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The Relational Model - theoretical foundation . The Relational Model. data structures constraints operations algebra (ISBL) tuple calculus (QUEL, SQL) domain calculus (QBE) views. Data Structures. let D 1 , D 2 , D 3 , ..., D n be sets (not necessarily distinct) of atomic values

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### The Relational Model- theoretical foundation

The Relational Model
• data structures
• constraints
• operations
• algebra (ISBL)
• tuple calculus (QUEL, SQL)
• domain calculus (QBE)
• views
Data Structures
• let D1, D2 , D3 , ..., Dnbe sets (not necessarily distinct) of atomic values
• relation, R, defined over D1, D2 , D3 , ..., Dn is a subset of the set of ordered n-tuples {<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
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)
Constraints
• keys
• primary keys
• entity integrity
• referential integrity

FLT-SCHEDULE

CUSTOMER

FLT#

CUST#

CUST-NAME

p

p

RESERVATION

FLT#

DATE

CUST#

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#

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)

FLT-SCHEDULE

flt# airline dtime from-airportcode atime to-airportcode miles price

Operations- insert, delete, update
• 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”);

FLT-SCHEDULE

flt# airline dtime from-airportcode atime to-airportcode miles price

FLT-WEEKDAY

flt# weekday

FLT-INSTANCE

flt# date plane#

#avail-seats

Operations- insert, delete, update
• “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

FLT-WEEKDAY

flt# weekday

Operations- insert, delete, update
• 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

FLT-WEEKDAY

flt# weekday

Operations- insert, delete, update
• 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
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Sdivideby

[A1 B1,.., An Bn] rename

FLT-WEEKDAY

flt# weekday

Selection
• “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: ( )

FLT-WEEKDAY

flt# weekday

Projection
• “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

FLT-WEEKDAY

flt# weekday

Union
• “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"

FLT-WEEKDAY

flt# weekday

Intersection
• “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"

FLT-WEEKDAY

flt# weekday

Set Difference
• “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)

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#

Cartesian Product

“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)

FLT-WEEKDAY

flt# weekday

FLT-INSTANCE

flt# date plane#

#avail-seats

Natural Join
• “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))

FLT-SCHEDULE

flt# airline dtime from-airportcode atime to-airportcode miles price

FLT-INSTANCE

flt# date plane#

#avail-seats

-join
• “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  ° =

FLT-INSTANCE

flt# date plane#

#avail-seats

RESERVATION

flt# date cust# seat# check-in-status ticket#

Divideby
• “list the cust# of customers that have reservations on all flight instances”

flt#, date, cust# RESERVATION

flt#, date (FLT-INSTANCE)

ISBL - an example algebra

R S R UNION S

R S R INTERSECT S

R \ S R MINUS S

A1, A2, ..., An (R) R[A1, A2, ..., An]

expression (R) R WHERE EXPRESSION

R S R JOIN S (no shared attributes)

R S R JOIN S (shared attributes)

R S via selection from 

RS R DIVIDEBY S

[A1 B1,..., An Bn](R)R[A1 B1,.., An Bn]

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 !

FLT-SCHEDULE

flt# airline dtime from-airportcode atime to-airportcode miles price

FLT-INSTANCE

flt# date plane#

#avail-seats

ISBL - an example query
• “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, TO-AIRPORTCODE AA1, ATIME AT1, DATE D1]) JOIN
• (FLT-SCHEDULE JOIN FLT-INSTANCE) WHERED1=DATE AA1=FROM-AIRPORTCODE  AT1< DTIME)[FL1, FLT#]
Relational Calculus
• the Relational Calculus is non-procedural. 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
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

CUSTOMER

cust# first middle last phone# street city state zip

Tuple Calculus
• { 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

FLT-WEEKDAY

flt# weekday

Selection
• “find (FLT#, WEEKDAY) for all flights scheduled for Mondays

{ t | FLT-WEEKDAY(t) t.WEEKDAY=MO}

FLT-WEEKDAY

flt# weekday

Projection
• “find FLT# for all flights scheduled for Mondays

{ t.FLT# | FLT-WEEKDAY(t) t.WEEKDAY = MO}

FLT-WEEKDAY

flt# weekday

Union
• “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)}

FLT-WEEKDAY

flt# weekday

Intersection
• “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)}

FLT-WEEKDAY

flt# weekday

Set Difference
• “find the FLT# for flights that are scheduled for Mondays, but not for Tuesdays”

{ t.FLT# | FLT-WEEKDAY(t) t.WEEKDAY=MO ((s) (FLT-WEEKDAY(s) t.FLT#=s.FLT# s.WEEKDAY=TU))}

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”

Cartesian Product

{s.FLT#, s.DATE, t.CUST#| FLT-INSTANCE(s) CUSTOMER(t) t.CITY=ROSWELLs.FLT#=DL212 s.DATE=1998-09-10rFLT-INSTANCE(r) r ° sr.FLT#=s.FLT#r.DATE=s.DATE r.CUST#=t.CUST#)}

FLT-WEEKDAY

flt# weekday

FLT-INSTANCE

flt# date plane#

#avail-seats

Natural Join
• “make a list with complete flight instance information”

{ s.FLT#, s.WEEKDAY, t.DATE, t.PLANE#, t.#AVAIL-SEATS | FLT-WEEKDAY(s) FLT-INSTANCE(t)  s.FLT#=t.FLT# }

FLT-SCHEDULE

flt# airline dtime from-airportcode atime to-airportcode miles price

FLT-INSTANCE

flt# date plane#

#avail-seats

-join
• “make a list of pairs of (FLT#1, FLT#2) that form possible connections”

{ s. FLT#, t.FLT# | FLT-SCHEDULE(s) FLT-SCHEDULE(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.FROM-AIRPORTCODEs.ATIME < t.DTIME) }

FLT-INSTANCE

flt# date plane#

#avail-seats

RESERVATION

flt# date cust# seat# check-in-status ticket#

Divideby
• “list the CUST# for customers that have reservations on all flight instances”

{ s.CUST# | RESERVATION(s)  (( t) FLT-INSTANCE(t) ((r) RESERVATION(r)  r.FLT#=t.FLT#  r.DATE=t.DATE r.CUST#=s.CUST#))}

FLT-SCHEDULE

flt# airline dtime from-airportcode atime to-airportcode miles price

FLT-INSTANCE

flt# date plane#

#avail-seats

QUEL - an example tuple calculus
• “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.FROM-AIRPORTCODE and s.ATIME < t.DTIME;

FLT-WEEKDAY

FLT#

WEEKDAY

P.

=MONDAY

QBE - Projection
• “find FLT# for all flights scheduled for Mondays

FLT-WEEKDAY

FLT#

WEEKDAY

P.

MONDAY

P.

TUESDAY

QBE - Union
• “find the FLT# for flights that are schedule for either Mondays, or Tuesdays, or both”

FLT-WEEKDAY

FLT#

WEEKDAY

P._SX

MONDAY

_SX

TUESDAY

QBE - Intersection
• “find the FLT# for flights that are schedule for both Mondays and Tuesdays”

FLT-WEEKDAY

FLT#

WEEKDAY

P._SX

MONDAY

_SX

TUESDAY

QBE - Set Difference
• “find the FLT# for flights that are scheduled for Mondays, but not for Tuesdays”

CUSTOMER

FLT-INSTANCE

#AVAIL-

SEATS

CUST#

CUST-NAME

CITY

FLT#

DATE

P._C

ROSWELL

P._F

P._D

_F

98-9-10

DL212

_D

RESERVATION

FLT#

DATE

CUST#

_F

_D

_C

QBE - Cartesian Product

“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”

FLT-WEEKDAY

FLT-INSTANCE

#AVAIL-

SEATS

FLT#

WEEKDAY

FLT#

DATE

P._SX

P.

_SX

P.

P.

QBE - Natural Join
• “make a list with complete flight instance information”

FLT-SCHEDULE

FROM-

AIRPORT

CODE

TO-

AIRPORT

CODE

FLT#

AIRLINE

DTIME

ATIME

PRICE

P._SX

_A

_AT

FLT-SCHEDULE

FROM-

AIRPORT

CODE

TO-

AIRPORT

CODE

FLT#

AIRLINE

DTIME

ATIME

PRICE

P._SY

_A

_DT

FLT-INSTANCE

CONDITION

FLT#

DATE

#SEATS

_AT < _DT

_SX

_D

_SY

_D

QBE-join
• “make a list of pairs of (FLT#1, FLT#2) that form possible same day connections”
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