Relational Schemas and Predicate Logic: Notation

Relations Let E1,..,En be sets of entities or objects (we may have Ei = Ej). The Cartesian product E1 x … x En is the set of tuples of the form (e1,..,en) where ei is in set Ei. A relation R on E1,..,En is a subset of the Cartesian product E1 x … x En. We say that n is the arity of the relation. Note that a relation may be unary.

Relational Schemas A schema specifies a finite set of relations R1,..Rn with additional structure: Each column/field in a relation gets a name (can also just use position) and a domain. A subset of fields is identified as the key. The non-key fields are often called (descriptive) attributes. The values of the attributes are determined by the values of the key field. Common Notation –Student(Name:string,GPA:numeric,Age:integer) –Registered(Name:string,Course:string,grade:nume ric)

ER Model: Entities Entity: Real-world object distinguishable from other objects. An entity is described (in DB) using a set of attributes. Entity Set: A collection of similar entities. E.g., all employees. – All entities in an entity set have the same set of attributes. – Each entity set has a key. – Each attribute has a domain. – A key defines the values of attributes---the attributes are functions of the keys. Employees ssn name lot

ER Model: Relationships Relationship: Association among two or more entities. E.g., Attishoo works in Pharmacy department. Relationship Set: Collection of similar relationships. – An n-ary relationship set R relates n entity sets E1... En; each relationship in R involves entities e1,..., en Same entity set could participate in different relationship sets, or in different “roles” in same set. lot dname budget did since name Works_In DepartmentsEmployees ssn Reports_To lot name Employees subor- dinate super- visor ssn

Notes on ER relationships Note that relationships can have descriptive attributes too. The values of the descriptive attributes are determined once we have identified the entities involved: grade(Joe,CMPT354). The fields that identify the entities involved are callled foreign key pointers.

Relational Instance and Finite Models A relational instance is a relational schema + set of tuples specified for each relation. The list of relations (with tuples specified, but without field names and key constraints) is called a finite model in logic.

Graphical Visualization A graph on set S is a binary symmetic relation over S x S. If the relation instance contains only one binary symmetric relation, it can be visualized as a graph whose edges and nodes are annotated with the values of descriptive attributes. Classic social network analysis considers only the graph structure, not the attributes.

Translating Schemas Into Logic: no functions Each relation of arity n  predicate symbol with n arguments. E.g., “A student with GPA 3.0 is younger than 40”  Student(S,G,A) AND G = 3.0  A < 40. Pros: –Simple Translation. –Simple logic. Cons: –Loses information about key fields. –Not always natural to read.

Translating Schemas Into Logic: Functions Introduce one function symbol for each descriptive attribute. The arguments are the key fields. E.g., Age(S), grade(S,C). Also: S.age,Registered.grade. E.g., “A student with GPA 3.0 is younger than 40”  GPA(S) = 3.0  Age(S) < 40. Pros: –Keeps information about keys and foreign keys. –Natural to read. Cons: a bit more complex mathematically.

Formal definitions in logic Use the function-free formulation, with predicate symbols R1,..,Rn. The language contains a set of constants and variables. A term is a constant or a variable. An atom is: –a predicate symbol with the required numbers of terms, e.g. Student(N,G,40), Student(Jack,3.0,40). –A comparison of terms, e.g. X > 3, X = Y, 5 > 1. A literal is an atom or a negated atom.

Clauses A clause is a set of literals. The negated literals are called the body, the positive ones the head. A clause is often written in implication form: b1 AND b2  h1. Also h1 :- b1,b2. A clause with a single positive literal is a Horn clause.