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Silberschatz Korth Sudarshan: Database System Concepts, Fourth Edition
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There are several uses of the attribute closure algorithm: To test if is a superkey, we compute + , and check if + contains all attributes of R We can check if a functional dependency holds (or, in other words, is in F + ), by checking if + That is, we compute + by using attribute closure, and then check if it contains This test is particularly useful, as we will see later in this chapter It gives us an alternative way to compute F + : For each R, we nd the closure + , and for each S + , we output a functional dependency S
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Suppose that we have a set of functional dependencies F on a relation schema Whenever a user performs an update on the relation, the database system must ensure that the update does not violate any functional dependencies, that is, all the functional dependencies in F are satis ed in the new database state The system must roll back the update if it violates any functional dependencies in the set F We can reduce the effort spent in checking for violations by testing a simpli ed set of functional dependencies that has the same closure as the given set Any database that satis es the simpli ed set of functional dependencies will also satisfy the original set, and vice versa, since the two sets have the same closure However, the simpli ed set is easier to test We shall see how the simpli ed set can be constructed in a moment First, we need some de nitions An attribute of a functional dependency is said to be extraneous if we can remove it without changing the closure of the set of functional dependencies The formal de nition of extraneous attributes is as follows Consider a set F of functional dependencies and the functional dependency in F Attribute A is extraneous in if A , and F logically implies (F { }) {( A) } Attribute A is extraneous in if A , and the set of functional dependencies (F { }) { ( A)} logically implies F For example, suppose we have the functional dependencies AB C and A C in F Then, B is extraneous in AB C As another example, suppose we have the functional dependencies AB CD and A C in F Then C would be extraneous in the right-hand side of AB CD Beware of the direction of the implications when using the de nition of extraneous attributes: If you exchange the left-hand side with right-hand side, the implication will always hold That is, (F { }) {( A) } always logically implies F, and also F always logically implies (F { }) { ( A)}
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II Relational Databases
7 Relational Database Design
The McGraw Hill Companies, 2001
Functional Dependencies
Here is how we can test ef ciently if an attribute is extraneous Let R be the relation schema, and let F be the given set of functional dependencies that hold on R Consider an attribute A in a dependency If A , to check if A is extraneous consider the set F = (F { }) { ( A)} and check if A can be inferred from F To do so, compute + (the closure of ) under F ; if + includes A, then A is extraneous in If A , to check if A is extraneous, let = {A}, and check if can be inferred from F To do so, compute + (the closure of ) under F ; if + includes all attributes in , then A is extraneous in For example, suppose F contains AB CD, A E, and E C To check if C is extraneous in AB CD, we compute the attribute closure of AB under F = {AB D, A E, and E C} The closure is ABCDE, which includes CD, so we infer that C is extraneous A canonical cover Fc for F is a set of dependencies such that F logically implies all dependencies in Fc , and Fc logically implies all dependencies in F Furthermore, Fc must have the following properties: No functional dependency in Fc contains an extraneous attribute Each left side of a functional dependency in Fc is unique That is, there are no two dependencies 1 1 and 2 2 in Fc such that 1 = 2 A canonical cover for a set of functional dependencies F can be computed as depicted in Figure 78 It is important to note that when checking if an attribute is extraneous, the check uses the dependencies in the current value of Fc , and not the dependencies in F If a functional dependency contains only one attribute in its right-hand side, for example A C, and that attribute is found to be extraneous, we would get a functional dependency with an empty right-hand side Such functional dependencies should be deleted The canonical cover of F , Fc , can be shown to have the same closure as F ; hence, testing whether Fc is satis ed is equivalent to testing whether F is satis ed However, Fc is minimal in a certain sense it does not contain extraneous attributes, and it Fc = F repeat Use the union rule to replace any dependencies in Fc of the form 1 1 and 1 2 with 1 1 2 Find a functional dependency in Fc with an extraneous attribute either in or in /* Note: the test for extraneous attributes is done using Fc , not F */ If an extraneous attribute is found, delete it from until Fc does not change Figure 78 Computing canonical cover
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