ch 10 Functional Dependencies and Normalization for Relational Databases

Normal Forms Defined Informally

1st normal form All attributes depend on the key 2nd normal form All attributes depend on the whole key 3rd normal form All attributes depend on nothing but the key

second normal form (2NF)

A relation schema R is in second normal form (2NF) if every non-prime attribute A in R is fully functionally dependent on the primary key -R can be decomposed into 2NF relations via the process of 2NF normalization

third normal form (3NF)

A relation schema R is in third normal form (3NF) if it is in 2NF and no non-prime attribute A in R is transitively dependent on the primary key R can be decomposed into 3NF relations via the process of 3NF normalization

Equivalence of Sets of FDs

Every FD in F can be inferred from G, and Every FD in G can be inferred from F Hence, F and G are equivalent if F+ =G+

Minimal Sets

A set of FDs is minimal if it satisfies the following conditions: Every dependency in F has a single attribute for its RHS. We cannot remove any dependency from F and have a set of dependencies that is equivalent to F. We cannot replace any dependency X -> A in F with a dependency Y -> A, where Y proper-subset-of X ( Y subset-of X) and still have a set of dependencies that is equivalent to F. Every set of FDs has an equivalent minimal set There can be several equivalent minimal sets There is no simple algorithm for computing a minimal set of FDs that is equivalent to a set F of FDs To synthesize a set of relations, we assume that we start with a set of dependencies that is a minimal set

Functional dependencies (FDs)

Are used to specify formal measures of the "goodness" of relational designs. keys are used to define normal forms for relations Are constraints that are derived from the meaning and interrelationships of the data attributes -A set of attributes X functionally determines a set of attributes Y if the value of X determines a unique value for Y - derived from the real-world constraints on the attributes

Closure

Closure of a set F of FDs is the set F+ of all FDs that can be inferred from F Closure of a set of attributes X with respect to F is the set X+ of all attributes that are functionally determined by X X+ can be calculated by repeatedly applying IR1, IR2, IR3 using the FDs in F

Normal form:

Condition using keys and FDs of a relation to certify whether a relation schema is in a particular normal form

Inference Rules for FDs

Decomposition: If X -> YZ, then X -> Y and X -> Z Union: If X -> Y and X -> Z, then X -> YZ Psuedotransitivity: If X -> Y and WY -> Z, then WX -> Z

Guideline to Redundant Information in Tuples and Update Anomalies

Design a schema that does not suffer from the insertion, deletion and update anomalies. If there are any anomalies present, then note them so that applications can be made to take them into account.

Key and attributes

If a relation schema has more than one key, each is called a candidate key. One of the candidate keys is arbitrarily designated to be the primary key, and the others are called secondary keys. A Prime attribute must be a member of some candidate key A Nonprime attribute is not a prime attribute—that is, it is not a member of any candidate key.

Semantics of the Relation Attributes GUIDELINE 1

Informally, each tuple in a relation should represent one entity or relationship instance. (Applies to individual relations and their attributes). -Attributes of different entities (EMPLOYEEs, DEPARTMENTs, PROJECTs) should not be mixed in the same relation -Only foreign keys should be used to refer to other entities -Entity and relationship attributes should be kept apart as much as possible.

Spurious Tuples two important properties of decompositions:

Non-additive or losslessness of the corresponding join Preservation of the functional dependencies

Armstrong's inference rules:

R1. (Reflexive) If Y subset-of X, then X -> Y IR2. (Augmentation) If X -> Y, then XZ -> YZ (Notation: XZ stands for X U Z) IR3. (Transitive) If X -> Y and Y -> Z, then X -> Z -IR1, IR2, IR3 form a sound and complete set of inference rules These are rules hold and all other rules that hold can be deduced from these

Null Values in Tuples

Relations should be designed such that their tuples will have as few NULL values as possible Attributes that are NULL frequently could be placed in separate relations (with the primary key) Reasons for nulls: Attribute not applicable or invalid Attribute value unknown (may exist) Value known to exist, but unavailable

General Normal Form Definitions (2)

Superkey of relation schema R - a set of attributes S of R that contains a key of R A relation schema R is in third normal form (3NF) if whenever a FD X -> A holds in R, then either: (a) X is a superkey of R, or (b) A is a prime attribute of R A relation schema R is in Boyce-Codd Normal Form (BCNF) if whenever an FD X -> A holds in R, then X is a superkey of R Each normal form is strictly stronger than the previous one Every 2NF relation is in 1NF Every 3NF relation is in 2NF Every BCNF relation is in 3NF There exist relations that are in 3NF but not in BCNF The goal is to have each relation in BCNF (or 3NF)

Normalization:

The process of decomposing unsatisfactory "bad" relations by breaking up their attributes into smaller relations -is carried out in practice so that the resulting designs are of high quality and meet the desirable properties The practical utility of these normal forms becomes questionable when the constraints on which they are based are hard to understand or to detect The database designers need not normalize to the highest possible normal form

Denormalization:

The process of storing the join of higher normal form relations as a base relation—which is in a lower normal form

Achieving the BCNF by Decomposition

Two FDs exist in the relation TEACH: fd1: { student, course} -> instructor fd2: instructor -> course {student, course} is a candidate key for this relation and that the dependencies shown follow the pattern in Figure 10.12 (b). So this relation is in 3NF but not in BCNF A relation NOT in BCNF should be decomposed so as to meet this property, while possibly forgoing the preservation of all functional dependencies in the decomposed relations.

Information is stored redundantly

Wastes storage Causes problems with update anomalies Insertion anomalies Deletion anomalies Modification anomalies

Transitive functional dependency

a FD X -> Z that can be derived from two FDs X -> Y and Y -> Z

Full functional dependency:

a FD Y -> Z where removal of any attribute from Y means the FD does not hold any more

2NF, 3NF, BCNF

based on keys and FDs of a relation schema

4NF

based on keys, multi-valued dependencies : MVDs; 5NF based on keys, join dependencies : JDs

First Normal Form dissallows

composite attributes multivalued attributes nested relations; attributes whose values for an individual tuple are non-atomic

X -> Y holds

if whenever two tuples have the same value for X, they must have the same value for Y, If t1[X]=t2[X], then t1[Y]=t2[Y] -specifies a constraint on all relation instances r(R)

key

is a superkey with the additional property that removal of any attribute from K will cause K not to be a superkey any more.

A superkey

of a relation schema R = {A1, A2, ...., An} is a set of attributes S subset-of R with the property that no two tuples t1 and t2 in any legal relation state r of R will have t1[S] = t2[S]

If K is a key of R,

then K functionally determines all attributes in R