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Silberschatz Korth Sudarshan: Database System Concepts, Fourth Edition
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II Relational Databases
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The McGraw Hill Companies, 2001
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empl(X, Y ) : manager(X, Y ) empl(X, Y ) : manager(X, Z), empl(Z, Y ) Figure 57 Recursive Datalog program
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523 Semantics of Nonrecursive Datalog
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We consider the formal semantics of Datalog programs For now, we consider only programs that are nonrecursive The semantics of recursive programs is somewhat more complicated; it is discussed in Section 526 We de ne the semantics of a program by starting with the semantics of a single rule
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A ground instantiation of a rule is the result of replacing each variable in the rule by some constant If a variable occurs multiple times in a rule, all occurrences of the variable must be replaced by the same constant Ground instantiations are often simply called instantiations Our example rule de ning v1, and an instantiation of the rule, are: v1(A, B) : account(A, Perryridge , B), B > 700 v1( A-217 , 750) : account( A-217 , Perryridge , 750), 750 > 700 Here, variable A was replaced by A-217, and variable B by 750 A rule usually has many possible instantiations These instantiations correspond to the various ways of assigning values to each variable in the rule Suppose that we are given a rule R, p(t1 , t2 , , tn ) : L1 , L2 , , Ln and a set of facts I for the relations used in the rule (I can also be thought of as a database instance) Consider any instantiation R of rule R: p(v1 , v2 , , vn ) : l1 , l2 , , ln where each literal li is either of the form qi (vi,1 , v1,2 , , vi,ni ) or of the form not qi (vi,1 , v1,2 , , vi,ni ), and where each vi and each vi,j is a constant We say that the body of rule instantiation R is satis ed in I if 1 For each positive literal qi (vi,1 , , vi,ni ) in the body of R , the set of facts I contains the fact q(vi,1 , , vi,ni ) 2 For each negative literal not qj (vj,1 , , vj,nj ) in the body of R , the set of facts I does not contain the fact qj (vj,1 , , vj,nj )
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Silberschatz Korth Sudarshan: Database System Concepts, Fourth Edition
II Relational Databases
5 Other Relational Languages
The McGraw Hill Companies, 2001
5
Other Relational Languages
account-number A-201 A-217 Figure 58
balance 900 750
Result of infer(R, I)
We de ne the set of facts that can be inferred from a given set of facts I using rule R as infer(R, I) = {p(t1 , , tni ) | there is an instantiation R of R, where p(t1 , , tni ) is the head of R , and the body of R is satis ed in I} Given a set of rules R = {R1 , R2 , , Rn }, we de ne infer(R, I) = infer(R1 , I) infer (R2 , I) infer(Rn , I) Suppose that we are given a set of facts I containing the tuples for relation account in Figure 54 One possible instantiation of our running-example rule R is v1( A-217 , 750) : account( A-217 , Perryridge , 750), 750 > 700 The fact account( A-217 , Perryridge , 750) is in the set of facts I Further, 750 is greater than 700, and hence conceptually (750, 700) is in the relation > Hence, the body of the rule instantiation is satis ed in I There are other possible instantiations of R, and using them we nd that infer(R, I) has exactly the set of facts for v1 that appears in Figure 58
5232 Semantics of a Program
When a view relation is de ned in terms of another view relation, the set of facts in the rst view depends on the set of facts in the second one We have assumed, in this section, that the de nition is nonrecursive; that is, no view relation depends (directly or indirectly) on itself Hence, we can layer the view relations in the following way, and can use the layering to de ne the semantics of the program: A relation is in layer 1 if all relations used in the bodies of rules de ning it are stored in the database A relation is in layer 2 if all relations used in the bodies of rules de ning it either are stored in the database or are in layer 1 In general, a relation p is in layer i + 1 if (1) it is not in layers 1, 2, , i, and (2) all relations used in the bodies of rules de ning p either are stored in the database or are in layers 1, 2, , i Consider the program in Figure 56 The layering of view relations in the program appears in Figure 59 The relation account is in the database Relation interest-rate is
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