292 MATEC Web of Conferences , 01001 (2019) https://doi.org/10.1051/matecconf/201929201001 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 2 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 2 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 = hu; tV (u) ; 1 − fV (u) i : u 2 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 2 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) 2 [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 f y; ; ; ; ; ; ; g = If it happens that tV (u) = 1 − fV (u) = 1, then the Em Ted Jim Katie S ara Tina Joe John , U2 f ; ; g = f ; g 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 2 [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 = fhu1; [0:2; 0:7]i; hu2; [1; 1]ig Table 1. instead of Name Int S ucc f g fh ; ; ig fh ; ; ig = fh ; : ; : i; h ; ; i; h ; ; ig ; t1 Ted 115 [1 1] 10 [1 1] V u1 [0 2 0 7] u2 [1 1] u3 [0 0] f g h ; : ; : i; h ; : ; : i; t2 S ara 115 [0 7 0 9] 5 [0 6 0 9] h130; [0:9; 0:95]i h10; [0:8; 0:95]i where U = fu ; u ; u g is some universe of discourse, and 1 2 3 t fJimg h130; [0:8; 0:9]i; fh10; [1; 1]ig V is some vague set in U. 3 h145; [0:85; 0:95]i Let R (A1; A2; :::; An) be a relation scheme on domains t4 fKatieg fh145; [1; 1]ig fh5; [0:9; 0:95]ig U1, U2,..., Un, where Ai is an attribute on the universe of discourse Ui, i 2 f1; 2; :::; ng = I. The vague sets fh115; [1; 1]ig and fh10; [1; 1]ig, 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 2 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/201929201001 CSCC 2019 by the sets f115; 130; 145g and f5; 10g, 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 00 f g fh ; ; g fh ; ; g t1 Joe 130 [1 1] 5 [1 1] fh115; [0:7; 0:9]i; h130; [0:9; 0:95]ig. Since the truth value 00 t fTinag fh145; [1; 1]g fh10; [1; 1]g 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 < fh115; [0:7; 0:9]i; h130; [0:9; 0:95]ig). 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 2 U1 and u2 2 U2, respectively, where son who is relatively successful. Finally, Jim is a pretty V = hu ; t (u ) ; 1 − f (u ) i : u 2 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 2 f1; 2g. 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 − · 0 2 t fEmyg h ; a i; h ; a i; q 1 130 1 5 3 h ; i h ; i − − + − : 145 a2 10 a4 1 tV1 (u1) tV2 (u2) fV1 (u1) fV2 (u2) 0 f g fh ; ig fh ; ig t2 John 115 a5 5 1 Note that SE (a1; a2) 2 [0; 1]. Moreover, SE (a ; a ) = SE (a ; a ), SE (a ; a ) = 1 if 2 ; 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 = 2 fh ; i; h ; ig 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 2 [0; 1] (a = 2 ; 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 2 [0; 1]). 2 fh ; ig 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 0 t fEmyg h130; [a1; a1]i; h5; [a3; a3]i; 1 A = hu; tA (u) ; 1 − fA (u) i : u 2 U h145; [a2; a2]i h10; [a4; a4]i 0 t fJohng fh115; [a ; a ]ig fh5; [1; 1]ig and 2 5 5 B = hu; tB (u) ; 1 − fB (u) i : u 2 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.