Informationslogistik Unit 4: The Relational Algebra

Transcription

1 Informationslogistik Unit 4: The Relational Algebra 26. III. 2012

2 Outline 1 SQL 2 Summary What happened so far? 3 The Relational Algebra Summary 4 The Relational Calculus

3 Outline 1 SQL 2 Summary What happened so far? 3 The Relational Algebra Summary 4 The Relational Calculus

4 SQL-Lesson 4 Today: Nachbesprechung Wissensüberprüfung Joining tables NULL values

5 Outline 1 SQL 2 Summary What happened so far? 3 The Relational Algebra Summary 4 The Relational Calculus

6 Reminder What happened so far: Entity (types) and relationship (types), ER model the relational model for describing databases (tables)

7 Outline 1 SQL 2 Summary What happened so far? 3 The Relational Algebra Summary 4 The Relational Calculus

8 Formal Languages for Information Extraction What this is about: The Relational Model: abstract model for (relational) databases Given a relational database, what information can be abstracted? describe all possible queries (resp. their result sets) on a relational database Two (abstract) languages for description: The relational algebra The relational calculus

9 Formal Languages for Information Extraction Relation calculus: can express what information we want to have Relational algebra: also includes how the information is extracted The relational algebra is more useful for realization of database systems (=theoretical basis for SQL). Note: All operations of the relation calculus as well as the relational algebra return a relation.

10 The Relational Algebra Selection A Selection chooses all tuples t of a relation R that satisfy the so-called selection-predicate F: The predicate F may consist of constants σ F (R) := {t R F(t)} attribute names of the relation R operators for comparison: <, >,,, =, logical connectives: (and), (or), (not)

11 The Relational Algebra Selection Example: Show all countries in Europe with an area larger than km 2. constants: Europe, attribute names of the relation R: country, region, area operators for comparison: =, > logical connectives: and Note: Selection corresponds to WHERE part in SQL query.

12 The Relational Algebra Projection Selection chooses tuples (=rows) Projection chooses attributes (=columns) Given a relation R and attributes A 1, A 2,..., A n, the projection Π A1,A 2,...,A n (R) := { (t.a 1, t.a 2,..., t.a n ) t R } contains for each tuple of the relation the respective attribute values. Usually, each tuple shall appear not more than once.

13 The Relational Algebra Projection Given a relation R and attributes A 1, A 2,..., A n, the projection Π A1,A 2,...,A n (R) := { (A 1 (t), A 2 (t),..., A n (t)) t R } contains for each tuple of the relation the respective attribute values. Usually, each tuple shall appear not more than once. Examples: Show all regions from cia. Show all year-month combinations from sowe. Note: Projection corresponds to SELECT part in SQL query.

14 The Relational Algebra Union and Set Difference Union ( ) chooses all tuples from two (or more) relations: R 1 R 2 = {t t R 1 or t R 2 } The Set Difference ( ) chooses all tuples from one relation that is not in the other relation: R 1 R 2 = {t t R 1 and t / R 2 } Examples: Show all regions and countries from cia. Show all regions that are not countries from cia. Note: Union corresponds to UNION in SQL, set difference to MINUS.

15 The Relational Algebra Cartesian Product The Cartesian Product ( ) chooses all combinations of tuples from two (or more) relations: R 1 R 2 = {(t 1, t 2 ) t 1 R 1 or t 2 R 2 } The attributes of the arising relation R 1 R 2 are a union of the attributes of R 1 and R 2. If R 1 and R 2 have attributes of the same name A, one has to use qualified attribute names: R 1.A or R 2.A Note: The Cartesian Product corresponds to a JOIN without ON condition in SQL.

16 The Relational Algebra Renaming Renaming is used to rename relations: ρ Rnewname (R) rename attribute names: ρ Anewname A(R) Note: Renaming in SQL is done with AS (but often can be skipped). Note: Renaming is not just for aesthetic reasons. Sometimes renaming is essential! (example follows below)

17 The Relational Algebra Definition For a formal definition of the relational algebra the previous operations are sufficient. Inductive Definition of the relational algebra Given a database D, all constant relations as well as all relations (tables) of D are expressions of the relational algebra. If E is an expression of the relational algebra, so is any selection σ F (E), any projection Π A (E) and any renaming ρ V (E) or ρ A B (E), respectively. If E 1 and E 2 are expressions of the relational algebra, so are their union E 1 E 2, their difference E 1 E 2, and their Cartesian product E 1 E 2. Intuitively: The relational algebra consists of all possible queries on database.

18 The Relational Algebra Definition For a formal definition of the relational algebra the previous operations are sufficient. Still, it is convenient to add some more operations to make expressions simpler (and writing queries easier). Note: Adding the following operations does not add anything to the relational algebra.

19 The Relational Algebra Natural Join The natural Join of relation R 1 with attributes A 1,..., A m, B 1,..., B k with relation R 2 with attributes B 1,..., B k, C 1,..., C n is defined as R 1 R 2 := Π A1,...,A m,b 1,...,B k,c 1,...,C n (σ R1.B 1 =R2.B 1...R 1.B k =R2.B k (R 1 R 2 )). Properties of the Join Operation: commutative: R 1 R 2 = R 2 R 1 associative: (R 1 R 2 ) R 3 = R 1 (R 2 R 3 ) If attributes that have the same meaning have different names, the renaming operator has to be applied. Note: The natural Join in SQL is done with JOIN and ON condition.

20 The Relational Algebra General Join In a general Join (θ-join) of relation R 1 having attributes A 1,..., A n with relation R 2 having attributes B 1,..., B m the join can be any predicate over the attributes A 1,..., A n and B 1,..., B m. Formally, for a predicate θ R 1 θ R 2 := σ θ (R 1 R 2 ). The most common join is an equi-join of the form R 1 R1.A i =R 2.B j R 2. (Difference to natural join: attributes may have different names.) However: It is possible to have joins with > or < in θ. ( in the ON part of a query examples in later unit)

21 The Relational Algebra Other Joins The joins considered so far are usually inner joins. tuples without partner in the other relation are deleted. Outer Joins: left (outer) join: all tuples from the left relation are kept right (outer) join: all tuples from the right relation are kept full outer join: all tuples from both relations are kept In Semi-Joins an inner join is conducted, but only attributes of one relation are selected (actually: projected). left semi-join ( ) right semi-join ( )

22 The Relational Algebra Set Intersection Set intersection is defined as R 1 R 2 := R 1 (R 1 R 2 ). It returns all tuples that are in both relations. Note: Intersection in SQL is done with INTERSECT.

23 The Relational Algebra Relational Division The Relational Division R 1 R 2 is defined as follows: t R 1 R 2 if for all t 2 R 2 there is a t 1 R 1 such that: For all attributes of R 2 : t 1.A = t 2.A, for all attributes of R 1 R 2 : t 1.A = t. (see example in handout)

24 Summary Summary: The Relational Algebra What is this relational algebra stuff about? Define basic operations: selection ( WHERE) projection ( SELECT) union, set difference ( UNION, MINUS) Cartesian product ( JOIN) renaming ( AS) we may extract from database everything that can be expressed with these operations (corresponds to SQL query)

25 Outline 1 SQL 2 Summary What happened so far? 3 The Relational Algebra Summary 4 The Relational Calculus

26 The Relational Algebra vs. The Relational Calculus the relational algebra: gives description how to extract information the relation calculus: describes information which can be extracted (based on so-called predicate logic)

27 The Main Idea of the Relational Calculus The Main Idea of the Relational Calculus Describe extractable information by a generalized selection. Reminder: The selection in the relational algebra was defined as: A Selection chooses all tuples t of a relation R that satisfy the so-called selection-predicate F: σ F (R) := {t R F(t)} The predicate F may consist of constants attribute names of the relation R operators for comparison: <, >,,, =, logical connectives: (and), (or), (not)

28 The Main Idea of the Relational Calculus A Selection chooses all tuples t of a relation R that satisfy the so-called selection-predicate F: σ F (R) := {t R F(t)} The predicate F may consist of constants attribute names of the relation R operators for comparison: <, >,,, =, logical connectives: (and), (or), (not) The Main Idea of the Relational Calculus Describe extractable information by a generalized selection. allow more general predicates in selection

29 The Relational Calculus Two distinct languages for relational calculus: relational tuple calculus: based on relations relational domain calculus: based on domains

30 The Relational Tuple Calculus Examples: all countries in Europe: { c c country and c.region= Europe } also allowed in selection of relational algebra

31 The Relational Tuple Calculus Examples: all countries in Europe: { c c country and c.region= Europe } also allowed in selection of relational algebra all countries that are larger than some other country in the same region: { c c country and there is c with c.region=c.region such that: c.area>c.area } more complex predicate than allowed in selection of rel. algebra

32 The Relational Tuple Calculus Examples: all countries in Europe: { c c country and c.region= Europe } also allowed in selection of relational algebra all countries that are larger than some other country in the same region: { c c country and there is c with c.region=c.region such that: c.area>c.area } more complex predicate than allowed in selection of rel. algebra all pairs of countries that lie in the same region: { (c, c ) c country and c country and c.region=c.region} tuples (c, c ) do not exist in table of database

33 The Relational Tuple Calculus: Formal Definition Similar to selection: { t F(t) } where F(t) is a formula as for selection, but now also quantifiers are allowed: all quantifier (symbol: ): for all existence quantifier (symbol: ): there is

34 The Relational Tuple Calculus: Safe Expressions It is possible to build expressions with infinite result set: { c c is not a country } Solution: define domain (=all possible values in database) consider only safe expressions where the result set is part of domain.

35 The Relational Domain Calculus very similar to relational tuple calculus only difference: not tuples but values { v P(v) } instead of { t P(t) } in tuple calculus, t has to be a tuple in domain calculus, v is just a value taken from a domain definition of safe expressions has to be modified

36 Expressiveness Theorem The following three languages have the same expressiveness: the relational algebra, the relational tuple calculus when restricted to safe expressions, the relational domain calculus when restricted to safe expressions. What does it mean? For each expression in one of the languages there is a corresponding expression in the two other languages.