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What is TRC?• it is a calculus that was introduced by Edgar F. Codd as part of the relational model, in order to provide a declarative database-query language for this data model.
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Tuple Relational Calculus• It is a nonprocedural query language.• A query is expressed as { t | cond }• Return all tuples T that satisfy the condition Cond.• t is a tuple variable, t[A] denotes the value of tuple t on attribute A
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Atoms• For the construction of the formula we will assume an infinite set V of tuple variables.• Examples of atoms are:• (t.age = s.age) — t has an age attribute and s has an age attribute with the same value• (t.name = “priyank”) — tuple t has a name attribute and its value is ”Priyank"• Book(t) — tuple t is present in relation Book.
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Formulas• Formulas are combined using operators ∧ (and), ∨ (or) and ¬ (not), and we can use the existential quantifier (∃) and the universal quantifier (∀) .• Here ∃ and ∀ quantifier are called bound variable..
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Predicate Calculus Formula• Set of comparison operatores: <, ≤, >, ≥ , = , ≠• Set of connectives: and ( ^), or (v), not (¬)• Implication (=>): x => y means: if x is true, then y is true.
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Predicate Calculus Formula (cont’d)• ∃ t ∈ r ( Q(t) ) ≡ There exists a tuple t in relation r such that predicate Q (t) is true• ∀ t ∈ r ( Q(t) ) ≡ Q is true for all tuples t in relation r
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Formal semantics• " f1 ∧ f2 " is true if and only if " f1 " is true and " f2 " is true,• " f1 ∨ f2 " is true if and only if " f1 " is true or " f2 " is true or both are true,• " ¬ f " is true if and only if " f " is not true,
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Example• student(rollNo, name, degree, year, sex, deptNo, advisor)• department(deptId, name, hod, phone)Question:Obtain the rollNo, name of all girl students in the Maths Dept (deptNo= 2)
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• Answer• {s.rollNo,s.name| student(s)^ s.sex=‘F’^ s.deptNo=2}• Here student(s) is true whenever value of s is a tuple in the student relation,• false otherwise
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Example Relational Schemes• student(rollNo, name, degree, year, sex, deptNo, advisor)• department (deptId, name, hod, phone)• course (courseId, cname, credits, deptNo)• enrollment (rollNo, courseId, sem, year, grade)
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Example Queries• 1)Determine the departments that do not have any girl students .
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Safety of Expressions• It is possible to write tuple calculus expressions that generate infinite relations.• For example, expression { t | ¬ t ∈ r } results in an infinite relation if the domain of any attribute of relation r is infinite.
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To solve this problem• We restrict the set of allowable expressions to safe expressions.• What is safe expression?
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Definition of Safe Expression• An expression { t | P(t) } in the tuple relational calculus is safe if every component of t appears in one of the relations, tuples, or constants that appear in tuple relational formula P.
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Example• { t | ¬ ( t ∈loan) } is NOT safe. It is possible to have a tuple t not in loan that contains values that do not appear in loan.• All examples in the previous expressions are safe.
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Limitations of TRC• It can not express queries involving aggregations,closure.• It can not express non-query operations like insert,delete,update.• because those involve modeling the changes in the state of a database.
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Summary• The relational calculus provides an• alternate way to express queries• • A formal model based on logical formulae and set theory • Equivalence with algebra means can use• either or both – but only one for formal• proofs
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