relational data model - west.uni-koblenz.de · steffen staab [email protected] advanced data...

Web Science & TechnologiesUniversity of Koblenz ▪ Landau, Germany

Relational Data Model

Steffen [email protected]

Advanced Data Modeling2 of 48

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Overview

Relational data model;Tuples and relations;Schemas and instances;Named vs. unnamed perspective;Relational algebra;



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Table

1971Bert

1988Mike

1985Liam

1975Wim

1980Andy

Birth YearPlayer



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Observations

The rows of the table contain pairs occurring in the relationplayer.There are two columns, labeled respectively by “name”and “birth year”.The values in each column belong to different domains of possible values.



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How to specify a relation

1. specifying the names of the columns (also called fields orattributes);

2. specifying a domain of possible values for each column;

3. enumerate all tuples in the relation.

(1)–(2) refer to the schema of this relation, while (3) to an instance.



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Domains and attributes

A set of domains (sets of values);

A set of corresponding domain names d1, d2, …

A set of attributes a1, a2, …

D1,D2



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Relation Schema and Instances

Tuple: any finite sequence (v1, … , vn).n is the arity of this tuple.

Relation schema: r(a1:d1, …, an:dn)

where n ≥ 0, r is a relation name, a1, …, an are distinctattributes, d1, …, dn are domain names.

Relation instance:finite set of tuples (v1, …, vn) of arity n such that vi ∈ Di forall i.



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Observation

1. The attributes in each column must be unique.

2. A relation is a set. Therefore, when we represent a relationby a table, the order of rows in the table does not matter.

Let us add to this:3. The order of attributes does not matter.



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New notation for tuples

A tuple is a set of pairs { (a1, v1), …, (an, vn) } denoted by{ a1=v1, …, an=vn } ,

Let d1, …, dn be domain names and be the corresponding domains.

The tuple conforms to a relation schemar(a1:d1, …, an:dn) if vi ∈ Di for all i.

D1, . . . ,Dn



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Relational data are structured

Note that in the relational data model tuples stored in a tableare structured:all tuples conform to the same relation schema;the values in the same column belong to the same domain.

Untyped perspective: there is a single domain, so thesecond condition can be dropped.



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Typed or untyped?

Consider the relation admire:

arsenalliam

andyliam

wimmike

andywim

admiredadmirer



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Database schema and instance

Relational database schema: a collection of relationschemas with distinct relation names.

Relational database instance conforming to a relational database schema S:

a mapping I from the relation names of S to relationinstances such that

• for every relation schema r(a1:d1, … , an:dn) in S the relation instance I(r) conforms to this relation schema.



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Unnamed perspective

No attributes

a tuple is simply a sequence (v_1, … , v_n) of values.

The components of tuples can therefore be identified bytheir position in the tuple.



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From Unnamed to Named Perspective

Introduce a collection of attributes #1,#2, …,

identify tuple (v1, … , vn) with the tuple{ #1 = v1, … ,#n = vn }.

Likewise, identify relation schema r(d1, … , dn) withr(#1:d1, … ,#n:dn).



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Relational Algebra and SQL

1. Can define new relations from existing ones;2. Uses a collection of operations on relations to do so.



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Constant

{ (v11, … , v1n),…..(vk1, … , vkn) }



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Union

R1 ∪ R2 = {(c1, … , ck) | (c1, … , ck) ∈ R1 or

(c1, … , ck) ∈ R2 }



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Set difference

R_1 - R2 = {(c1, … , ck) | (c1, … , ck) ∈ R1 and

(c1, … , ck) ∈ R2 }



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Cartesian product

R1 × R2 = {(c1, … , ck, d1, … , dm) |

(c1, … , ck) ∈ R1 and (d1, … , dm) ∈ R2 }.



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Projection

Let now R be a relation of arity k and i1, … , im be numbers in {1, … , k}.

πi1, … ,im (R) = {(ci1

, … , cim) | (c1, … , ck) ∈ R }.

We say that πi1, … ,im(R) is obtained from R by projection (on arguments i1, … , im).



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Selection

Assume formulas on domains with “variables” #1, #2, ….

For example, #1 = #2.

σF (R) = {(c1, …, ck) | (c1, …, ck) ∈ R andF holds on (c1, …, ck)}.



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Overview

Relational algebra, named perspectiveSQLIntegrity constraints(Aggregates and grouping)



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Constant

{ { a1 = v11, … , an = v1n },… … …

{ a1 = vk1, … , an = vkn } }



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Union

Let R1, R2 be relations with the same attributes.

R1 ∪ R2 = { t | t ∈ R1 or t∈ R2 }



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Union, example

1β

2α

1α

BA

3β

1β

2α

1α

BA

3β

2α

BA

R1 R2 R1 ∪ R2



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Renaming

Let R be a relation whose set of attributes is a1,…,an, c1,…,cm

Let b1, … , bn be distinct attributes such that{b1, … , bn} Å { c1, … , cm} = ∅

Thenρa1→b1, … , an→bn

(R) ={{b1 = v1, … , bn=vn , c1=w1 , … , cm=wm} |

{a1 = v1, … , an=vn , c1=w1 , … , cm=wm} ∈ R}



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SQL

SQL is based on set and relational operations with certainmodifications and enhancements

A typical SQL query has the form:select a1, … , an

from R1, … , Rm

where PThis query is equivalent to relational algebra expression:

πa1,…,an(σP (R1× … × Rm))

The result of an SQL query is a relation.

Exceptions?



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Integrity constraints

Domain constraints.Key constraints.Foreign key constraints.

More general, defined constraints.

How to translate them?



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Query language

Allow one to define:Relation and database schemas;Relations through our relations;Integrity constraints;Updates.

relational data model - west.uni-koblenz.de · steffen staab [email protected] advanced data...

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