| Abstract |
We augment every relation scheme in the Relational Data Base (RDB) with a special attribute F ( the flag attribute). The values for attribute F may only be non-negative integers. Values of F in the set (n:n>1, n is odd) are used for capturing inter-relational disjunctive information. Values of F in the set (n:n >1, n is even) are used for capturing intra-relational disjunctive information. Tuples of relations with flag value 1 represent information known to be true. Tuples of relations with flag value 0 represent information neither known to be true nor known to be false. We shall denote the predicate represented in a relation scheme by the name of the relation scheme. Below we show instances p and q of relation schemes P(X,Y,F) and Q(Y,Z,F) respectively and logical expressions representing the information contained in them: p(X Y F) q(Y Z F) P(x1,y1) x1 y1 1 y1 z1 0 Q(y1,z1) V~Q(y1,z1) x2 y2 2 y2 z2 2 Q(y2,z2) V Q(y3,z3) x3 y3 2 y3 z3 2 P(x2,y2) V P(x3,y3) x4 y4 3 y4 z4 3 P(x4,y4) V Q(y4,z4)Under the Domain Closure Assumption [1], a tuple (x1,@@@@) in an instance of the conventional relation scheme R(X,Y) represents the disjunction: R(x1,y1) V R(x1,y2) V R(x1,y3) when @@@@ denotes the null value and the domain of attribute y is the set (y1,y2,y3). Our scheme thus includes null values as a special case. In addition, our scheme captures many more kinds of incomplete information.We introduce the notion of an “alternative” under our definition the relation u(A B C F) a1 b1 c1 1 a1 b2 c2 2 a1 b1 c3 2 which contains only intra-relational incomplete information has the following conventional relations as alternatives: u1(A B C) u2(A B C) u3(A B C) a1 b1 c1 a1 b1 c1 a1 b1 c1 a1 b2 c2 a1 b2 c2 a1 b1 c3 a1 b1 c3 Also, in our treatment, the following RDB containing inter-relational incomplete information r(A B F) s(B C F) a1 b1 2 b2 c2 3 a2 b2 2 b1 c1 5 a1 b1 3 b1 c1 2 a2 b2 5 b2 c2 2 a3 b3 4 b3 c3 1 a4 b4 4 b4 c4 4 b4 c5 4 has seven alternatives; here each alternative is a pair (ri,si), i lying between 1 and 7 and each of the alternatives contains only intra-relational incompleteness. For this example, the alternatives are as follows: r1= r0 U {(a1,b1,1),(a2,b2,1)} s1= s0 U {(b2,c2,1),(b1,c1,1)} r2= r1, s2= s0 U {(b2,c2,1)} r3=r1, s3= s0 U {(b1,c1,1)} r4= r0 U {(a1,b1,1)}, s4= s1 r5= r4, s5= s3 r6=r0 U {(a2,b2,1)}, s6= s1 r7= r6, s7= s2 where r0 and s0 are as shown below: r0(A B F) s0(B C F) a3 b3 4 b3 c3 1 a4 b4 4 b4 c4 4 b4 c5 4We also introduce the notion of “minimality” of relation instances and based on the idea of an alternative, we give a specification and realization of an extended join operator; under our treatment, the extended join of r and s above is t, shown below: t(A B C F) a1 b1 c1 2 a2 b2 c2 2 a3 b3 c3 4 a4 b4 c4 4 a4 b4 c5 4In the context of our scheme, we also introduce an Extended Closed World Assumption, weak functional dependencies and inference rules and conditions for the lossless decomposition of a relation containing incomplete information. |
