The Meaning of Entity-Relationship Diagrams, part II

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1 (1/23) Logic The Meaning of Entity-Relationship Diagrams, part II Enrico Franconi franconi Faculty of Computer Science, Free University of Bozen-Bolzano

2 (2/23) Motivation Show how a Conceptual Data Model like temporal Entity-Relationship can be extended and mapped to an underlying logical formalism. Advantages: a clear semantics for the various ER constructs ability to express complex integrity constraints availability of decision procedures for consistency and logical implication in the enriched data model.

3 (3/23) Entity-Relationship and First Order Logic Entity-Relationship is a visual language to specify a set of constraints that should be satisfied by the relational database realising the ER diagram. The interpretation of an ER diagram is defined as the collection of all the legal databases i.e., all the (finite) relational structures which conform to the constraints imposed by the conceptual schema. In order to formally define the interpretation, an ER diagram is mapped into a set of closed First Order Logic (FOL) formulas. The legal databases of an ER diagram are all the finite relational structures in which the translated set of FOL formulas evaluate to true.

4 (4/23) ER Vs. FOL: The Alphabet The Alphabet of the FOL language will have the following set of Predicate symbols: 1-ary predicate symbols: E 1, E 2,..., E n for each Entity-set; D 1, D 2,..., D m for each Basic Domain. binary predicate symbols: A 1, A 2,..., A k for each Attribute. n-ary predicate symbols: R 1, R 2,..., R p for each Relationship-set.

5 FOL Notation Vector variables indicated as x stand for an n-tuple of variables: x = x 1,..., x n Counting existential quantifier indicated as n or n. n x. ϕ(x) x 1,..., x n, x n+1. ϕ(x 1 )... ϕ(x n ) ϕ(x n+1 ) (x 1 = x 2 )... (x 1 = x n ) (x 1 = x n+1 ) (x 2 = x 3 )... (x 2 = x n ) (x 2 = x n+1 ) (x n = x n+1 ) n x. ϕ(x) x 1,..., x n. ϕ(x 1 )... ϕ(x n ) (x 1 = x 2 )... (x 1 = x n ) (x 2 = x 3 )... (x 2 = x n ) (x n 1 = x n ) (5/23)

6 (6/23) ER: The Interpretation function Interpretation: I = D, I, where D is an arbitrary non-empty set such that: D = Ω B, where: B = m i=1b Di. B Di is the set of values associated with each basic domain (i.e., integer, string, etc.); and B Di B Dj =, i, j. i j Ω is the abstract entity domain such taht B Ω =.

7 (7/23) ER: The Formal Semantics for the Atoms I is the interpretation function that maps: Basic Domain Predicates to elements of the relative basic domain: D i I = B Di (e.g., String I = B String ). Entity-set Predicates to elements of the entity domain: E i I Ω. Attribute Predicates to binary relations such that: A i I Ω B. Relationship-set Predicates to n-ary relations over the entity domain: R i I Ω Ω... Ω = Ω n.

8 (8/23) The Relationship Construct E 1 R E n... The meaning of this constraint is: R I E 1 I... E n I

9 (8/23) The Relationship Construct E 1 R E n... The meaning of this constraint is: R I E 1 I... E n I The FOL translation is the formula: x 1,..., x n. R(x 1,..., x n ) E 1 (x 1 )... E n (x n )

10 (9/23) The Attribute Construct E A D The meaning of this constraint is: E I {e Ω (A I ({e} B D )) 1}

11 (9/23) The Attribute Construct E A D The meaning of this constraint is: E I {e Ω (A I ({e} B D )) 1} The FOL translation is the formula: x. E(x) y.a(x, y) D(y)

12 (10/23) The Cardinality Construct E 1 R E n E i (p,q) The meaning of this constraint is: E i I {e i Ω p (R I (Ω {e i } Ω)) q}

13 (10/23) The Cardinality Construct E 1 R E n (p,q) E i The meaning of this constraint is: E i I {e i Ω p (R I (Ω {e i } Ω)) q} The FOL translation is the formula: x i. E(x i ) p x 1,..., x i 1, x i+1,... x n. R(x 1,..., x n ) q x 1,..., x i 1, x i+1,... x n. R(x 1,..., x n )

14 (11/23) The Cardinality Construct: An Example Professor (2,3) (1,1) Supervises Student A valid Database is: Professor Student Supervises professorid studentid professorid studentid Alex John Alex John Bob Mary Bob Laura Nick Alex Mary Paul Bob Nick Laura Alex Paul

15 (12/23) The Cardinality Construct: An Example Professor (2,3) (1,1) Supervises Student An invalid Database is: Professor Student Supervises professorid studentid professorid studentid Alex John Alex John Bob Mary Bob Laura Nick Alex Mary Paul Bob Nick Laura Alex Paul Alex Laura

16 (13/23) The Cardinality Construct: An Example Professor (2,3) (1,1) Supervises Student The FOL translation is: x, y. Supervises(x, y) Professor(x) Student(y) x. Professor(x) 2 y. Supervises(x, y) 3 y. Supervises(x, y) y. Student(y) =1 x. Supervises(x, y)

17 (14/23) ISA Relations The ISA relation is a constraint that specifies subentity sets. Subentity-set = contains entities with more properties both more attributes and different participation in relationships not pertinent to the Superentity-set. A Subentity-set inherits all the properties of its Subentity-sets. We distinguish between the following different ISA relations: Overlapping Partial; Overlapping Total; Disjoint Partial; Disjoint Total.

18 (15/23) The Overlapping Partial Construct E E 1... E n The meaning of this constraint is: E i I E I, for all i = 1,..., n.

19 (15/23) The Overlapping Partial Construct E E 1... E n The meaning of this constraint is: E i I E I, for all i = 1,..., n. The FOL translation is the formula: x. E i (x) E(x), for all i = 1,..., n.

20 (16/23) The Overlapping Total Construct E E 1... E n The meaning of this constraint is: E i I E I, for all i = 1,..., n E I E 1 I... E n I

21 (16/23) The Overlapping Total Construct E E 1... E n The meaning of this constraint is: E i I E I, for all i = 1,..., n E I E 1 I... E n I The FOL translation is the set of formulas: x. E i (x) E(x), for all i = 1,..., n x. E(x) E 1 (x)... E n

22 (17/23) The Disjoint Partial Construct E E 1... E n The meaning of this constraint is: E i I E I E i I E j I = for all i = 1,..., n for all i j

23 (17/23) The Disjoint Partial Construct E E 1... E n The meaning of this constraint is: E i I E I E i I E j I = for all i = 1,..., n for all i j The FOL translation is the set of formulas: x. E 1 (x) E(x) E 2 (x)... E n (x) x. E 2 (x) E(x) E 3 (x)... E n (x) x. E n 1 (x) E(x) E n (x) x. E n (x) E(x)

24 (18/23) The Disjoint Total Construct E E 1... E n The meaning of this constraint is: E I i E I for all i = 1,..., n E I i E I j = for all i j E I E I I 1... E n

25 (18/23) The Disjoint Total Construct E The meaning of this constraint is: E I i E I E I i E I j = E I E I I 1... E n E 1... E n for all i = 1,..., n for all i j The FOL translation is the set of formulas: x. E(x) E 1 (x)... E n x. E 1 (x) E(x) E 2 (x)... E n (x) x. E 2 (x) E(x) E 3 (x)... E n (x) x. E n 1 (x) E(x) E n (x) x. E n (x) E(x)

26 (19/23) FOL Translation: An Example Employee Manager AreaManager TopManager (1,1) Works-for (1,n) Project (1,1) Manages x, y. Works-for(x, y) Employee(x) Project(y) x, y. Manages(x, y) Top-Manager(x) Project(y) y. Project(y) x. Works-for(x, y) y. Project(y) =1 x. Manages(x, y) x. Top-Manager(x) =1 y. Manages(x, y) x. Manager(x) Employee(x) x. Manager(x) Area-Manager(x) Top-Manager(x) x. Area-Manager(x) Manager(x) Top-Manager(x) x. Top-Manager(x) Manager(x)

27 (20/23) Additional (integrity) constraints Employee Works-for (1,n) Manager Project Resp-for OrganisationalUnit (1,1) AreaManager TopManager Manages Department InterestGroup (1,1) Managers do not work for a project (she/he just manages it). x. Manager(x) y. WORKS-FOR(x, y)

28 (20/23) Additional (integrity) constraints Employee Works-for (1,n) Manager Project Resp-for OrganisationalUnit (1,1) AreaManager TopManager Manages Department InterestGroup (1,1) Managers do not work for a project (she/he just manages it). x. Manager(x) y. WORKS-FOR(x, y) If the minimum cardinality for the participation of employees to the works-for relationship is increased, then...

29 (20/23) Additional (integrity) constraints Employee Works-for (1,n) Manager Project Resp-for OrganisationalUnit (1,1) AreaManager TopManager Manages Department InterestGroup (1,1) Managers do not work for a project (she/he just manages it). x. Manager(x) y. WORKS-FOR(x, y) If the minimum cardinality for the participation of employees to the works-for relationship is increased, then... If an ISA link is added stating that Interest Groups are Departments, then...

30 (21/23) Key constraints A key is a set of attributes of an entity whose value uniquely identify elements of the entity itself. PaySlipNumber(Integer) Salary(Integer) Employee Manager Works-for (1,n) Project ProjectCode(String) AreaManager TopManager (1,1) (1,1) Manages x. Project(x) =1 y. ProjectCode(x, y) String(y) y. x. ProjectCode(x, y) =1 x. ProjectCode(x, y) Project(x)

31 (22/23) Key constraints and relational schema A key is specified for each entity. There is a one-to-one correspondence between (tuple) values of key attribute(s) and instances of an entity. This is why entities are mapped into the relational schema directly with the keys (which have concrete values) rather than with the abstract entity instances. Key values are the concrete representative for the instance of the entity.

32 (23/23) Standard mappings to the relational schema Explain the following: No abstract instance is found ever in a database. Entities are mapped through their attributes. If there is a mandatory one-to-many binary relationship, the key of the dependant entity is added to the attributes of the main entity.

Entity-Relationship Diagrams and FOL

Entity-Relationship Diagrams and FOL Free University of Bozen-Bolzano Faculty of Computer Science http://www.inf.unibz.it/ artale Descrete Mathematics and Logic BSc course Thanks to Prof. Enrico Franconi for provoding the slides What is a