3.1 tuple relational_calculus

Prof P Sreenivasa Kumar,
Department of CS&E, IITM.
1
Tuple Relational Calculus ( TRC )
Introduction
Procedural Query language
query specification involves giving a step by step process of
obtaining the query result
e.g., relational algebra
usage calls for detailed knowledge of the operators involved
difficult for the use of non-experts
Declarative Query language
query specification involves giving the logical conditions the
results are required to satisfy
easy for the use of non-experts

2
TRC – a declarative query language
Tuple variable – associated with a relation
( called the range relation )
• takes tuples from the range relation as its values
• t: tuple variable over relation r with scheme R(A,B,C )
t.A stands for value of column A etc
TRC Query – basic form:
{ t1.Ai1
, t2.Ai2
,…tm.Aim
| θ }
predicate calculus expression
involving tuple variables
t1, t2,…, tm, tm+1,…,ts
- specifies the condition to be satisfied

3
An example TRC query
student (rollNo, name, degree, year, sex, deptNo, advisor )
department (deptId, name, hod, phone )
Obtain the rollNo, name of all girl students
in the Maths Dept (deptId = 2)
{s.rollNo,s.name| student(s)^ s.sex=‘F’^ s.deptNo=2}
attributes
required in
the result
This predicate is true whenever
value of s is a tuple from the
student relation, false otherwise
In general, if t is a tuple variable with range
relation r, r( t ) is taken as a predicate which
is true if and only if the value of t is a tuple in r

4
Atomic expressions are the following:
1. r ( t ) -- true if t is a tuple in the relation instance r
2. t1. Ai <compOp> t2 .Aj compOp is one of {<, ≤ , >, ≥, =, ≠ }
3. t.Ai <compOp> c c is a constant of appropriate type
Composite expressions:
1. Any atomic expression
2. F1 ∧ F2 ,, F1 ∨ F2 , ¬ F1 where F1 and F2 are expressions
3. (∀t) (F), (∃t) (F) where F is an expression
and t is a tuple variable
Free Variables
Bound Variables – quantified variables
General form of the condition in TRC queries

5
All possible tuple assignments to the free variables in the query are
considered.
For any specific assignment,
if the expression to the right of the vertical bar evaluates to true,
that combination of tuple values
would be used to produce a tuple in the result relation.
While producing the result tuple, the values of the attributes for the
corresponding tuple variables as specified on the left side of the
vertical bar would be used.
Note: The only free variables are the ones that appear to the left
of the vertical bar
Interpretation of the query in TRC

6
Obtain the rollNo, name of all girl students in the
Maths Dept
{s.rollNo,s.name | student(s) ^ s.sex=‘F’ ^
(∃ d)(department(d) ^ d.name=‘Maths’
^ d.deptId = s.deptNo)}
s: free tuple variable d: existentially bound tuple variable
Existentially or universally quantified tuple variables can be used
on the RHS of the vertical bar to specify query conditions
Attributes of free (or unbound ) tuple variables can be used on LHS
of vertical bar to specify attributes required in the results
Example TRC queries

7
Example Relational Scheme
student (rollNo, name, degree, year, sex, deptNo, advisor)
department (deptId, name, hod, phone)
professor (empId, name, sex, startYear, deptNo, phone)
course (courseId, cname, credits, deptNo)
enrollment (rollNo, courseId, sem, year, grade)
teaching (empId, courseId, sem, year, classRoom)
preRequisite (preReqCourse, courseID)
Q2
Q3
Q4
Q5

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Example queries in TRC (1/5)
1)Determine the departments that do not have
any girl students
student (rollNo, name, degree, year, sex, deptNo, advisor)
department (deptId, name, hod, phone)
{d.name|department(d) ^
¬(∃ s)(student(s) ^
s.sex =‘F’ ^ s.deptNo = d.deptId)

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2)Obtain the names of courses enrolled by student
named Mahesh
{c.name | course(c) ^
(∃s) (∃e) ( student(s) ^ enrollment(e)
^ s.name = “Mahesh”
^ s.rollNo = e.rollNo
^ c.courseId = e.courseId }
Examples queries in TRC (2/5) Schema

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3)Get the names of students who have scored ‘S’ in all
subjects they have enrolled. Assume that every
student is enrolled in at least one course.
{s.name | student(s) ^
(∀e)(( enrollment(e) ^
e.rollNo = s.rollNo) → e.grade =‘S’)}
person P with all S grades:
for enrollment tuples not having her roll number, LHS is false
for enrollment tuples having her roll number, LHS is true, RHS also true
so the implication is true for all e tuples
person Q with some non-S grades:
for enrollment tuples not having her roll number, LHS is false
for enrollment tuples having her roll number, LHS is true, but RHS is false for
at least one tuple.
So the implication is not true for at least one tuple.

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4) Get the names of students who have taken at least
one course taught by their advisor
{s.name | student(s) ^
(∃e)(∃t)(enrollment(e) ^ teaching(t) ^
e.courseId = t.courseId ^
e.rollNo = s.rollNo ^
t.empId = s.advisor}
5) Display the departments whose HODs are teaching
at least one course in the current semester
{d.name | department(d) ^(∃t)(teaching(t) ^
t.empid = d.hod
^ t.sem = ‘odd’ ^ t.year = ‘2008’)}

12
6)Determine the students who are enrolled for every
course taught by Prof Ramanujam. Assume that Prof
Ramanujam teaches at least one course.
1. {s.rollNo | student (s) ^
2. (∀c)(course (c) ^
3. ((∃t),(∃p)( teaching(t) ^ professor(p) ^
4. t.courseId = c.courseId ^
5. p.name = “Ramanujam” ^
6. p.empId = t.empId )) →
7. (∃e) (enrollment(e) ^
8. e.courseId = c.courseId ^
9. e.rollNo = s.rollNo)
10. )
11. }

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What is the result of the query:
{s.rollNo | ¬ student(s)} ?
Infinite answers !!
Unsafe TRC expression :
Any expression whose result uses “constants / values” that do not
appear in the instances of any of the database relations.
Unsafe expressions are to be avoided while specifying TRC queries.
Problem with unrestricted use of Negation

14
It can be shown that
both Tuple Relational Calculus and Relational Algebra
have the same expressive power
A query can be formulated in (safe) TRC
if and only if it can be formulated in RA
Both can not be used to formulate queries involving
transitive closure
-- find all direct or indirect pre-requisites of a course
-- find all subordinates of a specific employee etc.
Expressive power of TRC and Relational Algebra

3.1 tuple relational_calculus

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