A Type-2 Fuzzy Relational Database Model

Research and Development on Information and Communication Technology A Type-2 Fuzzy Relational Database Model Nguyen Hoa Saigon University, Ho Chi Minh City, Vietnam E-mail: nguyenhoa@sgu.edu.vn Communication: received 11 January 2017, revised 17 July 2017, accepted 31 July 2017 Abstract: This paper introduces a type-2 fuzzy relational database model (T-2FRDB) as an extension of type-1 fuzzy relational database models with a full set of basic fuzzy relational algebraic operations that can represent and query uncertain and imprecise information in real world applica- tions. In this model, the membership degree of tuples in a fuzzy relation is represented by fuzzy numbers on [0, 1], and fuzzy relational algebraic operations are defined by using the extension principle for computing minimum and maximum values of such fuzzy numbers. Some properties of the type- 2 fuzzy relational algebraic operations in T-2FRDB are also formulated and proven as extensions of their counterpart in the type-1 fuzzy relational database models. Keywords: Fuzzy set, fuzzy relation, type-2 fuzzy relational database, type-2 fuzzy relational algebraic operation. I. INTRODUCTION As we know, the classical relational database model [1] is very useful for modeling, designing and implementing large-scale systems. However, it is restricted to representing and handling uncertain and imprecise information about objects in practice [1, 2]. For example, applications of the classical relational database model can not deal with a query like “find all patients who are young and have lung cancer and a high treatment cost”, where young and high are the vague notion and imprecise value [3, 4]. So far, there have been many relational database models studied and built based on the fuzzy theory for modeling objects about which information may be vague and impre- cise to overcome the limitations of the classical relational database model such as [5–11]. Such models are called fuzzy relational database models. There are two main approaches to represent fuzzy rela- tions in fuzzy relational database models. The first approach represents each fuzzy relation as a set of tuples whose each attribute may take a fuzzy set (or a possibility distribution inferred from a fuzzy set) [5–7, 12–14], whereby the membership degree of tuples for the relation is hidden in that of their attribute values. The second one represents each fuzzy relation as a fuzzy set of tuples whose each attribute only takes a single and precise value [3, 8–11, 15–18], whereby the membership degree of tuples for the relation also is that for the fuzzy set. Fuzzy relational database models that are built based on one of above approaches are extensions of the classical re- lational database model with fuzzy sets. They have different capabilities for expressing and dealing with uncertain and imprecise information. There are two types of the models based on the second approach, including the type-1 fuzzy relational database model (T-1FRDB), or the (ordinary) fuzzy rela- tional database model, whereby the membership degree of tuples is assigned to a number in [0, 1], and the type-2 fuzzy relational database model (T-2FRDB), whereby the membership degree of tuples is expressed as a fuzzy number on [0, 1]. Many type-1 fuzzy relational database models have been proposed such as [3, 8–10, 17, 18]. However, since the membership degree of tuples is ex- pressed as a number in [0, 1], these models were restricted in representing the associated imprecise degree of attribute values. In real world relational databases, since attribute values of tuples may be imprecise, there are many situations in which we do not know exactly the membership degree of tuples as a number in [0, 1] but only can estimate it as an approximate number (or a fuzzy number) on [0, 1]. Some type-2 fuzzy relational database models have been introduced to overcome the shortcoming of type-1 fuzzy relational database models [11, 15, 16, 19]. However, in [19], only notions of the relational schema and instance were defined but relational algebraic operations were not introduced. In [16], data representative notions were not formally defined and some fuzzy relational algebraic oper- ations were missing. Also, in [11, 15], the set of basic fuzzy relational algebraic operations was not complete. Thus, the abilities to express and deal with imprecise information of those models were limited. In this paper, we propose a type-2 fuzzy relational database model (T-2FRDB) as an extension of the type-1 fuzzy relational database models to overcome the short- comings of the models in [11, 15, 16]. In our T-2FRDB, the notions of the data representative model are completely 19 Research and Development on Information and Communication Technology defined formally, the full set of basic type-2 fuzzy rela- tional algebraic operations is built based on the extension principle for computing the minimum and maximum values of fuzzy numbers. Some properties of these algebraic operations are also formulated as theorems and proven coherently. T-2FRDB allows representation of soft queries associated with fuzzy sets to handle imprecise information in practice. The mathematics that we used to develop T-2FRDB is presented in Section II. Schemas and relations of T- 2FRDB are introduced in Section III. Type-2 fuzzy rela- tional algebraic operations and their properties in T-2FRDB are presented in Sections IV and V, respectively. Finally, conclusions and future research directions are given in Section VI. II. FUZZY SETS AND FUZZY RELATIONS In this section, we present some notions about fuzzy sets and fuzzy relations as a mathematical basis for developing the T-2FRDB model. Fuzzy sets are used to represent and execute soft queries while relations in T-2FRDB are defined by fuzzy relations. 1. Fuzzy Sets For a classical set, an element is to be or not to be in the set or, in other words, the membership degree of an element in the set is binary. For a fuzzy set, the membership degree of an element in the set is expressed by a real number in the interval [0, 1]. The fuzzy set is extended from the classical set as in [4] and is defined as follows. Definition 1: A fuzzy set A on a domain X is defined by a membership function µA from X to the closed inter- val [0, 1]. For each x ∈ X, µA(x) is the membership degree of x for A. We note that a classical set A on X is also a fuzzy set [3] with the membership function µA(x) = 1 for all x ∈ A and µA(x) = 0 for all x < A. Even an element e in X is also considered as a special fuzzy set on X with the membership function µe(e) = 1 and µe(x) = 0, for all x ∈ X and x , e. The fuzzy set A on X as in the above definition is called the ordinary fuzzy set and can be denoted by A = {x: µA(x) | x ∈ X}. In addition, the notation A(x) can be used to replace µA(x). The support of a fuzzy set A on X is a classical set containing all elements of X that have nonzero membership degrees in A. The height h(A) of a fuzzy set A on X is the largest membership degree obtained from all elements in the set. It means that h(A) = supx∈X A(x). A fuzzy set A is said to be normal when h(A) = 1 and subnormal when h(A) < 1. A fuzzy set A on the set of real number R is said to be convex if for any elements x, y, z in the support of A, the relation x < y < z implies that µA(y) ≥ min(µA(x), µA(z)). Operations on fuzzy sets are generally defined based on functions from the Cartesian product of closed intervals [0, 1] to the closed interval [0, 1]. However, the section only presents standard operations in [4] which are applied in computing the relations of T-2FRDB. Definition 2: Let A and B be two fuzzy sets on X and have the membership functions µA and µB, respectively. The complement of A, union, intersection and difference of A and B are defined by their membership functions, for all x ∈ X, as follows: 1) µAc (x) = 1 − µA(x); 2) µA∪B(x) = max(µA(x), µB(x)); 3) µA∩B(x) = min(µA(x), µB(x)); 4) µA−B(x) = min(µA(x), 1 − µB(x)). Fuzzy numbers are special fuzzy sets that are used to represent the fuzzy relations in T-2FRDB, and were defined in [20], as follows. Definition 3: A fuzzy number A is a fuzzy set on the set of real number R such that 1) A is a normal and convex fuzzy set; 2) The support of A is bounded. Example 1: The fuzzy set about 0.5, given by a mem- bership function and its graph as shown in Figure 1, is a fuzzy number. about 0.5(x) =  2x, ∀x ∈ [0, 0.5], 2(1 − x), ∀x ∈ (0.5, 1], 0, ∀x < [0, 1]. 1 0 1 0.5 Figure 1. Fuzzy number about 0.5. For computing and combining the membership degrees of tuples in type-2 fuzzy relational algebraic operations, we use two operations MIN and MAX, defined by using the extension principle in [3], as follows. Definition 4: Let A and B be two fuzzy numbers. The minimum value and maximum value of A and B are fuzzy numbers that are defined for all z ∈ R by 1) MIN(A, B)(z) = supz=min(x,y)min[A(x), B(y)]; 2) MAX(A, B)(z) = supz=max(x,y)min[A(x), B(y)]. 20 Vol. E–3, No. 14, Sep. 2017 Example 2: Let A = {1 : 1, 0.9 : 0.8, 0.8 : 0.3} and B = {0.6 : 0.3, 0.5 : 1, 0.4 : 0.4} be two fuzzy numbers, then MIN(A, B) = {0.6: 0.3, 0.5:1, 0.4:04}. The fuzzy set of type 2 is extended from the ordinary fuzzy set in [3] as below. Definition 5: Let =([0, 1]) be the set of all ordinary fuzzy sets on [0, 1]. A type-2 fuzzy set A on a domain X is defined by a membership function µA from X to =([0, 1]). For each x ∈ X, µA(x) is the membership degree of x for A. We note that since each number in [0, 1] is considered as a special ordinary fuzzy set, each ordinary fuzzy set is also considered as a special type-2 fuzzy set. 2. Fuzzy Relations The ordinary fuzzy relation and type-2 fuzzy relation are the foundation for fuzzy relational database models. The fuzzy relation and type-2 fuzzy relation are defined in [21] by extending the notion of the classical relation based on fuzzy sets as follows. Definition 6: Let A1, A2, . . . , Ak be non-empty sets. A k- ary ordinary fuzzy relation R on these sets is an ordinary fuzzy set on the Cartesian product A1 × A2 × · · · × Ak . Definition 7: Let A1, A2, . . . , Ak be non-empty sets. A k-ary type-2 fuzzy relation R on these sets is a type-2 fuzzy set on the Cartesian product A1 × A2 × · · · × Ak . We note that the ordinary fuzzy relation is also called a type-1 fuzzy relation. The membership functions of the ordinary fuzzy relation and type-2 fuzzy relation are µR : A1 × A2 × · · · × Ak → [0, 1] and µR : A1 × A2 × · · · × Ak → =([0, 1]), respectively. III. T-2FRDB SCHEMAS AND RELATIONS 1. Type-2 fuzzy Relational Schemas In the following, we will define a special schema in T- 2FRDB, which consists of a set of attributes associated with a membership function of a type-2 fuzzy set that is used as a basis for determining type-2 fuzzy relations. Definition 8: A type-2 fuzzy relational schema is a pair R = (U, µ), where 1) U = {A1, A2, . . . , Ak} is a set of pairwise different attributes; 2) µ is a function that maps each (ν1, ν2, . . . , νk) ∈ D1 × D2 × · · · × Dk to a fuzzy number on [0, 1], where Di are the domains of the attributes Ai (i = 1, . . . , k). As in the classical relational database model, the nota- tions R(U, µ) and R can be used to replace R = (U, µ). In addition, each t = (ν1, ν2, . . . , νk) is called a tuple on the set of attributes {A1, A2, . . . , Ak}. TABLE I RELATION PATIENT P NAME AGE DISEASE D COST µ L. V. A 53 Lung cancer 350 0.9 L. T. B 65 Cirrhosis 40 about 0.5 N. T. C 29 Bronchitis 70 1.0 T. T. D 21 Hepatitis 30 high Example 3: A type-2 fuzzy relational schema PATIENT in T-2FRDB describing patients can be given as PATIENT(P NAME,AGE,DISEASE,D COST, µ), where µ : string × integer × string × real → ℘([0, 1]); ℘([0, 1]) is the set of all fuzzy numbers on [0, 1], string, integer and real are domains of the attributes P NAME, DISEASE, AGE and D COST, respectively. 2. Type-2 Fuzzy Relations The following definition extends the notion of ordinary fuzzy relation in T-1FRDB to T-2FRDB. Definition 9: Let U = {A1, A2, . . . , Ak} be a set of k pairwise different attributes. A type-2 fuzzy relation r over the type-2 fuzzy relational schema R(U, µ), is a finite set of tuples {t1, t2, . . . , tn} on the set of {A1, A2, . . . , Ak} in which each tuple ti is associated with the fuzzy number µ(ti) representing the membership degree of ti in r , for every i = 1, 2, . . . , k. The notation t .A or t[A] is used to denote the value of attribute A of tuple t in r . The membership degree of ti in r is denoted by µr (ti). For each set of attributes X ⊆ {A1, A2, . . . , Ak}, the notation t[X] is used to denote the rest of t after eliminating the value of attributes not belonging to X . Note that, as in the classical database model, if we only care about a unique relation over a schema, we can unify its symbol name with its schema’s name. Example 4: A type-2 fuzzy relation over the schema PATIENT in Example 3 can be given as shown in Table I. In the relation, the attributes P NAME, AGE, DISEASE, and D COST provide information about name, age, disease and daily treatment cost of each patient, respectively. In reality, the disease of each patient is not always exactly determined by physicians. Similarly, the daily treatment cost for patients is also not known even as the patients know about their diseases. Here, the conventional unit for the treatment cost is 1000 VND. We note that µ(t) represents the membership degree of each tuple t in the relation (by Definition 9). It means that the precise degree of information about the attribute values is expressed by t. For example, let consider the 21 Research and Development on Information and Communication Technology first tuple t1 in the relation PATIENT and assume that information about the patient’s name (L. V. A) expressed by t1 is correct. The µ(t1) = 0.9 of t1 represents the aggregated precise degree of information about the age (53), diagnosed disease (Lung cancer) and daily treatment cost (350,000 VND) of the patient. With t4, we do not know precisely both information about the attribute values that it represents and its membership degree in the relation PATIENT. We are able to just estimate that µ(t4) is high where high = {0.5:0, 0.6:0.5, 0.7:0.8, 0.8:0.9, 0.9:1.0, 1:1.0} is a fuzzy number on [0, 1]. In real world applications, fuzzy sets that represent the membership degrees of tuples in a fuzzy relation, like high and about 0.5 as mentioned above, will be defined ade- quately and consistently based on the meaning and precise degree of information about that these tuples express. The fuzzy sets high and about 0.5 given in this example are simply meant to give illustration for Definition 9. Definition 10: A type-2 fuzzy relational database over a set of attributes is a set of type-2 fuzzy relations corresponding with the set of their type-2 fuzzy rela- tional schemas. We note that when µ(t) ∈ [0, 1] for every tuple t in a type-2 fuzzy relation, this relation becomes a type-1 fuzzy relation. In other words, a type-1 fuzzy relational database is a particular case of an T-2 FRDB by Definition 10. IV. SELECTION OPERATION ON T-2FRDB 1. Syntax of Selection Conditions The syntax of selection conditions in T-2FRDB is ex- tended from those in [17] with type-2 fuzzy relations as the following definition. Definition 11: Let R be a schema in T-2FRDB, X be a set of relational tuple variables and θ be a binary relation from {=,,, ≤, , ≥}. Then selection conditions are inductively defined and have one of the following forms: 1) x.A θ ν, where x ∈ X, A is an attribute in R and ν a precise value; 2) x.A → ν, where x ∈ X, A is an attribute in R, → a binary fuzzy relation and ν a fuzzy set value; 3) x.A1 θ x.A2, where x ∈ X, A1 and A2 are two different attributes in R; 4) ¬E , if E is a selection condition; 5) E1 ∧ E2, if E1 and E2 are selection conditions on the same relational tuple variable; 6) E1 ∨ E2, if E1 and E2 are selection conditions on the same relational tuple variable. Example 5: Consider the schema PATIENT in Exam- ple 4, the selection of “all patients who are young and diagnosed hepatitis” can be expressed by the selection condition x.AGE→ young ∧ x.DISEASE = hepatitis. 2. Semantics of Selection Conditions The semantics of selection conditions is the satisfied degree measure for tuples in a fuzzy relation and is defined as follows. Definition 12: Let R(U, µ) be a fuzzy relational schema in T-2FRDB, r a relation over R, x be a tuple variable and t a tuple in r . The interpretation of selection conditions with respect to R, r and t, denoted by IntR,r,t , is a partial mapping from the set of all selection conditions to the set of all fuzzy numbers on [0, 1] that is inductively defined as follows: 1) IntR,r,t (x.A θ ν) = µr (t) if t .A θ ν, and IntR,r,t (x.A θ ν) = 0, otherwise; 2) IntR,r,t (x.A→ ν) = MIN(µr (t), µϕ(t)), with ϕ = x.A→ ν; 3) IntR,r,t (x.A1 θ x.A2) = µr (t) if t .A1 θ t .A2, and IntR,r,t (x.A1 θ x.A2) = 0, otherwise; 4) IntR,r,t (¬E) = 1 − IntR,r,t (E); 5) IntR,r,t (E1 ∧ E2) = MIN(IntR,r,t (E1), IntR,r,t (E2)); 6) IntR,r,t (E1 ∨ E2) = MAX(IntR,r,t (E1), IntR,r,t (E2)). We note that ν is a fuzzy set in x.A→ ν, so ϕ = x.A→ ν is a binary fuzzy relation. Consequently, ϕ is also a fuzzy set. In particular, ϕ is the fuzzy set whose elements are tuples in r . For each t ∈ r , µϕ(t) = ν(t .A). Intuitively, IntR,r,t (x.A θ ν) and IntR,r,t (x.A → ν) are respectively the satisfied degrees of the conditions t .A θ ν and t.A→ ν for the tuple t in r while IntR,r,t (x.A1 θ x.A2) is the satisfied degree of the condition t .A1 θ t .A2 for the tuple t in r . In the classical relational database model, for each tuple t and a relation r , µr (t) ∈ {0, 1}, so the interpretation of selection conditions with respect to r and t always takes one of two values 0 or 1. It also means that the concept of interpretation of selection conditions in the classical relational database model is a particular case of that in type-1 fuzzy relational database models and T-2FRDB. Example 6: Let the fuzzy set young represent the young age of a patient whose membership function is defined by µyoung(x) =  1, ∀x ∈ [0, 20], (35 − x)/15, ∀x ∈ (20, 35), 0, ∀x ≥ 35. The interpretation of selection conditions E1 = “x.AGE→ young”, E2 = “x.DISEASE = hepatitis”, E = “x.AGE→ young ∧ x.DISEASE = hepatitis”, 22 Vol. E–3, No. 14, Sep. 2017 with respect to r = PATIENT and t4 (the fourth tuple) in Example 4, are IntR,r,t4 (E1) = MIN(µr (t4), young(21)) = MIN(high, 0.93) = {0.5:0, 0.6:0.5, 0.7:0.8, 0.8:0.9, 0.9:1.0, 0.93:1.0}, IntR,r,t4 (E2) = µr (t4) = high, IntR,r,t4 (E) = MIN(IntR,r,t4 (E1), IntR,r,t4 (E2)) = MIN({0.5:0, 0.6:0.5, 0.7:0.8, 0.8:0.9, 0.9:1.0, 0.93:1.0}, high) = {0.5:0, 0.6:0.5, 0.7:0.8, 0.8:0.9, 0.9:1.0, 0.93:1.0}. Let approx 0.9 = {0.5 : 0, 0.6 : 0.5, 0.7 : 0.8, 0.8 : 0.9, 0.9 : 1.0, 0.93:1.0}. We then have IntR,r,t4 (E) = approx 0.9. Now, the selection operation in T-2FRDB is extended from the selection operation in [17] as follows. Definition 13: Let R(U, µ) be a relational schema in T- 2FRDB, r a type-2 fuzzy relation over R and φ a selection condition. The selection on r with respect to φ, denoted by σφ(r), is the type-2 fuzzy relation r ′ over R, including all tuples defined by r ′ = {t ∈ r | IntR,r,t (φ) , 0 ∧ µr′(t) = IntR,r,t (φ)}. We note that the number 0 in IntR,r,t (φ) , 0 is also the fuzzy number 0 on [0, 1]. Example 7: Consider the relation r = PATIENT in Example 4, the query “Find all patients who are young and diagnosed hepatitis” can be done by the selection operation r ′ = σφ(PATIENT), where the selection condition φ = “x.AGE→ young ∧ x.DISEASE = hepatitis”. The selection is implemented by checking the satisfaction of all tuples in PATIENT for φ. From the result computed in Example 6, we can see that only the tuple t4 satisfies φ with the value of membership function being approx 0.9 above. Therefore, the result of the selection is the relation r ′ = σφ(PATIENT), as shown in Table II. TABLE II RELATION σφ (PATIENT) P NAME AGE DISEASE D COST µ T. T. D 21 Hepatitis 30 approx 0.9 V. OTHER OPERATIONS ON T-2FRDB As for the classical relational database and T-1FRDB, other basic relational algebraic operations on T-2FRDB are the projection, Cartesian product, join, intersection, union, and difference. We now extend these operations of T-1FRDB for T-2FRDB to take into account the fuzzy membership degree of tuples in relations. 1. Projection A projection of a type-2 fuzzy relation on a set of attributes is a new type-2 fuzzy relation in which only the attributes in that set are considered for each tuple of the new relation as in the following definition. Definition 14: Let R = (U, µ) be the type-2 fuzzy relational schema, r be the relation over R and L = {A1, A2, . . . , Ak} be a subset of U. The projection of r on L, denoted by ΠL(r), is a type-2 fuzzy relation r ′ over the schema R′, and is determined by 1) R′ = (L, µ′), where µ′ is a mapping from D1 × D2 × · · · × Dk to a set of all fuzzy numbers on [0, 1], Di are the value domains of Ai (i = 1, . . . , k); 2) r ′ = {t ′ = t[L] | t ∈ r, µ′r′(t ′) = MAXt∈r {µr (t) | t ′ = t[L]}}. Example 8: The projection of the relation PATIENT in Table I on L = {P NAME,DISEASE} is the relation ΠL(PATIENT) as shown in Table III. TABLE III RELATION ΠL(PATIENT) P NAME DISEASE µ′ L. V. A Lung cancer 0.9 L. T. B Cirrhosis about 0.5 N. T. C Bronchitis 1.0 T. T. D Hepatitis high 2. Cartesian Product For the Cartesian product of two type-2 fuzzy relations, as in the classical relational database and T-1FRDB, we assume that the sets of attributes of their schemas are disjoint as in the definition below. Definition 15: Let U1, U2 be two sets of attributes that do not have any common element and r1, r2 be two fuzzy relations over two type-2 fuzzy relational schemas R1 = (U1, µ1) and R2 = (U2, µ2), respectively. The Cartesian product of r1 and r2, denoted by r1 × r2, is a type-2 fuzzy relation r over R, and is determined by 1) R = (U, µ), where U = U1 ∪ U2, µ is the mapping from D1 × D2 × · · · × Dk+m to the set of all fuzzy numbers on [0, 1], k = |U1 |, m = |U2 |, Di are the value domains of Ai ∈ U1 ∪ U2; 2) r = {t = (ν1, ν2, . . . , νk, νk+1, νk+2, . . . , νk+m) | t1 = (ν1, ν2, . . . , νk), t2 = (νk+1, νk+2, . . . , νk+m), t1 ∈ r1, t2 ∈ r2, µr (t) = MIN(µ1r1 (t1), µ2r2 (t2))}. 23 Research and Development on Information and Communication Technology 3. Join The join of two fuzzy relations in T-2FRDB is extended from the natural join of two relations in a classical relational database and the join in T-1FRDB. The join in T-2FRDB is defined as follows. Definition 16: Let U1 and U2 be two sets of attributes such that value domains of two attributes of the same name A in U1 and U2, respectively, are identical. Let r1 and r2 be two fuzzy relations over the type-2 fuzzy relational schemas R1 = (U1, µ1) and R2 = (U2, µ2), respectively, and {Ak, . . . , Al} = U1 ∩ U2. The natural join of r1 and r2, denoted by r1 ./ r2, is a type-2 fuzzy relation r over the schema R, and is determined by 1) R = (U, µ), where U = U1 ∪ U2, µ is the mapping from D1 ×D2 × · · · ×Dn to the set of all fuzzy numbers on [0, 1], n = |U|, Di are the value domains of Ai ∈ U1 ∪ U2; 2) r = {t = (ν1, . . . , νj, νk, . . . , νl, νm, . . . , νn) | t1 = (ν1, . . . , νj, νk, . . . , νl), t2 = (νk, . . . , νl, νm, . . . , νn), t1 ∈ r1, t2 ∈ r2 such that νk = t1[Ak] = t2[Ak], . . . , νl = t1[Al] = t2[Al] and µr (t) = MIN(µ1r1 (t1), µ2r2 (t2))}. Example 9: Let U1 = {P ID, DISEASE} and U2 = {P NAME, DISEASE} be two sets of attributes, PATIENT1 and PATIENT2 two type-2 fuzzy relations over two schemas R1 = (U1, µ1) and R2 = (U2, µ2), as shown in Tables IV and V, respectively. It is easy to see that MIN({1 : 1, 0.9 : 0.8, 0.8 : 0.3}, {0.6 : 0.3, 0.5 : 1, 0.4 : 0.4}) = {0.6 : 0.3, 0.5:1, 0.4:0.4}. So, the result of the join of PATIENT1 and PATIENT2 is the relation PATIENT over the schema R = (U1 ∪ U2, µ) computed as in Table VI. TABLE IV RELATION PATIENT1 P ID DISEASE µ1 PT005 Bronchitis 0.8 PT006 Gall-stone {1:1, 0.9:0.8, 0.8:0.3} TABLE V RELATION PATIENT2 P NAME DISEASE µ2 L. V. E Bronchitis 0.9 N. T. F Gall-stone {0.6:0.3, 0.5:1, 0.4:0.4} TABLE VI RELATION PATIENT = PATIENT1 ./ PATIENT2 P ID P NAME DISEASE µ PT005 L. V. E Bronchitis 0.8 PT006 N. T. F Gall-stone {0.6:0.3, 0.5:1, 0.4:0.4} 4. Intersection, Union and Difference By extending the operations of ordinary fuzzy sets in Definition 2, the set operations on the type-2 fuzzy relations in T-2FRDB are defined in turn below. Definition 17: Let r1 and r2 be two type-2 fuzzy relations over the same schema R(U, µ). The intersection of r1 and r2, denoted by r1 ∩ r2, is a type-2 fuzzy relation r over R including tuples t’s, and is defined as r ∩ s = {t | µr∩s(t) = MIN(µr (t), µs(t))}. Definition 18: Let r1 and r2 be two type-2 fuzzy relations over the same schema R(U, µ). The union of r1 and r2, denoted by r ∪ s, is a type-2 fuzzy relation r over R including tuples t’s, and is defined as r ∪ s = {t | µr∪s(t) = MAX(µr (t), µs(t))}. Definition 19: Let r1 and r2 be two type-2 fuzzy relations over the same schema R(U, µ). The difference of r1 and r2, denoted by r − s, is a type-2 fuzzy relation r over R including tuples t’s, and is defined as r − s = {t | µr∩¬s(t) = MIN(µr (t), 1 − µs(t))}. 5. Properties of Algebraic Operations In this section, we propose some properties of the fuzzy relational algebraic operations in T-2FRDB as an extension from those in the classical relational database and T- 1FRDB. Clearly, these properties say that T-2FRDB model is coherent and consistent. Theorem 1: Let r be a fuzzy relation over the schema R in T-2FRDB, φ1 and φ2 be two selection conditions. Then σφ1 (σφ2 (r)) = σφ2 (σφ1 (r)) = σφ1∧φ2 (r), (1) where, the expressions φ1 and φ2 in φ1 ∧ φ2 are assumed to have the same tuple variable. Proof: Let s = σφ2 (r). We have σφ1 (σφ2 (r)) = {t ∈ s | IntR,s,t (φ1) , 0} (by Definition 13) = {t ∈ r | IntR,r,t (φ2) , 0 ∧ IntR,s,t (φ1) , 0} = {t ∈ r | IntR,r,t (φ2) , 0 ∧ IntR,r,t (φ1) , 0)} (because s ⊆ r) = {t ∈ r |MIN(IntR,r,t (φ2), IntR,r,t (φ1)) , 0)} (by Definition 12) = {t ∈ r | IntR,r,t (φ2 ∧ φ1) , 0)} = σφ1∧φ2 (r). So, σφ1 (σφ2 (r)) = σφ1∧φ2 (r). Similarly, we have σφ2 (σφ1 (r)) = σφ2∧φ1 (r). Since φ1 ∧ φ2 if and only if φ2 ∧ φ1 (i.e., the logical conjunction of selection condi- tions is commutative), we have σφ1∧φ2 (r) = σφ2∧φ1 (r). 24 Vol. E–3, No. 14, Sep. 2017 Therefore, it results in σφ1 (σφ2 (r)) = σφ2 (σφ1 (r)) and so σφ1 (σφ2 (r)) = σφ2 (σφ1 (r)) = σφ1∧φ2 (r).  Theorem 2: Let r be a fuzzy relation over the schema R in T-2FRDB, A and B be two subsets of attributes of R, and A ⊆ B. Then ΠA(ΠB(r)) = ΠA(r). (2) Proof: Because A ⊆ B, so A∩B = A. By Definition 14, it is easy to see that both sides of (2) are relations over the same schema with the set of attributes A ∩ B = A. By the property of the projection of classical relations (Definitions 9 and 14), it follows that two classical sets of tuples which are collected from two relations ΠA(ΠB(r)) and ΠA(r) are the same. Also by Definition 14, the opera- tion MAX in both sides of (2) is executed on the same value set of the membership degrees of tuples of r . Therefore, ΠA(ΠB(r)) = ΠA∩B(r) = ΠA(r).  Theorem 3: Let R1, R2 and R3 be the schemas in T- 2FRDB such that if they have attributes of the same name, such attributes have the same value domain. Let r1, r2 and r3 be the fuzzy relations over R1, R2 and R3 respectively. Then r1 ./ r2 = r2 ./ r1, (3) (r1 ./ r2) ./ r3 = r1 ./ (r2 ./ r3). (4) Proof: Clearly, r1 ./ r2 and r2 ./ r1 are two relations over the same schema. By the property of the join of classical relations (Definitions 9 and 16), it follows that two classical sets of tuples which are collected from two relations r1 ./ r2 and r2 ./ r1 are the same. In addition, the operation MIN of two fuzzy numbers (two membership degrees of two tuples in r1 and r2, respectively) has com- mutativity. From that the join of tuples has commutativity. So, by Definition 16 we have r1 ./ r2 = r2 ./ r1. By Definition 16, clearly (r1 ./ r2) ./ r3 and r1 ./ (r2 ./ r3) are two relations over the same schema. By the property of the join of classical relations (Definitions 9 and 16), it follows that two classical sets of tuples which are collected from two relations (r1 ./ r2) ./ r3 and r1 ./ (r2 ./ r3) are the same. Let A be a common attribute in U1, U2 and U3 of R1, R2 and R3. Because the operation MIN of two fuzzy numbers and the identical operation of attribute values have associativity, the join of tuples has associativity. Thus, by Definition 16, we have (r1 ./ r2) ./ r3 = r1 ./ (r2 ./ r3).  Because the Cartesian product (Definition 15) is a par- ticular case of the join, we have the straight corollary of Theorem 3 as follows. Corollary 1: Let R1, R2 and R3 be schemas in T-2FRDB such that each pair of them does not have any common attribute, r1, r2 and r3 be fuzzy relations over R1, R2 and R3 respectively. Then r1 × r2 = r2 × r1, (5) (r1 × r2) × r3 = r1 × (r2 × r3). (6) Theorem 4: Let r1, r2 and r3 be fuzzy relations over the same schema R in T-2FRDB. Then r1 ∩ r2 = r2 ∩ r1, (7) (r1 ∩ r2) ∩ r3 = r1 ∩ (r2 ∩ r3), (8) r1 ∪ r2 = r2 ∪ r1, (9) (r1 ∪ r2) ∪ r3 = r1 ∪ (r2 ∪ r3). (10) Proof: Because the intersection and union operations of sets and the MIN and MAX operations of fuzzy numbers have commutativity and associativity. So, by Definitions 17 and 18, Equations (7), (8), (9) and (10) then follow.  VI. CONCLUSIONS In this paper, we have proposed a type-2 fuzzy relational database model, abbreviated by T-2FRDB, as an extension of the type-1 fuzzy relational database models. In T- 2FRDB, the membership degrees of tuples in a relation are represented by the fuzzy numbers on the interval [0, 1]. The data model and fuzzy relational algebraic operations in T-2FRDB have been defined formally and consistently. Computing and associating the membership degrees of tuples in manipulating of the algebraic operations are implemented by the operations MAX and MIN using the extension principle. 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Parsons, “Current approaches to handling imperfect infor- mation in data and knowledge bases,” IEEE Transactions on knowledge and data engineering, vol. 8, no. 3, pp. 353–372, 1996. [19] A. Konar, Computational intelligence: principles, techniques and applications. Springer, 2005. [20] T. J. Ross, Fuzzy logic with engineering applications, 3rd ed. John Wiley & Sons, 2010. [21] N. N. Karnik and J. M. Mendel, “Operations on type-2 fuzzy sets,” Fuzzy sets and systems, vol. 122, no. 2, pp. 327–348, 2001. Nguyen Hoa is a lecturer at Saigon Univer- sity. He received the B.Sc. degree in Math- ematics in 1982 from Vinh Pedagogical University, the M.Eng. degree in Computer Science in 2003 from Ho Chi Minh City University of Technology, and the Ph.D. degree in Computer Science in 2008 from Vietnam National University, Ho Chi Minh City. His research interests include imprecise and uncertain knowl- edge representation, fuzzy databases and probabilistic databases. 26