The Relational Model - theoretical foundation

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## The Relational Model - theoretical foundation

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**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 RSdivideby [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 ofA1, 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: RS = 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-INSTANCEFLT-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 RS 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: tR 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: tR, 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 ), P1P2, P1 P2, and P1 P2 • if P(t) is a predicate, t is a free variable in P, and R is a relation then tR(P(t)) andtR (P(t)) are predicates**CUSTOMER**cust# first middle last phone# street city state zip Tuple Calculus • { r |(rCUSTOMER} 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 sFLT-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=ROSWELLs.FLT#=DL212 s.DATE=1998-09-10rFLT-INSTANCE(r) r ° sr.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-AIRPORTCODEs.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