Functional dependencies 1. 2 Outline motivation: update anomalies cause: not expressed constraints on data (FDs) functional dependencies (FDs) definitions

Functional dependencies

1



2

Outline

motivation: update anomalies cause: not expressed constraints on data (FDs) functional dependencies (FDs)

• definitions• examples• concepts and terminology• (more) advanced theoretical issues (in brief)


3

Redundancy

Student-Modules

S_id S_name P_ tutor Module Val Res

50021 G. Zolla MFU Database Systems 1cu A

50021 G. Zolla MFU Software Engineering 1cu null

50021 G. Zolla MFU Advanced Programming 1/2cu B

50012 D. Petrescu MFU Computing 1cu null

50012 D. Petrescu MFU Mathematics 1/2cu A

41002 T.A. Flo MFU HCI 1/2cu null

50033 D. Wise MH Database Systems 1cu null

50033 D. Wise MH Algorithms 1cu null


4

Update anomalies - insertion

insert the fact that 50012 takes “Networks - Introduction”; the name of the student and the name of the personal tutor have to be entered as well; this is prone to errors inconsistent data





50012 D. Pwtrwscu MFFU Networks 1/2cu null


can you identify other kinds of update anomalies on this relation?


5

Update anomalies - deletion

delete the fact that 41002 takes “HCI”, in the original table; relevant information will be also deleted - about “T.A Flo” and about “HCI”.




50021 G. Zolla MFU Advanced Programming 1/2cu B






6

Update anomalies - modification

it is possible to modify an attribute and to bring the relation in an inconsistent state; e.g. it is possible (e.g. by mistake) to modify the value of “Database Systems” to “1/2cu” in just some rows; such situations must be avoided


50021 G. Zolla MFU Database Systems 1/2cu A


41002 T.A. Flo MFU HCI 1/2cu null




7

Cause

… of anomalies• redundancy

… of redundancy• constraints on data that are not considered or expressed

in the relational model (or, more specifically, on for a particular relation)

• (some of) these constraints are called Functional Dependencies (FDs)

– (others: multiple dependencies, …)


8

To eliminate the problem

express the existing functional dependencies• this is done by choosing particular forms for relations -

normal forms• a relation is in a certain normal form if if it does not

contain certain kinds of functional dependencies• a relation in a certain normal form eliminates certain

kinds of update anomalies• a relation that is not in a certain normal form can be

transformed (decomposed) into an equivalent set of relations - this process is called normalisation


9

Functional dependency (FD)

R - relation, X and Y - subset of attributes of RX Y iffin every possible legal value of R each X-value has a single Y-value associated

time independent (for all time)


10

Examples

S_id S_nameS_id P_TutorS_id S_id(S_id, P_tutor) P_tutor(S_id, S_name, P_tutor) P_tutorModule Val(S_id, Module) Res(S_id, S_name, P_tutor, Module, Val) Res


11

Concepts

FD is a semantic concept determinant / dependent trivial / non-trivial FD left-irreducible

• yes: (S_id, P_tutor) P_tutor• no: (S_id, Module) Res

closure irreducible set


12

Functional diagram

S_id

City

P_tutor

S_name

S_id

ModuleRes

Module


13

Closure

all FDs that can be derived from a given set S• notation S+

Armstrong’s inference rules• if B A then A B• if A B then AC BC• if A B and B C then A C• sound and complete

– other rules can be inferred (make the derivation process easier)

– A BC iff A B and A C

– A B and C D then AC BD


14

Irreducible set

S1 covers S2 iff S2+ S1

+

S is irreducible iff• RHS of every FD is non-composite• all FDs in S are left-irreducible• no FD ca be discarded from S without changing S+

a database that enforces S enforces, in fact, S+

the irreducible set of S is S’ iff• S’ - irreducible• S’+ = S+

more efficient to work with the irreducible set


15

Conclusion

2NF, 3NF and BCNF will be defined based on FDs

Documents

Functional dependencies 1. 2 Outline motivation: update anomalies cause: not expressed constraints on data (FDs) functional dependencies (FDs) definitions