Chapter 3
Tuple and Domain Relational
Calculus
Tuple Relational Calculus
What is Tuple Relational
Calculus?

It is a non procedural query language: Describes
the desired information without giving a specific
procedure for obtaining that information.
 A query in tuple relational calculus is expresses as:
{t | P(t)}
This represents a set of all tuples t such that
predicate P is true for t.
Sample Queries
Finding the branch – name, loan – number,
and amount for loans of over $ 1200
{t | t € loan ٨ t[amount] > 1200}
 In English this query would mean: The set
of tuples t where t belongs to the loan
relation and the loan amount for each t is
greater than $ 1200.

Sample queries(cont….)

Finding the loan – number for each loan of an
amount greater than $1200.
{t | З s € loan (t[loan – number] = s[loan –
number] ٨ s[amount > 1200)}
In English we would read the preceding statement
as “ The set of all tuples t such that there exists a
tuple s in relation loan for which the values of t
and s for the loan – number attribute are equal,
and the value of s for the amount attribute is
greater than $1200.”
Sample Queries(cont….)

The expression:
{t | -] r € customer (r[customer – name] =
t[customer – name]) ٨ (Џ u Є branch (u[branch –
city] = Brooklyn” => Э s Є depositor (t[customer
– name] = s[customer – name] ٨ З w Є account
(w[account – number- = s[account – number] ٨
w[branch – name] = u[branch – name]))))}
In English, would mean “The set of all customers( i.e
customer name tuples t) such that for all tuples u in the
branch relation, if the value of u on attribute branch – city
is Brooklyn, then the customer has an account at the
branch whose name appears in the branch name attribute of
u”
Formal Definition

As shown earlier, a tuple relational calculus expression is
of the form
{t | P(t)}
Where P is a formula.
 A formula may contain several tuple variables.
 A tuple variable is said to be a free variable unless it is
quantified by a З(there exists) or Џ(for all).
 Tuple variables that are quantified by З or Џ are called
bound variables.
 For ex, in the expression:
t Є loan ٨ З s Є customer(t[branch – name] =
s[branch – name])
t is a free variable and s is a bound variable.
Formal definition(cont….)


1.
2.
3.
A formula in relational calculus is made up of
atoms.
An atom has one of the following forms.
S Є r, where s is a tuple variable and r is a relation.
S[x] © u[y], where s and u are tuple variables, x is an
attribute on which s is defined, y is an attribute on which
u is defined, and © is the comparison operator. It is
required that x and y have domains that can be compared
using ©.
S[x] © c, where s is a tuple variable, x is an attribute on
which s is defined, © is a comparison operator, and c is a
constant in the domain of x.
Building a formula
We build up a formula from atoms by using the
following rules.
 An atom is a formula.
 If P1 is a formula, then so are ¬P1 and (P1).
 If P1 and P2 are formulae, then so are P1 ٧ P2, P1
٨ P2, and P1 => P2.
 If P1(s) is a formula containing a free tuple
variable s, and r is a relation, then
Э s Є r(P1(s)) and Џ s Є (P1(s))
are also formulae
Safety of Expressions – The
concept of domain

A tuple relational calculus may generate an infinite
relation. For ex:
{t | ¬(t Є loan)}
There are infinitely many tuples that are not in loan.
 To address this issue, we use the concept of domain of
tuple relational formula, P.
The domain of P, denoted dom(P) , is a set of all value
referenced by P.These include values mentioned in P itself,
as well as the values that appear in a tuple of a relation
mentioned in P.
For ex: dom(t Є loan ٨ t[amount] > 1200) is the set
containing 1200 as well as set of all values appearing in
loan.
Domain Relational Calculus
This is the second form of relational calculus
 Uses domain variables that take on values
from an attribute domain, rather than values
for an entire tuple.
 Closely related to tuple relational calculus.
Formal Definition

A general expression in Domain relational
calculus is of the form
{<x1, x2,…., xn> | P(x1,x2,….,xn)}
where x1, x2,….,xn represent domain
variables. P represents a formula composed
of atoms, as was the case in tuple relational
calculus.
An atom in Domain Relational
Calculus
An atom in domain relational calculus has one of
the following forms:
 <x1,x2,….,xn> Є r, where r is a relation on n attributes

and x1, x2, ….,xn are domain values or domain
constraints.
X © y, where x and y are domain values, and © is a
comparison operator (<, <=,=,>,>=). We require that
attributes x and y have domains that can be compared by
©.

X © c, where x is a domain variable, © is a comparison
operator, and c is a constant in the domain of the attribute
for which x is a domain variable.
Example Queries

Find the loan number, branch name, and amount
for loans over $1200.
{<l,b,a> | <l,b,a> Є loan ٨ a > 1200}

Find all loan numbers for loans with an amount > 1200:
{<l> | Э b,a (<l,b,a> Є loan ٨ a > 1200)}

When we write Э b in domain calculus, b refers not
to a tuple, but rather to a domain value.
 The domain of b is unconstrained(unlike as in
relational tuple calculus) until the sub formula
<l,b,a> Є loan constraints b to branch names that appear
in the loan relation.
Example queries(cont….)

Find the names of all customers who have an
account at all branches located in Brooklyn:
{<c> | Э n(<c,n> Є customer) ٨ Џ x,y,z (<x,y,z>
Є branch ٨ y = “Brooklyn” => Э a,b(<a,x,b> Є
account ٨ <c,a> Є depositor))}
In English, we interpret this expression as “The set of
all(customer name) tuples c such that, for all (branch –
name, branch – city, assets) tuples, x,y,z, if the branch
city is Brooklyn, then the following is true:
1.
There exists a tuple in the relation account with account
number a and branch name x.
2. There exists a tuple in the relation depositor with
customer c and account number a.”
Safety of Expressions

As noted in tuple relational calculus
expressions in the domain relational
calculus may also generate an infinite
relation. For ex:
{{<l,b,a> | ¬ (<l,b,a> Є loan)}
is unsafe, because it allows values in the result
that are not in the domain of the expression.
Safety(cont….)
An expression:
{<x1,x2,….,xn> | P(x1,x2,….,xn)}
is safe if all of the following hold.
 All values that appear in tuples of the expression are values
from dom(P).
 For every “there exists” subformula of the form Э
x(P1(x)), the sub formula is true only if and only if there is
a value x in dom(P1) such that P1(x) is true.
 For every “for all” subformula of the form Џx(P1(x)), the
subformula is true if and only if P1(x) is true for all values
of x from dom(P1).