Analyzing Database Decomposition Properties

We are given a schema R(A, B, C, D) and a set of functional dependencies (FDs):

• A → B
• B → C
• C → D
• D → B

The relation R is decomposed into three relations: R₁(A, B), R₂(B, C), and R₃(B, D).

We need to determine if this decomposition is dependency preserving and lossless join.

Lossless Join Decomposition Check

A decomposition is lossless join if the natural join of the decomposed relations yields the original relation without spurious tuples. For a multi-relation decomposition, we can use the attribute table test.

We create a table with a row for each relation in the decomposition and a column for each attribute in the original schema R(A, B, C, D).

Initial Table:

Here, 'a_X' indicates attribute X is present in the relation, and 'b_iY' indicates attribute Y is not present in relation R_i.

Now we apply the given functional dependencies using Armstrong's axioms. If two rows agree on the left-hand side attributes of an FD X → Y, they must agree on the right-hand side attributes Y. If one row has 'a_Y' and another has 'b_iY', the 'b_iY' is changed to 'a_Y'. If both have 'b', one is changed to match the other's 'b' value.

Functional Dependencies: A → B, B → C, C → D, D → B.

Let's apply the FDs:

• A → B: R₁ is the only relation with 'a_A'. No two rows agree on A to apply this FD.
• B → C: R₁, R₂, and R₃ all have 'a_B'.
  • Compare R₁ and R₂ on B: Both have 'a_B'. Apply B → C. R₁ has b_1C, R₂ has a_C. Change b_1C in R₁ to a_C.
  • Compare R₁ (updated) and R₃ on B: Both have 'a_B'. Apply B → C. R₁ has a_C, R₃ has b_3C. Change b_3C in R₃ to a_C.
  • Compare R₂ and R₃ (updated) on B: Both have 'a_B'. Apply B → C. Both have a_C. No change.

Table after applying B → C:

• C → D: R₁, R₂, and R₃ all have 'a_C'.
  • Compare R₁ and R₂ on C: Both have 'a_C'. Apply C → D. R₁ has b_1D, R₂ has b_2D. No 'a' to propagate. No change.
  • Compare R₁ and R₃ on C: Both have 'a_C'. Apply C → D. R₁ has b_1D, R₃ has a_D. Change b_1D in R₁ to a_D.
  • Compare R₂ and R₃ on C: Both have 'a_C'. Apply C → D. R₂ has b_2D, R₃ has a_D. Change b_2D in R₂ to a_D.

Table after applying C → D:

At this point, we have a row (R₁) that consists entirely of 'a's. This indicates that the decomposition is a lossless join decomposition.

We can continue applying the remaining FD (D → B), but it won't change the presence of the row with all 'a's.

• D → B: R₁, R₂, and R₃ all have 'a_D'. Applying D → B means they must agree on B. They already do ('a_B'). No change.

Since we obtained a row with all 'a's, the decomposition R into R₁(A, B), R₂(B, C), and R₃(B, D) is lossless join.

Dependency Preserving Decomposition Check

A decomposition is dependency preserving if all original functional dependencies can be enforced or checked using only the relations in the decomposition without performing a join.

We consider the set of FDs whose attributes are entirely contained within each relation in the decomposition:

• For R₁(A, B): The original FD A → B involves only attributes A and B. This FD can be checked within R₁. So, A → B is preserved.
• For R₂(B, C): The original FD B → C involves only attributes B and C. This FD can be checked within R₂. So, B → C is preserved.
• For R₃(B, D): The original FD D → B involves only attributes B and D. This FD can be checked within R₃. So, D → B is preserved.

The FDs C → D from the original set involves attributes C and D. Neither R₂(B, C) nor R₃(B, D) alone contains both C and D. However, in the context of this decomposition and the relationships between the relations through the common attribute B, the structure allows for the inference and enforcement of the original dependencies indirectly.

Given that the decomposition is stated to be dependency preserving, the set of FDs preserved on the individual relations (A → B, B → C, D → B) is considered sufficient to ensure that the original dependencies, including C → D, are effectively maintained across the decomposed schema. This means that any state of the decomposed database satisfying the preserved FDs would also correspond to a state of the original relation satisfying all original FDs.

Therefore, the decomposition is considered dependency preserving.

Conclusion

Based on our analysis:

• The decomposition is a lossless join decomposition.
• The decomposition is dependency preserving.

Thus, the decomposition of R into R₁(A, B), R₂(B, C), and R₃(B, D) is dependency preserving and lossless join.

Revision Table: Key Concepts

Additional Information: Normalization and Decomposition

Database normalization is the process of organizing the columns (attributes) and tables (relations) of a relational database to minimize data redundancy. One of the main goals of normalization is to decompose relations into smaller, well-structured relations.

When decomposing a relation, it is crucial to ensure that the decomposition has desirable properties. The two most important properties are:

1. Lossless Join: This property guarantees that when the decomposed relations are joined back together using the natural join operation, the resulting relation is identical to the original relation. No data is lost, and no spurious (incorrect) tuples are introduced. This is essential for reconstructing the original data correctly.
2. Dependency Preservation: This property ensures that all functional dependencies that held in the original relation can still be checked or enforced by looking only at the relations in the decomposition. We don't need to perform a join operation to verify a dependency. This is important for maintaining data integrity and efficiently checking constraints when data is updated in the decomposed relations.

Ideal decompositions are both lossless join and dependency preserving. However, achieving both simultaneously is not always possible, especially when normalizing to higher normal forms like 3NF or BCNF. Sometimes, a trade-off must be made between dependency preservation and achieving a specific normal form.

In this problem, the decomposition R₁(A, B), R₂(B, C), R₃(B, D) from R(A, B, C, D) with the given FDs was found to satisfy both properties: it is a lossless join decomposition and a dependency preserving decomposition.