Soundness and Completeness of Inference Rules for New Vague Functional Dependencies

292 MATEC Web of Conferences , 01001 (2019) https://doi.org/10.1051/matecconf/2019 29201001 CSCC 2019

Soundness and Completeness of Inference Rules for New Vague Func- tional Dependencies

Dženan Gušic´1,∗ 1University of Sarajevo, Faculty of Sciences and Mathematics, Department of Mathematics, Zmaja od Bosne 33-35, 71000 Sarajevo, Bosnia and Herzegovina

Abstract. In this paper we introduce a new deﬁnition of vague functional dependency based on application of appropriately chosen similarity measures. The deﬁnition is adjusted in order to be applicable to both, the imprecise and precise vague functional dependencies. Ultimately, the set of inference rules for new vague functional dependencies is given, and is proven to be sound and complete.

1 Introduction A tuple t of r is then of the form

Let U be the universe of discourse. (t [A1] , t [A2] , ..., t [An]) , Suppose that V is a vague set in U. where t [A ] is a vague set in U , i ∈ I. Now, there exist functions tV : U → [0, 1], fV : U → i i Note that we may (more freely speaking) consider [0, 1], such that tV (u) + fV (u) ≤ 1 for u ∈ U. We shall write t [Ai] the value of the attribute Ai on t. A vague relation r on R (A , A , ..., A ) can be visibly 1 2 n V = ⟨u, tV (u) , 1 − fV (u) ⟩ : u ∈ U , represented as a two-dimensional table with n columns and the table headings A1, A2,..., An, where each horizontal where tV (u) , 1 − fV (u) ⊆ [0, 1] is the vague value joined row of the table is a tuple of r, and each column of the ∈ table contains the attribute values under the corresponding to u U. Recall that the vague value tV (u) , 1 − fV (u) reduces heading. , , to the fuzzy value tV (u) = 1 − fV (u) ∈ [0, 1] if it happens Let R (Name Int S ucc) be a relation scheme on do- = that tV (u) = 1 − fV (u) in (0, 1). mains U1 { y, , , , , , , } = If it happens that tV (u) = 1 − fV (u) = 1, then the Em Ted Jim Katie S ara Tina Joe John , U2 { , , } = { , } vague value tV (u) , 1 − fV (u) reduces to the ordinary 115 130 145 , U3 5 10 , where Int (as intelligence) value tV (u) = 1 − fV (u) = 1 ∈ [0, 1]. and S ucc (as success) are vague attributes on universes U2 and U , respectively, and Name as ordinary attribute on The ordinary case tV (u) = 1 − fV (u) = 0, we read as: 3 the element u does not belong to the vague set V. In such the universe of discourse U1. scenario, we write Let r be the vague relation instance on R (Name, Int, S ucc) given by Table 1.

V = {⟨u1, [0.2, 0.7]⟩, ⟨u2, [1, 1]⟩} Table 1.

instead of Name Int S ucc { }{⟨ , , ⟩} {⟨ , , ⟩} = {⟨ , . , . ⟩, ⟨ , , ⟩, ⟨ , , ⟩} , t1 Ted 115 [1 1] 10 [1 1] V u1 [0 2 0 7] u2 [1 1] u3 [0 0] { } ⟨ , . , . ⟩, ⟨ , . , . ⟩, t2 S ara 115 [0 7 0 9] 5 [0 6 0 9] ⟨130, [0.9, 0.95]⟩ ⟨10, [0.8, 0.95]⟩ where U = {u , u , u } is some universe of discourse, and 1 2 3 t {Jim} ⟨130, [0.8, 0.9]⟩, {⟨10, [1, 1]⟩} V is some vague set in U. 3 ⟨145, [0.85, 0.95]⟩ Let R (A1, A2, ..., An) be a relation scheme on domains t4 {Katie}{⟨145, [1, 1]⟩} {⟨5, [0.9, 0.95]⟩} U1, U2,..., Un, where Ai is an attribute on the universe of discourse Ui, i ∈ {1, 2, ..., n} = I. The vague sets {⟨115, [1, 1]⟩} and {⟨10, [1, 1]⟩}, given in Suppose that V (Ui) is the family of all vague sets in the ﬁrst row of the Table 1, mean that the knowledge about Ui, i ∈ I. Ted’s intelligence and success is very accurate. More pre- A vague relation r on R (A1, A2, ..., An) is a subset of cisely, one knows that his intelligence and success are ex- the cross product V (U1) × V (U2) × ... × V (Un). actly 115 and 10, respectively. Having in mind that the ∗ e-mail: [email protected] ranges of person’s intelligence and success are determined

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). 292 MATEC Web of Conferences , 01001 (2019) https://doi.org/10.1051/matecconf/2019 29201001 CSCC 2019

by the sets {115, 130, 145} and {5, 10}, we may say that Table 5. Ted is very successful person with regard to his intelli- Name Int S ucc gence. Sara’s intelligence is determined by the vague set ′′ { }{⟨ , , }{⟨ , , } t1 Joe 130 [1 1] 5 [1 1] {⟨115, [0.7, 0.9]⟩, ⟨130, [0.9, 0.95]⟩}. Since the truth value ′′ t {Tina}{⟨145, [1, 1]}{⟨10, [1, 1]} 0.7 is quite high, the false value 0.1 = 1 − 0.9 is pretty 2 small, and the difference 0.9 − 0.7 = 0.2 is also very small, we conclude that Sara’s intelligence must be close to 115. The aforementioned examples show clearly that the However, 0.9 > 0.7, 0.05 = 1 − 0.95 < 0.1, and 0.95 − 0.9 vague relation concept represents a natural generalization = 0.05 < 0.2 = 0.9 − 0.7, so Sara’s intelligence is defini- of the ordinary relation concept and the fuzzy relation con- tively closer to 130 (from bellow) than to 115 (note that cept. While the relation theory is not able to handle im- 115 < {⟨115, [0.7, 0.9]⟩, ⟨130, [0.9, 0.95]⟩}). Reasoning in precise data almost at all, and the knowledge about fuzzy the same way, we conclude that Sara’s success is between data has its own limitations, the quality of the information 5 and 10, and it is closer to 10 than to 5. The data about about vague data is obviously much more refined. = , − ⊆ , = Katie are quite precise. As opposed to Ted, however, she Let a1 tV1 (u1) 1 fV1 (u1) [0 1] and a2 , − ⊆ , is a very intelligent person who is not so successful. Com- tV2 (u2) 1 fV2 (u2) [0 1] be the vague values joined pared to Ted and Katie, Sara is a relatively intelligent per- to u1 ∈ U1 and u2 ∈ U2, respectively, where son who is relatively successful. Finally, Jim is a pretty V = ⟨u , t (u ) , 1 − f (u ) ⟩ : u ∈ U intelligent person who is also very successful. i i Vi i Vi i i i For the basic relational concepts, see, e.g., [10]. is a vague set in the universe of discourse Ui, i ∈ {1, 2}. Let r1 be the fuzzy relation instance on We define the similarity measure SE (a1, a2) between R (Name, Int, S ucc) given by Table 2 (now, we assume that the vague values a1 and a2 following Lu-Ng [12]. Int and S ucc are fuzzy attributes on U and U , respec- 2 3 SE (a , a ) tively). 1 2 = SE t (u ) , 1 − f (u ) , t (u ) , 1 − f (u ) s V1 1 V1 1 V2 2 V2 2 Table 2. − − − tV1 (u1) tV2 (u2) fV1 (u1) fV2 (u2) Name Int S ucc = 1 − · ′ 2 t {Emy} ⟨ , a ⟩, ⟨ , a ⟩, q 1 130 1 5 3 ⟨ , ⟩ ⟨ , ⟩ − − + − . 145 a2 10 a4 1 tV1 (u1) tV2 (u2) fV1 (u1) fV2 (u2) ′ { }{⟨ , ⟩}{⟨ , ⟩} t2 John 115 a5 5 1 Note that SE (a1, a2) ∈ [0, 1]. Moreover, SE (a , a ) = SE (a , a ), SE (a , a ) = 1 if ∈ , 1 2 2 1 1 2 In Table 2, a1 (0 1) denotes the membership value of and only if a = a , and SE (a , a ) = 0 if and only if a = ∈ {⟨ , ⟩, ⟨ , ⟩} 1 2 1 2 1 the element 130 U2 to the fuzzy set 130 a1 145 a2 , [0, 0], a = [1, 1] or a = [0, 1], a = [a, a], a ∈ [0, 1] (a = ∈ , 2 1 2 1 etc., a5 (0 1) denotes the membership value of the ele- [1, 1], a = [0, 0] or a = [a, a], a = [0, 1], a ∈ [0, 1]). ∈ {⟨ , ⟩} 2 1 2 ment 115 U2 to the fuzzy set 115 a2 . Note that several authors, including Chen [5], [6], The authors in [8] and [1], for example, apply fuzzy Hong-Kim [7], Li-Xu [11], Szmidt-Kacprzyk [15], Grze- membership values to incorporate fuzzy data into rela- gorzewski [9], proposed various definitions of similarity tional database theory. measures between vague sets and distances between intu- Note that the fuzzy relation instance r1 may be repre- itionistic fuzzy sets. According to Lu-Ng [12], however, sented as the vague relation instance given by Table 3. the similarity measure given above, reflects reality in a more appropriate manner when it comes to more general Table 3. cases. Name Int S ucc Let ′ t {Emy} ⟨130, [a1, a1]⟩, ⟨5, [a3, a3]⟩, 1 A = ⟨u, tA (u) , 1 − fA (u) ⟩ : u ∈ U ⟨145, [a2, a2]⟩ ⟨10, [a4, a4]⟩ ′ t {John}{⟨115, [a , a ]⟩}{⟨5, [1, 1]⟩} and 2 5 5 B = ⟨u, tB (u) , 1 − fB (u) ⟩ : u ∈ U Similarly, the relation instance r2 on R (Name, Int, S ucc) given by Table 4 (now, we assume that be two vague sets in some universe of discourse U. We define the similarity measure SE (A, B) between the attributes Int and S ucc are ordinary attributes on U2 the vague sets A and B as follows: and U3, respectively), may be represented as the vague relation instance given by Table 5. SE (A, B) X 1 Table 4. = SE t (u) , 1 − f (u) , t (u) , 1 − f (u) |U| A A B B u∈U Name Int S ucc r ′′ X | − − − | t {Joe}{130}{5} = 1 − (tA (u) tB (u)) ( fA (u) fB (u)) × 1′′ 1 t {Tina}{145}{10} |U| 2 2 q u∈U 1 − t (u) − t (u) + ( f (u) − f (u)) , For the ordinary relational database theory, see [16]. VA B A B

2 292 MATEC Web of Conferences , 01001 (2019) https://doi.org/10.1051/matecconf/2019 29201001 CSCC 2019

where |U| denotes the number of elements in U. instance on R (A1, A2, ..., An). If Y ⊆ X ⊆ {A1, A2, ..., An}, As it is usual, we write A ⊆ B (and say that the vague then set A is contained in the vague set B), if tA (u) ≤ tB (u) and , ≥ , 1 − fA (u) ≤ 1 − fB (u) for all u ∈ U. SEY (t1 t2) SEX (t1 t2) Hence, A ⊆ B if and only if tA (u) ≤ tB (u), fA (u) ≥ for any t1 and t2 in r. fB (u) for u ∈ U. Since A = B if A ⊆ B and B ⊆ A, we obtain that A = Proof. Let t1, t2 ∈ r. Since, B if and only if tA (u) ≤ tB (u), 1 − fA (u) ≤ 1 − fB (u) and tB (u) ≤ tA (u), 1 − fB (u) ≤ 1 − fA (u) for u ∈ U, i.e., if and {SE (t1 [A] , t2 [A]) : A ∈ Y} only if tA (u) = tB (u), 1 − fA (u) = 1 − fB (u) for u ∈ U, ⊆ {SE (t1 [A] , t2 [A]) : A ∈ X} , i.e., if and only if tA (u) = tB (u), fA (u) = fB (u) for u ∈ U. Note that SE (A, B) ∈ [0, 1]. it immediately follows that Furthermore, SE (A, B) = SE (B, A), SE (A, B) = 1 if and only if A = B, and SE (A, B) = 0 if and only if SEY (t1, t2) = min {SE (t1 [A] , t2 [A])} , − = , , − = , A∈Y tA (u) 1 fA (u) [0 0], tB (u) 1 fB (u) [1 1] for all u ∈ U or tA (u) , 1 − fA (u) = [0, 1], tB (u) , 1 − fB (u) ≥ min {SE (t1 [A] , t2 [A])} A∈X = [a, a], a ∈ [0, 1], for all u ∈ U. =SE (t , t ) . Now, we are able to calculate the similarity X 1 2 , measures SE ti [Int] tj [Int] and the similarity measures ⊓⊔ SE ti [S ucc] , t j [S ucc] for i, j ∈ {1, 2, 3, 4}, where ti, i ∈ {1, 2, 3, 4} are tuples of the vague relation instance r on Lemma 2 Let R (A , A , ..., A ) be a relation scheme on R (Name, Int, S ucc) given by Table 1. 1 2 n domains U , U ,..., U , where A is an attribute on the We obtain the following results: 1 2 n i universe of discourse Ui, i ∈ I. Let r be a vague re- , , ..., = lation instance on R (A1 A2 An). Suppose that X   , , ..., { , , ..., }  1 0.69 0.22 0.33  Ai1 Aih2 i Aik h, wherei X is a subset of A1 A2 An . If    . . .  SE t A , t A ≥ θ for all j ∈ {1, 2, ..., k}, where t  0 69 1 0 78 0 22  1 i j 2 i j 1 I =   , , ≥ θ  0.22 0.78 1 0.46  and t2 are some two tuples in r, then SEX (t1 t2) . 0.33 0.22 0.46 1   Proof. Since, . .  1 0 64 1 0 13   . . .   0 64 1 0 64 0 55  SEX (t1, t2) = min {SE (t1 [A] , t2 [A])} , S =   , A∈X  1 0.64 1 0.13  . . . 0 13 0 55 0 13 1 the assertion follows. ⊓⊔ where, for example, 0.78 means that Lemma 3 Let R (A1, A2, ..., An) be a relation scheme on 0.78 =SE (t3 [Int] , t2 [Int]) = SE (t2 [Int] , t3 [Int]) domains U1, U2,..., Un, where Ai is an attribute on the ∈ 1 universe of discourse Ui, i I. Let r be a vague rela- = SE ([0.7, 0.9] , [0, 0]) + , , ..., , ≥ θ 3 tion instance on R (A1 A2 An). If SEX (t1 t2) and , ≥ θ 1 SEX (t2 t3) , where t1, t2 and t3 are some three, mutu- SE ([0.9, 0.95] , [0.8, 0.9]) + { , , ..., } 3 ally distinct tuples in r, and X is a subset of A1 A2 An , , ≥ θ 1 then the inequality SEX (t1 t3) does not necessarily SE ([0, 0] , [0.85, 0.95]) . hold true. 3 Proof. Let r be the vague relation instance on Let R (A1, A2, ..., An) be a relation scheme on domains R (Name, Int, S ucc) given by Table 1. U1, U2,..., Un, where Ai is an attribute on the universe Suppose that X = {Int, S ucc}. We have (see matrices I of discourse Ui, i ∈ I. Suppose that r is a vague rela- and S ), tion instance on R (A1, A2, ..., An). Let t1 and t2 be any two (vague) tuples in r. Finally, let X ⊆ {A , A , ..., A } be some 1 2 n , set of attributes. SEX (t1 t2) = { , , , } We deﬁne the similarity measure SEX (t1, t2) between min SE (t1 [Int] t2 [Int]) SE (t1 [S ucc] t2 [S ucc]) the tuples t1 and t2 on the attribute set X as = min {0.69, 0.64} = 0.64,

SEX (t2, t3) SEX (t1, t2) = min {SE (t1 [A] , t2 [A])} . A∈X = min {SE (t2 [Int] , t3 [Int]) , SE (t2 [S ucc] , t3 [S ucc])} Now, we are able to derive some auxiliary results. = min {0.78, 0.64} = 0.64,

SEX (t1, t3) Lemma 1 Let R (A1, A2, ..., An) be a relation scheme on = min {SE (t1 [Int] , t3 [Int]) , SE (t1 [S ucc] , t3 [S ucc])} domains U1, U2,..., Un, where Ai is an attribute on the = min {0.22, 1} = 0.22. universe of discourse Ui, i ∈ I. Let r be a vague relation

3 292 MATEC Web of Conferences , 01001 (2019) https://doi.org/10.1051/matecconf/2019 29201001 CSCC 2019

Now, SEX (t1, t2) > θ and SEX (t2, t3) > θ, where θ = Let R (A1, A2, ..., An) be a relation scheme on domains 0.5 for example. U1, U2,..., Un, where Ai is an attribute on the universe ∈ However, SEX (t1, t3) < θ. This completes the proof. ⊓⊔ of discourse Ui, i I. A fuzzy relation instance r U1 on R (A1, A2, ..., An) is a subset of the cross product 2 U2 Un Note that the fact that SE (t1 [A] , t2 [A]) ∈ [0, 1] for all × 2 × ... × 2 of the power sets of the domains A ∈ X, yields that SEX (t1, t2) ∈ [0, 1]. of the attributes. A tuple t of r is then of the form Moreover, (t [A1] , t [A2] , ..., t [An]), where t [Ai] ⊆ Ui for i ∈ I, and where we assume that t [Ai] , ∅ for i ∈ I. SEX (t1, t2) = min {SE (t1 [A] , t2 [A])} As we already noted, any fuzzy attribute value is de- A∈X scribed by some set of crisp values, where each of the crisp = min {SE (t2 [A] , t1 [A])} A∈X values is similar to the fuzzy value. More precisely, each attribute domain U , j ∈ I is equipped with some simi- =SEX (t2, t1) . j larity relation s j : U j × U j → [0, 1], where s j : U j × U j → , Furthermore, SEX (t1, t2) = 1 if and only [0 1] is said to be a similarity relation on U j, if for ev- y ∈ , = , y = if min {SE (t1 [A] , t2 [A])} = 1 if and only if ery x, , z U j, the conditions: s j (x x) 1, s j (x ) A∈X y, , ≥ , y , y, SE (t [A] , t [A]) = 1 for all A ∈ X if and only if t [A] = s j ( x), and s j (x z) max min s j (x ) s j ( z) hold 1 2 1 y∈U j t2 [A] for all A ∈ X. true. Thus, while in the case of relational database model , = Finally, SEX (t1 t2) 0 if and only if we were able to check if t j [Al] = tk [Al], now, in the case of min {SE (t1 [A] , t2 [A])} = 0 if and only if there exists A ∈ fuzzy relational database model, we are able to define how A∈X ⊆ { , , ..., } , = conformant t [A ] and t [A ] are. In particular (see, [14]), X A1 A2 An , such that SE (t1 [A] t2 [A]) 0 if and j l k l ∈ , − if sl : Ul × Ul → [0, 1] is a similarity relation on Ul, and only if there exists A X such that tt1[A] (u) 1 ft1[A] (u) h i = , , − = , ∈ t [A ] = d , t [A ] = d , then the conformance φ A t , t [0 0], tt2[A] (u) 1 ft2[A] (u) [1 1] for all u UA or j l j k l k l j k , − = , , − = tt1[A] (u) 1 ft1[A] (u) [0 1], tt2[A] (u) 1 ft2[A] (u) of the attribute Al on tuples t j and tk is defined by [a, a], a ∈ [0, 1] for all u ∈ UA, where UA ∈ {U1, U2, ..., Un} h i is the universe of discourse that corresponds to the at- φ Al t j, tk tribute A ∈ X. ( ( ) = min min max {sl (x, y)} , x∈d y∈d ( j k )) 2 Vague functional dependencies min max {sl (x, y)} . x∈dk y∈d j Let R (A , A , ..., A ) be a relation scheme on domains U , 1 2 n 1 h i U2,..., Un, where Ai is an attribute on the universe of dis- φ , Hence, we calculate Al t j tk instead of calculating course Ui, i ∈ I. Suppose that r is a relation instance on i t [A ] , t [A ] , i.e., instead of checking whether or not R (A1, A2, ..., An). Furthermore, let X and Y be subsets of l j l k l h i {A1, A2, ..., An}. t j [Al] = tk [Al]. Consequently, we calculate φ X t j, tk = Relation instance r is said to satisfy the functional de- instead of checking whether or not t j[X]h tki[X], where X → pendency X Y, if for every pair of tuples t1 and t2 in r, is a subset of {A1, A2, ..., An}, and φ X t j, tk is the con- = = = t1 [X] t2 [X] implies that t1 [Y] t2 [Y]. Here, t1 [X] formance of the attribute set X on tuples t j and tk, defined t2 [X] means that t1 [A] = t2 [A] for every A ∈ X. by As it is known, the relational model restricts the at- h i n h io tribute values to be atomic (if the attribute value is precise φ X t j, tk = min φ A t j, tk . A∈X and crisp, then the value is atomic), i.e., t [Ai] ∈ Ui, i ∈ I ∈ ∈ for every t r. Moreover, each U j, j I is equipped with For the similarity-based fuzzy relational database ap- × → { , } , y the identity relation i j : U j U j 0 1 , such that i j (x ) proach, we refer to [2]-[4]. = 1 if and only if x = y, and i j (x, y) = 0 if and only if x , Now, the condition: t1 [X] = t2 [X] implies that t1 [Y] y. In other words, the crisp relational model compares two = t2 [Y], could be read as: if φ (X [t1, t2]) resp. φ (Y [t1, t2]) attribute values by checking whether or not the two values is the conformance of the attribute set X resp. Y on tu- are equal. Thus, i t [A ] , t [A ] = 1 if and only if t [A ] l j l k l j l ples t1 and t2, then φ (Y [t1, t2]) ≥ φ (X [t1, t2]). More pre- = tk [Al], and il t j [Al] , tk [Al] = 0 if and only if t j [Al] , cisely, we could say that some fuzzy relation instance r , , ..., tk [Al], where t j, tk ∈ r, and Al ∈ {A1, A2, ..., An}. on R (A1 A2 An) satisfies the fuzzy functional depen- Unfortunately, the ordinary relational database model dency X →F Y, if for every pair of tuples t1 and t2 in r, is far from being enough to capture all of the information φ (Y [t1, t2]) ≥ φ (X [t1, t2]). about the real-world facts. Namely, the attribute values are Consider the following example. usually imprecise ones, i.e., fuzzy. In order to be able to Let R (Tea, Exp, S al) be a relation scheme on do- store such fuzzy attribute value, one stores a set of crisp mains U1 = {Grace, Harry, Oscar}, U2 = {low, high}, U2 values in place of the fuzzy value, where the crisp values = {3800USD, 4500USD}, where Exp (as experience) and are some, mutually distinct elements from the attribute do- S al (as salary) are fuzzy attributes on universes U2 and U3, main, and are similar to the fuzzy value. Therefore, the respectively, and Tea (as teachers) is ordinary attribute on following definition is more than justified. the universe of discourse U1.

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Let s2 : U2 × U2 → [0, 1] be the similarity relation Assume for a moment that t2 Exp = {low, high}. on U2 defined by s2 (low, high) = 0.3, and s3 : U3 × Now, U3 → [0, 1] be the similarity relation on U3 defined by s (3800USD, 4500USD) = 0.5. Let r be the fuzzy rela- 3 3 φ , tion instance on R (Tea, Exp, S al) given by Table 6. (X [t1 t2]) =φ (Exp [t , t ]) ( 1 2 Table 6. n = min min max {s2 (low, low) , s2 (low, high)} , Tea Exp S all o t1 {Grace}{low, high}{3800USD, 4500USD} max {s2 (high, low) , s2 (high, high)} , t2 {Oscar}{low} {3800USD} n min max {s2 (low, low) , s2 (low, high)} , ) Consider the dependency: teachers with similar expe- o { g , w , g , g } riences should have similar salaries. Note that the values max s2 (hi h lo ) s2 (hi h hi h) of the attributes: experience and salary my be imprecise. ( This fact, as well as the fact that the word similar is ap- = min min {max {1, 0.3} , max {0.3, 1}} , plied within dependency, imply that this dependency can ) be taken as an example of fuzzy functional dependency. It min {max {1, 0.3} , max {0.3, 1}} can be written in the form X →F Y, where X = {Exp} and = { } Y S al . = min {min {1, 1} , min {1, 1}} Let’s check if the fuzzy relation instance r satisfies X 3 = min {1, 1} = 1. →F Y. We obtain,

In this case, φ (Y [t1, t2]) = 0.5 < 1 = φ (X [t1, t2]), i.e., φ (X [t1, t2]) r3 violates X →F Y. = min {φ (A [t1, t2])} = φ (Exp [t1, t2]) A∈X ( Note that the dependency: teachers with similar experiences should have similar salaries, tells the truth about = min min {max {s (low, low)} , max {s (high, low)}} , 2 2 the real-world. Obviously, both scenarios t Exp = {low} ) 2 and t2 Exp = {low, high} are possible in reality. In min {max {s2 (low, low) , s2 (low, high)}} the first case, Grace and Oscar have similar experiences ( ) and similar salaries, where the salaries are more simi- = min min {max {1} , max {0.3}} , min {max {1, 0.3}} lar than the experiences are. In the second case, their salaries are similar, but not identical, although their ex- = min {min {1, 0.3} , min {1}} periences are identical. This discussion shows that the dependency: teachers with similar experiences should have = min {0.3, 1} = 0.3, similar salaries, makes sense by itself, and that the instance r3 should satisfy this dependency in both cases, t2 Exp = {low} and t2 Exp = {low, high}. The inequali- φ (Y [t1, t2]) ties φ (Y [t1, t2]) ≥ φ (X [t1, t2]), φ (Y [t1, t2]) < φ (X [t1, t2]), = min {φ (A [t , t ])} = φ (S al [t , t ]) ∈ 1 2 1 2 where t Exp = {low}, t Exp = {low, high}, respec- A Y ( 2 2 n tively, tell us, however, that the condition φ (Y [t1, t2]) ≥ = { , } , min min max s3 (3800USD 3800USD) φ (X [t1, t2]) is not adequate for determining whether or not o the fuzzy relation instance r satisfies the fuzzy functional { , } , max s3 (4500USD 3800USD) dependency X →F Y. If this condition is satisfied, the instance r satisfies the dependency X →F Y for sure. Other- wise, if the condition fails, the instance r may or may not satisfy X → Y. n n F min max s3 (3800USD, 3800USD) , In order to overcome these difficulties and correct the oo) irregularities, Sozat and Yazici [14] introduced the follow- s3 (3800USD, 4500USD) ing definition.

( ) Let R (A1, A2, ..., An) be a relation scheme on domains = min min {max {1} , max {0.5}} , min {max {1, 0.5}} U1, U2,..., Un, where Ai is an attribute on the universe of discourse Ui, i ∈ I. Suppose that r is a fuzzy relation in- = min {min {1, 0.5} , min {1}} stance on R (A1, A2, ..., An). Furthermore, let X and Y be { , , ..., } θ ∈ , = min {0.5, 1} = 0.5. subsets of A1 A2 An , and [0 1]. Fuzzy relation instance r is said to satisfy the fuzzy functional depen- θ The condition φ (Y [t1, t2]) ≥ φ (X [t1, t2]) is satisﬁed. dency X −→F Y, if for every pair of tuples t1 and t2 in r, This means that r3 satisﬁes X →F Y. φ (Y [t1, t2]) ≥ min {θ, φ (X [t1, t2])}.

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Thus, if it happens that φ (Y [t1, t2]) ≥ φ (X [t1, t2]) for scenario presented by the vague relation instance r (Table t1, t2 ∈ r, then 1), makes sense in reality. This actually means that the instance r should somehow satisfy the dependency X → φ , ≥φ , V (Y [t1 t2]) (X [t1 t2]) Y. ≥ min {θ, φ (X [t1, t2])} Reasoning as in the case of fuzzy functional dependencies, we conclude that the condition SEY (t1, t2) ≥ θ SE (t , t ), t , t ∈ r, must be adapted. We introduce the for every t1, t2 ∈ r, and θ ∈ [0, 1], i.e., r satisfies X −→F Y X 1 2 1 2 for every θ ∈ [0, 1]. following definition. Let R (A , A , ..., A ) be a relation scheme on domains More generally, if it happens that for every t1, t2 ∈ r, 1 2 n U , U ,..., U , where A is an attribute on the universe of either φ (Y [t1, t2]) ≥ φ (X [t1, t2]) or φ (Y [t1, t2]) ≥ θ, then 1 2 n i discourse Ui, i ∈ I. Suppose that r is a vague relation instance on R (A1, A2, ..., An). Furthermore, let X and Y be φ (Y [t1, t2]) ≥ min {θ, φ (X [t1, t2])} subsets of {A1, A2, ..., An}, and θ ∈ [0, 1]. Vague relation instance r is said to satisfy the vague functional depen- θ ∈ −→ θ for every t1, t2 r, i.e., r satisfies X F Y. dency X −→V Y, if for every pair of tuples t1 and t2 in r, ∈ Consequently, if it happens that for some t1, t2 r, SEY (t1, t2) ≥ min {θ, SEX (t1, t2)}. φ , < φ , φ , < θ (Y [t1 t2]) (X [t1 t2]), and (Y [t1 t2]) , then the Thus, if SEY (t1, t2) ≥ SEX (t1, t2) for t1, t2 ∈ r, the in- θ −→ θ instance r violates the dependency X F Y. stance r satisfies X −→V Y for θ ∈ [0, 1]. If for every t1, t2 In particular, the instance r3 satisfies the dependency ∈ r, either SEY (t1, t2) ≥ SEX (t1, t2) or SEY (t1, t2) ≥ θ, the θ θ −→ θ ∈ , −→ ∈ X F Y for every [0 1] (see, Table 6). Furthermore, instance r satisfies X V Y. Finally, if for some t1, t2 r, if t2 Exp = {low, high}, then the instance r3 satisfies resp. the conditions SEY (t1, t2) < SEX (t1, t2) and SEY (t1, t2) < θ θ violates the dependency X −→F Y if θ ∈ [0, 0.5] resp. θ ∈ θ hold true, the instance r violates X −→V Y. (0.5, 1]. Now, the vague relation instance r given by Table 1, θ θ ∈ , −→ satisfies resp. violates the vague functional dependency X The value [0 1] that appears in the notation X F θ θ Y is called the linguistic strength of the fuzzy functional −→V Y ({Int} −→V {S ucc}), if θ ∈ [0, 0.13] resp. θ ∈ (0.13, 1]. θ dependency. If θ = 1, the fuzzy functional dependency X θ = −→ θ If 1, the vague functional dependency X V Y be- −→F Y becomes X →F Y. comes X →V Y. Now, one could try to say that some vague relation in- For yet another definition of vague functional depen- stance r on R (A1, A2, ..., An) satisfies the vague functional dency, called α-vague functional dependency, see [13]. dependency X →V Y, if for every pair of tuples t1 and t2 in r, SEY (t1, t2) ≥ SEX (t1, t2) (see, e.g., [12], [17]). Recall the vague relation instance r given by Table 1. Consider the dependency: the intelligence level of a 3 Soundness of inference rules for person more or less determines the degree of success. vague functional dependencies Since the values of the attributes intelligence and suc- The following rules are the inference rules for vague func- cess may be imprecise, we may consider this dependency tional dependencies (VFDs). as a vague functional dependency. We can write it in the θ form X →V Y, where X = {Int} and Y = {S ucc} (see, Table 1 VF1 Inclusive rule for VFDs: If X −→V Y holds, and 1). θ θ ≥ θ , then X −→2 Y holds. Let’s check if the vague relation instance r satisfies X 1 2 V → V Y. VF2 Reflexive rule for VFDs: If X ⊇ Y, then X →V Since (see, matrices I and S ), Y holds. θ VF3 Augmentation rule for VFDs: If X −→V Y holds, θ SEY (t3, t4) = min {SE (t3 [A] , t4 [A])} A∈Y then XZ −→V YZ holds. = , = . SE (t3 [S ucc] t4 [S ucc]) 0 13 θ1 VF4 Transitivity rule for VFDs: If X −→V Y and Y θ2 min(θ1,θ2) and −→V Z hold true, then X → V Z holds true.

SEX (t3, t4) = min {SE (t3 [A] , t4 [A])} Here, XZ means X ∪ Z. A∈X =SE (t [Int] , t [Int]) = 0.46, 3 4 Theorem 4 The inference rules: VF1, VF2, VF3 and VF4

it follows that the instance r violates X →V Y. The vague are sound. functional dependency: the intelligence level of a person Proof. (proof for VF1) We must prove that r more or less determines the degree of success, however, θ −→2 tells the truth about the real-world in the same way the satisﬁes X V Y, if r is any vague relation instance on , , ..., fuzzy functional dependency: teachers with similar expe- R (A1 A2 An) which satisﬁes the vague functional de- θ1 riences should have similar salaries does. Moreover, the pendency X −→V Y.

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θ1 θ2 Suppose that r satisﬁes X −→V Y, but violates X −→V Now, there exist tuples t1 and t2 in r, such that Y. θ , < θ , θ , , . 2 SEYZ (t1 t2) min (min ( 1 2) SEX (t1 t2)) Since r violates X −→V Y, we know that there are tuples

t1 and t2 in r, such that It follows that SEYZ (t1, t2) < θ1, SEYZ (t1, t2) < θ2, SE (t , t ) < SE (t , t ). , < {θ , , } . YZ 1 2 X 1 2 SEY (t1 t2) min 2 SEX (t1 t2) θ1 θ2 Since r satisﬁes X −→V Y and X −→V Z, we have that Therefore, SE (t , t ) < θ and SE (t , t ) < SE (t , t ). Y 1 2 2 Y 1 2 X 1 2 , ≥ θ , , , θ1 SEY (t1 t2) min ( 1 SEX (t1 t2)) The instance r satisﬁes the dependency X −→ Y. V , ≥ θ , , . Hence, SEZ (t1 t2) min ( 2 SEX (t1 t2)) We deduce, SEY (t1, t2) ≥ min {θ1, SEX (t1, t2)} .

SEYZ (t1, t2) If min {θ1, SEX (t1, t2)} = θ1, then = min {SE (t1 [A] , t2 [A])} A∈YZ SEY (t1, t2) ≥ θ1 ≥ θ2. = min min {SE (t1 [A] , t2 [A])} , A∈Y This contradicts the fact that SEY (t1, t2) < θ2. {θ , , } = , min {SE (t1 [A] , t2 [A])} If min 1 SEX (t1 t2) SEX (t1 t2), then A∈Z

= min (SEY (t1, t2) , SEZ (t1, t2)) SEY (t1, t2) ≥ SEX (t1, t2) . ≥ min (min (θ1, SEX (t1, t2)) , min (θ2, SEX (t1, t2))) This contradicts the fact that SEY (t1, t2) < SEX (t1, t2). = min (θ1, θ2, SEX (t1, t2)) θ2 Consequently, r satisﬁes X −→V Y. = min (min (θ1, θ2) , SEX (t1, t2)) . The cases VF2-VF4 are discussed similarly. This completes the proof. ⊓⊔ The obtained inequality contradicts the fact that

SEYZ (t1, t2) < min (min (θ1, θ2) , SEX (t1, t2)) .

θ ,θ 4 Soundness of additional inference min( 1 2) Therefore, r satisﬁes X → V YZ. rules for vague functional dependencies (proof II for VF5) We have:

The following inference rules are additional inference θ1 1) X −→ Y (input), rules for vague functional dependencies. We shall prove V θ that these rules follow from the rules: VF1, VF2, VF3 and 2 2) X −→ Z (input), VF4. This will mean that the vague functional dependen- V θ cies obtained by the additional rules can certainly be ob- 1 3) X −→ XY (VF3, 1) augment with X), tained by successive application of the rules: VF1, VF2, V θ VF3 and VF4. The additional inference rules, however, 2 4) XY −→V YZ (VF3, 2) augment with Y), can make such an effort much shorter and easier. min(θ ,θ ) θ θ 1 2 1 2 5) X → V YZ (VF4, 3), 4)). VF5 Union rule for VFDs: If X −→V Y and X −→V min(θ1,θ2) Z hold true, then X → V YZ holds also true. The cases VF6 and VF7 are discussed similarly. θ 1 ⊓⊔ VF6 Pseudo-transitivity rule for VFDs: If X −→V Y This completes the proof. θ2 min(θ1,θ2) and WY −→V Z hold true, then WX → V Z holds true. θ 5 Completeness of inference rules for −→ VF7 Decomposition rule for VFDs: If X V Y vague functional dependencies θ holds, and Z ⊆ Y, then X −→V Z also holds. Let R (A1, A2, ..., An) be a relation scheme on domains U1, U2,..., Un, where Ai is an attribute on the universe of discourse U , i ∈ I. Theorem 5 The inference rules: VF5, VF6 and VF7 are i Suppose that V is some set of vague functional depen- sound. + dencies on {A1, A2, ..., An}. The closure V of V is the set Proof. (proof I for VF5) Let r be any vague relation in- of all vague functional dependencies that can be derived V θ1 from by repeated applications of the inference rules: stance (on R (A1, A2, ..., An)) which satisfies X −→V Y and θ θ ,θ VF1, VF2, VF3 and VF4. 2 min( 1 2) + X −→V Z. We shall prove that r satisfies X → V YZ. Note that the set V is infinite one regardless of min(θ1,θ2) θ Suppose that r violates X → V YZ. whether the set V is finite or not. Namely, if X −→V Y

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θ ′ ′′ 1 + θ ∈ θ , θ belongs to V, then, by VF1, X −→V Y belongs to V for Fix some , where θ ∈ , θ all 1 [0 ). ′′ θ θ = max {1θl (V)} . Let X −→ Y be some vague functional dependency on θ V 1 + θ A−→V B∈V : 1θl(V)<θ {A1, A2, ..., An}. The dependency X −→V Y may or may not θ θ V+ −→ 1 + belong to . The limit strength of X V Y (with respect Now, if A −→V B ∈ V is a vague functional depen- θ V l( ) θ V θ to V) is the number θl (V) ∈ [0, 1], such that X −−−−→V dency whose limit strength 1 l ( ) is less than , then ′ ′ ′ ′ θ θ (V) < θ < θ. Otherwise, if θ (V) ≥ θ, then θ < θ Y V+ θ ≤ θ (V) X −→ Y 1 l 1 l belongs to , and l for each V that ≤ θ (V). belongs to V+. 1 l Suppose that U1 = U2 = ... = Un = {u} = U. θ + If X −→V Y belongs to V , then the limit strength θl (V) Let θ obviously exists. Otherwise, if X −→V Y does not belong to + V1 = ⟨u, tV (u) , 1 − fV (u) ⟩ : u ∈ U V , the limit strength θl (V) does not necessarily exist. 1 1 = ⟨u, t (u) , − f (u) ⟩ = {⟨u, a⟩} Let X be a subset of {A1, A2, ..., An}, and θ be a number V1 1 V1 in [0, 1]. The closure X+ (θ, V) of X (with respect to V) is θ and the set of attributes A ∈ {A1, A2, ..., An}, such that X −→V A V+ belongs to . V2 = ⟨u, tV (u) , 1 − fV (u) ⟩ : u ∈ U + 2 2 Now, if A ∈ X, then, by VF2, X →V A belongs to V . = ⟨u, tV (u) , 1 − fV (u) ⟩ = {⟨u, b⟩} θ + 2 2 Hence, by VF1, X −→V A belongs to V . Therefore, A ∈ X+ (θ, V). Since A ∈ X, we conclude that X ⊆ X+ (θ, V). be two vague sets in U, such that

′ SE (a, b) = θ . Theorem 6 Let R (A1, A2, ..., An) be a relation scheme on domains U1, U2,..., Un, where Ai is an attribute on the Hence, universe of discourse U , i ∈ I. Let V+ be the closure of i , = , = θ′ . V, where V is some set of vague functional dependencies SE (V1 V2) SE (a b) θ on {A1, A2, ..., An}. Suppose that X −→V Y is some vague + θ, V θ Let X ( ) be the closure of X. functional dependency on {A1, A2, ..., An}. Then, X −→V Y Now, let r = {t1, t2} be the vague relation instance on + + belongs to V if and only if Y ⊆ X (θ, V). R (A1, A2, ..., An) given by Table 7.

θ + Table 7. Proof. (⇒) Suppose that X −→V Y belongs to V . θ + + θ, V Now, by VF7, X −→V A belongs to V for every A ∈ Y. attributes of X ( ) other attributes + Hence, A ∈ X (θ, V) for every A ∈ Y, i.e., we have that t1 V1, V1, ..., V1 V1, V1, ..., V1 + Y ⊆ X (θ, V). t2 V1, V1, ..., V1 V2, V2, ..., V2 ⇒ ⊆ + θ, V ( ) Let Y X ( ). θ + −→1 Now, A ∈ X (θ, V) for every A ∈ Y. It is not so hard to prove that r satisﬁes A V B if A θ θ θ + −→1 V+ −→ Therefore, X −→ A belongs to V for every A ∈ Y. V B belongs to , and violates X V Y. V ⊓⊔ θ + This completes the proof. Hence, by V5, X −→V Y belongs to V . This completes the proof. ⊓⊔ References Theorem 7 The set {VF1, VF2, VF3, VF4} is complete set. [1] J. F. Baldwin, Knowledge engineering using a fuzzy relational inference language, IFAC Proceedings

Proof. Let R (A1, A2, ..., An) be a relation scheme on do- Volumes 16, 14-20 (1983) mains U1, U2,..., Un, where Ai is an attribute on the uni- [2] B. P. Buckles, F. E. Petry, A fuzzy representation of verse of discourse Ui, i ∈ I. data for relational databases, Fuzzy Sets and Systems Let V+ be the closure of V, where V is some set of 7, 213-216 (1982) vague functional dependencies on {A1, A2, ..., An}. [3] B. P. Buckles, F. E. Petry, Fuzzy databases and their θ applications, Fuzzy Inform. Decision Processes 2, Suppose that X −→V Y is some vague functional depen- + 361-371 (1982) dency on {A1, A2, ..., An} which is not a member of V . [4] B. P. Buckles, F. E. Petry, Uncertainty models in in- In order to prove the theorem, it is enough to prove that formation and database systems, J. Inform. Sci. 11, there exists a vague relation instance r on R (A1, A2, ..., An) θ θ 1 1 + 77-87 (1985) which satisﬁes A −→V B if A −→V B belongs to V , and θ [5] S. M. Chen, Similarity Measures Between Vague violates X −→V Y. Sets and Between Elements, IEEE Transactions on r can be constructed in the following way. Systems, Man and Cybernetics 27, 153-159 (1997)

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[6] S. M. Chen, Measures of Similarity Between Vague Lu, H., Zhou, S., Ling, T.-W. (eds.) ER 2004 LNCS, Sets, Fuzzy Sets and Systems 74, 217-223 (1995) vol. 3288, pp. 259-272, Springer-Verlag, Berlin Hei- [7] D. H. Hong, C. Kim, A note on Similarity Measures delberg, (2004) Between Vague Sets and Between Elements, Infor- [13] J. Mishra, S. Ghosh, A New Functional Dependency mation Sciences 115, 83-96 (1999) in a Vague Relational Database Model, International [8] C. Giardina, I. Sack, D. Sinha, Fuzzy Field Rela- Journal of Computer Applications 39, 29-36 (2012) tional Database Tech. Report 8332 (Elect. Engng. [14] M. Sozat, A. Yazici, A complete axiomatization for and Computer Science Dept., Stevens Institute of fuzzy functional and multivalued dependencies in Technology, Hoboken, NJ, 1983) fuzzy database relations, Fuzzy Sets and System 117, [9] P. Grzegorzewski, Distances Between Intuitionistic 161-181 (2001) Fuzzy Sets and/or Interval-valued Fuzzy Sets Based [15] E. Szmidt, J. Kacprzyk, Distances Between Intu- on the Hausdorff Metric, Fuzzy Sets and Systems itionistic Fuzzy Sets, Fuzzy Sets and Systems 114, 148, 319-328 (2004) 505-518 (2000) [10] M. Levene, G. Loizou, A Guided Tour of Relational [16] J. D. Ullman, Primciples of Database Systems (Com- Databases and Beyond (Springer-Verlag, London, puter Science Press, Rockville, MD, 1982) 1999) [17] F. Zhao, Z. M. Ma, Functional Dependencies in [11] F. Li, Z. Xu, Measures of Similarity Between Vague Vague Relational Databases, In: IEEE International Sets, Journal of Software 12, 922-927 (2001) Conference on Systems, Man, and Cybernetics, vol. [12] A. Lu, W. Ng, Managing Merged Data by Vague 5, pp. 4006-4010, Taipei, Taiwan, (2006) Functional Dependencies, In: Atzeni, P., Chu, W.,