Advanced Database Systems Fuzzy Orderings in Flexible Query Answering Systems Lê Hồng Dũng – 7140819 Âu Mậu Dương – 7140820 Lê Nguyên Dũng – 7140224 Ngô Đình Dũng – 1570203 Content Introduction VQS Conditions based on fuzzy orderings The aggregation issue Summary 1/ The vague query system (VQS) VQS is an add-on to conventional relational databases which acts as a proxy between the user and the database Since VQS communicates with the underlying database only on the basis of standard SQL which allows easy integration into existing applications In case of numeric attributes, considering Euclidean distances is most often sufficient For non-numeric attributes, VQS using a so-called NCR table (numeric coordinate representation) 1/ The vague query system (VQS) The syntax of VQL The operator ‘‘IS’’ should be understood in the sense of ‘‘is similar to’’ 1/ The vague query system (VQS) There is one single ‘‘IS’’ condition in the query, VQS retrieves all records from the data source and ranks them according to the distance from the query value In case that the column contains numeric values, the distance between two values x; y can easily be computed as the absolute value of the difference |x - y| (Euclidean norm for the one-dimensional real space R) If the column under consideration is non-numeric, the distance is computed as the distance of the associated values in the corresponding NCR table VQS works with normalized distances Every condition, therefore, is assigned a distance value normalized to the unit interal [0;1] 1/ The vague query system (VQS) In case that two or more ‘‘IS’’ conditions are combined with ‘‘AND’’, a weighted average of the distances in the different columns is used to rank the results (equal weights are used by default, which can be overridden using the optional ‘‘WEIGHTED BY’’ expression) VÍ DỤ Xét bảng liệu VÍ DỤ Bảng Ma trận khoảng cách Location bảng (bảng NCR) VÍ DỤ Consider the following query: SELECT FROM HotelTable WHERE Location IS ‘Salzburg Center’ AND Price IS 70 AND Category IS INTO ResultSet Assume that the distances between locations are given as in Table (as the result of computing a distance measure for corresponding values in an NCR table) To compute the result set for this query which means that the distance of any two locations is divided by 147.8, each discrepancy in the price is divided by 60, and each discrepancy in the category is divided by VÍ DỤ Using equal weights, we obtain the result set shown in Table sorted by the closeness to the query Example TL (2) n   ⇒Aw ( x1 , , xn ) =1 −min 1,1 −∑wi (1 − x )  i =1   n n   = max0, ∑wi xi −∑wi +1 i =1  i =1  n  n  With : ∑wi =1 ⇒Aw ( x1 , , xn ) = max0, ∑xi  i =1  n i =1  f P ( x ) = −ln x n  n  ⇒Aw ( x1 , , xn ) = exp∑wi ln x  =∏xiwi  i =1  i =1 n With : ∑wi =1 ⇒Aw ( x1 , , xn ) = n i =1 n ∏x i =1 i ● If not specify any weight, all weights are equal: w1 = = wn = n ● If specify weight for all conditions: ~ wi = wi n ~ w ∑i i =1 ● If specify weight some conditions, filled up with 1’s n TL = ∑ti n i =1 # Location TP = n n ∏t i =1 i t1 t2 t3 t TL Salzburg Liefering 0.7800 1.0000 1.0000 0.9267 Salzburg Aigen 0.7900 0.5333 1.0000 0.7744 TP Salzburg Liefering 0.8025 1.0000 1.0000 0.9293 Salzburg Aigen 0.8106 0.6271 1.0000 0.7981 Content Introduction VQS Conditions based on fuzzy orderings The aggregation issue Summary Weight Does not specify any weight at all If the weight was defined W(Location) = W(Price) = W(Star) = W1 = 4/6 = 2/3 W2 = 1/6 W3 = 1/6 The choice of the underlying t-norm With equal weights With Weights The choice of the underlying t-norm We have two choices Leads to the same results, there is no point in choosing Condition, radius C has a clear and unambiguous interpretation However, T(L) is a pragmatic and justifiable choice Summary NCR table for the no-numeric attributes and corresponding distance measures T-norm T with additive generator f Tolerance Radius C Operation of Fuzzy System Crisp Input Fuzzification Input Membership Functions Fuzzy Input Rule Evaluation Rules / Inferences Fuzzy Output Defuzzification Crisp Output Output Membership Functions Extensions Create a new type condition, such as “AT LEAST MEDIUM” or “AT MOST AROUND” CÂU HỎI THẢO LUẬN 1/ Trong hệ thống VQS, thuộc tính phi số tính nào? 2/ Cách tính khoảng cách lớn phần tử hệ thống VQS? [...]... fuzzy orderings  Definition 2: fuzzy ordering   A fuzzy relation L: X2 → [0,1] is called fuzzy ordering with respect to a t-norm T and a T-equivalence E, for brevity T-E-ordering, if and only if it is T-transitive and fulfills the following two axioms for all x, y 1 E-Reflexivity: E(x,y) L(x,y) 2 T-E-antisymmetry: T(L(x,y),L(y,x)) Conditions based on fuzzy orderings Definition 3: A crisp ordering... based on fuzzy orderings Some important formula be deduced from theorem 1&2: LC(x,y) = (TL-Ed,C-ordering) L’C(x,y) = (TP-Ed,C-ordering) Conditions based on fuzzy orderings Formula is applied following as: t(“x IS q” | x0) = EC(x0,q) t(“x IS AT LEAST q” | x0) = LC(q, x0) t(“x IS AT MOST q” | x0) = LC(x0, q) t(“x IS WITHIN (a,b)” | x0) = min(LC(min(a, b), x0), LC(x0, max(a, b))) Content Introduction... an explicit distinction between numeric and non-numeric attributes For non-numeric ones, only the ‘‘IS’’ condition is defined like in VQL For numeric ones, three new types of conditions ‘‘IS AT LEAST’’, ‘‘IS AT MOST’’, and ‘‘IS WITHIN’’ are added Content Introduction VQS Conditions based on fuzzy orderings The aggregation issue Summary Conditions based on fuzzy orderings The single conditions... Conditions based on fuzzy orderings  Theorem 2: Consider a continuous Archimedean t-norm T with additive generator f  For any pseudo-metric d: X2 → [0,] the mapping Ed: X2 → [0,] defined as Ed(x,y) = f-1 ( min (d(x,y), f(0) ) (1) Provided that E: X2 → [0,] is a T-equivalence, we can define a pseudo-metric dE: X2 → [0,] as dE (x,y) = f (E(x,y)) (2) Conditions based on fuzzy orderings  Proposition... ordering on a domain X and a T-equivalence E: X2 → [0,1] are called compatible if and only if the following holds for all x, y, z X: x y z => E(x,z) Conditions based on fuzzy orderings Theorem 1: [1, 2] Consider a fuzzy relation L on a domain X and a T-equivalence E Then the following two statements are equivalent: L is a strongly complete T-E-ordering There exists a linear ordering follows: • L(x,y)... orderings  Proposition 1: Let T be a continuous Archimedean t-norm with an additive generator f and let be an ordering of the domain X If a pseudo-metric d: X2 → [0,], fulfills x y z => d(x,z) (3) If a fuzzy equivalence relation E: X2 → [0,1] is compatible with , , its induced pseudo-metric dE defined as in (2), fulfills property (3) Conditions based on fuzzy orderings  Some important formula be deduced... conditions ‘‘IS’’, ‘‘IS AT LEAST’’, ‘‘IS AT MOST’’, and ‘‘IS WITHIN’’ are modeled Example: Price IS AT MOST 70 Conditions based on fuzzy orderings  Definition 1: Fuzzy equivalence relation   A binary fuzzy relation E: X2 → [0,1] is called fuzzy equivalence relation with respect to a t-norm T, for brevity T-equivalence, if and only if the following three axioms are fulfilled for all x, y, z X: 1 Reflexivity:... distance between two values in the column The result is that two distance values for different columns may be difficultly comparable This paper is to tackle the first problem Ngôn ngữ oVQL (Ordering-enriched vague query language) Ngôn ngữ oVQL (Ordering-enriched vague query language) Ngôn ngữ oVQL (Ordering-enriched vague query language) It is obvious that oVQL differs from VQL in the respect that there... shortcomings of VQS: 1 VQS is restricted to ‘‘IS’’ queries that are interpreted with a certain tolerance for imprecision For the price column, however, this is a painful limitation, as the user is not necessarily interested in a price that is as close to 70 as possible, but more likely in a price that exceeds 70 as little as possible 2 The normalization of distances is done for all columns independently... min(LC(min(a, b), x0), LC(x0, max(a, b))) Content Introduction VQS Conditions based on fuzzy orderings The aggregation issue Summary The aggregation issue Problem (2) ( x1 , , xn ) ∈[0,1]n ( y1 , , yn ) ∈[0,1]n T ( A( x1 , , xn ), A( y1 , , y n )) ≤ A(T ( x1 , y1 ), , T ( xn , y n )) n    −1 Aw ( x1 , , xn ) = f  min f (0), ∑wi f ( xi )   i =1    f ( Aw ( x1 , , xn )) = ∑i =1 wi f ( xi ... MOST’’, and ‘‘IS WITHIN’’ are added Content Introduction VQS Conditions based on fuzzy orderings The aggregation issue Summary Conditions based on fuzzy orderings The single conditions ‘‘IS’’,... ‘‘IS WITHIN’’ are modeled Example: Price IS AT MOST 70 Conditions based on fuzzy orderings  Definition 1: Fuzzy equivalence relation   A binary fuzzy relation E: X2 → [0,1] is called fuzzy equivalence... Conditions based on fuzzy orderings Some important formula be deduced from theorem 1&2: LC(x,y) = (TL-Ed,C-ordering) L’C(x,y) = (TP-Ed,C-ordering) Conditions based on fuzzy orderings Formula is