The Relational Model
- theoretical foundation
Database Group, Georgia Tech
© Leo Mark
Relational Model 1
The Relational Model
• data structures
• constraints
• operations
– algebra (ISBL)
– tuple calculus (QUEL, SQL)
– domain calculus (QBE)
• views
Database Group, Georgia Tech
© Leo Mark
Relational Model 2
Data Structures
• let D1, D2 , D3 , ..., Dn be sets (not
necessarily distinct) of atomic values
• relation, R, defined over D1, D2 , D3 , ..., Dn
is a subset of the set of ordered ntuples {<d1, d2, d3, ..., dn | di  Di, i=1, ...,n};
D1, D2 , D3 , ..., Dn are called domains
• the number, n, is the degree of the
relation (unary, binary, ternary, n-ary).
• the number of tuples, |R|, in R is called
the cardinality of R
• if D1, D2 , D3 , ..., Dn are finite then there
are 2|D1||D2| ... |Dn| possible relation
states
Database Group, Georgia Tech
© Leo Mark
Relational Model 3
Data Structures
• an attribute name refers to a
position in a tuple by name rather
than position
• an attribute name indicate the role
of a domain in a relation
• attribute names must be unique
within relations
• by using attribute names we can
forget the ordering of field values in
tuples
• a relation definition includes the
following R( A1:D1, A2 :D2 , ..., An :Dn)
Database Group, Georgia Tech
© Leo Mark
Relational Model 4
Constraints
•
•
•
•
keys
primary keys
entity integrity
referential integrity
FLT-SCHEDULE
CUSTOMER
FLT#
CUST# CUST-NAME
p
p
RESERVATION
FLT#
DATE CUST#
Database Group, Georgia Tech
© Leo Mark
Relational Model 5
AIRPORT
airportcode
name city state
FLT-SCHEDULE
flt# airline dtime from-airportcode atime to-airportcode miles price
FLT-WEEKDAY
flt# weekday
FLT-INSTANCE
flt# date plane# #avail-seats
AIRPLANE
plane# plane-type total-#seats
CUSTOMER
cust# first middle last phone# street city state zip
RESERVATION
flt# date cust# seat# check-in-status ticket#
Database Group, Georgia Tech
© Leo Mark
Relational Model 6
Operations
• classes of relational DMLs:
– relational algebra (ISBL)
– tuple calculus (QUEL, SQL)
– domain calculus (QBE)
• a relational DML with the same
“retrieval power” as the relational
algebra is said to be relationally
complete
• all relational DMLs have syntax for:
– change (insert, delete, update)
– queries (retrieval)
Database Group, Georgia Tech
© Leo Mark
Relational Model 7
Operations
- insert, delete, update
FLT-SCHEDULE
flt# airline dtime from-airportcode atime to-airportcode miles price
• constructs for insertion are very
primitive:
INSERT INTO FLT-SCHEDULE
VALUES (“DL212”, “DELTA”, 11-15-00, “ATL”,
13-05-00, ”CHI”, 650, 00351.00);
INSERT INTO FLT-SCHEDULE
VALUES (FLT#:“DL212”, AIRLINE:“DELTA”);
Database Group, Georgia Tech
© Leo Mark
Relational Model 8
Operations
- insert, delete, update
FLT-SCHEDULE
flt# airline dtime from-airportcode atime to-airportcode miles price
FLT-WEEKDAY
flt# weekday
FLT-INSTANCE
flt# date plane# #avail-seats
• “insert into FLT-INSTANCE all
flights scheduled for Thursday,
9/10/98”
INSERT INTO FLT-INSTANCE(flt#, date)
(SELECT S.flt#, 1998-09-10
FROM FLT-SCHEDULE S, FLT-WEEKDAY D
WHERE S.flt#=D.flt# AND weekday=“TH”);
• interesting only because it involves
a query
Database Group, Georgia Tech
© Leo Mark
Relational Model 9
Operations
- insert, delete, update
FLT-WEEKDAY
flt# weekday
• constructs for deletion are very
primitive:
• “delete flights scheduled for
Thursdays”
DELETE
FROM FLT-WEEKDAY
WHERE weekday=“TH”;
• interesting only because it involves a
query
Database Group, Georgia Tech
© Leo Mark
Relational Model 10
Operations
- insert, delete, update
FLT-WEEKDAY
flt# weekday
• constructs for update are very
primitive:
• “update flights scheduled for
Thursdays to Fridays”
UPDATE FLT-WEEKDAY
SET weekday=“FR”
WHERE weekday=“TH”;
• interesting only because it involves
a query
Database Group, Georgia Tech
© Leo Mark
Relational Model 11
Relational Algebra
• the Relational Algebra is procedural;
you tell it how to construct the result
• it consists of a set of operators
which, when applied to relations,
yield relations (closed algebra)
R S
union
R S
intersection
R\S
set difference
R S
Cartesian product
A1, A2, ..., An (R)
projection
expression (R)
selection
R S
natural join
R S
theta-join
RS
divideby
[A1 B1,.., An Bn] rename
Database Group, Georgia Tech
© Leo Mark
Relational Model 12
Selection
FLT-WEEKDAY
flt# weekday
• “find (flt#, weekday) for all flights
scheduled for Mondays”
weekday=MO (FLT-WEEKDAY)
• the expression in expression (R) involves:
• operands: constants or attribute
names of R
• comparison operators: Š •° =
• logical operators: 
• nesting: ( )
Database Group, Georgia Tech
© Leo Mark
Relational Model 13
Projection
FLT-WEEKDAY
flt# weekday
• “find flt# for all flights scheduled for
Mondays
flt#(weekday=MO (FLT-WEEKDAY))
• the attributes in the attribute list
ofA1, A2, ..., An (R) must be attributes of
the operand R
Database Group, Georgia Tech
© Leo Mark
Relational Model 14
Union
FLT-WEEKDAY
flt# weekday
• “find the flt# for flights that are
schedule for either Mondays, or
Tuesdays, or both”
flt#(weekday=MO (FLT-WEEKDAY))
flt#(weekday=TU (FLT-WEEKDAY))
• the two operands must be "type
compatible"
Database Group, Georgia Tech
© Leo Mark
Relational Model 15
Intersection
FLT-WEEKDAY
flt# weekday
• “find the flt# for flights that are
schedule for both Mondays and
Tuesdays”
flt#(weekday=MO (FLT-WEEKDAY))
flt#(weekday=TU (FLT-WEEKDAY))
• the two operands must be "type
compatible"
Database Group, Georgia Tech
© Leo Mark
Relational Model 16
Set Difference
FLT-WEEKDAY
flt# weekday
• “find the flt# for flights that are
scheduled for Mondays, but not for
Tuesdays”
flt#(weekday=MO (FLT-WEEKDAY))
\ flt#(weekday=TU (FLT-WEEKDAY))
• the two operands must be "type
compatible"
• Note: R  S = R \ (R \ S)
Database Group, Georgia Tech
© Leo Mark
Relational Model 17
Cartesian Product
FLT-INSTANCE
flt# date plane# #avail-seats
CUSTOMER
cust# first middle last phone# street city state zip
RESERVATION
flt# date cust# seat# check-in-status ticket#
“make a list containing (flt#, date, cust#)
for DL212 on 9/10, 98 for all customers in
Roswell that are not booked on that flight”
(cust#(city=ROSWELL(CUSTOMER)) 
flt#,date (flt#=DL212  date=1998-09-10
(FLT-INSTANCE))) \ flt#,date ,cust#(RESERVATION)
Database Group, Georgia Tech
© Leo Mark
Relational Model 18
Natural Join
FLT-WEEKDAY
flt# weekday
FLT-INSTANCE
flt# date plane# #avail-seats
• “make a list with complete flight
instance information”
FLT-INSTANCE
FLT-WEEKDAY
• natural join joins relations on
attributes with the same names
• all joins can be expressed by a
combination of primitive operators:
FLT-INSTANCE.flt#, date, weekday, #avail-seats
(FLT-INSTANCE.flt#=FLT-WEEKDAY.flt#
(FLT-INSTANCE FLT-WEEKDAY))
Database Group, Georgia Tech
© Leo Mark
Relational Model 19
-join
FLT-SCHEDULE
flt# airline dtime from-airportcode atime to-airportcode miles price
FLT-INSTANCE
flt# date plane# #avail-seats
• “make a list of pairs of (FLT#1, FLT#2)
that form possible connections”
fl1, flt#(([flt# fl1, from-airportcode da1,dtime dt1, to-airportcode aa1,
atime at1, date d1]
(FLT-SCHEDULE
FLT-INSTANCE ))
d1=date  aa1=from-airportcode  at1< dtime
(FLT-SCHEDULE
FLT-INSTANCE))
• the-operators: Š •° =
Database Group, Georgia Tech
© Leo Mark
Relational Model 20
Divideby
FLT-INSTANCE
flt# date plane# #avail-seats
RESERVATION
flt# date cust# seat# check-in-status ticket#
• “list the cust# of customers that
have reservations on all flight
instances”
flt#, date, cust# RESERVATION
flt#, date (FLT-INSTANCE)
Database Group, Georgia Tech
© Leo Mark
Relational Model 21
ISBL - an example algebra
R S
R S
R\S
A1, A2, ..., An (R)
expression (R)
R S
R S
R S
RS
[A1 B1,..., An Bn] (R)
R UNION S
R INTERSECT S
R MINUS S
R[A1, A2, ..., An]
R WHERE EXPRESSION
R JOIN S (no shared attributes)
R JOIN S (shared attributes)
via selection from 
R DIVIDEBY S
R[A1 B1,.., An Bn]
Database Group, Georgia Tech
© Leo Mark
Relational Model 22
Features of ISBL
• the Peterlee Relational Test Vehicle,
PRTV, has a query optimizer for ISBL
• Naming results: T = R JOIN S
• Lazy evaluation: T = N!R JOIN N!S
• LIST T
• 2-for-1 JOIN:
– Cartesian product if no shared attribute
names
– natural join if shared attribute names
• ISBL is relationally complete !
Database Group, Georgia Tech
© Leo Mark
Relational Model 23
ISBL - an example query
FLT-SCHEDULE
flt# airline dtime from-airportcode atime to-airportcode miles price
FLT-INSTANCE
flt# date plane# #avail-seats
• “make a list of pairs of (FLT#1, FLT#2)
that form possible connections”
LIST(((FLT-SCHEDULE JOIN FLT-INSTANCE )
[FLT# FL1, FROM-AIRPORTCODE DA1,DTIME DT1, TOAIRPORTCODE AA1, ATIME AT1, DATE D1]) JOIN
(FLT-SCHEDULE JOIN FLT-INSTANCE) WHERE
D1=DATE AA1=FROM-AIRPORTCODE  AT1<
DTIME)[FL1, FLT#]
Database Group, Georgia Tech
© Leo Mark
Relational Model 24
Relational Calculus
• the Relational Calculus is nonprocedural. It allows you to express
a result relation using a predicate on
tuple variables (tuple calculus):
{ t | P(t) }
or on domain variables (domain
calculus):
{ <x1, x2, ..., xn> | P(<x1, x2, ..., xn>) }
• you tell the system which result you
want, but not how to construct it
Database Group, Georgia Tech
© Leo Mark
Relational Model 25
Tuple Calculus
• query expression: { t | P(t) } where P
is a predicate built from atoms
• range expression: t  R denotes that t
is a member of R; so does R(t)
• attribute value: t.A denotes the value
of t on attribute A
• constant: c denotes a constant
• atoms: t R, r.A  s.B, or r.A  c
• comparison operators: Š •
<>°=
• predicate: an atom is a predicate; if P1
and P2 are predicates, so are ¬(P1 )
and (P1 ), P1  P2, P1  P2, and P1  P2
• if P(t) is a predicate, t is a free
variable in P, and R is a relation then
t R (P(t)) andt R (P(t)) are
predicates
Database Group, Georgia Tech
© Leo Mark
Relational Model 26
Tuple Calculus
CUSTOMER
cust# first middle last phone# street city state zip
• { r |(rCUSTOMER} is infinite, or
unsafe
• a tuple calculus expression { r | P(r) }
is safe if all values that appear in the
result are from Dom(P), which is the
set of values that appear in P itself or
in relations mentioned in P
Database Group, Georgia Tech
© Leo Mark
Relational Model 27
Selection
FLT-WEEKDAY
flt# weekday
• “find (FLT#, WEEKDAY) for all flights
scheduled for Mondays
{ t | FLT-WEEKDAY(t)  t.WEEKDAY=MO}
Database Group, Georgia Tech
© Leo Mark
Relational Model 28
Projection
FLT-WEEKDAY
flt# weekday
• “find FLT# for all flights scheduled for
Mondays
{ t.FLT# | FLT-WEEKDAY(t)  t.WEEKDAY =
MO}
Database Group, Georgia Tech
© Leo Mark
Relational Model 29
Union
FLT-WEEKDAY
flt# weekday
• “find the FLT# for flights that are
schedule for either Mondays, or
Tuesdays, or both”
{ t.FLT# | FLT-WEEKDAY(t) 
(t.WEEKDAY=MO t.WEEKDAY=TU)}
Database Group, Georgia Tech
© Leo Mark
Relational Model 30
Intersection
FLT-WEEKDAY
flt# weekday
• “find the FLT# for flights that are
schedule for both Mondays and
Tuesdays”
{ t.FLT# | FLT-WEEKDAY(t)
t.WEEKDAY=MO 
sFLT-WEEKDAY(s) t.FLT#=s.FLT# 
s.WEEKDAY=TU)}
Database Group, Georgia Tech
© Leo Mark
Relational Model 31
Set Difference
FLT-WEEKDAY
flt# weekday
• “find the FLT# for flights that are
scheduled for Mondays, but not for
Tuesdays”
{ t.FLT# | FLT-WEEKDAY(t) 
t.WEEKDAY=MO  ((s) (FLTWEEKDAY(s) t.FLT#=s.FLT# 
s.WEEKDAY=TU))}
Database Group, Georgia Tech
© Leo Mark
Relational Model 32
Cartesian Product
FLT-INSTANCE
flt# date plane# #avail-seats
CUSTOMER
cust# first middle last phone# street city state zip
RESERVATION
flt# date cust# seat# check-in-status ticket#
“make a list containing (FLT#, DATE, CUST#)
for DL212 on 9/10, 98 for all customers in
Roswell that are not booked on that flight”
{s.FLT#, s.DATE, t.CUST# | FLT-INSTANCE(s) 
CUSTOMER(t) t.CITY=ROSWELL
s.FLT#=DL212 s.DATE=1998-0910rFLT-INSTANCE(r)  r ° s 
r.FLT#=s.FLT# r.DATE=s.DATE
r.CUST#=t.CUST#)}
Database Group, Georgia Tech
© Leo Mark
Relational Model 33
Natural Join
FLT-WEEKDAY
flt# weekday
FLT-INSTANCE
flt# date plane# #avail-seats
• “make a list with complete flight
instance information”
{ s.FLT#, s.WEEKDAY, t.DATE, t.PLANE#, t.#AVAILSEATS | FLT-WEEKDAY(s)  FLT-INSTANCE(t) 
s.FLT#=t.FLT# }
Database Group, Georgia Tech
© Leo Mark
Relational Model 34
-join
FLT-SCHEDULE
flt# airline dtime from-airportcode atime to-airportcode miles price
FLT-INSTANCE
flt# date plane# #avail-seats
• “make a list of pairs of (FLT#1, FLT#2)
that form possible connections”
{ s. FLT#, t.FLT# | FLT-SCHEDULE(s)  FLTSCHEDULE(t)  ((u)(v) FLT-INSTANCE(u) 
FLT-INSTANCE(v)  u.FLT#=s.FLT#  v.FLT#=t.FLT#
 u.DATE=v.DATE  s.TO-AIRPORTCODE=t.FROMAIRPORTCODE  s.ATIME < t.DTIME) }
Database Group, Georgia Tech
© Leo Mark
Relational Model 35
Divideby
FLT-INSTANCE
flt# date plane# #avail-seats
RESERVATION
flt# date cust# seat# check-in-status ticket#
• “list the CUST# for customers that
have reservations on all flight
instances”
{ s.CUST# | RESERVATION(s)  (( t) FLTINSTANCE(t) ((r) RESERVATION(r) 
r.FLT#=t.FLT#  r.DATE=t.DATE 
r.CUST#=s.CUST#))}
Database Group, Georgia Tech
© Leo Mark
Relational Model 36
QUEL - an example tuple
calculus
FLT-SCHEDULE
flt# airline dtime from-airportcode atime to-airportcode miles price
FLT-INSTANCE
flt# date plane# #avail-seats
• “make a list of pairs of (FLT#1, FLT#2)
that form possible connections”
range s is FLT-SCHEDULE
range t is FLT-SCHEDULE
range u is FLT-INSTANCE
range v is FLT-INSTANCE
retrieve into CON( s.FLT#, t.FLT#)
where u.FLT#=s.FLT# and v.FLT#=t.FLT# and
u.DATE=v.DATE and s.TO-AIRPORTCODE=t.FROMAIRPORTCODE and s.ATIME < t.DTIME;
Database Group, Georgia Tech
© Leo Mark
Relational Model 37
QBE - Projection
FLT-WEEKDAY
FLT# WEEKDAY
P.
=MONDAY
• “find FLT# for all flights scheduled for
Mondays
Database Group, Georgia Tech
© Leo Mark
Relational Model 38
QBE - Union
FLT-WEEKDAY
FLT# WEEKDAY
P.
MONDAY
P.
TUESDAY
• “find the FLT# for flights that are
schedule for either Mondays, or
Tuesdays, or both”
Database Group, Georgia Tech
© Leo Mark
Relational Model 39
QBE - Intersection
FLT-WEEKDAY
FLT# WEEKDAY
P._SX MONDAY
_SX
TUESDAY
• “find the FLT# for flights that are
schedule for both Mondays and
Tuesdays”
Database Group, Georgia Tech
© Leo Mark
Relational Model 40
QBE - Set Difference
FLT-WEEKDAY
FLT# WEEKDAY
P._SX MONDAY

_SX
TUESDAY
• “find the FLT# for flights that are
scheduled for Mondays, but not for
Tuesdays”
Database Group, Georgia Tech
© Leo Mark
Relational Model 41
QBE - Cartesian Product
CUSTOMER
FLT-INSTANCE
#AVAIL-
CUST#
P._C
CUST-NAME
CITY
FLT#
DATE
ROSWELL
P._F
P._D
_F
DL212
SEATS
98-9-10
_D
RESERVATION

FLT#
_F
DATE
_D
CUST#
_C
“make a list containing (FLT#, DATE, CUST#)
for DL212 on 9/10, 98 for all customers in
Roswell that are not booked on that flight”
Database Group, Georgia Tech
© Leo Mark
Relational Model 42
QBE - Natural Join
FLT-WEEKDAY
FLT-INSTANCE
#AVAIL-
FLT# WEEKDAY
P._SX P.
FLT#
DATE
_SX
P.
SEATS
P.
• “make a list with complete flight
instance information”
Database Group, Georgia Tech
© Leo Mark
Relational Model 43
QBE-join
FLT-SCHEDULE
FLT# AIRLINE
FROMAIRPORT
DTIME
CODE
P._SX
TOAIRPORT
CODE
_A
ATIME
PRICE
_AT
FLT-SCHEDULE
FLT# AIRLINE
P._SY
FROMAIRPORT
DTIME
CODE
_A
TOAIRPORT
CODE
ATIME
PRICE
_DT
FLT-INSTANCE
CONDITION
_AT < _DT
FLT#
DATE
_SX
_D
_SY
_D
#SEATS
• “make a list of pairs of (FLT#1, FLT#2)
that form possible same day
connections”
Database Group, Georgia Tech
© Leo Mark
Relational Model 44
Views
• relational query languages are
closed, i.e., the result of a query is a
relation
• a view is a named result of a query
• a view is a snapshot relation
• views can be used in other queries
and view definitions
• queries on views are evaluated by
query modification
• some views are updatable
• some views are not updatable
• more on views when we look at SQL
Database Group, Georgia Tech
© Leo Mark
Relational Model 45
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