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**5NF and other normal forms**

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**Outline n-decomposability 3D constraint join dependency 5NF**

non-5NF - update anomalies problems in bringing a relation to 5NF other normal forms

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**Always two projections?**

so far every relation was non-loss decomposable into two projections is this always possible? n-decomposable relations

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**Courses - tutors - levels (CTL)**

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**CTL - 2 attribute projections**

CL

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**CTL - 3-decomposable the join of any two projections is not CTL; e.g:**

join(CT, TL) Extra!

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**Constraint 3D Let R be a degree 3 relation. IF (a, b, x) R**

AND (a, y, c) R AND (z, b, c) R THEN (a, b, c) R

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**Constraint 3D illustrated on the CTL relation**

IF tutor t1 teaches subject s1 AND level l1 studies subject s1 AND tutor t1 teaches level l1 THEN tutor t1 teaches subject s1 for level l1 note: this constraint is not expressed in CTL

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**Constraint 3D and Join Dependency**

4NF does not express the constraint 3D the constraint 3D is a facet of a more general constraint: join dependency

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Join dependency Let R be a relation. Let A, B, ..., Z be arbitrary subsets of R’s attributes. R satisfies the JD ( A, B, ..., Z ) if and only if R is equal to the join of its projections on A, B, ..., Z

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5 NF R is in 5NF if and only if every join dependency in R is implied by the candidate keys of R 5NF is always achievable

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Explanation a join dependency, (A, B, …, Z), is implied by the candidate keys, K1, …, Km of R if the fact that K1, …, Km are candidate keys for R determine the fact that R has the JD (A, B, …, Z)

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**Illustration - positive example**

consider R (S_id, S_name, Status, City) with S_id and S_name candidate keys ({S_id, S_name, Status}, {S_id, City}) is a JD because S_id is a candidate key in R ({S_id, S_name}, {S_id, Status}, {S_name, City}) is a JD because S_id and S_name are both candidate keys in R

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**Illustration - negative example**

consider CTL (Course, Tutor, Level) with (Course, Tutor, Level) - candidate key (and an extra constraint : constraint 3D) ({Course, Tutor}, {Course, Level}, {Tutor, Level}) is a JD, but this is not due to the CK, but to the constraint 3D if CTL had not had constraint 3D, would it have been in 5NF?

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**Not 5NF - update anomalies**

CTL satisfies ( {Course, Tutor}, {Tutor, Level}, {Course, Level} ) insert (Programming, M. Ursu, Level2) what else must be done?

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**Not 5NF - update anomalies**

CTL satisfies the same JD as before delete (Databases, M. Ursu, Level2) what else must be done?

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**JDs and MVDs Fagin’s theorem restated**

R ( A, B, C ) satisfies ( AB, AC ) if and only if it satisfies the MVDs A B | C JD is the most general form of dependency (read as determination) possible between the attributes of a relation (in the relational model)

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Activity Is 4NF subsumed by 5NF? Can you prove this using Fagin’s theorem and the definitions for 4 and 5 NF?

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**Problems in bringing a relation to 5NF**

check whether all JDs are implied by the candidate keys Fagin : provides an algorithm for doing this for any relation, given all its JDs and all candidate keys discover all JDs for a given relation they do not have that intuitive meaning as FDs or MVDs

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Concluding remarks 5NF is the ultimate normal form with respect to projection / join 5NF is guaranteed to be free of all anomalies that can be eliminated via projections determining whether a relation is in 4NF but not in 5NF is still fuzzy very rare in practice

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**Recap JD - a more general constraint than MD**

a relation can be in 4NF and have un-expressed JDs this results in update anomalies such a relation can be decomposed (via projection) into an equivalent set of 5NF relations a relation is 5NF if all its JDs are deducible from its candidate keys for a relation in 4NF but not in 5NF, an unexpressed JD is a possible decomposition (towards 5NF)

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**Other normal forms FDs, MVDs or JDs are not used**

domain-key normal form R is in DK/NF if and only if every constraint of R is a logical consequence of domain constraints and (candidate) key constraints restriction-union normal form decomposing operator: restriction abusing the language it can be said that: this normalisation theory is orthogonal on the “projection” normalisation theory

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