Write down the sequence of steps needed to transform ? 2 A ((R B=C S) D=E T) into ? A ((? E (T) E=D ? ACD (S)) C=BR). List the attributes that each of the schemas R, S, and T must have and the attributes that each (or some) of these schemas must not have in order for the above transformation to be correct.
What will be an ideal response?
A. R must have: B (because of the join)
B. S must have: ACD (because of ?ACD)
C. T must have: E (because of ?E)
D. These schemas should not have identically named attributes, because otherwise it will not be clear which of the two identically named attributes will be renamed in the joins. In particular, T should not have A and C , because ? ACD(S) clearly suggests that it is expected that S will have the attribute A that will survive for the outermost projection ?A to make sense, and the attribute C, which should survive in order for the join with R to make sense.
1. Associativity of the join: ?A(R B=C(S D=ET))
2. Commutativity of the join: ?A((S D=ET)C=BR)
3. Commutativity of the join: ?A((T E=DS) C=BR)
4. Pushing projection ?A to the ?rst operand of the join: ?A(?AC((T E=DS)) C=BR).
5. Pushing projection to the ?rst operand in the innermost join: ?A (?AC((?E(T) E=DS)) C=BR). This is possible if AC are the attributes of S and not of T.
6. Pushing projection to the second operand in the innermost join: ?A(?AC((?E(T)E=D ?ACD(S))) C=BR).
7. Reverse of pushing a projection: ?A((?E(T) E=D?ACD(S)) C=BR)
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