data dependencies, Fuzzy Sets and Systems 92 (1997) 341–348.