Relational Schemas and Predicate Logic: Notation

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Relational Schemas and Predicate Logic Notation
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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.

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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(Namestring,GPAnumeric,Ageinteger)
• Registered(Namestring,Coursestring,gradenumeric
  )

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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.

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ER Model Relationships
name
ssn
lot
Employees
since
name
dname
super-visor
subor-dinate
budget
ssn
lot
did
Reports_To
Works_In
Departments
Employees

• 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.

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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.

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Relational Instance and Finite Models

• A relational instance is a relational schema
  set of tuples specified for each relation.
• The list ltR1,..,Rngt of relations (with tuples
  specified, but without field names and key
  constraints) is called a finite model in logic.

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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.

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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 lt 40.
• Pros
• Simple Translation.
• Simple logic.
• Cons
• Loses information about key fields.
• Not always natural to read.

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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) lt 40.
• Pros
• Keeps information about keys and foreign keys.
• Natural to read.
• Cons a bit more complex mathematically.

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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 gt 3, X Y, 5 gt 1.
• A literal is an atom or a negated atom.

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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.