MINIMAL BASIS


EQUIVALENCY: Two functional dependencies F1 and F2 over relation R are equivalent if

     F1+ = F2+

MINIMAL BASIS: A set of functional dependencies F is minimal is we cannot remove any
   f.d.s or any attributes from an f.d. without changing its meaning (closure).

Algorithm for converting a set F to a minimal basis
----------------------------------------------------

Step 1: Convert F to a basis by using splitting rule


AB->CD
C->AC
D->AF
AC->G
AB->BFH

-----

AB->C
AB->D
C->A
C->C
D->A
D->F
AC->G
AB->B
AB->F
AB->H

Step 2: Remove all trivial functional dependencies

AB->C
AB->D
C->A
     C->C -- remove
D->A
D->F
AC->G
     AB->B  -- remove
AB->F
AB->H

----

AB->C
AB->D
C->A
D->A
D->F
AC->G
AB->F
AB->H

Step 3: Suppose X->Y is in F, create F' by removing X->Y. If X+ is the same
in F and F', then X->Y can be removed.

F=
AB->C
AB->D  ** removed in F'
C->A
D->A
D->F
AC->G
AB->F
AB->H

F'=
AB->C
C->A
D->A
D->F
AC->G
AB->F
AB->H

in F , AB+={A,B,C,D,F,G,H}
in F', AB+={A,B,C,G,H}
We cannot remove AB->D, not equal

--------------

F=
AB->C
AB->D 
C->A
D->A
D->F
AC->G
AB->F   ** removed in F'
AB->H

F'=
AB->C
AB->D 
C->A
D->A
D->F
AC->G
AB->H


in F :  AB+={A,B,C,D,F,G,H}
in F':  AB+={A,B,C,D,F,G,H}
Same! I can AB->F

-----

AB->C
AB->D 
C->A
D->A
D->F
AC->G
AB->H


Step 4: If XZ->Y in F, then replace it with X->Y to get F'. If X+ in F and F' is the same,
then F' can become F.

F =
AB->C
AB->D 
C->A
D->A
D->F
AC->G
AB->H

F' =
AB->C
A->D    ** changed fd
C->A
D->A
D->F
AC->G
AB->H

in F , A+={A}
in F', A+={A,D,F}
Not the same, cannot remove B

----------

F =
AB->C
AB->D 
C->A
D->A
D->F
AC->G
AB->H

F' =
AB->C
B->D    ** changed fd
C->A
D->A
D->F
AC->G
AB->H

in F , B+={B}
in F', B+={B,D,F}
Not the same, cannot remove A

---------

F =
AB->C
AB->D 
C->A
D->A
D->F
AC->G
AB->H

F' =
AB->C
AB->D 
C->A
D->A
D->F
C->G   ** changed fd
AB->H

in F , C+={C,A,G}
in F', C+={C,A,G}
Same, I can remove A from AC->G.

------------

final result: minimal basis

AB->C
AB->D 
C->A
D->A
D->F
C->G  
AB->H


combining rule: minimal cover

AB->CDH
C->AG
D->AF