CS457 Syllabus & Progress

Finding the closure set X⁺: an abstract example

An abstract example of finding X⁺

The database design theory using functional dependencies (i.e., the theory of normal forms ) does not use semantical informations
In other words, the method must work with abstract attribute names

Example:

R(A, B, C, D, E, F) ℉ = { A → BC BD → EF F → A }

Computing A⁺:

Initialize: A⁺ := A;
For every FP in ℉ do: 1. A → BC ---> A⁺ := A,B,C DONE = FALSE; 2. BD → EF ---> no change 3. F → A ---> no change Result at end of loop: A⁺ := A,B,C
There were changes, so loop again... For every FP in ℉ do: 1. A → BC ---> no change 2. BD → EF ---> no change 3. F → A ---> no change Result at end of loop: A⁺ = A,B,C
There were NO changes, so DONE A⁺ = A,B,C

Computing BD⁺:

Initialize: BD⁺ := BD;
For every FP in ℉ do: 1. A → BC ---> no change 2. BD → EF ---> BD⁺ := B,D,E,F DONE = FALSE; 3. F → A ---> BD⁺ := A,B,D,E,F DONE = FALSE; Result at end of loop: BD⁺ := A,B,D,E,F
There were changes, so loop again... For every FP in ℉ do: 1. A → BC ---> BD⁺ := A,B,C,D,E,F DONE = FALSE; 2. BD → EF ---> no change 3. F → A ---> no change Result at end of loop: BD⁺ := A,B,C,D,E,F
There were changes, so loop again... For every FP in ℉ do: 1. A → BC ---> no change 2. BD → EF ---> no change 3. F → A ---> no change Result at end of loop: BD⁺ = A,B,C,D,E,F
There were NO changes, so DONE BD⁺ = A,B,C,D,E,F

NOTE: : BD is a superkey
We can easily determine that BD satisfies the minimality property:
So BD is a key

Computing AD⁺:

Initialize: AD⁺ := AD;
For every FP in ℉ do: 1. A → BC ---> AD⁺ := A,B,C,D DONE = FALSE; 2. BD → EF ---> BD⁺ := A,B,C,D,E,F DONE = FALSE; 3. F → A ---> no change Result at end of loop: AD⁺ := A,B,C,D,E,F
For every FP in ℉ do: 1. A → BC ---> no change 2. BD → EF ---> no change 3. F → A ---> no change Result at end of loop: AD⁺ = A,B,C,D,E,F
There were NO changes, so Done AD⁺ = A,B,C,D,E,F

NOTE:
We can also determine that AD satisfies the minimality property by computing:
Therefore:
So: AD is also a key

Finding the closure set X+: an abstract example

Finding the closure set X⁺: an abstract example