Database normalization

Database normalization is a series of steps followed to obtain a database design that allows for consistent storage and efficient access of data in a relational database. These steps reduce data redundancy and the risk of data becoming inconsistent.

However, many relational DBMSs lack sufficient separation between the logical database design and the physical implementation of the data store, such that queries against a fully normalized database often perform poorly. In this case denormalization is sometimes used to improve performance, at the cost of reduced consistency guarantees.

Informal overview

A table in a relational database is said to be in a certain normal form if it satisfies certain constraints. Edgar F. Codd's original work defined three such forms but there are now other generally accepted normal forms. We give here a short informal overview of the most common ones. Each normal form below represents a stronger condition than the previous one (in the order below). For most practical purposes, databases are considered normalized if they adhere to third normal form.

First normal form (or 1NF) requires that all column values in a table are atomic (e.g., a number is an atomic value, while a list or a set is not). For example, normalization eliminates repeating groups by putting each into a separate table and connecting them with a primary key-foreign key relationship.

Second normal form (or 2NF) requires that there are no non-trivial functional dependencies of a non-key attribute on a part of a candidate key.

Third normal form (or 3NF) requires that there are no non-trivial functional dependencies of non-key attributes on something else than a superset of a candidate key.

Boyce-Codd normal form (or BCNF) requires that there are no non-trivial functional dependencies of attributes on something else than a superset of a candidate key. At this stage, all attributes are dependent on a key, a whole key and nothing but a key (excluding trivial dependencies, like A->A).

Fourth normal form (or 4NF) requires that there are no non-trivial multi-valued dependencies of attribute sets on something else than a superset of a candidate key.

Fifth normal form (or 5NF or PJ/NF) requires that there are no non-trivial join dependencies that do not follow from the key constraints.

Domain-key normal form (or DK/NF) requires that all constraints follow from the domain and the key constraints.

Formal treatment

Before we can talk about normalization we first need to fix some terms from the relational model and define them in set theory. These definitions will sometimes be simplifications of their proper definitions in this model because normalization only concerns certain aspects of the relational model.

Basic notions in the relational model are relation names and attribute names. We will represent these as strings such as "Person" and "name" and we will usually use the variables r, s, t, ... and a, b, c to range over them. Another basic notion is the set of atomic values that contains values such as numbers and strings.

Our first definition concerns the notion of tuple, which formalizes the notion of row or record in a table:

Def. A tuple is a partial function from attribute names to atomic values.

Def. A header is a finite set of attributes names.

Def.- The projection of a tuple t on a finite set of attributes A is t[A] = { (a, v) : (a, v) ∈ t, a ∈ A }.

The next definition defines relation which formalizes the contents of a table as it is defined in the relational model.

Def. A relation is a tuple (H, B) with H, the header, a header and B, the body, a set of tuples that all have the domain H.

Such a relation closely corresponds to what is usually called the extension of a predicate in first-order logic except that here we identify the places in the predicate with attribute names. Usually in the relational model a database schema is said to consist of a set of relation names, the headers that are associated with these names and the constraints that should hold for every instance of the database schema. For normalization we will concentrate on the constraints that hold for individual relations, i.e., the relation constraints. The purpose of these constraints is to describe the relation universe, i.e., the set of all relations that are allowed to be associated with a certain relation name.

Def. A relation universe U over a header H is a non-empty set of relations with header H.

Def. A relation schema (H, C) consists of a header H and a predicate C(R) that is defined for all relations R with header H.

Def. A relation satisfies the relation schema (H, C) if it has header H and satisfies C.

Key constraints and functional dependencies

One of the simplest and most important type of relation constraint is the key constraint. It tells us that in every instance of a certain relational schema the tuples can be identified by their values for certain attributes.

Def. A superkey is written as a finite set of attribute names.

Def. A superkey K holds in a relation (H, B) if K ⊆ H and there are no two distinct tuples t₁ and t₂ in B such that t₁[K] = t₂[K].

Def. A superkey holds in a relation universe U over a header H if it holds in all relations in U.

Def. - A superkey K holds as a candidate key for a relation universe U over H if it holds as a superkey for U and there is no proper subset of K that also holds as a superkey for U.

Def. A functional dependency (or FD for short) is written as X->Y with X and Y finite sets of attribute names.

Def. A functional dependency X->Y holds in a relation (H, B) if X and Y are subsets of H and for all tuples t₁ and t₂ in B it holds that if t₁[X] = t₂[X] then t₁[Y] = t₂[Y]

Def. A functional dependency X->Y holds in a relation universe U over a header H if it holds in all relations in U.

Def. A functional dependency is trivial under a header H if it holds in all relation universes over H.

Theorem A FD X->Y is trivial under a header H iff Y ⊆ X ⊆ H.

Theorem A superkey K holds in a relation universe U over H iff K ⊆ H and K->H holds in U.

Def. (Armstrong's rules) Let S be a set of FDs then the closure of S under a header H, written as S⁺, is the smallest superset of S such that:

(reflexivity) if Y ⊆ X ⊆ H then X->Y in S⁺

(transitivity) if X->Y in S⁺ and Y->Z in S⁺ then X->Z in S⁺

(augmentation) if X->Y in S⁺ and Z ⊆ H then X∪Z -> Y∪Z in S⁺

Theorem Armstrong's rules are sound and complete, i.e., given a header H and a set S of FDs that only contain subsets of H then the FD X->Y is in S⁺ iff it holds in all relation universes over H in which all FDs in S hold.

Def. If X is a finite set of attributes and S a finite set of FDs then the completion of X under S, written as X⁺, is the smallest superset of X such that:

if Y->Z in S and Y ⊆ X⁺ then Z ⊆ X⁺

The completion of an attribute set can be used to compute if a certain dependency is in the closure of a set of FDs.

Theorem Given a header H and a set S of FDs that only contain subsets of H it holds that X->Y is in S⁺ iff Y ⊆ X⁺.

Algorithm (deriving candidate keys from FDs)

      INPUT: a set S of FDs that contain only subsets of a header H
      OUTPUT: the set C of superkeys that hold as candidate keys in
              all relation universes over H in which all FDs in S hold
      begin
        C := ∅;          // found candidate keys
        Q := { H };      // superkeys that contain candidate keys
        while Q <> ∅ do
          let K be some element from Q;
          Q := Q - { K };  
          minimal := true;
          for each X->Y in S do 
            K' := (K - Y) ∪ X;   // derive new superkey
            if K' ⊂ K
            then
              minimal := false;
              Q := Q ∪ { K' };
            fi
          od
          if minimal and there is not a subset of K in C
          then
            remove all supersets of K from C;
            C := C ∪ { K };
          fi
        od
      end

Def. Given a header H and a set of FDs S that only contain subsets of H an irreducible cover of S is a set T of FDs such that

S⁺ = T⁺
there is no proper subset U of T such that S⁺ = U⁺,
if X->Y in T then Y is a singleton set and
if X->Y in T and Z a proper subset of X then Z->Y is not in S⁺.

First normal form

Def. A relation is in 1NF if and only if all underlying domains contain scalar values only. (Note: that relations as defined above are necessarily in 1NF)
define NFNF relations
how to transform NFNF relations (also called UNF relations) to 1NF relations
- how to transform the key constraints of nested relations
- how to transform the functional dependencies of nested relations

Second normal form

Def. A relation is said to be in the 2NF if and only if it is in the 1NF and every non-key attribute is irreducibly dependent on the primary key (i.e.,not partially dependent on candidate key).

example1: - If you have a table STUDENTCOURSE where the primary key is the two fields student_id and course_id, the field student_address does not belong because it has nothing to do with course_id. The student_address field would be in the STUDENT table, whose key is student_id only. The final_grade, year_taken, and semester_taken fields would belong in the STUDENTCOURSE table.
only interesting for history's sake

example2:

       Consider relation schema 2nf(A#,B#,C#,D,E,F)
       Here if combination A#,B# is acting as a Candidate-Key(it might be P-Key)     and if A#(or B#) alone determines to D(or E or F i.e.,non-key attributes) then the schema is not in 2nd NF.

Third normal form

Def. A relation is said to be in the 3NF if and only if it is in the 2NF and all non-key attributes are mutually independent(i.e. there should not be transitive dependencies).
Third Normal Form is the most common level of normalization.
example: Given this relation (#A, B, C) in 2NF where #A->B, #A->C and B->C (first two are fixed by 2NF). This relation isn't in 3NF because C is functionaly dependent from B. This relation should be descomposed into (#A, B),(#B, C) or (#A, B), (#A, C) to be in 3NF.
(rissanen's rule ? )
how to transform from 2NF to 3NF
def: dependency preserving
can always be reached while staying dependency preserving

Boyce-Codd nrmal form

Def. A relation is said to be in the BCNF if and only if it is in the 3NF and every non-trivial, left-irreducible functional dependency has a candidate key as its determinant. In more informal terms, a relation is in BCNF if it is in 3NF and the only determinants are the candidate keys.
example
how to transform from 3NF to BCNF
can not always be reached in a dependency preserving way.

Multi-valued and join dependencies

def Given a a relation (A,B,C) there is a multi-valued dependency between A and B noted by A->>B if and only if the set of values of B, given a fixed value (a: A, c: C), only depends on a and is independent of c.
example
- trivial multi-value dependency (X->>Y is trivial if X+Y contains all attributes or Y is a subset of X)
reasoning rules for MVDs (it only can happen if the relation have at least tree attributes, all functional dependency implies multi-valued dependeny but not in the other direction)
def join dependency
example
reasoning rules for JDs
when is join dependency implied by key constraints?
Fagin's theorem: A relation (A,B,C) can be decomposed lossless in (A,B) and (A,C) if and only if A->> B | C is true in (A,B,C).
relationship between JDs and MVDs

Fourth normal form

Def. A relation is said to be in 4NF if and only if it is in the BCNF and multi-valued dependencies are functional dependencies.
Notes: The 4NF removes unwanted data structures: multi-valued dependencies. Ronald Fagin demonstrated that it is always possible to achive 4NF (but not always desirable). Rissanen's theorem is also applicable on multi-valued dependencies.
example
how to transform from BCNF to 4NF

Fifth normal form

Def. A relation R is said to be in the 5NF if and only if it is in 4NF and every join dependency in R is implied by the candidate keys.
example
from 4NF to 5NF
explain that ultimate normal form that can be reached with projections

Domain-key normal form

1NF, 2NF, 3NF, BCNF, 4NF, 5NF are all subsets of the general constraints (G) and thus are all special cases of DK/NF
ultimate normal form as any general constraint on the Universe of Discourse can be included in the set G
emphasizes on separating the attributes according to themes
have to take into account all the logical consequences of domains and keys
has no modificational anomalies

Other dependencies

embedded dependencies
dependencies as statements in first-order logic

Implementation notes

At 3NF it may become necessary to create additional entities in order to normalize relations beyond 3NF. From an implementation standpoint, this is not always desirable due to possible performance issues, and sheer complexity. See Denormalization.

References

This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008 and incorporated under the "relicensing" terms of the GFDL, version 1.3 or later.