5 the Relational Calculus

The Relational Calculus

(Based on Chapter 9 in Fundamentals of Database Systems

by Elmasri and Navathe, Ed. 3)

The Relational Calculus 2

Contents Introduction to Relational Calculus Tuple Relational Calculus

Tuple Variables and Range Relations Formal Specification of Tuple Relational Calculus Queries Using the Existential Quantifier Transforming Universal and Existential Quantifiers Universal Quantifiers and Safe Expressions

Domain Relational Calculus


Introduction to Relational Calculus A formal language based on first-

order predicate calculus. Many commercial relational

languages based on some aspects of relational calculus, including SQL.

QUEL, QBE(Chapter 9) closer to relational calculus than SQL


Difference from relational algebra: One declarative calculus expression

specifies a retrieval query. A sequence of operations is used in

relational algebra. Relational algebra more procedural. Relational calculus more declarative

(less procedural). Expressive power of the two languages

is identical.


Relational Completeness: A relational query language L is

relationally complete if we can express in L any query that can be expressed in the relational calculus (or algebra)

Most relational query languages are relationally complete.

More expressive power is provided by operations such as aggregate functions, grouping, and ordering.


Tuple Variable and Range Relations The tuple relational calculus is

based on specifying a number of tuple variables.

A tuple variable ranges over the tuples of a particular relation.

Such relation is called a range relation.


A Form of Tuple Relational Calculus Query A simple tuple relational calculus

query is of the from{t | COND(t)} t is a tuple variable COND(t) is a conditional expression

involving t. The result of such a query is set of

all tuples t that satisfies COND(t).


Example: Find all employees whose salary is

above $50000: { t | EMPLOYEE(t) and t.SALARY > 50000

} EMPLOYEE(t) specifies the range relation

EMPLOYEE for the tuple variable t Each tuple t satisfying t.SALARY > 50000

is retrieved Retrieves the whole tuple t


Example: To retrieve only some attributes of

t: { t.FNAME, t.LNAME | EMPLOYEE(t)

and t.SALARY>50000 } Similar to the SQL query:

SELECT T.FNAME, T.LNAMEFROM EMPLOYEE TWHERE T.SALARY > 50000


A Tuple Calculus Expression Need to specific the following

information For each tuple variable t, the range

relation R of t. This value is specified by a condition of the form R(t).

A condition to select particular combinations of tuples.

A set of attributes to be retrieved, the requested attribute.


A general expression of tuple relational calculus is of the form{(t1.A1, t2.A2,…tn.An | COND(t1,t2,…,tn,tn+1,tn+2,

…,tn+m)} Where t1.A1, t2.A2,…tn.An are tuple variable

s, each Ai is an attribute of the relation on w

hich ti ranges, and COND is a condition or formula

Expressions of Tuple Relational Calculus


Formulas of Tuple Relational Calculus A formula is made up of predicate calculus

atoms, which can be one of the following: An atom of the form R(ti), where R is a relation

name and ti is a tuple variable. An atom of the form ti.A op tj.B, where op is on

e of the comparison operators in the set {=,>,,<,,}, ti and tj are tuple variables, A is an attribute of the relation on which ti ranges, and B is an attribute of the relation on which tj ranges.


Formulas of Tuple Relational Calculus A formula is made up of predicate calculus

atoms, which can be one of the following: An atom of the form ti.A op c or c op tj.B, where op is one of the comparison operators in the set {=,>,,<,,}, ti and tj are tuple variables, A is an attribute of the relation on which ti ranges, and B is an attribute of the relation on which tj ranges, and c is a constant value.


Truth Value Each of the preceding atoms

evaluates to ether TRUE or FALSE for a specific combination of tuples.

This is called the truth value of an atom.

In general, a tuple variable ranges over all possible tuples “in the universe.”


Formula Atoms connected via and, or and not.

Every atom is a formula If F1 and F2 are formulas, so are :

(F1 and F2) (F1 or F2) not(F1) not(F2)


The Existential and Universal Quantifiers Universal quantifier()

Read for all Existential quantifiers()

Read their exists


Free and Bound Tuple Variables

Informally, A tuple variable t is bound if it is quantified, mean that it appears in an (t) or (t) clause; otherwise, it is free.



An occurrence of a tuple variable in a formula F that is an atom is free in F.

An occurrence of a tuple variable t is free or bound in a formula made up of logical connectives – (F1 and F2), (F1 or F2), not(F1), and not(F2) – depending on whether it is free or bound in F1 or F2 (if it occur in either).



All free occurrences of a tuple variable t in F are bound in a formula F’ of the form F’ = (t)(F) or F’ = (t)(F) .

The tuple variable is bound to the quantifier specified in F’.


For example F1 : d.DNAME=‘Research’ F2 : (t)(d.DNUMBER=t.DNO) F3 : (t)(d.MGRSSN=‘333445555’)

d is free in F1, F2 and F3 t is bound to the quantifier in F2 t is bound to the quantifier in F3


Formulas Every atom is a formula. If F1 and F2 are formulas, then so

are (F1 and F2), (F1 or F2), not(F1), and not(F2).

If F is a formula, then so is (t)(F), where t is a tuple variable.

If F is a formula, then so is (t)(F), where t is a tuple variable.


Truth Value for Existential Quantifiers The formula (t)(F) is TRUE if the form

ula F evaluates to TRUE for some (at least one) tuple assigned to free occurrences of t in F; otherwise (t)(F) is FALSE.


Existential Quantifiers called existential quantifier

because (t)(F) is TRUE if “there exists” some tuple t that make F TURE.


Truth Value for Universal Quantifiers The formula (t)(F) is TRUE if the

formula F evaluates to TRUE for every tuple (in the universe) assigned to free occurrences of t in F; otherwise (t)(F) is FALSE.


Universal Quantifiers called universal quantifier

because every tuple in “the universe of” tuples must make F TRUE if (t)(F) is to be TRUE.


Example Queries Using the

Existential Quantifiers (1)

Query 1 : Retrieve the name and address of all

employees who work for the ‘Research’ department.

Query 1 : Retrieve the name and address of all

employees who work for the ‘Research’ department.




Q1: { t.FNAME, t.LNAME, t.ADDRESS |

EMPLOYEE(t) and( d)(DEPARTMENT(d) and d.NAME=‘Research’ and d.DNUMBER=t.DNO) }

Q1: { t.FNAME, t.LNAME, t.ADDRESS |

EMPLOYEE(t) and( d)(DEPARTMENT(d) and d.NAME=‘Research’ and d.DNUMBER=t.DNO) }



Existential Quantifiers (1) The only free tuple variables in a

relational calculus expression should be those that appear to the left of the bar (|).

Each free variable is bound successively to each tuple that satisfies the condition to the right of the bar (|).

The bar (|) read as “such that”



Existential Quantifiers (1) EMPLOYEE(t), DEPARTMENT(d) specify

range relations for t. The condition d.NAME=“Research” is

selection condition. (corresponds to SELECT in relational algebra)

The condition d.DNUMBER=t.DNO is a join condition. (serves a similar purpose to EQUIJOIN in

relational algebra)




Query 2 For every project located in ‘Stafford’,

list the project number, the controlling department number, and the department manage’s last name, address, and birthdate.

Query 2 For every project located in ‘Stafford’,

list the project number, the controlling department number, and the department manage’s last name, address, and birthdate.




Q2: {p.PNUMBER, p. DNUM, m.LNAME,

m.BDATE, m.ADDRESS |PROJECT(p) and EMPLOYEE(m)

and PLOCATION='Stafford‘ and (d)

(DEPARTMENT(d) and p.DNUM=d.DNUMBER

and d.MGRSSN=m.SSN) }

Q2: {p.PNUMBER, p. DNUM, m.LNAME,

m.BDATE, m.ADDRESS |PROJECT(p) and EMPLOYEE(m)

and PLOCATION='Stafford‘ and (d)

(DEPARTMENT(d) and p.DNUM=d.DNUMBER

and d.MGRSSN=m.SSN) }




Query 8: For each employee, retrieve the

employee's name, and the name of his or her immediate supervisor.

Query 8: For each employee, retrieve the

employee's name, and the name of his or her immediate supervisor.




Q8: {e.FNAME, e.LNAME, s.FNAME,s.LNAME |

EMPLOYEE(e) and EMPLOYEE(s) and e.SUPERSSN=s.SSN}

Q8: {e.FNAME, e.LNAME, s.FNAME,s.LNAME |

EMPLOYEE(e) and EMPLOYEE(s) and e.SUPERSSN=s.SSN}



Existential Quantifiers (3’)

Query 3’ Find the names of employees who

work on some projects controlled by department number 5.

Query 3’ Find the names of employees who

work on some projects controlled by department number 5.



Existential Quantifiers (3’)

Q3’: {e.FNAME, e.LNAME |

EMPLOYEE(e) and((x)(w)(PROJECT(x)

and WORKS_ON(w) and x.DNUM=5 and e.SSN=w.ESSN

and p.PNO=x.PNUMBER))}

Q3’: {e.FNAME, e.LNAME |

EMPLOYEE(e) and((x)(w)(PROJECT(x)

and WORKS_ON(w) and x.DNUM=5 and e.SSN=w.ESSN

and p.PNO=x.PNUMBER))}




Query 4 Make a list of project numbers for

projects that involve an employee whose last name is ‘Smith’, either as a worker or a manager of the department that controls the project.

Query 4 Make a list of project numbers for

projects that involve an employee whose last name is ‘Smith’, either as a worker or a manager of the department that controls the project.



Existential Quantifiers (4) Q4:

{p.PNUMBER | PROJECT(p) and (((e)( w)(EMPLOYEE(e) and WORKS_ON(w)

and p.PNUMBER=w.PNO and e.LNAME='Smith‘

and w.ESSN=e.SSN ))or

(( m)( d)(EMPLOYEE(m) and DEPARTMENT(d) and p.DNUM=d.DNUMBER and d.MGRSSN=m.SSN and m.LNAME='Smith')))}

Q4: {p.PNUMBER | PRO

JECT(p) and (((e)( w)(EMPLOYEE(e) and WORKS_ON(w)

and p.PNUMBER=w.PNO and e.LNAME='Smith‘

and w.ESSN=e.SSN ))or

(( m)( d)(EMPLOYEE(m) and DEPARTMENT(d) and p.DNUM=d.DNUMBER and d.MGRSSN=m.SSN and m.LNAME='Smith')))}



Existential Quantifiers (4) In general, UNION in relational

algebra corresponds to an or connective in relational calculus.

INTERSECTION corresponds to an and connective.


Transforming Universal and Existential Quantifiers The not connective can be used to

transform universal and existential quantifiers to equivalent formulas.


Well-known transformations from mathematical logic.

( x) (P(x)) (not x) (not(P(x))) ( x) (P(x)) not ( x) (not(P(x)))

( x) (P(x)) (not x) (not(P(x))) ( x) (P(x)) not ( x) (not(P(x)))


Well-known transformations from mathematical logic. The following is also true, where

stands for implies: ( x) (P(x)) ( x) (P(x)) (not x) (P(x)) not ( x) (P(x))

The following is not true: not( x) (P(x)) (not x) (P(x))


Example Queries Using Universal Quantifiers (3)

Query 3 Find the names of employees who

work on all the projects controlled by department number 5.

Query 3 Find the names of employees who

work on all the projects controlled by department number 5.


Example Queries Using Universal Quantifiers (3) Q3:

{e.FNAME, e.LNAME | EMPLOYEE(e) and((x)(not(PROJECT(x)) or

(not(x.DNUM=5) or ((w)(WORKS_ON(w)

and e.SSN=w.ESSNand p.PNO=x.PNUMBER)

)) )

}

Q3: {e.FNAME, e.LNAME | EMPLOYEE(e) and

((x)(not(PROJECT(x)) or (not(x.DNUM=5) or ((w)(WORKS_ON(w)

and e.SSN=w.ESSNand p.PNO=x.PNUMBER)

)) )

}


Example Queries Using Universal Quantifiers (3)

Q3 : For every tuple x in the project relation

with x.DUM = 5, there must exist a tuple w in WORK_ON such that w.ESSN=e.SSN and w.PNO=x.PNUMBER.

Q3 : For every tuple x in the project relation

with x.DUM = 5, there must exist a tuple w in WORK_ON such that w.ESSN=e.SSN and w.PNO=x.PNUMBER.


Basic Components of Q3


F’} F’= ((x)(not(PROJECT(x)) or F1)) F1 = not(x.DNUM=5) or F2

F2 = ((w)(WORKS_ON(w) and e.SSN=w.ESSNand p.PNO=x.PNUMBER))


F’} F’= ((x)(not(PROJECT(x)) or F1)) F1 = not(x.DNUM=5) or F2

F2 = ((w)(WORKS_ON(w) and e.SSN=w.ESSNand p.PNO=x.PNUMBER))


Basic Components of Q3 Must exclude all tuples not of

interest from the universal quantification by making the condition TRUE for all such tuples.

Universally quantified variable x must evaluate to TRUE for every possible tuple in the universe.


Basic Components of Q3 In F’, not(PROJECT(x)) makes x TRUE for al

l tuples not in the relation of interest “PROJECT”. F’= ((x)(not(PROJECT(x)) or F1))

In F1, not(x.DNUM=5) makes x TRUE for those PROJECT tuples we are not interested in “whose DNUM is not 5” F1 = not(x.DNUM=5) or F2


Basic Components of Q3 F2 specifies the condition that must

hold on all remaining tuples “ all PROJECT tuples controlled by department 5” F2 = ((w)(WORKS_ON(w)

and e.SSN=w.ESSNand p.PNO=x.PNUMBER))


Safe Expressions A safe expression in relational

calculus is one that is guaranteed to yield a finite number of tuples as its result; otherwise, the expression is called unsafe.

Unsafe expression may yield infinite number of tuples, and the tuples may be different types.


Safe Expressions For example:

{t | not(EMPLOYEE(t))} is unsafe.

Yields all non-EMPLOYEE tuples in the universe.


Universal Quantifiers and Safe Expressions One must be careful when

specifying universal quantification. Judicious to follow rules to ensure

expression make sense. Otherwise, unsafe expressions

may result.


Safe Expressions We discuss rules for safe expressions

using the universal quantifier by looking at query Q3.

Following the rules for Q3 discussed above guarantees safe expressions when using universal quantifiers.

Using transformations from universal to existential quantifiers, can rephrase Q3 as Q3A


Example of Transforming Universal and Existential Quantifiers

Q3A: {e.FNAME, e.LNAME |

EMPLOYEE(e) and (not ( x)(PROJECT(x) and (x.DNUM=5)

and (not ( w)(WORKS_ON(w) and e.SSN=w.ESSN

and p.PNO=x.PNUMBER)))) }

Q3A: {e.FNAME, e.LNAME |

EMPLOYEE(e) and (not ( x)(PROJECT(x) and (x.DNUM=5)

and (not ( w)(WORKS_ON(w) and e.SSN=w.ESSN

and p.PNO=x.PNUMBER)))) }


Additional Examples (6)

Query 6: Retrieve the names of employees who

have no dependents.

Query 6: Retrieve the names of employees who

have no dependents.



Existential Quantifiers (6) Q6:

{e.FNAME, e.LNAME |EMPLOYEE(e) and(not(d)(DEPENDENT(d)

and e.SSN=d.ESSN))}



Universal Quantifiers (6)

Q6: {e.FNAME, e.LNAME |

EMPLOYEE(e) and((d)(not (DEPENDENT(d))

or not (e.SSN=d.ESSN))) }


EMPLOYEE(e) and((d)(not (DEPENDENT(d))

or not (e.SSN=d.ESSN))) }


Additional Examples (7)

Query 7: List the names of managers who have

at least one dependent.

Query 7: List the names of managers who have

at least one dependent.





EMPLOYEE(e) and((d)(p)

(DEPARTMENT(d) and (DEPENDENT(p)

and e.SSN=d.MGRSSN and p.ESSN=e.SSN)) }


EMPLOYEE(e) and((d)(p)

(DEPARTMENT(d) and (DEPENDENT(p)

and e.SSN=d.MGRSSN and p.ESSN=e.SSN)) }


Quantifiers in SQL The EXISTS function in SQL is

similar to the existential quantifier of the relational calculus.


Quantifiers in SQL

SELECT …FROM …WHERE EXISTS (SELECT *

FROM R AS XWHERE P(X))

SELECT …FROM …WHERE EXISTS (SELECT *

FROM R AS XWHERE P(X))


Quantifiers in SQL SQL does not include a universal

quantifier. Use of a negated existential

quantifier not (x) by writing NOT EXISTS is how SQL supports universal quantification.

5 the Relational Calculus

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