1. Dr. Alexandra I. Cristea http://www.dcs.warwick.ac.uk/~acristea/ CS 319: Theory of Databases

2. 2 Content 1.Generalities DB 2.Temporal Data 3.Integrity constraints (FD revisited) 4.Relational Algebra (revisited) 5.Query optimisation 6.Tuple calculus 7.Domain calculus 8.Query equivalence 9.LLJ, DP and applications 10.The Askew Wall 11.Datalog

3. 3 … previous FD revisited; proofs with FD with definition & counter- example

4. 4 FD Part 2: Proving with FDs: Proving with Armstrong axioms (non)Redundancy of FDs

5. 5 Armstrongs Axioms Axioms for reasoning about FDs reflexivity F1: reflexivity if Y X then X Y augmentation F2: augmentation if X Y then XZ YZ transitivity F3: transitivityif X Y and Y Z then X Z

6. 6 Theorems Additional rules derived from axioms: Union F4. Union if A B and A C, then A BC Decomposition F5. Decomposition if A BC, then A B and A C Prove them ! AB C A B C

7. 7 Union Rule if A B and A C, then A BC Let A B and A C A B, augument (F2) with A: A AB A C, augument (F2) with B: AB BC A AB and AB BC, apply transitivity (F3): A BC q.e.d.

8. 8 Decomposition Rule if A BC, then A B and A C Let A BC B BC, apply reflexivity (F1) : BC B A BC and BC B, apply transitivity (F3): A B Idem for A C q.e.d.

9. 9 Rules hold vs redundant? Armstrong Rules hold – but are they all necessary? Can we leave some out? –How do we check this?

10. 10 Redundancy DEF: An inference rule inf in a set of inference rules Rules for a certain type of constraint C is redundant (superfluous) when for all sets F of constraints of type C it holds that: F + {Rules –{inf}} = F + {Rules}

11. 11 F+, F* F+ = {fd | F |= fd} closure of F F* = {fd | F |- fd} cover of F

12. 12 Exercises 1.Show that Armstrongs inference rules for FDs (F1-3) are not redundant. 2.Show that Rules = {F1, F2, F3, F4} is redundant.

13. 13 Hint (Ex. 1) Show with the help of an example that, if one of the three axioms is omitted, the remaining set of functional dependencies is not complete. Take therefore an appropriate set of constraints and compute with the help of Rules – {inf} all possible consequences. Show then that there is another consequence to be computed with the help of inf.

14. 14 Solution We start from a relation scheme R and an arbitrary legal instance r(R). Let, and be sets of attributes (headers), so that Attr(R), Attr(R) and Attr(R). We have the following axioms: F1: (Reflexivity) Let be valid (holds). Then we also have. F2: (Augmentation) Let be valid. Then we also have. F3: (Transitivity) Let and be valid. Then we also have. Now we omit in turn one of the axioms. –Why in turn? –Why not just one?

15. 15 Case 1: F1 is not superfluous: Let Attr(R) = {X} and F =. Because F is empty, neither F2 nor F3 can be used to deduce new fds. Therefore, F+ = F =. From F1 we could however deduce that X X is valid, which is not present in the above set.

16. 16 Case 3: F2 is not superfluous: Let R = {X, Y} and F = {X Y}. With the help of F1 and F3 we deduce: F+ = {, X X, Y Y, X, Y, XY XY, XY Y, XY X, XY } However, with X Y and with the help of F2 we can infer that X XY is valid, which is not present in the above set.

17. 17 Case 3: F3 is not superfluous: Let R = {X, Y, Z} and F = {X Y, Y Z }. F+ = { XYZ XYZ,XY XY, YZ YZ,X Y,Y Z, XYZ XY,XY X,YZ Y,X XY,Y YZ, XYZ XZ,XY Y,YZ Z,XY Y,XY XZ, XYZ YZ,XY,YZ,XZ YZ,YZ Z, XYZ X,XZ XZ, X X, X, Y Y, Y, Z Z, Z XYZ Y, XYZ Z,XYZ,XZ, } With the help of F3 we can also infer X Z, which is not in F+.

18. 18 How do we show something is redundant (superfluous)? Show that it is inferable from the other axioms

19. 19 F4 is superfluous: F4 (union rule) : Let and be valid. Then is also valid. We show now that F = {F1, F2, F3, F4} is redundant is by, e.g., inferring F4 from the other three. By using augumentation, from we deduce that also is valid (augmentation with ). By using transitivity, from and, we deduce that also is valid. Note that to prove that a set of rules (axioms) is redundant we can use normal calculus; however, to prove that a set of rules is not redundant, we need to know the meaning of the rules.

20. 20 Summary We have learned how to prove fds based on the Armstrong axioms –and also why & when its ok to do so We have learned how to prove that a set of axioms is redundant or not We have learned that the Armstrong axioms are not redundant

21. 21 … to follow Constrains revisited: Soundness and Completeness of Armstrong Axioms