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

Transcription

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

2 Overview Relational data model; Tuples and relations; Schemas and instances; Named vs. unnamed perspective; Relational algebra; 2of 48

3 Table Player Birth Year Andy 1980 Wim 1975 Liam 1985 Mike 1988 Bert of 48

4 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. 4of 48

5 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. 5of 48

6 Domains and attributes A set of domains D 1, D 2 (sets of values); A set of corresponding domain names d 1, d 2, A set of attributes a 1, a 2, 6of 48

7 Relation Schema and Instances Tuple: any finite sequence (v 1,, v n ). n is the arity of this tuple. Relation schema: r(a 1 :d 1,, a n :d n ) where n 0, r is a relation name, a 1,, a n are distinct attributes, d 1,, d n are domain names. Relation instance: finite set of tuples (v 1,, v n ) of arity n such that v i D i for all i. 7of 48

8 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. 8of 48

9 New notation for tuples A tuple is a set of pairs { (a 1, v 1 ),, (a n, v n ) } denoted by { a 1 =v 1,, a n =v n }, Let d 1,, d n be domain names and be the corresponding domains. D 1,...,D n The tuple conforms to a relation schema r(a 1 :d1,, a n :d n ) if v i D i for all i. 9of 48

10 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. 10 of 48

11 Typed or untyped? Consider the relation admire: admirer admired wim andy mike wim liam andy liam arsenal 11 of 48

12 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(a 1 :d 1,, a n :d n ) in S the relation instance I(r) conforms to this relation schema. 12 of 48

13 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. 13 of 48

14 From Unnamed to Named Perspective Introduce a collection of attributes #1,#2,, identify tuple (v 1,, v n ) with the tuple { #1 = v 1,,#n = v n }. Likewise, identify relation schema r(d 1,, d n ) with r(#1:d 1,,#n:d n ). 14 of 48

15 Relational Algebra and SQL 1. Can define new relations from existing ones; 2. Uses a collection of operations on relations to do so. 15 of 48

16 Constant { (v 11,, v 1n ),.. (v k1,, v kn ) } 16 of 48

17 Union R 1 R 2 = {(c 1,, c k ) (c 1,, c k ) R 1 or (c 1,, c k ) R 2 } 17 of 48

18 Set difference R_1 - R2 = {(c 1,, c k ) (c 1,, c k ) R 1 and (c 1,, c k ) R 2 } 18 of 48

19 Cartesian product R1 R2 = {(c 1,, c k, d 1,, d m ) (c1,, c k ) R 1 and (d 1,, d m ) R 2 }. 19 of 48

20 Projection Let now R be a relation of arity k and i 1,, i m be numbers in {1,, k}. π i1,, im (R) = {(c i1,, c im ) (c 1,, c k ) R }. We say that π i1,, im (R) is obtained from R by projection (on arguments i 1,, i m ). 20 of 48

21 Selection Assume formulas on domains with variables #1, #2,. For example, #1 = #2. σ F (R) = {(c 1,, c k ) (c 1,, c k ) R and F holds on (c 1,, c k )}. 21 of 48

22 Overview Relational algebra, named perspective SQL Integrity constraints (Aggregates and grouping) 22 of 48

23 Constant { { a 1 = v 11,, a n = v 1n }, { a 1 = v k1,, a n = v kn } } 23 of 48

24 Union Let R1, R2 be relations with the same attributes. R1 R2 = { t t R 1 or t R 2 } 24 of 48

25 Union, example R 1 R 2 R 1 R 2 A B A B A B α 1 α 2 α 1 α 2 β 3 α 2 β 1 β 1 β 3 25 of 48

26 Renaming Let R be a relation whose set of attributes is a 1,,a n, c 1,,c m Let b 1,, b n be distinct attributes such that {b 1,, b n } Å { c 1,, c m } = Then ρ a1 b 1,, a n b (R) = n {{b 1 = v 1,, b n =v n, c 1 =w 1,, c m =w m } {a 1 = v 1,, a n =v n, c 1 =w 1,, c m =w m } R} 26 of 48

27 SQL SQL is based on set and relational operations with certain modifications and enhancements A typical SQL query has the form: select a 1,, a n from R 1,, R m where P This query is equivalent to relational algebra expression: π a1,,a n (σ P (R 1 R m )) The result of an SQL query is a relation. Exceptions? 27 of 48

28 Integrity constraints Domain constraints. Key constraints. Foreign key constraints. More general, defined constraints. How to translate them? 28 of 48

29 Query language Allow one to define: Relation and database schemas; Relations through our relations; Integrity constraints; Updates. 29 of 48