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

Steffen Staab Advanced Data Modeling 2 of 48 WeST Overview  Relational data model;  Tuples and relations;  Schemas and instances;  Named vs. unnamed perspective;  Relational algebra;

Steffen Staab Advanced Data Modeling 3 of 48 WeST Table PlayerBirth Year Andy1980 Wim1975 Liam1985 Mike1988 Bert1971

Steffen Staab Advanced Data Modeling 4 of 48 WeST 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.

Steffen Staab Advanced Data Modeling 5 of 48 WeST 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.

Steffen Staab Advanced Data Modeling 6 of 48 WeST Domains and attributes  A set of domains (sets of values);  A set of corresponding domain names d 1, d 2, …  A set of attributes a 1, a 2, …

Steffen Staab Advanced Data Modeling 7 of 48 WeST 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 2 D i for all i.

Steffen Staab Advanced Data Modeling 8 of 48 WeST 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.

Steffen Staab Advanced Data Modeling 9 of 48 WeST 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.  The tuple conforms to a relation schema r(a 1 :d1, …, a n :d n ) if v i 2 D i for all i.

Steffen Staab Advanced Data Modeling 10 of 48 WeST 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.

Steffen Staab Advanced Data Modeling 11 of 48 WeST Typed or untyped?  Consider the relation admire: admireradmired wimandy mikewim liamandy liamarsenal

Steffen Staab Advanced Data Modeling 12 of 48 WeST 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.

Steffen Staab Advanced Data Modeling 13 of 48 WeST 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.

Steffen Staab Advanced Data Modeling 14 of 48 WeST 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 ).

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

Steffen Staab Advanced Data Modeling 16 of 48 WeST Constant { (v 11, …, v 1n ), ….. (v k1, …, v kn ) }

Steffen Staab Advanced Data Modeling 17 of 48 WeST Union R 1 [ R 2 = {(c 1, …, c k ) | (c 1, …, c k ) 2 R 1 or (c 1, …, c k ) 2 R 2 }

Steffen Staab Advanced Data Modeling 18 of 48 WeST Set difference R_1 - R2 = {(c 1, …, c k ) | (c 1, …, c k ) 2 R 1 and (c 1, …, c k ) 2 R 2 }

Steffen Staab Advanced Data Modeling 19 of 48 WeST Cartesian product R1 £ R2 = {(c 1, …, c k, d 1, …, d m ) | (c1, …, c k ) 2 R 1 and (d 1, …, d m ) 2 R 2 }.

Steffen Staab Advanced Data Modeling 20 of 48 WeST Projection Let now R be a relation of arity k and i 1, …, i m be numbers in {1, …, k}. ¼ i 1, …, i m (R) = {(c i 1, …, c i m ) | (c 1, …, c k ) 2 R }. We say that ¼ i 1, …, i m (R) is obtained from R by projection (on arguments i 1, …, i m ).

Steffen Staab Advanced Data Modeling 21 of 48 WeST Selection Assume formulas on domains with “variables” #1, #2, …. For example, #1 = #2. ¾ F (R) = {(c 1, …, c k ) | (c 1, …, c k ) 2 R and F holds on (c 1, …, c k )}.

Steffen Staab Advanced Data Modeling 22 of 48 WeST Overview  Relational algebra, named perspective  SQL  Integrity constraints  (Aggregates and grouping)

Steffen Staab Advanced Data Modeling 23 of 48 WeST Constant { { a 1 = v 11, …, a n = v 1n }, … …… { a 1 = v k1, …, a n = v kn } }

Steffen Staab Advanced Data Modeling 24 of 48 WeST Union Let R1, R2 be relations with the same attributes. R1 [ R2 = { t | t 2 R 1 or t 2 R 2 }

Steffen Staab Advanced Data Modeling 25 of 48 WeST Union, example AB ® 1 ® 2 ¯ 1 AB ® 1 ® 2 ¯ 1 ¯ 3 AB ® 2 ¯ 3 R1R1 R2R2 R 1 [ R 2

Steffen Staab Advanced Data Modeling 26 of 48 WeST 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 ½ a 1 ! b 1, …, a n ! b n (R) = {{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 } 2 R}

Steffen Staab Advanced Data Modeling 27 of 48 WeST 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: ¼ a 1,…, a n ( ¾ P (R 1 £ … £ R m )) The result of an SQL query is a relation. Exceptions?

Steffen Staab Advanced Data Modeling 28 of 48 WeST Integrity constraints  Domain constraints.  Key constraints.  Foreign key constraints.  More general, defined constraints.  How to translate them?

Steffen Staab Advanced Data Modeling 29 of 48 WeST Query language Allow one to define:  Relation and database schemas;  Relations through our relations;  Integrity constraints;  Updates.