Web Science & Technologies University of Koblenz ▪ Landau, Germany Relational Data Model

Web Science & Technologies

University 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

Player Birth Year

Andy 1980

Wim 1975

Liam 1985

Mike 1988

Bert 1971



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Observations

The rows of the table contain pairs occurring in the relation player.

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 or attributes);

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 distinct attributes, d1, …, dn are domain names.

Relation instance: finite set of tuples (v1, …, vn) of arity n such that vi 2 Di for all 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 relation by 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 schema r(a1:d1, …, an:dn) if vi 2 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 table are 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 the second condition can be dropped.



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Typed or untyped?

Consider the relation admire: admirer admired

wim andy

mike wim

liam andy

liam arsenal



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Database schema and instance

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

Relational database instance conforming to a relational database schema S: a mapping I from the relation names of S to relation

instances 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 by their 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) with r(#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) 2 R1 or

(c1, … , ck) 2 R2 }



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Set difference

R_1 - R2 =

{(c1, … , ck) | (c1, … , ck) 2 R1 and

(c1, … , ck) 2 R2 }



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Cartesian product

R1 £ R2 =

{(c1, … , ck, d1, … , dm) |

(c1, … , ck) 2 R1 and

(d1, … , dm) 2 R2 }.



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Projection

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

¼i1, … ,im

(R) = {(ci1, … , cim

) | (c1, … , ck) 2 R }.

We say that ¼i1, … ,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.

¾F (R) = {(c1, …, ck) |

(c1, …, ck) 2 R and

F holds on (c1, …, ck)}.



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Overview

Relational algebra, named perspective SQL Integrity 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 2 R1 or t2 R2 }



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Union, example

A B

® 1

® 2

¯ 1

A B

® 1

® 2

¯ 1

¯ 3

A B

® 2

¯ 3

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} = ;

Then

½a1!b1, … , an!bn(R) =

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

{a1 = v1, … , an=vn , c1=w1 , … , cm=wm} 2 R}



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SQL

SQL is based on set and relational operations with certain modifications and enhancements

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

from R1, … , Rm

where P

This query is equivalent to relational algebra expression:

¼a1,…,an (¾P (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.

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Web Science & Technologies University of Koblenz ▪ Landau, Germany Relational Data Model