Computing Canonical Cover

1. Computing Canonical Cover Souhad M. Daraghma

2. Canonical Cover • A canonical coverfor F is a set of dependencies Fc such that • F logically implies all dependencies in Fc, and • Fc logically implies all dependencies in F, and • No functional dependency in Fccontains an extraneous attribute, and • Each left side of functional dependency in Fcis unique • Intuitively, a canonical cover of F is a “minimal” set of functional dependencies equivalent to F, having no redundant dependencies or redundant parts of dependencies

3. Extraneous Attributes • Consider F, and a functional dependency, A  B. • “Extraneous”: Are there any attributes in A or B that can be safely removed ? • Without changing the constraints implied by F

4. Testing if an Attribute is Extraneous • Consider a set F of functional dependencies and the functional dependency   in F. • To test if attribute A   is extraneousin • compute ({} – A)+ using the dependencies in F • check that ({} – A)+ contains A; if it does, A is extraneous • To test if attribute A  is extraneous in  • compute + using only the dependencies in F’ = (F – {})  {(– A)}, • check that + contains A; if it does, A is extraneous

5. Computing Canonical Cover R = {A,B,C,D,E,F,G,H} F = {ACG, DEG, BCD, CGBD, ACDB, CEAG} Find the canonical cover of F. 1. Union 2. Find a redundant (extraneous) attribute Consider ACDB, Dis extraneous, because ACG and CGBD imply AC  CGB So change ACDB into ACB {ACG, DEG, BCD, CGBD, ACB, CEAG}

6. Computing Canonical Cover {ACG, DEG, BCD, CGBD, ACB, CEAG} 1. Union {ACBG, DEG, BCD, CGBD, CEAG} 2. Find a redundant (extraneous) attribute Consider ACBG, G is extraneous, because ACB, BCD, and DEG imply AC D G So change ACBG into ACB {ACB, DEG, BCD, CGBD, CEAG}

7. Computing Canonical Cover {ACB, DEG, BCD, CGBD, CEAG} 1. Union 2. Find a redundant (extraneous) attribute Consider CGBD, B is extraneous, because CGD, D EG, CEAG, and AC B imply CG CD  CE  AC  B So change CG BD into CG D {ACB, DEG, BCD, CGD, CEAG}

8. Compute Canonical Cover {ACB, DEG, BCD, CGD, CEAG} 1. Union 2. Find a redundant (extraneous) attribute Consider CE AG, G is extraneous, because CE A, AC B, BCD, and D EG imply CE  AC  BC  D G So change CE  AG into CE  A {ACB, DEG, BCD, CGD, CEA}

9. Compute Canonical Cover {ACB, DEG, BCD, CGD, CEA} No more extraneous attributes FC ={ACB, DEG, BCD, CGD, CEA} * Different order of considering the extraneous attributes can result in different FC

10. Example2: Computing a Canonical Cover • R = (A, B, C)F = {A BC B C A BABC} • Combine A BC and A B into A BC • Set is now {A  BC, B  C, AB  C} • A is extraneous in ABC • Check if the result of deleting A from AB  C is implied by the other dependencies • Yes: in fact, BC is already present! • Set is now {A  BC, B  C} • C is extraneous in ABC • Check if A  C is logically implied by A  B and the other dependencies • Yes: using transitivity on A B and B  C. • Can use attribute closure of A in more complex cases • The canonical cover is: A B B C

11. Example3: Computing a Canonical Cover • Given F = {AC, ABC } • B is extraneous in AB C because {AC, AB C} is equivalent to {AC, AC } = {AC} • Given F = {AC, ABCD} • C is extraneous in ABCD because {AC, ABCD} is equivalent to {AC, ABD}